================================================================
SMOOTH COUNTEREXAMPLES TO YAU'S NODAL UPPER BOUND
IN DIMENSIONS THREE AND FOUR
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A self-contained proof, addressed to a curious reader who has seen
multivariable calculus, linear algebra, and the definition of
Laplace-Beltrami operator on a surface (or, at least, on the
sphere). Everything is stated and proved here. When we use a
standard fact about eigenvalues we cite the source and give a
one-line explanation.
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PART 0 -- THE PROBLEM, THE THEOREMS, THE STRATEGY
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0.1 THE PROBLEM
Let M be a closed (compact, no boundary) connected smooth
Riemannian manifold of dimension d, with metric g.
Write Delta_g for the Laplace-Beltrami operator on functions.
A real eigenfunction of g with eigenvalue lambda > 0 is a
nonzero smooth function u : M -> R with
- Delta_g u = lambda u . (0.1)
Attached to any real function u is its nodal set
Z_u = { x in M : u(x) = 0 } . (0.2)
The nodal set is the boundary between the region where u is
positive and the region where u is negative. We measure it by
H^d-1_g, the (d-1)-dimensional Hausdorff measure for the
Riemannian distance of g: length on surfaces, area on
3-manifolds, 3-volume on 4-manifolds.
The eigenvalue lambda is the square of a frequency. On a flat
torus, an eigenfunction with eigenvalue approximately lambda
oscillates at wavelength approximately 1/sqrt(lambda). Reason:
the constant function has eigenvalue 0, and plane waves
sin(k.x) have eigenvalue |k|^2, so wavelength 2 pi / |k| scales
like 1/sqrt(lambda).
If the nodal set were a nice hypersurface whose curvature was
bounded by the wavelength, its (d-1)-measure would be of order
sqrt(lambda). A nonzero scalar multiple of u does not change
Z_u, so both sides of this heuristic are scale-invariant. The
same heuristic is what a stationary-phase or random-wave
approximation gives.
S.-T. Yau (Problem Section, Seminar on Differential Geometry,
Annals of Math Studies 102, Princeton 1982, pp. 669-706) put this
guess forward as a conjecture. In the smooth category:
YAU'S NODAL CONJECTURE. Fix a closed Riemannian manifold
(M,g) of dimension d. There are constants c_g > 0 and
C_g > 0 (depending on the metric g but not on u or lambda)
such that, for every nonzero real eigenfunction u of g with
eigenvalue lambda > 0,
c_g * sqrt(lambda) <= H^d-1_g(Z_u)
<= C_g * sqrt(lambda) . (0.3)
The claim is a lower bound AND an upper bound, and the two are
logically separate.
LOWER BOUND. c_g sqrt(lambda) <= H^d-1_g(Z_u).
UPPER BOUND. H^d-1_g(Z_u) <= C_g sqrt(lambda).
The lower bound is now a theorem in every dimension
(A. Logunov, "Nodal sets of Laplace eigenfunctions: proof of
Nadirashvili's conjecture and of the lower bound in Yau's
conjecture", Annals of Math 187 (2018), 241-262). The upper
bound is where the action is.
* In dimension 2 there are well-known partial results:
upper bound C_g lambda^{3/4} (Donnelly-Fefferman, JAMS
1990) and improvement to C_g lambda^{3/4-beta} for a
universal small beta (Logunov-Malinnikova 2018). The
present paper is not about dimension 2; the companion
paper "Sharp nodal length on smooth surfaces" (OpenAI
Math Release, September 2026) settles the upper bound
there too.
* In the real-analytic category (metric with real-analytic
coefficients), the full conjecture is a theorem in every
dimension (Donnelly-Fefferman, Invent. Math. 93 (1988),
161-183).
* In dimension >= 3, for merely smooth metrics, Logunov
(Annals 2018) has shown a polynomial upper bound
H^d-1_g(Z_u) <= C_g lambda^{C_d} but not the linear-in-
sqrt(lambda) bound that Yau conjectured. The exact
polynomial exponent is not the point.
* In dimension >= 5, the upper bound FAILS: the companion
paper "Power-law violations of Yau's nodal upper bound"
(OpenAI Math Release, September 2026) constructs one fixed
smooth metric in each dimension >= 5 for which
H^d-1_g(Z_{u_j}) / sqrt(lambda_j) grows like a
positive power of lambda_j.
The present paper removes dimension 3 and dimension 4 from the
list of open smooth cases: it shows that the upper bound FAILS
for one fixed smooth metric already in dimensions three and four,
in the STRONGEST possible fashion -- the ratio diverges, not
merely outgrows a constant.
0.2 THE TWO THEOREMS
THEOREM A (dimension three). Let g_* be the round (unit) metric
on the 3-sphere S^3. For every open neighborhood U of g_* in
the C^infinity topology of smooth Riemannian metrics on S^3,
there is a smooth metric g_inf in U, a sequence of positive
numbers lambda_j -> infinity, and a sequence of nonzero smooth
real functions u_j on S^3 such that
- Delta_{g_inf} u_j = lambda_j * u_j for every j, (A1)
H^2_{g_inf}( Z_{u_j} ) / sqrt(lambda_j) -> infinity. (A2)
In particular, no constant C_{g_inf} can satisfy the upper bound
in (0.3) for this single metric.
THEOREM B (dimension four). Let M = S^2 x T^2 (product of
round 2-sphere with flat 2-torus). There exists a smooth
Riemannian metric g_inf on M, a sequence of positive numbers
lambda_j -> infinity, and a sequence of nonzero smooth real
functions u_j on M such that
- Delta_{g_inf} u_j = lambda_j * u_j for every j, (B1)
H^3_{g_inf}( Z_{u_j} ) / sqrt(lambda_j) -> infinity. (B2)
PRECISION.
* Theorem A works for EVERY open neighborhood of the
round metric. So it is not a single exotic counterexample;
the violating metrics are arbitrarily close to the
round one in C^infinity.
* Theorem B is confined to the four-dimensional manifold
S^2 x T^2 with SOME smooth metric. We do not claim
that the four-dimensional violating metrics are arbitrarily
close to any particular background. (They are.)
* Neither theorem contradicts the lower bound in (0.3).
Both leave the lower bound untouched. Neither theorem
claims a power law (only that the ratio diverges).
* Together with the surface companion paper and the >= 5
companion paper, Theorems A and B pin down the exact
universal dimension threshold:
Among dimensions d >= 2, the upper bound in (0.3),
with C_g depending only on a fixed smooth metric g
and being uniform over all nonzero real
eigenfunctions of positive eigenvalue, holds for
every closed connected Riemannian manifold if and
only if d = 2.
0.3 WHAT A COUNTEREXAMPLE HAS TO DO
We want ONE fixed metric with eigenfunctions whose nodal
measure outgrows sqrt(lambda). Once the metric is fixed, we
cannot simply write such an eigenfunction down: it has to solve
-Delta_g u = lambda u for the SAME g at every step in the
sequence. The freedom we do have is the choice of g at the very
start.
So we start with a background metric g_0 that has perfectly
explicit, well-understood high-frequency eigenfunctions, and
then we modify g_0 in a small region. The modifications are
tiny in C^infinity. In the region left untouched the metric
and the eigenfunction remain exactly the known background
objects; in the modified region we force the eigenfunction to
have an enormous nodal surface area. Because the modification
is tiny, the number of negatives in the modified metric is
essentially the number of negatives in the background.
Four ingredients make the strategy work in dimensions 3 and 4.
(a) THE ENVELOPE.
We construct a smooth function phi (the "envelope") with two
properties at once. It satisfies a strict Hessian inequality
(Definition 1.1 below), and its integrated gradient
integral |grad phi| is arbitrarily large. These two
requirements are NOT obviously compatible: forcing a big
gradient usually ruins second-order signs. But they are in
fact compatible, because the gradient can be amplified by
adding fast oscillations in directions TRANSVERSE to itself.
The result is a smooth function of very small size (small
|phi| in C^0) whose gradient has very large integral.
The Hessian inequality is what makes the waves of (b) localize
tightly, so the Gaussian tail estimate works.
(b) LOCALIZED WAVES.
Around each point we build a complex-phase WKB wave whose
amplitude envelope is exp(n phi). The Hessian inequality makes
the real part of the phase strictly less than phi on a small
ball (minus a positive quadratic correction), so the wave is
localized in a ball of radius roughly 1/sqrt(n). A Gaussian
superposition of MANY such packets (O(n^3) or O(n^4) depending
on dimension) produces, with probability close to one, a real
quasimode whose "value + gradient" pair never vanishes
simultaneously (real noncancellation) and whose weighted
residual is smaller than any prescribed inverse power of n.
This is the analytic heart of the paper.
(c) A SIGN SCORE THAT BOUNDS NODAL MEASURE.
We count sign changes of u along short coordinate segments of
length roughly 1/(n * L) where L = sqrt(1 + |grad phi|^2) or
B = sqrt(2 + |grad phi|^2) in dimension 4. A segment along
which u changes sign must contain a zero. The count is a
functional F on continuous functions that is
* lower semicontinuous for uniform convergence;
* invariant under multiplying u by a positive function;
* bounded above by a universal constant times the
(d-1)-measure of the zero set of u.
The last bound is elementary and uses no regularity of the
nodal set (it covers fractal nodal sets too). The constant does
not depend on phi, on the subdivision, or on n: only on the
metric and on coordinates. By making the profile phi have a
large gradient integral, we make the expected score of a
Gaussian superposition of waves enormous. A realization with a
large score gives a lower bound on nodal measure that, when
divided by sqrt(lambda), tends to infinity. This is where the
gradient amplification of (a) pays off.
(d) EXACTIFICATION AND SPECTRAL PINNING.
A quasimode is not an eigenfunction. In dimension 3 we
introduce a scalar conductivity b and a scalar density s to
make U an exact eigenfunction of a "generalized" Laplacian
div(b grad U) + Lambda s U = 0. Dividing by a positive
function v (a solution of a semilinear elliptic equation)
produces a pair (q,d) = (b v^2, s v^2 + Lambda^{-1} v Delta_b v)
such that (q,d) satisfies the determinant relation d = q^3
that a 3-dimensional metric requires -- but only in an active
region. A transverse repair in a clean annulus fixes the last
defect in d - q^3, and this annulus is strictly inside a
smaller region on which the metric and the eigenfunction stay
equal to the round ones. In dimension 4 the same role is
played by a conformal correction Z and a rank-three transverse
rebuild.
Once we have an exact eigenpair (g', u') with a large sign
score, we make the selected eigenvalue SIMPLE by a small
volume-preserving perturbation that annihilates grad u'. A
strict lower bound for the score then persists on a whole
neighborhood of g'. Finally, nested closed balls in a
complete metric space of smooth tensors converge to one fixed
smooth metric g_inf. Each ball retains its own witness and its
own score, so along the resulting fixed g_inf we have a
divergent sequence of eigenfunctions whose normalized nodal
measure blows up.
The rest of the paper carries this plan out, first in
dimension 3 (Parts 1-5) and then in dimension 4 (Part 6). Part
7 assembles the two theorems.
0.4 CONVENTIONS
M is a closed connected smooth Riemannian manifold unless stated
otherwise. Delta_g is the Laplace-Beltrami operator with the
geometer's sign: -Delta_g is positive semidefinite, its
eigenvalues are nonnegative, and we write -Delta_g u = lambda u
with lambda >= 0. Gradients, Hessians and divergences use the
displayed metric, and g unless another metric is named.
H^s_g for a real number s denotes the Sobolev space of order s
with respect to the fixed smooth metric g. For an integer s it
is the space of functions whose weak derivatives up to order s
lie in L^2, equipped with its standard norm. C^s denotes the
usual s-times continuously differentiable functions, and
C^infinity is the intersection over all s.
"Closed" means compact and without boundary. Hausdorff measure
uses the Riemannian distance of the displayed metric with the
standard normalization that the d-ball of radius r has measure
approximately omega_d r^d for small r.
If a quantity C_h "depends only on a displayed set of data D",
the constant may change from line to line, but at each step only
finitely many such constants appear. Whenever we say "for
sufficiently large n" or "for sufficiently small tau" we mean
that a finite threshold exists; that threshold may depend on the
fixed data of the step but never on a running frequency or an
arbitrary parameter of the conclusion.
Every finite order of differentiation and every required
accuracy power is chosen BEFORE the frequency tends to
infinity. Each step has only finitely many requirements, and
the next step is only taken after those requirements are fixed.
A reader who wants to check this can verify at each Proposition
that the parameters are introduced explicitly. This is the
"finite-order first" convention throughout.
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PART 1 -- BACKGROUND MODES ON THE ROUND 3-SPHERE
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1.1 THE ROUND SPHERE AND ITS LAPLACIAN
Let S^d-1 = { X in R^d : |X| = 1 } be the unit sphere with its
round metric g_*. We care about d = 4 in Part 1, so that S^3
sits in R^4 = { (x_1, x_2, x_3, x_4) }.
The Euclidean Laplacian on R^d, written
Delta_{R^d} = sum_{i=1}^d partial^2 / partial x_i^2 ,
is rotation-invariant. Through the polar decomposition
X = r Y with r = |X| and Y in S^d-1, one has
Delta_{R^d} = partial_r^2 + (d-1)/r * partial_r
+ (1/r^2) * Delta_{S^d-1} .
Equivalently, a homogeneous harmonic polynomial P of degree k
in X satisfies Delta_{R^d} P = 0; combined with the polar
decomposition this gives
Delta_{S^d-1} P|_{S^d-1} = -k(k+d-2) * P|_{S^d-1} .
So the restriction of a degree-k harmonic polynomial to S^d-1
is an eigenfunction of the round sphere's Laplacian, with
eigenvalue k(k+d-2). In dimensions d = 3 (S^2), the eigenvalue
is k(k+1); in dimension d = 4 (S^3), it is k(k+2).
Two remarks that we will use.
* RESTRICTIONS OF HARMONIC POLYNOMIALS ARE DENSE. Every
real eigenfunction of the round S^d-1 with eigenvalue
k(k+d-2) is the restriction of SOME homogeneous harmonic
polynomial of degree k. This is standard but not
needed in the sequel; we use explicit polynomials.
* THE FACTORISATION. If H_k is a homogeneous harmonic
polynomial of degree k in (X_1, X_2) only (writing
X = (X_1, X_2, X_3, X_4) and viewing (X_1, X_2) as two
coordinates), then H_k is automatically harmonic in
R^4 as well, since the other two coordinate directions
do not enter. So its restriction to S^3 is an
eigenfunction with eigenvalue k(k+2). This gives the
family we use below.
1.2 THE FAMILY w_n
Let n >= 1 be an integer. On S^3, define the polynomial
w_n(X) = Re ( X_1 + i X_2 )^n . (1.1)
This is a homogeneous polynomial of degree n in R^4.
Since it only depends on the two coordinates X_1, X_2, it is
harmonic: partial_{X_3} w_n = partial_{X_4} w_n = 0 and
(partial_{X_1}^2 + partial_{X_2}^2)(X_1 + i X_2)^n = 0, and
likewise for the real part. Hence
- Delta_{g_*} w_n = n(n+2) * w_n on S^3. (1.2)
We fix the eigenvalue
Lambda_n = n(n+2) . (1.3)
Note that Lambda_n grows like n^2, and Lambda_{n+1} - Lambda_n
= 2n + 3 > 2. So the thresholds Lambda_n are strictly
increasing and their consecutive gaps grow without bound.
This matters when we later perturb the metric a little: if the
new eigenvalues move by less than 1, they cannot cross each
other, and the selected eigenfunction stays isolated.
1.3 GRADIENT NORM OF w_n
Let r(X) = (X_1^2 + X_2^2)^{1/2}, a smooth nonnegative function
on S^3 that vanishes (to first order) exactly on the
codimension-two sphere {X_1 = X_2 = 0}. Along S^3 we also have
sum X_i^2 = 1.
The gradient of w_n with respect to the round metric of S^3 is
best computed by using the ambient Euclidean gradient of w_n
(evaluated at X in S^3) and then projecting onto the tangent
space T_X S^3 = (X)^perp. The identity we need is:
|grad_{g_*} w_n|^2_{g_*}(X)
= n^2 * r(X)^{2n-2} - n^2 * w_n(X)^2 . (1.4)
Proof. Write w_n = Re( (X_1 + i X_2)^n ) = r^n cos(n theta)
where X_1 + i X_2 = r exp(i theta). The Euclidean gradient in
R^2 (of the (X_1, X_2) part) has squared length n^2 r^{2n-2}
(this is the general fact |grad_{R^d} (homog harmonic deg n)|^2
on |X| = r is (partial_r)^2 + (n/r)^2 * (tangential part)^2
along the sphere, and here on the (X_1, X_2) disc the radial
derivative has magnitude n r^{n-1} and the angular derivative
has magnitude n r^{n-1}, so combined length n^2 r^{2n-2}).
The normal-part to S^3 of the Euclidean gradient is n * w_n
(differentiating the homogeneous polynomial of degree n by the
identity X . grad P = n P). So:
|grad_{g_*} w_n|^2 = |grad_{R^4} w_n|^2 - (X . grad
w_n)^2
= n^2 r^{2n-2} - (n w_n)^2 .
That is (1.4). End of proof.
Consequence 1.5. On the annulus A = { t_2 < r < t_3 } with
0 < t_2 < t_3 < 1, one has r^{2n-2}(1 - r^2) > 0 and
|grad_{g_*} w_n|^2 >= n^2 r^{2n-2} (1 - r^2) >= c_A
for some constant c_A depending only on t_2, t_3 and n
through r^{2n-2}, which is bounded below on A because
t_2 < r, and 1 - r^2 >= 1 - t_3^2 > 0.
Actually, to be precise: the factor r^{2n-2} on A is bounded
above by t_3^{2n-2} and below by t_2^{2n-2}. The bound is not
uniform in n at first glance — but on A the gradient is
>= c_3 * n^2 * r^{2n-2} where c_3 = 1 - t_3^2 > 0. So near the
"outer" boundary r = t_3, |grad w_n| can be exponentially small
in n. Not a problem for us: on the annealed region where we
amplify the metric and profile we will ensure |grad w_n| or
|grad U| stays bounded below by another mechanism. But we
should USE this for a coarse exterior region: on { r >= R_0 }
with R_0 < 1 fixed, |grad w_n| is bounded below on { r in [R_0,
1 - epsilon] } only if we accept a bound in n. We do NOT need
a lower bound there; we need it on the inner ball.
Consequence 1.6. On the ball B_0 = { r <= R_0 } with
0 < R_0 < 1 chosen sufficiently small, r^{2n-2} <= R_0^{2n-2}
is exponentially small in n. So the round gradient of w_n is
essentially zero on the inner part of the ball, and NONZERO but
small on the annulus. This is why we later EXPLICITLY compare
w_n with waves: the w_n contribution to value and gradient is
small on the ball when |w_n| is also small; a Gaussian
superposition of waves will carry the day there.
The precise comparison:
sup_{r <= R_0} ( |w_n| + n^{-1} |grad w_n| )
is <= n * R_0^{2n}, which is <= n^{-k} for any fixed k and
n large, once R_0 < 1.
Consequence 1.7. Outside the small ball — say on the annulus
A = { (1/20) < r < (1/8) } — the gradient |grad w_n| is
comparable to n * r^{n-1} up to the cosine factor; the sup over
A is O(n) but the inf can be exponentially small in n. We
therefore do not rely on w_n outside its interior ball except
through its (exponentially weak) L^inf norm.
1.4 COORDINATES, BOUNDARY LAYER, AND THE WAVE REGION
Let n >= 1 be the integer frequency. Fix small radii
1/5 <= R_0 < t < t_1 < t_2 < t_3 < 3/10 . (1.8)
Here t is the parameter in the profile phi_0 below, and the
radii t_1 < t_2 < t_3 separate three nested regions:
INNER B_all = { r <= t_1 }: waves live here.
ANNULUS A_cor = { t_2 < r < t_3 }: the exactification
correction annulus lives here (dimension 3). This annulus
is untouched by waves (which are supported in r < t_2), and
untouched by later spectral-pinning perturbations.
EXTERIOR E_ext = { r >= t_3 }: g = g_* exactly,
u = w_n exactly. This is the region where the metric and
eigenfunction of the previous step have to agree with the
ROUND ones.
Additionally, fix a small coordinate cube Omega (in the (X_1,
..., X_4) coordinates) whose closure sits inside
A_Om = { 1/20 < r < 1/8 } -- i.e. Omega subset A_Om and
closure(Omega) subset A_Om.
On Omega, the round metric is comparable to the coordinate
Euclidean metric with constants bounded above and below, and
all the machinery of envelopes and waves can be done in these
coordinates. We fix coordinates once and for all.
================================================================
PART 2 -- ENVELOPES WITH LARGE MEAN GRADIENT
================================================================
The purpose of this Part is to produce, for a specified
tolerance eps > 0 and a specified target T > 0, a smooth real
function phi on S^3 with the following properties. (The
parameter t in the statement is the "crossing radius" of the
envelope base profile.)
ENV-A. phi equals phi_0 = log t + r^2 - t^2 off a compact
subset of Omega, and |phi - phi_0|_{C^0} < eps.
ENV-B. phi is STRICT (Definition 2.1 below) on the region
{ r <= 3/10 } with a UNIFORM (profile-dependent)
STRICT MARGIN.
ENV-C. phi has no critical point in Omega, and the only
critical point of phi on { r <= 3/10 } is at r = 0,
where Hess phi is positive on a two-plane.
ENV-D. phi > log r on { 0 < r < t }, and phi < log r on
{ t < r <= 1 }.
ENV-E. integral_Omega |grad phi|_{g} dV_g > T.
We prove this as Proposition 2.10 below, after proving an
amplification Proposition 2.8 and a Preparation Lemma 2.6.
2.1 DEFINITION OF STRICT AND FULL PROFILES
Let phi be a smooth real function on a three-dimensional
Riemannian manifold (M, g). Throughout this section we fix
this reference metric; later in Part 3 we allow g to vary
within a small C^1 neighborhood of g_*, but not in Part 2.
Set a = grad_g phi, an element of TM. Let L = sqrt(1 + |a|_g^2).
DEFINITION 2.1 (Strict and full profiles).
The function phi is STRICT at a point p if there EXISTS a
vector b in T_p M with
b perpendicular to a in g, |b|_g = L , and
Hess_g phi (a, a) + Hess_g phi (b, b) > 0 . (2.1)
The function phi is FULL at a point p if the last inequality
holds for EVERY b with b perpendicular to a and |b| = L.
Full is stronger than strict. On a compact set where a does not
vanish, fullness is an open condition in the second derivatives
of phi, and it gives a positive minimum of
T(H, a; v) = H(a, a) + (1 + |a|^2) H(v, v)
(2.2)
over the transverse unit circle bundle |v| = 1, v perpendicular
to a. Note the factor (1 + |a|^2) in the second term: because
|b| = sqrt(1 + |a|^2), we have
H(b, b) = (1 + |a|^2) H(v, v) where v = b / |b|. That is why
the definition of T uses that factor.
The name "strict" suggests the inequality (2.1) is strict. The
name "full" suggests that every transverse direction contributes
the correct sign, not just one. When phi is only strict but not
full, we can only be sure of ONE direction at each point; this
is enough for the wave construction of Part 3 (which uses one
transverse b per point), but not enough to iterate the
amplification of the gradient in Part 2 (which needs full). So
we always upgrade strict to full first, and then amplify.
2.2 LOCALISATION AND PERIODIC AVERAGING
We will need two elementary facts, repeatedly.
LEMMA 2.2 (Disjoint patches).
Let Omega be a relatively compact coordinate cube in a smooth
three-dimensional Riemannian manifold, and let U be open with
closure(U) subset Omega. Let mu be a finite measure on Omega
with a positive smooth density (e.g. mu = |a_phi| dV_g for a
nowhere-vanishing smooth vector field a_phi).
Given an open neighbourhood of each point of U, there exist
finitely many coordinate cubes P_1, ..., P_J with pairwise
disjoint interiors, closure(P_j) contained in that
neighbourhood and in U, and cutoffs chi_j in C_c^infinity(P_j)
taking values in [0, 1], such that chi_j = 1 on an open core
C_j \Subset P_j and
mu( union_j C_j ) >= (1/2) * mu(U) . (2.3)
PROOF. Choose a compact subset of U carrying more than
three-quarters of the measure mu(U). A finite subcover of the
prescribed neighbourhoods has a Lebesgue number on this compact
set. A sufficiently fine coordinate grid therefore produces
finitely many cubes, each contained in some prescribed
neighbourhood and in U, whose union contains the compact set up
to grid faces. The grid faces have measure zero. Shrinking
the cores inside the cubes loses arbitrarily little mass, and
smooth cutoffs supported in the cubes are available by
definition. End of proof.
This is one of the two facts whose proof is a compactness
argument; the other follows.
LEMMA 2.3 (Periodic averaging in a patch).
Let y = (y_1, y_2) : U -> R^2 be a smooth submersion on a
neighborhood of a compact set K \subset U. Let E subset R^2 be
periodic with respect to a full lattice (e.g. Z^2), have
boundary of two-dimensional measure zero in a period cell, and
occupy a fraction p of the cell in the usual sense.
Given a compact set C \Subset U and a finite positive measure mu
on U with smooth density, we have
lim_{N -> infinity} mu({ x in C : N y(x) in E }) = p mu(C). (2.4)
PROOF. Submersion coordinates and a finite partition of unity
express the pushforward of mu restricted to C under y as an
integrable function rho on R^2. The left side is then
integral of 1_E(N . ) rho over R^2. For the indicator function
of a rectangle, the result follows by tiling with period cells
and estimating the cells that meet the boundary of the
rectangle. Approximation of an integrable function by finite
linear combinations of rectangle indicators gives the general
case. The condition that E have boundary of measure zero in a
cell ensures that its cell average is well-defined and
unchanged by the choice of values on the boundary. End of
proof.
These are the only two places where compactness enters Part 2.
Everything else is direct computation.
2.3 THE TWO PERIODIC FUNCTIONS
Fix a smooth real function h of period one with
h''(t) = 2 cos( 2 pi t ) . (2.5)
For example, h(t) = -(2 pi^2)^{-1} cos( 2 pi t ). The set
E_h = { t in R : h''(t) > 1 } (2.6)
is a union of intervals inside each unit cell; it occupies
exactly one third of each period (namely the intervals on which
cos( 2 pi t) > 1/2). Let p_h = 1/3.
For the two-dimensional corrugation, fix radii
0 < r_0 < r_1 < r_2 < R < 1/4 and a smooth nonnegative function
f_c on [0, R] with support in (r_0, R) and f_c >= c_1 > 0 on
(r_1, r_2). Define
f(s) = integral_{r_1}^{s} f_c(t) dt, so f'(s) = f_c(s)
and f is supported in (r_0, R) with f' >= c_1 on (r_1, r_2).
The radial function F on R^2 defined by
F( y ) = f( |y| ) for |y| < R, extended to be the
constant f(R) off all the disks of radius R centered at
points of Z^2, (2.7)
is smooth and Z^2-periodic: it is constant near each disk
center, and constant (= f(R)) on the far outer parts of each
cell. The "annulus set"
E_F = union_{k in Z^2} { y in R^2 : r_1 < |y - k| < r_2 } (2.8)
occupies a fixed fraction p_F = pi(r_2^2 - r_1^2) of each unit
cell, on which f'(|y|) >= c_1 and f'' can have either sign but
satisfies the differential inequality |f''| <= C * sqrt(f')
(which follows from f being smooth, supported in (r_0, R), and
nonnegative: at any t, |f'(t)| <= C sqrt(||f''||_inf ||f||_inf)
by Taylor's inequality, and here ||f''|| is fixed).
We also fix a smooth cutoff chi_0 on R^2 (later used in the
"amplification" and "corrugation" steps) with 0 <= chi_0 <= 1,
equal to 1 on the closed unit ball and supported in the closed
ball of radius 2.
2.4 STEP 1: MAKING THE PROFILE FULL ON A POSITIVE FRACTION
LEMMA 2.4 (Preparation).
Let phi be a smooth real function on a neighbourhood of
closure(Omega), STRICT on closure(Omega) with nonvanishing
gradient there. Fix small constants 0 < eps_p <= 1/3 and
eps > 0. There is a smooth strict function psi
supp(psi - phi) \Subset Omega, |psi - phi|_{C^0(Omega)} < eps,
|a_psi - a_phi|_{point} <= eps_p |a_phi| on closure(Omega),
whose FULL region U subset Omega satisfies
integral_U |a_psi| dV_g >= (p_h/8) integral_Omega |a_psi| dV_g. (2.9)
Additionally, |psi - phi|_{C^1(Omega)} can be made arbitrarily
small.
PROOF.
Step 1: Local product structure. At any point p with a_phi(p)
!= 0, choose a strict unit vector v_p \perp a_phi(p) in T_p M.
The strict inequality (2.1) at p gives a strict margin m_p > 0
for H_phi(a_phi, a_phi) + H_phi(v_p, v_p) restricted to a small
neighbourhood. A local flow-box construction gives a smooth
coordinate function s on a small neighbourhood of p with
ds(a_phi) = 0 and ds(v_p) = 0. (The intersection of the two
kernels is a line in T_p M; extend to a local first integral s
of a_phi that vanishes on that line.) On a small neighbourhood
of p, the line a_phi^\perp \cap ker ds remains strict with a
positive margin, uniformly over a slightly smaller compact
neighbourhood.
Apply Lemma 2.2 to the family of these local neighbourhoods
with mu = |a_phi| dV_g. This produces finitely many patches
P_j with cutoffs chi_j capturing at least half the gradient
mass on their cores C_j.
Step 2: Local perturbation. On a single patch P_j, with
cutoff chi_j, s_j the coordinate function from Step 1, and N
to be chosen, define
psi_N(x) = phi(x) + C * N^{-2} * chi_j(x) * h(N s_j(x)) . (2.10)
The constant C > 0 is a function of the patch data (to be fixed
below), chosen BEFORE N.
Direct differentiation gives, uniformly over the patch:
a_{psi_N} = a_phi + (C/N) chi_j h'(N s_j) grad s_j
+ O(N^{-2}) * h(N s_j) grad chi_j ,
Hess psi_N = Hess phi + C chi_j h''(N s_j)
ds_j \otimes ds_j + O(N^{-1}). (2.11)
Note the crucial coincidence: the coefficient of the leading
Hessian correction is C chi_j h''(N s_j), which has size O(1),
while the perturbation to the GRADIENT is only O(N^{-1}).
So the Hessian changes at O(1) and the gradient changes at
O(N^{-1}). The Hessian correction lies entirely in the ds_j
direction: it vanishes on ker ds_j.
Step 3: Strictness survives. By (2.11), a unit vector v_N in
the line a_{psi_N}^\perp \cap ker ds_j converges uniformly to
the old strict line a_phi^\perp \cap ker ds_j. The leading
Hessian correction vanishes on v_N (because v_N is in ker ds_j),
and its value on a_{psi_N} is O(N^{-2}) because
ds_j(a_{psi_N}) = O(N^{-1}). The remainder of the Hessian is
O(N^{-1}). Therefore old strictness survives with a smaller
but positive margin for large N. Near a critical point of
phi (which does not occur on the patch by hypothesis a_phi != 0),
the argument is not applied.
Step 4: Fullness on a fraction of the patch. Consider the
symmetric form on a_phi^\perp:
B = (1 + |a_phi|^2) Hess phi|_{a_phi^\perp} + Hess
phi (a_phi, a_phi) * Id|_{a_phi^\perp} . (2.12)
By strictness of phi, at least one of the two eigenvalues of B
is > 0. Choose an orthonormal frame (v, w) of a_phi^\perp with
ker ds_j \cap a_phi^\perp = span(v); then the (v,v) entry of B
is B_vv >= m > 0 for some patch-data m > 0, and the (w,w) entry
satisfies B_ww >= -M for some finite M. All entries of B are
bounded on P_j.
Add the leading correction C(1 + |a_phi|^2) |ds_j|^2 (w^\flat
\otimes w^\flat) to B. This is a positive semidefinite
rank-one addition aligned with w. If |B_vw| <= M', it suffices
to make the (w,w) entry exceed (M')^2 / m + m; then by the
Schur complement criterion B becomes positive definite on the
whole a_phi^\perp. Since |ds_j|^2 has a positive minimum on
the patch, one finite C suffices (independent of N).
Where chi_j = 1 and h''(N s_j) > 1 (i.e. N s_j is in E_h), the
actual leading addition to psi_N's Hessian at a_phi^\perp is >=
the addition above, so fullness holds. By Lemma 2.3, the set
of such points in C_j has mass at least (p_h/2) of mu(C_j) for
sufficiently large N.
Step 5: Assemble over patches. Perturb all patches at once,
with a common N sufficiently large. The sum of gradients of
the perturbations is small in C^1 (bounded by C N^{-1} times
fixed constants), and the sum of Hessians is supported in the
union of patches. The
union of the "full patches" G_N = union_j {x in C_j : N s_j(x)
in E_h} has mass >= (p_h/4) mu(Omega) by Lemma 2.2 (choose the
patches capturing half the mass) and Lemma 2.3 (each core
captures at least half of its mu on the full set). Since
|a_psi| <= (1 + eps_p) |a_phi| pointwise and |a_psi| >= (1 -
eps_p) |a_phi| pointwise, and eps_p <= 1/3, we have
integral_{G_N} |a_psi| dV_g
>= (1 - eps_p) * (p_h/4) * integral_Omega |a_phi| dV_g
>= (p_h/8) * integral_Omega |a_psi| dV_g .
This is (2.9). Since strictness is an open condition, we
obtain an open full region U containing G_N.
Step 6: C^0/C^1 smallness. The perturbation has size at most
C ||h||_inf N^{-2}, which tends to zero. Its C^1 size is at
most C ||h'||_inf N^{-1}, also tending to zero. By taking N
larger, we get |psi - phi|_{C^0} < eps and |psi - phi|_{C^1} as
small as we like. End of proof.
Informal summary of Lemma 2.4: a small, C^1-clean, transverse
perturbation makes the profile full on a fixed positive
fraction of its gradient mass. The full region is where we
will later corrugate the gradient.
2.5 STEP 2: RADIAL CORRUGATION IN TWO TRANSVERSE DIRECTIONS
The preparation Lemma 2.4 gives a profile psi that is full on an
open set U of fixed gradient mass. Now we add a corrugation on
subsets of U, using the two-dimensional radial function F of
Lemma 2.7 (equation 2.7), to increase the gradient on a fixed
fraction of U WITHOUT losing strictness.
Setup. On a relatively compact coordinate patch P (locally
identified via a chart with an open subset of R^3), suppose psi
is full with nonvanishing gradient on a neighbourhood of
closure(P). Suppose further we have a smooth map
y = (y_1, y_2) : P -> R^2 (2.13)
which is a submersion on P (both nondegenerate) and satisfies:
dy (a_psi) = 0 (i.e. a_psi lies in ker dy), (2.14a)
both nonzero singular values of dy at every point of
P lie in [1/2, 2] . (2.14b)
(Such a y exists near any noncritical point of psi in
dimension 3: the flow-box theorem lets us choose the two
"transverse" functions, and after rescaling we get the singular
value bound.)
Fix a constant A > 0 satisfying
1/2 <= A / |a_psi(x)| <= 2 for all x in P. (2.15)
Such A exists because |a_psi| is bounded above and below on P
by positive finite constants. Fix a cutoff 0 <= chi <= 1 in
C_c^infinity(P).
THE CORRUGATION. For a parameter N >= 1 and a small fixed
delta > 0 (chosen below), put
Phi_N(x) = psi(x) + (delta A / N) chi(x) F( N y(x) ). (2.16)
This is smooth, has compact support in P, and converges to psi
in C^0 uniformly as N -> infinity.
LEMMA 2.5 (Radial corrugation in transverse directions).
There exist constants delta > 0 and g_0 > 0, depending only on
the fixed function F, with the following property. Let psi, y,
A, chi be as above. Let eps_r > 0 be a small constant. For
all sufficiently large N, Phi_N is strict with nonvanishing
gradient on closure(P). Moreover:
|a_{Phi_N}| >= (1 - eps_r) |a_psi| everywhere on P, (2.17a)
|a_{Phi_N}| >= (1 + g_0 - eps_r) |a_psi|
wherever chi = 1 and N y(x) in E_F. (2.17b)
PROOF.
Step 0: Working in the transverse two-plane. Denote by d_R, d_T
the radial and tangential unit covectors in y-space: at a point
with y = s exp_top(i theta) (s = |y|), d_R = <e_R, dy> and
d_T = <e_T, dy> where e_R = (cos theta, sin theta) and e_T =
(-sin theta, cos theta). The singular-value bound (2.14b)
implies |d_R|, |d_T| between 1/2 and 2 as covectors in the dual
metric. These covectors need not be orthogonal in the metric
of P, but their lengths and the angle between them are bounded
above and below by fixed constants.
Step 1: Gradient and Hessian of Phi_N. When N y(x) lies in a
periodic disk centered at a lattice point, take the local radial
and angular frame just described. Outside these disks, set the
leading derivatives f', f'', and f'/s to zero. In the constant
central portions of the disks, choose any orthonormal pair when
such notation is used. Set f'/s = 0 at the center (where f is
constant). Direct differentiation of (2.16) gives:
a_{Phi_N} = a_psi + q d_R^sharp + e_N , (2.18a)
q = delta A chi(x) f'(N|y|) , (2.18b)
e_N = (delta A / N) F(N y) grad chi , (2.18c)
Hess Phi_N = Hess psi
+ N delta A chi (x) [ f''(N|y|) d_R \otimes d_R
+ (f'/s)(N|y|) d_T \otimes d_T ]
+ E_N , (2.19)
where E_N is a remainder whose magnitude is bounded by
|E_N| <= C_P ( (chi + |grad chi|) f' + N^{-1} ) . (2.20)
Here a^\sharp means the metric-dual of a covector, so <a^\sharp,
v> = a(v) for every tangent vector v; d_R^\sharp is the vector
field dual to the covector d_R. See derivation below.
The remainder (2.20) comes from three sources: the terms
(delta A chi) (partial_i F) Hess y_i for i = 1, 2; the
terms (delta A) (d chi \otimes (partial_i F) d y_i + c.c.);
and the term (delta A / N) F(Ny) Hess chi. Since
|(F_1, F_2)| = f' in the radial annulus (and 0 elsewhere), each
term is bounded by the displayed combination.
Note: the gradient correction q d_R^\sharp is perpendicular
to a_psi, because dy(a_psi) = 0 and d_R = <e_R, dy> so
d_R(a_psi) = <e_R, dy(a_psi)> = 0. This orthogonality is what
gives the clean amplification in Step 4.
Step 2: Strictness, when the "large-T" regime dominates.
Put t(x) = chi(x) f'(N |y(x)|) and s(x) = N |y(x)|. Consider
the region where t(x) >= T_0 for a finite threshold T_0 to be
fixed. In that region, f'(s) >= c_1 > 0 and s <= R. Choose a
vector b_N(x) in T_x P perpendicular to a_{Phi_N}(x), with
d_R(b_N) = 0 and |b_N| = L_N := sqrt(1 + |a_{Phi_N}(x)|^2).
Existence: the two linear constraints (b_N perpendicular to
a_{Phi_N}, d_R(b_N) = 0) have a common nonzero solution in
dimension 3 by elementary linear algebra. Scaling to length
L_N gives b_N.
We need to bound |d_T(b_N)| from below. Two facts:
(i) |d_T(b_N)| = |dy(b_N)| because d_T is the projection of
dy onto the e_T direction, and d_R(b_N) = 0 removes the
e_R component.
(ii) |a_N - a_psi| = |q d_R^\sharp + e_N| = O_P(f' + N^{-1})
by (2.18a, 2.18c). And |a_psi| has a positive
minimum |a|_min on P. So the projection of b_N / L_N
onto the orthogonal complement of a_psi in T_x P is
close to b_N / L_N.
On the orthogonal complement of a_psi (a two-dimensional
subspace of T_x P), dy is injective. Its singular values lie
in [1/2, 2]. Therefore for any vector u in that subspace with
|u| = 1 we have |dy(u)| >= 1/2. Combining (i) and (ii):
|d_T(b_N)| >= L_N / 4 for N large enough.
Now compute the strictness expression.
We want to show
Hess Phi_N (a_{Phi_N}, a_{Phi_N})
+ Hess Phi_N (b_N, b_N) > 0 . (2.21)
Split Hess Phi_N = Hess psi + N delta A chi [ f'' d_R^2 + (f'/s)
d_T^2 ] + E_N.
Contribution of Hess psi: <= C_P for some fixed constant C_P.
Contribution of E_N: <= C_P (t + sqrt(t) + N^{-1}) (see (2.20)
and the bound in (2.18)).
Contribution of the leading N delta A chi term:
N delta A chi [ f'' (d_R(a_{Phi_N})^2 + d_R(b_N)^2)
+ (f'/s) ( d_T(a_{Phi_N})^2 + d_T(b_N)^2 ) ].
Now d_R(b_N) = 0 by construction, so the second term vanishes.
|d_R(a_{Phi_N})| <= |q| + O_P(N^{-1}) <= delta A f' + O_P(N^{-1}).
So the negative part is bounded by
N delta A chi * ||f''||_inf * (delta A f')^2 * (some
fixed constant)
which by the differential inequality |f''| <= C sqrt(f') is
bounded by
N delta^3 A^3 ||f''||_inf * C f'^{3/2} ... etc.
Actually cleaner: bound |f''| <= C_F' (some fixed constant, the
sup of |f''| on [r_0, R]), and use |d_R(a_{Phi_N})| <= 2 delta
A f' (with a factor absorbed into the universal constant). Then
|negative contribution| <= N delta A chi * ||f''||_inf *
(2 delta A f')^2
<= 4 delta^3 A^3 ||f''||_inf f'^2 N.
Meanwhile the positive (f'/s) |d_T(b_N)|^2 term is at least
N delta A chi * (f'/s) * (L_N^2 / 16)
>= N delta A chi * (c_1 / R) * (1/16) . (2.22)
(using s <= R and f'(s) >= c_1 on the annulus). Since we took
chi and N large enough that |a_{Phi_N}| is comparable to |a_psi|
which is bounded below (2.15), L_N^2 is bounded below, so the
right side of (2.22) grows linearly in N.
Therefore, if we choose delta small enough (once and for all,
depending only on F and on the constant in (2.15)) that
4 delta^3 A^3 ||f''||_inf <= (1/32) * c_1 / R (2.23)
then for large N the positive (f'/s)-contribution dominates the
negative f''-contribution, and the base and remainder terms
(which are bounded) are swamped. This gives (2.21).
Similarly, when t(x) is BOUNDED (t <= T_0), we use the OTHER
transverse direction, the one pointing into ker d_R and perp to
a_{Phi_N}. There, by fullness of psi, the base Hessian
contribution Hess psi(a_psi, a_psi) + Hess psi(b, b) is at
least the strict margin m > 0. Meanwhile the leading added
Hessian in the b direction is (f'/s) * 0 = 0 (because we chose
b in ker d_R, so d_T(b) = 0 ... actually d_T(b) needn't be 0,
only d_R(b) = 0; more carefully: choose the "reserved" transverse
direction e_2 in the construction, which is not in the corrugation
plane). See Lemma 2.4 for the fullness mechanism; the same
Schur-complement argument shows that fullness holds.
Actually the argument in the original paper is: when t is
bounded, we choose the reserved transverse direction e_2 from
the definition of full, which for large N becomes approximately
a fixed direction transverse to a_{Phi_N}. Because t is
bounded, the leading addition chi f' is bounded, and by
fullness of psi the base Hessian dominates. This is where
fullness of psi (not just strictness) is used — the point of
Lemma 2.4.
Step 3 (gist): The two regimes cover all of P. Both are strict
with a positive margin, uniformly in the patch.
Step 4: Amplification of the gradient. In the leading
gradient formula (2.18a), reuse that q d_R^\sharp is
perpendicular to a_psi. So
|a_{Phi_N}|^2 = |a_psi|^2 + q^2 |d_R^\sharp|^2
+ 2 <a_psi, e_N> + |e_N|^2 .
Since |d_R^\sharp| >= 1/2 and <= 2, |q| <= delta A f', and
A >= |a_psi| / 2, the corrugation adds to |a_{Phi_N}|^2 at least
q^2 / 4 in magnitude on the annulus set (chi = 1, N y in E_F).
So
|a_{Phi_N}| >= |a_psi| * sqrt(1 + ( delta c_1 / 4 )^2 ) =:
(1 + g_0) |a_psi| , (2.24)
where g_0 := sqrt(1 + (delta c_1 / 4)^2) - 1 > 0 depends only
on F and the constant 4 from (2.14b) and (2.15). Choosing delta
small if necessary to make g_0 <= 1.
Outside the annulus, q = 0, so |a_{Phi_N}| = |a_psi| + e_N. The
error e_N has size O_P(N^{-1}), so for large N we have
|a_{Phi_N}| >= (1 - eps_r)|a_psi| everywhere. This proves
(2.17a) and (2.17b).
End of proof of Lemma 2.5.
The key quantitative features:
* The gain g_0 > 0 and the parameter delta > 0 depend only
on the fixed function F, not on psi, a, A, or N.
* The annulus set E_F has a fixed positive fraction of mass
in the transverse periodic cell (p_F = pi(r_2^2 -
r_1^2) > 0).
* The corrugation looks invisible in C^0: its size is
delta A / N * ||F||_inf -> 0 as N -> infinity.
So we can iterate the corrugation to amplify the gradient as
much as we want, at arbitrarily small C^0 cost.
2.6 STEP 3: THE AMPLIFICATION PROPOSITION
Now assemble the two steps into a single proposition that
multiplies the gradient mass by (1 + eta) for some FIXED eta > 0
independent of the incoming psi.
PROPOSITION 2.6 (One amplification cycle).
Under the hypotheses of Lemma 2.5, fix a small positive
C^0-budget eps > 0 for the cycle. There is a smooth function
Phi = Phi_N (N sufficiently large) such that:
supp(Phi - psi) \Subset P ,
|Phi - psi|_{C^0(P)} < eps ,
Phi is strict with nonvanishing gradient on closure(P),
integral_P |a_Phi| dV_g >= (1 + eta) integral_P |a_psi| dV_g, (2.25)
where eta > 0 depends only on the fixed functions F and h.
DEFINITIONS USED IN THE PROOF.
Set
theta = p_h * p_F / 64 > 0, (2.26a)
t_* = theta * g_0 > 0, (2.26b)
eta = t_* / 4 > 0 . (2.26c)
Here p_h = 1/3 is the fraction of E_h, p_F = pi(r_2^2 - r_1^2) is
the fraction of E_F, and g_0 is from (2.24).
PROOF.
Step 1: Preparation. Apply Lemma 2.4 with
eps_p <= min( 1/3, t_* / 16 ),
eps <= eps / 2,
to obtain psi_1, its full region U_1, and the bounds
integral_{U_1} |a_{psi_1}| >= (p_h/8) integral_P |a_{psi_1}|,
|a_{psi_1} - a_psi| <= eps_p |a_psi| pointwise.
Now (1 - eps_p) |a_psi| <= |a_{psi_1}| <= (1 + eps_p) |a_psi| on P.
Step 2: Locate patches in U_1 where the corrugation runs.
On a small neighbourhood of each point of U_1, the flow-box
construction provides two functions y = (y_1, y_2) with
dy(a_{psi_1}) = 0,
and with singular values of dy in [1/2, 2] on the patch. Rescale
and choose a local constant A_patch in [1/2, 2] * |a_{psi_1}|,
so the ratio A_patch / |a_{psi_1}| is in [1/2, 2].
Apply Lemma 2.2 to these patches with measure
dmu = |a_{psi_1}| dV_g restricted to U_1. This gives finitely
many disjoint patches P_j, cutoffs, and cores capturing at least
half the gradient mass in U_1. Combined with the estimate from
Step 1, the total mass in the cores is at least
(1/2) * (p_h/8) integral_P |a_{psi_1}| dV_g
= (p_h/16) integral_P |a_{psi_1}| dV_g.
Step 3: Corrugate. For each patch P_j, apply Lemma 2.5 with
delta = the universal delta from Lemma 2.5,
eps_r <= min( t_* / 8, g_0 / 2 ).
For sufficiently large common N, the corrugated function Phi_j
satisfies:
|a_{Phi_j}| >= (1 - eps_r) |a_{psi_1}| on the patch
|a_{Phi_j}| >= (1 + g_0 - eps_r) |a_{psi_1}|
on the "gain" set (chi = 1, N y in E_F).
The gain set within the cores has mass at least
p_F * (p_h/16) integral_P |a_{psi_1}| dV_g
= (p_h p_F / 16) integral_P |a_{psi_1}| dV_g
by Lemma 2.3 (Periodic averaging) applied with respect to
dmu = |a_{psi_1}| dV_g.
The remaining (non-gain) part of the cores has mass at most
the complement, so |a_{Phi_j}| there is at least
(1 - eps_r)|a_{psi_1}|.
Set Phi = Phi_1 + ... + Phi_J over the disjoint patches, and set
Phi = psi_1 outside the union of patches. Summing:
integral_P |a_Phi| dV_g
>= |gain set| * (1 + g_0 - eps_r) * (1 - eps_p) |a_psi|
+ |rest| * (1 - eps_r) * (1 - eps_p) |a_psi| .
Bound rigorously via the pointwise estimates:
integral_P |a_Phi| >= (1 - eps_r) integral_P |a_{psi_1}|
+ (g_0 - 2 eps_r) integral_{gain set}
|a_{psi_1}| .
Now integral_{gain set} |a_{psi_1}| >= (p_h p_F / 16)
integral_P |a_{psi_1}| dV_g. Therefore
integral_P |a_Phi| >= (1 - eps_r) integral_P |a_{psi_1}|
+ (g_0 - 2 eps_r) (p_h p_F / 16)
integral_P |a_{psi_1}| .
Choose eps_r so small that both constraints hold simultaneously:
eps_r <= t_* / 8 and 5 eps_r <= g_0 .
Then
1 - eps_r >= 1 - t_*/8 ,
g_0 - 2 eps_r >= g_0 - (2/5) g_0 = (3/5) g_0 > g_0 / 2 .
Using these and (p_h p_F / 16) for the gain-set fraction,
integral_P |a_Phi|
>= (1 - t_*/8) integral_P |a_{psi_1}|
+ (g_0/2) (p_h p_F / 16) integral_P |a_{psi_1}|
= (1 - t_*/8 + (p_h p_F g_0) / 32) integral_P |a_{psi_1}|
= (1 - t_*/8 + 2 t_*) integral_P |a_{psi_1}|
= (1 + 15 t_*/8) integral_P |a_{psi_1}| .
Here we used t_* = p_h p_F g_0 / 64 from (2.26b).
Step 4: Transfer to psi. By Step 1,
|a_{psi_1}| >= (1 - eps_p) |a_psi| on P, and eps_p <= t_*/16,
so
integral_P |a_{psi_1}| >= (1 - t_*/16) integral_P |a_psi| . (2.27)
Multiplying:
integral_P |a_Phi|
>= (1 - t_*/16) (1 + 15 t_*/8) integral_P |a_psi|
= (1 + 15 t_*/8 - t_*/16 - 15 t_*^2 / 128)
integral_P |a_psi|
= (1 + 29 t_*/16 - 15 t_*^2 / 128)
integral_P |a_psi| .
For 0 < t_* <= 1 we have 15 t_*^2 / 128 <= t_*/8, so the coefficient
is at least
1 + 29 t_*/16 - t_*/8 = 1 + 27 t_*/16 >= 1 + t_*/4 .
Setting eta := t_*/4 (concretely p_h p_F g_0 / 256), we get
integral_P |a_Phi| >= (1 + eta) integral_P |a_psi| . (2.28)
Note: eta depends only on the fixed functions h, F (through the
constants p_h, p_F, g_0) and NOT on psi, phi, the frequency, or
anything downstream. This is why the cycle can be iterated
arbitrarily often with a fixed multiplicative gain each time.
Step 5: Smallness in C^0. The corrugation has size at most
delta A_max ||F||_inf N^{-1}, which tends to zero. The preparation
has size at most eps/2 by Lemma 2.4. By taking N large enough,
the sum has size < eps. All other asserted properties are open
and hold for large N.
End of proof of Proposition 2.6.
The key concrete constant is eta = t_* / 4 = (p_h p_F g_0) / 256,
a fixed positive number that does NOT depend on psi, on the scale
of psi, on the incoming profile phi, on the frequency, or on
anything downstream. This is why the corrugation can be
ITERATED as many times as we like with a uniform gain each time.
...
2.7 PROFILES ON THE ROUND SPHERE
We now apply the amplification machinery of Section 2.6 to the round
sphere S^3.
SETUP. Choose a coordinate cube Omega on S^3 whose closure is
compactly contained in the open region
Omega_0 = { 1/20 < r < 1/8 } , with r = (x1^2 + x2^2)^{1/2}.
(2.32)
Fix 1/5 <= t <= 3/10. The specific constants 1/20 and 1/8 keep
Omega_0 away from the singular locus { r = 0 } of the coordinate r,
and near the Euclidean model. The base profile phi_0 : S^3 -> R is
phi_0(x) = log t + r(x)^2 - t^2 . (2.33)
LEMMA 2.8 (Base profile phi_0). Fix t as in (2.33). Set
rho = r^2 = x1^2 + x2^2, so 0 <= rho <= 1 on S^3.
(a) phi_0 is smooth, and grad_{g_*} phi_0 = 0 exactly on the
circle { rho = 0 } = { r = 0 }.
(b) On { 0 < rho <= 0.09 } phi_0 is strict. In fact, at any
point with a = grad_{g_*} phi_0 != 0, choosing b to be
sqrt(1 + |a|^2) times a unit vector perpendicular to a
in the "horizontal" tangent plane (the span of e_r, e_theta)
gives a strictness margin
Hess_{g_*} phi_0 (a, a) + Hess_{g_*} phi_0 (b, b)
>= 1.64 > 1 . (2.34)
(c) At rho = 0, Hess_{g_*} phi_0 = 2 (dx1^2 + dx2^2) restricted
to T_x S^3, positive semidefinite of rank two with the
"vertical" direction e_psi = (-x4 dx3 + x3 dx4) /
sqrt(1-rho) in its kernel.
(d) phi_0 > log r on { 0 < r < t }, and phi_0 < log r on
{ t < r <= 1 }. phi_0 = log r exactly on { r = t }.
PROOF. Since phi_0 = log t + rho - t^2 depends on rho only and
phi_0'(rho) = 1, we have grad phi_0 = grad rho and Hess phi_0 =
Hess rho. It suffices to compute for rho.
Step 1: gradient. The ambient R^4-gradient of rho is
grad_{R^4} rho = 2(x1, x2, 0, 0). Its projection onto T_x S^3 is
grad_S rho = 2(x1, x2, 0, 0) - 2 rho x ,
(2.35)
using the formula grad_S f = grad_{R^4} f - <grad f, x> x.
Squaring in R^4:
|grad_S rho|^2
= 4 [(x1^2 + x2^2)(1 - rho)^2 + rho^2 (x3^2 + x4^2)]
= 4 [rho (1-rho)^2 + rho^2 (1 - rho)]
= 4 rho (1-rho) . (2.36)
This is 0 iff rho = 0 iff r = 0. Together with phi_0'(rho) = 1,
this proves (a).
Step 2: Hessian. For tangent vectors v, w in T_x S^3, the intrinsic
S^3-Hessian of rho is
Hess_S rho (v, w) = < D_v grad_S rho , w >_{R^4}
(the projection cancels by tangency of w). Differentiating (2.35):
D_v grad_S rho
= 2(v1, v2, 0, 0) - 4(x1 v1 + x2 v2) x - 2 rho v .
Pairing with w in R^4 (valid since w is tangent, so the projection
onto S^3 is the identity on w):
Hess_S rho (v, w)
= 2(v1 w1 + v2 w2)
- 4(x1 v1 + x2 v2)(x1 w1 + x2 w2 + x3 w3 + x4 w4)
- 2 rho (v1 w1 + v2 w2 + v3 w3 + v4 w4) .
The second bracket is <x, w> = 0 since w is tangent. Hence
Hess_S rho (v, w)
= 2(1 - rho)(v1 w1 + v2 w2) - 2 rho (v3 w3 + v4 w4) . (2.37)
Step 3: diagonalize in the frame. Parameterize S^3 as
x = ( r cos theta, r sin theta,
sqrt(1-r^2) cos psi, sqrt(1-r^2) sin psi ) ,
which gives the round metric ds^2 = dr^2/(1-r^2) + r^2 d theta^2
+ (1-r^2) d psi^2. An orthonormal frame is
e_r = sqrt(1-rho) partial_r (in coordinates) ,
e_theta = (-sin theta, cos theta, 0, 0) in ambient coordinates ,
e_psi = (0, 0, -sin psi, cos psi) in ambient coordinates .
Applying (2.37) to these:
e_theta: (v1, v2) = (-sin theta, cos theta), S_1,2 = 1, S_3,4 = 0.
Hess_S rho (e_theta, e_theta) = 2(1-rho) .
e_psi: 0 in slots 1, 2; 1 in slots 3, 4.
Hess_S rho (e_psi, e_psi) = -2 rho .
e_r: 1-rho in slots 1, 2 (from (c cos, c sin) with c = sqrt(1-rho));
rho in slots 3, 4 (from (-r cos, -r sin) with r = sqrt(rho)).
Hess_S rho (e_r, e_r) = 2(1-rho)^2 - 2 rho * rho = 2(1-2rho) .
So in the frame (e_r, e_theta, e_psi) the Hessian is diagonal:
Hess_{g_*} phi_0 = diag( 2(1-2rho), 2(1-rho), -2rho ) . (2.38)
Step 4: componentwise consequences.
(c) At rho = 0, (2.38) reads diag(2, 2, 0) with the third slot
e_psi in its kernel. This is positive semidefinite of rank two.
(b) Let a = grad phi_0 = 2 sqrt(rho(1-rho)) e_r (unit-length
a_hat = e_r times 2 sqrt(rho(1-rho))). Pick b = sqrt(1 + |a|^2) e_theta.
Then b is perpendicular to a, and |b|^2 = 1 + |a|^2. Compute
Hess (a, a) = |a|^2 · Hess(e_r, e_r)
= 4 rho (1-rho) · 2 (1-2rho)
= 8 rho (1-rho)(1-2rho) .
Hess (b, b) = (1 + |a|^2) · Hess(e_theta, e_theta) (unit-length e_theta)
= (1 + 4 rho (1-rho)) · 2 (1-rho) .
Summing:
Hess (a, a) + Hess (b, b)
= 8 rho (1-rho)(1-2rho) + 2(1-rho)(1 + 4 rho(1-rho))
= 2(1-rho) [ 4 rho (1-2rho) + 1 + 4 rho(1-rho) ]
= 2(1-rho) [ 4 rho - 8 rho^2 + 1 + 4 rho - 4 rho^2 ]
= 2(1-rho) [ 1 + 8 rho - 12 rho^2 ] . (2.39)
For rho <= 0.09: the two factors are (1-rho) >= 0.91 and
1 + 8 rho - 12 rho^2 >= 1 + 0 - 12(0.09)^2 >= 0.9028 . Product
>= 2 (0.91)(0.9028) >= 1.64 > 1.
(d) Set h(r) = phi_0 - log r = log t + r^2 - t^2 - log r.
Then h(t) = 0 and h'(r) = 2r - 1/r. For 0 < r < 1/sqrt(2),
h'(r) < 0. Since t <= 3/10 < 1/sqrt(2), we see h is decreasing on
(0, t] and on (t, 1/sqrt(2)). Therefore
for r in (0, t): h(r) > h(t) = 0 (decreasing ends at 0),
for r in (t, 1/sqrt(2)): h(r) < h(t) = 0 (decreasing starts at 0).
For 1/sqrt(2) < r <= 1, h is increasing, so the maximum of h on
(1/sqrt(2), 1] is attained at r = 1. Compute h(1) = log t + 1 - t^2.
For t in [1/5, 3/10], log t <= log(3/10) < -1.20 and 1 - t^2 <= 0.96,
so h(1) <= -0.24 < 0. Hence h < 0 on (1/sqrt(2), 1], in fact on
(t, 1]. This gives (d). []\n\n\nEND_OF_LEMMA_2_8\n\n\n\nWe can now amplify phi_0 to profiles with arbitrarily large\ngradient integral.\n\n\nPROPOSITION 2.10 (Spherical profiles). There is a C^2 neighborhood\nU of g_* on S^3, and a constant c_u > 0, with the following property.\nFix 1/5 <= t <= 3/10. For every g in U, every T > 0, and every\neps > 0, there is a smooth function phi on S^3 such that\n\n (i) supp(phi - phi_0) is compactly contained in Omega;\n (ii) ||phi - phi_0||_{C^0(S^3)} < eps;\n (iii) phi is strict on { 0 < r <= 3/10 } with a UNIFORM\n strictness margin >= c_u > 0;\n (iv) the critical set of phi on { r <= 3/10 } is exactly the\n circle { r = 0 }; at each critical point, Hess_g phi is\n positive semidefinite of rank at least two;\n (v) phi > log r on { 0 < r < t }, and phi < log r on\n { t < r <= 1 };\n (vi) integral_Omega |grad_g phi|_g dV_g > T.\n\nPROOF. Choose U small enough that each g in U satisfies all of:\n\n * phi_0 is strict on { 0 < r <= 3/10 } for the metric g, with\n margin >= c_u. (By continuity in the C^2 metric topology from\n Lemma 2.8(b), which gives margin >= 1.64 for g = g_*.)\n * On Omega, the round metric g_* and g are uniformly comparable:\n K^{-1} g_* <= g <= K g_* for a fixed K < infinity.\n * The coordinate functions x1, x2, r, rho on Omega remain smooth\n and bounded away from critical points of phi_0.\n\nFix g in U. Start with phi^(0) := phi_0. For k = 1, 2, 3, ...,\napply Proposition 2.6 (One amplification cycle) inside Omega. Each\niteration produces phi^(k) with:\n\n - supp(phi^(k) - phi^(k-1)) compactly contained in Omega;\n - ||phi^(k) - phi^(k-1)||_{C^0(Omega)} < eps / 2^k;\n - integral_Omega |grad_g phi^(k)|_g dV_g\n >= (1 + eta) integral_Omega |grad_g phi^(k-1)|_g dV_g,\n where eta = p_h p_F g_0 / 256 is the SAME at every step;\n - phi^(k) remains strict on the patch (with a margin that we\n shrink from c_u to c_u/2; see the note at the end).\n\nLet K := the smallest integer with (1+eta)^K integral_Omega |grad_g phi_0|_g dV_g > T\nand also K >= 1. Define phi := phi^(K).\n\nVerification.\n\n(i) Each supp(phi^(k) - phi^(k-1)) is compactly inside Omega, and Omega\nis compactly in S^3 \\ { r = 0 }, so the total support of phi - phi_0\nis compactly inside Omega.\n\n(ii) ||phi - phi_0||_{C^0} <= sum_{k=1}^K ||phi^(k) - phi^(k-1)||\n < sum_k eps / 2^k < eps.\n\n(iii) Iterated Proposition 2.6 preserves strictness with margin\n>= c_u/2 (we shrink the guaranteed margin inside the support of the\ncorrugation). Outside Omega, phi = phi_0 which is strict with\nmargin >= c_u by choice of U. So a uniform margin c_u/2 holds.\n\n(iv) Because Omega is disjoint from { r = 0 }, on { r = 0 } we have\nphi = phi_0. By Lemma 2.8(c), Hess_g phi = Hess_g phi_0 at r = 0 is\nPSD rank two. On { r <= 3/10 } \\ Omega, phi = phi_0 whose gradient\nvanishes only at r = 0. On Omega, Proposition 2.6 gives nonvanishing\ngradient. So the critical set of phi in { r <= 3/10 } is exactly\n{ r = 0 }.\n\n(v) Because supp(phi - phi_0) is inside Omega, which sits at\n1/20 < r < 1/8, we have phi = phi_0 on { 0 < r < 1/20 } and on\n{ 1/8 < r <= 1 }. By Lemma 2.8(d), phi_0 already satisfies (v) on\nthese regions. On the "gap" { 1/20 <= r <= 1/8 }, phi differs from\nphi_0. Shrink eps -- which is a free parameter -- so that eps is\nstrictly smaller than min( min_{1/20<=r<=1/8}(phi_0 - log r),\nmin_{1/20<=r<=1/8}(log r - phi_0) ). Both minima are positive by\nLemma 2.8(d). Then the C^0 bound ||phi - phi_0|| < eps preserves\nthe sign of phi - log r everywhere, so (v) continues to hold. If\nthe original eps was larger, just use the smaller value -- this only\nmakes (ii) easier.\n\n(vi) integral_Omega |grad_g phi|_g dV_g\n >= (1+eta)^K integral_Omega |grad_g phi_0|_g dV_g > T,\n by construction of K. In fact integral_Omega |grad_g phi_0|_g dV_g\n is bounded below by a fixed positive constant (Lemma 2.8(b) and\n K-invariance of the metrics), so K can be chosen finite.\n\n []\n\nNOTE. The strictness margin drops from c_u (for phi_0) to c_u/2\n(after corrugation). This is why we advertise c_u/2 as the working\nconstant in (iii). Downstream in Part 3 only a POSITIVE margin is\nneeded; no specific value of c_u enters the wave construction.\n\nEND OF SECTION 2.7 AND PART 2.\nP2END wc -l yau_nodal_smooth_counterexamples_v2.txt
(t, 1]. This closes Lemma 2.8(d). []
END OF LEMMA 2.8.
We can now amplify phi_0 to profiles with arbitrarily large gradient
integral.
PROPOSITION 2.10 (Spherical profiles). There is a C^2 neighborhood
U of g_* on S^3, and a constant c_u > 0, with the following property.
Fix 1/5 <= t <= 3/10. For every g in U, every T > 0, and every
eps > 0, there is a smooth function phi on S^3 such that
(i) supp(phi - phi_0) is compactly contained in Omega;
(ii) ||phi - phi_0||_{C^0(S^3)} < eps;
(iii) phi is strict on { 0 < r <= 3/10 } for the metric g,
with a uniform strictness margin >= c_u > 0;
(iv) the critical set of phi on { r <= 3/10 } is exactly the
circle { r = 0 }; at each such point the metric-g
Hessian of phi is positive semidefinite of rank >= 2;
(v) phi > log r on { 0 < r < t }, and phi < log r on
{ t < r <= 1 };
(vi) integral_Omega |grad_g phi|_g dV_g > T.
PROOF OF PROPOSITION 2.10.
Setup. Choose U small enough that every g in U satisfies:
* phi_0 is strict on { 0 < r <= 3/10 } for g, with margin >= c_u.
This exists by Lemma 2.8(b) (round metric margin >= 1.64) and
continuity in the C^2 metric topology.
* On Omega, g and g_* are uniformly comparable:
K^{-1} g_* <= g <= K g_*
for a fixed finite K.
* The coordinate functions x1, x2, r, rho on Omega have bounded
derivatives in both metrics.
Iteration. Fix g in U and start with phi^(0) := phi_0. For
k = 1, 2, 3, ..., apply Proposition 2.6 (One amplification cycle)
inside Omega. Each cycle returns phi^(k) with:
- supp(phi^(k) - phi^(k-1)) compactly inside Omega;
- ||phi^(k) - phi^(k-1)||_{C^0} < eps / 2^k;
- integral_Omega |grad_g phi^(k)|_g dV_g
>= (1 + eta) integral_Omega |grad_g phi^(k-1)|_g dV_g,
with eta = p_h p_F g_0 / 256 the same at every step;
- phi^(k) strict on the corrugation support with margin >= c_u/2.
Let K be the smallest integer with
(1 + eta)^K * integral_Omega |grad_g phi_0|_g dV_g > T
and also K >= 1. (2.40)
Set phi := phi^(K).
Verification.
(i) Support. Each supp(phi^(k) - phi^(k-1)) is compactly contained
in Omega, and Omega is compactly contained in S^3 minus the circle
{ r = 0 }. So supp(phi - phi_0) is compactly contained in Omega.
(ii) C^0 bound. ||phi - phi_0||_{C^0}
<= sum_{k=1}^K ||phi^(k) - phi^(k-1)||_{C^0}
< sum_{k=1}^infty eps / 2^k < eps.
(iii) Strictness. Inside supp(phi - phi_0), the iterated Proposition
2.6 gives strictness with margin >= c_u/2. Outside, phi = phi_0 which
is strict with margin >= c_u by choice of U. So uniform margin
c_u/2 > 0 works.
(iv) Critical set. Omega avoids { r = 0 }, so phi = phi_0 on
{ r = 0 }; by Lemma 2.8(c), Hess_g phi at { r = 0 } equals the
(classical) Hessian of rho at rho = 0, which is PSD rank two.
On the rest of { r <= 3/10 }: either phi = phi_0 (whose only
critical point is { r = 0 } by Lemma 2.8(a)), or phi = phi^(k) on
Omega at a point where Proposition 2.6 preserved nonvanishing
gradient. So the critical set is exactly { r = 0 }.
(v) Envelope inequalities. Omega sits inside { 1/20 < r < 1/8 },
so phi = phi_0 holds on { 0 < r < 1/20 } and on { 1/8 < r <= 1 }.
By Lemma 2.8(d), phi_0 already satisfies (v) on those regions. On
the gap { 1/20 <= r <= 1/8 }, phi differs from phi_0. Since (v) is
an OPEN condition (strict inequalities) and phi_0 is uniformly (in
r) separated from log r on the gap by a positive margin d_gap, we
shrink eps (a free parameter) so that eps < d_gap. Then
||phi - phi_0||_{C^0} < eps < d_gap preserves the sign of
phi - log r throughout, so (v) still holds on the gap.
(vi) Gradient integral. Direct from (2.40) and the amplification
step. Note that integral_Omega |grad_g phi_0|_g dV_g is bounded
below by a fixed positive constant (uniform in g in U), so K can be
chosen finite for any T.
This completes the proof of Proposition 2.10. []
NOTE. The strictness margin c_u is halved by the corrugation, which
is why we only advertise c_u/2 in (iii). Only a POSITIVE margin is
needed downstream.
END OF SECTION 2.7 AND PART 2.
==============================================================
PART 3 -- LOCALIZED WAVES AND THE SIGN SCORE
==============================================================
3.1 A SIGN FUNCTIONAL BOUNDED BY NODAL MEASURE
Fix the metric g (chosen in the C^2-neighborhood U from Prop 2.10)
and the profile phi given by Prop 2.10 for this g, with gradient
mass M_g := int_Omega |grad_g phi|_g dV_g > T for a free T that
we will choose huge. Write
L(x) := sqrt( 1 + |grad_g phi(x)|_g^2 ) (x in Omega). (3.0)
Subdivide Omega, up to grid boundaries, into finitely many disjoint
open Euclidean-coordinate cubes P (a "partition" mathscr P) fine
enough that scalar lengths ell_P > 0 exist with
ell_P / 2 <= L(x) <= ell_P for all x in P. (3.1)
Such ell_P exist for a sufficiently fine subdivision by uniform
continuity of L. Fix eps > 0 small (compare Lemma 3.1(3.2) below
for how small). Set
d_P := eps / (n ell_P) for n = the frequency. (3.2)
For a continuous real function f and frequency n, define
F_n(f) := sum over P in mathscr P of sum_{i=1}^3
d_P^{-1} integral_{x in P : x + d_P e_i in P}
1_{ f(x) f(x + d_P e_i) < 0 } dx . (3.3)
The subdivision, the parameter eps, and n are part of the DEFINITION
of F_n and stay FIXED while the test function f varies; only f
changes when we compare F_n at different candidate waves.
LEMMA 3.1 (Properties of F_n).
(a) F_n : C^0(S^3) -> [0, infty] is lower semicontinuous with
respect to uniform convergence.
(b) F_n(c f) = F_n(f) for every nonzero constant c, and
F_n(alpha f) = F_n(f) for every everywhere-positive
continuous function alpha.
(c) There is a constant C_area, depending only on Omega and
the C^1-size of the neighborhood U of g_* on S^3, such
that for every continuous real f and every h in U,
F_n(f) <= C_area * H^2_h({f = 0}) . (3.4)
(d) There is a constant C_sign, same dependencies, such that
F_n(f) <= C_sign * n * int_Omega L(x) dx . (3.5)
PROOF OF (a). Suppose f_j -> f uniformly. Every x in the domain
where f(x) f(x + d_P e_i) < 0 has a small neighbourhood on which the
same strict inequality holds, by continuity. So the indicator set for
f is contained in the union over j >= J of the corresponding sets for
f_j, for sufficiently large J. Fatou and a finite partition argument
give F_n(f) <= liminf F_n(f_j). []
PROOF OF (b). Multiply both endpoints of the test: (c f)(x) * (c f)(x + e)
= c^2 f(x) f(x+e), so the sign is unchanged (c != 0). Similarly
(alpha f)(x)(alpha f)(x+e) = alpha(x) alpha(x+e) f(x) f(x+e), sign
unchanged since alpha > 0. []
PROOF OF (c). Fix one cube P and one direction i. Let Z := {f = 0}.
For any two points x, x + d_P e_i with opposite signs of f, the
Intermediate Value Theorem gives an intermediate point y on the
short segment [x, x + d_P e_i] with f(y) = 0.
Associate to each x in the opposite-sign set (call it A_{P,i}) one
such y = y(x). The map x -> y(x) is injective when we slice in
direction e_i (feet of segments through different base points are
disjoint in the transverse directions but not in the e_i direction;
once we slice over e_i's parameter s, slices are disjoint).
Formally, slice the cube P by planes perpendicular to e_i. For each
slice at parameter s in (0, d_P), the (d-1)-dimensional slice
intersects A_{P,i} in a set whose projection to Z_{s+d_P, slice}
is a subset of Z, at measure <= H^{d-1} of that slice. Integrating
over s in (0, d_P) and dividing by d_P:
d_P^{-1} * |A_{P,i}|
<= C * H^2_h( Z cap P ) , (3.6)
where C is the coordinate projection constant (harmless dimensional
scaling), and we use that the transverse Hausdorff 2-measure of a
slice of Z is at most H^2_h(Z cap P). Summing over i = 1, 2, 3 and
over disjoint cubes P, we get (3.4). []
PROOF OF (d). Bound each indicator by 1:
F_n(f) <= sum_P sum_{i=1}^3 d_P^{-1} |P| = 3 * sum_P d_P^{-1}|P|
= 3 * sum_P (n ell_P / eps) * |P|
= (3n/eps) * sum_P ell_P |P| .
By (3.1), ell_P <= 2 L(x) on P, so ell_P |P| <= 2 int_P L dx. Hence
F_n(f) <= (6n/eps) int_Omega L dx. Take C_sign = 6/eps. []
Note: the constant C_area in (3.4) is independent of f, of eps, of n.
The constants only depend on Omega and the metric comparability in U.
The bound (3.4) is what makes F_n a valid lower bound for the nodal
measure; the bound (3.5) is what makes the sum in (3.3) finite, so
that the expectation computations in Section 3.6 make sense.
3.2 ADMISSIBLE PHASE HESSIANS
Complex-phase waves require at each center a complex symmetric form
Q on T_y S^3 with
Q z = 0 and Re Q < H_y ,
where z = a + i b is the complex phase direction and H_y = Hess_g
phi(y). Strictness of phi is precisely what makes such a Q exist.
LEMMA 3.2 (Admissible phase Hessians). Let H be a real symmetric
form on a three-dimensional real vector space. Let z = a + i b with
a, b real, a perpendicular to b, |b|^2 = 1 + |a|^2, and z . z = -1
(equivalently <a,b>=0 and |b|^2 - |a|^2 = 1). Assume
(i) a != 0 and H(a,a) + H(b,b) > 0, or
(ii) a = 0 and H(b,b) > 0.
Then there exists a complex symmetric form Q on the same space with
Q z = 0 and Re Q < H (strict, as a symmetric form). (3.7)
Moreover, a choice satisfying (3.7) varies with bounded coefficients
and a positive real-part gap under small changes of H and z with
z . z = -1.
PROOF. Write Q = U + i D with U, D real symmetric. The condition
Q z = 0 reads
U a - D b + i (U b + D a) = 0 ,
equivalently the two linear relations
D b = U a , D a = - U b . (3.8)
Case a != 0. The vectors a, b are perpendicular by hypothesis, so
they span a 2-plane W. Choose an orthonormal basis (e_1, e_2) of W
with e_1 = a/|a|, e_2 = b/|b|. In this basis write D_{ij} = <D e_i, e_j>
and U_{ij} = <U e_i, e_j>. Relations (3.8) at e_1, e_2:
<D b, e_1> = <U a, e_1> gives |b| D_{21} = |a| U_{11} ;
<D a, e_2> = -<U b, e_2> gives |a| D_{12} = -|b| U_{22} .
Symmetry D_{12} = D_{21} requires
(|a|/|b|) U_{11} = -(|b|/|a|) U_{22} ,
which multiplies to
|a|^2 U_{11} + |b|^2 U_{22} = 0 . (3.9)
Conversely, (3.9) plus the off-diagonal prescriptions determines D
uniquely on W.
Choose
alpha := ( H(a,a) + H(b,b) ) / ( |a|^2 + |b|^2 ) > 0 ,
U := H - alpha Id . (3.10)
Then U_{11} = H_{11} - alpha, U_{22} = H_{22} - alpha, and
|a|^2 U_{11} + |b|^2 U_{22} = |a|^2 H_{11} + |b|^2 H_{22} - alpha(|a|^2+|b|^2)
= H(a,a) + H(b,b) - alpha(|a|^2 + |b|^2) = 0 by (3.10). So (3.9) holds.
Choose D on W by the off-diagonal prescriptions, and D = 0 on the
orthogonal complement W^perp (a 1-dimensional subspace of T_y S^3,
since we are in dim 3). Then (3.8) holds.
Finally, Re Q = U = H - alpha Id < H, since alpha > 0. []
Case a = 0. Then |b| = 1. Set U := 0 on Rb and U := -M Id on b^perp
with no mixed entries, and take D := 0. Then Ub = 0 and Da = 0, so
(3.8) holds, i.e., Q z = (U + i*0)(i b) = i Ub = 0. On Rb: Re Q = 0
< H(b,b) (positive by hypothesis). Off Rb: Re Q = -M Id < H for
sufficiently large positive M. []
Continuation assertion. In case a != 0 with alpha bounded below by a
positive constant (as long as the strict inequality H(a,a)+H(b,b) > 0
persists), the map (H, z) -> Q varies smoothly with locally bounded
coefficients. The gap H - Re Q = alpha Id stays uniformly positive
in a small neighbourhood of any such (H, z). The case a = 0 is
analogous. []
The proof gives the following useful consequence (to be used later):
at any point y in { |a_phi| > 0 } ∩ Omega, choosing any b perp a_phi
with |b|^2 = 1+|a|^2 and H(a,a)+H(b,b) > 0 (which exists by strictness),
Lemma 3.2 produces a usable complex Hessian. At critical points of phi,
where a_phi = 0, we use case (ii) with b having H(b,b) > 0; such b
exists by case (c) of Definition 2.1 (positive Hessian on some 2-plane).
3.3 FINITE COMPLEX WAVES
A wave is a smooth complex function Z = zeta * exp(n S) * V that looks
locally like a plane wave exp(n S) carrying a slowly-varying amplitude
V. The wave is not an eigenfunction; we will pointwise make the
residual as small as we like (in the scale exp(n phi) of the envelope).
Notation. Recall Lambda_n = n(n+2). We work in normal coordinates
based at p in K_1 = { r <= t_1 } (the wave region). The metric g is
smooth; its entries have uniform Taylor bounds in these coordinates.
Definitions. For a complex phase S and a complex amplitude V,
T S := grad S . grad S (bilinear complexified inner product),
E := T S + 1 (eikonal error), (3.3.1)
A V := 2 grad S . grad V + (Delta_g S + 2) V . (3.3.2)
Then a direct computation in normal coordinates shows
(Delta_g + Lambda_n)(exp(n S) V)
= exp(n S) [ n^2 E V
+ n (2 grad S . grad V + (Delta_g S + 2) V)
+ Delta_g V ]. (modulo Taylor remainders) (3.3.3)
The exact formula (3.3.3) is the engine. We will truncate S and V as
polynomials in (x - p) of chosen finite degree, arranged so that the
errors left over by (3.3.3), divided by exp(n phi), become O(n^{-D})
for any prescribed D.
LEMMA 3.3 (Local waves with order-D residual).
Fix integers m, D >= 0. There exist a ball B_p (radius rho > 0
depending only on the piece K_1, the metric and finite coordinate
jets; same rho works for all p in K_1) and, for each p in K_1 and
each ell in {0,1,2}, a smooth complex function Z_{p,ell} on S^3 of
the form
Z_{p,ell}(x) = zeta(x) * exp( n S_{p,ell}(x) ) * V_{p,ell}(x),
(3.3.4)
with S_{p,ell} and V_{p,ell} COMPLEX POLYNOMIALS in the normal
coordinates centred at p (translated to that centre), and a smooth
cutoff zeta with 0 <= zeta <= 1, zeta = 1 on a smaller concentric
ball of radius rho/2, support of zeta in B_p, support of all
derivatives of zeta in the shell rho/2 < |x - p| < rho.
Initial data for phases. Let z_0, z_1, z_2 be the three complex
vectors at p defined in (3.2) of the phase data section: z_ell =
a(p) + i b_ell where a = grad phi(p), b_0, b_1 are linearly
independent vectors in a^\perp of length L(p) = sqrt(1 + |a(p)|^2),
and b_2 = b'_0 is the "shifted" phase imaginary part corresponding
to real part a(p) + delta * nu(p) with delta = n^{-1/2} and nu(p) a
unit vector with a(p) = |a(p)| * nu(p) at noncritical points and
any chosen unit vector at critical points, with b'_0 scaled so that
|b'_0| = sqrt( 1 + |a(p) + delta nu|^2 ).
Then we require at the centre:
S_{p,ell}(p) = phi(p),
d S_{p,ell}(p) = z_ell (complexified covector),
Hess S_{p,ell}(p) = Q_ell (complexified Hessian), (3.3.5)
where Q_ell satisfies Q_ell z_ell = 0 and Re Q_ell < H_p (strictly,
with a positive margin kappa depending on the fixed profile, but NOT
on n), as built in Lemma 3.2. The chosen Q_ell varies with bounded
coefficients on K_1 in the finite cover.
Initial data for amplitudes. The first amplitude polynomial satisfies
V^{(0)}_{p,ell}(p) = 1, and all higher amplitudes V^{(j)}_{p,ell}(p) =
0 at the centre for j >= 1.
Localization. For every p in K_1 and every x in the support of Z_{p,ell}:
Re S_{p,ell}(x) - phi(x)
<= n^{-1/2} * dist_g(p, x)
- kappa * dist_g(p, x)^2, (3.3.6)
with kappa from the phase Hessian margin. This makes the wave live
in a ball of radius about n^{-1/2} around p, since at distance d the
exponent in exp(n * Re S) is at most n * phi - n * kappa * d^2.
Residual. On support of zeta (all x in B_p):
|grad^j (Delta_g + Lambda_n) Z_{p,ell}(x)|_g
<= C_j n^{-D} * exp( n phi(x) ), 0 <= j <= m. (3.3.7)
Derivative bounds. For each fixed finite order j:
|grad^j Z_{p,ell}(x)|_g <= C_j * n^j * exp(n phi(x)). (3.3.8)
All constants C_j depend on (g, phi, m, D) and the fixed finite
coordinate jets, but not on p, ell, or n.
PROOF. (a) Eikonal recursion. Write S_{p,ell} = sum_{d=0}^{N}
S_d * x^d where x is the coordinate tuple centred at p and S_d
denotes the degree-d-to-multilinear coefficient. We have already
prescribed S_0, S_1, S_2 via (3.3.5). We now identify higher S_d so
that the eikonal error E = g^{ij} S_i S_j + 1 vanishes to high
Taylor order at p. This is a finite system of polynomial equations
with polynomial right sides, and the coefficient of the new
homogeneous term S_{d+1} at degree d >= 2 can only enter through
2 * D_z S_{d+1} where D_z = z * partial_x is the derivative along
z (this is exactly Lemma 3.2's compatibility used to solve the
linear-algebraic structure of the eikonal equation). Since
D_z : (monomials of degree d+1) -> (homogeneous polynomials of
degree d) is onto (the explicit right inverse
(I_z f)(x) = ell_z(x) * int_0^1 f( pi_z(x) + s * ell_z(x) z ) ds ,
ell_z(x) = Re(sum conj(z_i) * x_i) / sum |z_i|^2,
pi_z(x) = x - ell_z(x) z,
is standard: D_z ell_z = 1, D_z (pi_z x) = 0), we can greedily
determine S_3, S_4, ..., S_N so that E vanishes to Taylor order at
least L_S - 1 at p, where L_S = N+1 is a chosen length. The
constructed polynomial coefficients are bounded uniformly in
(p, ell) for these finitely many degrees.
The convention that z * z = -1 is our standing one; it implies the
constant term of E is exactly 0 (good) and the linear term vanishes
automatically (because Q z = 0).
(b) Transport recursion. With S_{p,ell} fixed through degree N+1,
we determine V^{(0)}, V^{(1)}, ..., V^{(J)} as polynomials in the
centred coordinate x with DECREASING degrees (taylored to fit the
scheduling below). Explicitly: from the identity (3.3.3),
A V^{(0)} = O(|x|^{n_0}) at p,
A V^{(j)} = - Delta_g V^{(j-1)} + O(|x|^{n_j}) at p. (3.3.9)
Choose the polynomial degrees N_j so that at each homogeneous level
d in V^{(j)} we can solve using the same surjectivity D_z and thus
cancel the forcing term of order d. Concretely, one sets
J = D + m + 1,
K = 2D + 3m + 6,
N_j = K + J - j for 0 <= j <= J,
N_phi = K + J + 1 (phase degree),
and then inductively determines the coefficients of V^{(j)} through
homogeneous level N_j, canceling (3.3.9) through order N_j - 1. The
crucial scheduling fact: at level d of A V^{(j)} the coefficient of
V^{(j)} contributes only a 2 D_z * V^{(j)}_{d+1} term (linear in the
new coefficient), while Delta_g V^{(j-1)} contributes via V^{(j-1)}
through degree d+2 <= N_j + 1 = N_{j-1}, which has already been
fixed. So the recursion is well-founded. All constructed
coefficients are bounded uniformly over (p, ell) for these finitely
many degrees.
(c) Retained remainders. With N_phi, N_0, ..., N_J chosen as in
(b), the leftover terms in (3.3.3) after polynomial truncation are:
eikonal: n^2 E * V, E = O(|x|^{N_phi}), contribution
O(n^2 |x|^{N_phi}) in the exp(n S) scale,
transport: n * sum_{j=0}^J n^{-j} R_j * (unchanged),
R_j = O(|x|^{N_j}) = O(|x|^K),
contribution O(n^{1-j} |x|^K),
top amplitude term: n^{-J} Delta_g V^{(J)}, O(n^{-J} |x|^K)
in the exp(n S) scale.
(d) Gaussian-envelope moment. For fixed q >= 0 and c > 0 there is
C_{q,c} < infinity with
sup_{r >= 0} r^q * exp( sqrt(n) * r - c * n * r^2 )
<= C_{q,c} * n^{-q/2}, uniformly in n >= 1. (3.3.10)
(Substitute s = sqrt(n) r; the resulting sup_s (s/sqrt(n))^q e^{s - c s^2}
is C_{q,c} n^{-q/2}.)
Combined with Taylor's theorem (which bounds the m derivatives of a
function vanishing to order K by |x|^{K-m} times a fixed constant in
the considered compact region) we obtain after at most m derivatives
of (3.3.3), then dividing by exp(n phi(x)) and using the localization
inequality (3.3.6) to absorb the exp(n (Re S - phi)) factor, the
following weights (2n exponents in each factor):
eikonal term: 2 + m - (N_phi - m)/2 (E)
transport: (1 - j) + m - (N_j - m)/2, j = 0..J (R)
top amplitude: - J + m - (N_J - m)/2 (G)
We want each of these to be at most -D - 1 (then the residual bound
(3.3.7) follows by the extra exp(n phi) factor, with C = 2 or so).
Choosing J = D + m + 1, K = 2D + 3m + 6 and N_phi = K + J + 1 gives
(E): = 2 + m - (N_phi - m)/2
= 2 + m - (3D + 3m + 8)/2 (N_phi = 3D+4m+8)
= -3D/2 - m/2 - 2 (simplify)
<= -D - 1 (since -D/2 - m/2 - 1 <= 0 whenever D,m >= 0).
Good.
(R): for 0 <= j <= J,
= (1 - j) + m - (K - m)/2 - (J - j)/2 (exact split)
= -D - 2 - j - (J - j)/2 (K - m = 2D+2m+6)
= -3D/2 - m/2 - 5/2 - j/2 (J = D+m+1)
<= -3D/2 - m/2 - 5/2 (j >= 0)
<= -D - 1 (since -D/2 - m/2 - 3/2 <= 0 when D,m >= 0).
The maximal R occurs at j = 0 and equals -3D/2 - m/2 - 5/2,
which is <= -D - 2 for all D, m >= 0.
Good.
(G): = -J + m - (K - m)/2
= -(D+m+1) + m - (D+m+3) (K-m = 2D+2m+6)
= -2D - m - 4 (simplify)
<= -D - 1 (since -D - m - 3 <= 0 when D,m >= 0).
Good.
(e) Cutoff errors. The support of any derivative of zeta lies in
the shell rho/2 < |x - p| < rho, where dist_g(p, x) >= rho/2 is
bounded away from zero. The localization inequality (3.3.6) then
gives a fixed NEGATIVE gap between Re S and phi on this shell:
Re S - phi <= (const) - kappa rho^2/4 < 0, uniformly in n and p.
Finitely many derivatives of zeta introduce multiplication by fixed
coefficients, and the exp(n(Re S - phi)) factor is at most exp(-cn)
for a positive c on the shell. These cutoff errors are therefore
exponentially small compared to the interior polynomial terms and do
not affect (3.3.7). (This is where (3.3.6) earns its keep: a
STRICTLY negative gap at positive distance is what kills the cutoff
error.)
(f) Derivative bounds. Direct differentiation of Z_{p,ell} =
zeta * exp(n S) V, using |grad S| <= C (uniformly in p; see that S
varies with bounded coefficients and phi has bounded derivatives on
K_1), |grad V| <= C (the fixed polynomial coefficients are uniformly
bounded), and the Gaussian moment (3.3.10), gives (3.3.8). Increasing
the number of finite derivative orders increases the polynomial
degrees and the constants C_j, but never introduces a new power of n
in (3.3.8).
This closes Lemma 3.3.
End of section 3.3.
3.4 REAL GAUSSIAN SUPERPOSITION AND NONCANCELLATION
The waves of Lemma 3.3 are complex. We now combine them into a real
quasimode with two properties at once:
(N1) the value and first derivative do not vanish simultaneously,
and in fact |U_n| + n^{-1} |grad U_n| stays bounded below by
a fixed inverse power of n in the scale W_n, and
(N2) the weighted residual is smaller than any prescribed inverse
power of n.
Property (N1) is the input we need later in Part 4 to divide pointwise
by either the value of U or the squared gradient of U. (N2) is the
input we need for the metric correction.
3.4.1 The construction.
Let S_n be a maximal set of points in K_1 = { r <= t_1 } that are
n^{-1}-separated: | p - q |_g >= n^{-1} for p != q. Maximality means
every x in K_1 is within n^{-1} of some point of S_n. Volume
comparison gives |S_n| <= C n^3.
For every p in S_n and ell in { 0, 1, 2 }, let Z_{p,ell} be the wave
of Lemma 3.3 built at p with those choices. Each wave has support in
B_p (a fixed small ball around p) and is normalised so that at the
centre p, Z_{p,ell}(p) = |Z_{p,ell}(p)| e^{i arg} has modulus at least
c n^{-6} * e^{n phi(p)} (the lower bound follows from the phase
data (3.3.5) combined with the Gaussian envelope (3.3.6) and the
amplitude bound A^{(0)}_{p,ell}(p) = 1).
Introduce independent standard real Gaussian variables
Re gamma_{p,ell} and Im gamma_{p,ell}, for p in S_n, ell in {0,1,2}.
So there are 2 |S_n| * 3 = O(n^3) real Gaussian variables total. For
each p in S_n we may equivalently use a complex Gaussian
gamma_{p,ell} = Re gamma + i Im gamma. Define
U_n(x) := w_n(x) +
Re sum_{p in S_n} sum_{ell=0,1,2} gamma_{p,ell} Z_{p,ell}(x),
(3.4.1)
a smooth real function on S^3 (the sum is finite). Define the weight
W_n(x) := e^{n phi(x)} + r(x)^n, r(x) = (x_1^2 + x_2^2)^{1/2},
(3.4.2)
and the normalised jet
J_n(x) := W_n(x)^{-1} ( U_n(x), n^{-1} grad U_n(x) )
in R x T_x S^3 approximately R x R^3 = R^4,
(3.4.3)
where the identification T_x S^3 ~ R^3 uses the local orthonormal frame
we fixed in section 3.3. Since frames are finite and their overlaps
bounded, the choices below produce equivalent norms up to fixed
constants; we suppress these from now on.
3.4.2 The coefficient event.
G_n^coef := { |Re gamma_{p,ell}| <= n and |Im gamma_{p,ell}| <= n for
every (p, ell) }. (3.4.4)
Since there are O(n^3) real coefficients and each is standard Gaussian,
P( (G_n^coef)^c ) <= C n^3 e^{-n^2/2} -> 0. (3.4.5)
LEMMA 3.4 (Real noncancellation and residual).
Fix integers m, D >= 0. Build the waves Z_{p,ell} of Lemma 3.3 with
residual order D + 4 and with derivative bounds through order
max(m, 2). There is a universal exponent B = 112 (independent of the
profile, the requested orders, and n) with the following property.
On the event G_n^coef, the following pointwise derivative and residual
bounds hold on all of S^3:
|grad^j U_n|_g <= C_j n^{j+4} W_n, 0 <= j <= m, (3.4.6)
|grad^j (Delta_g + Lambda_n) U_n|_g <= C_j n^{-D} W_n,
0 <= j <= m, (3.4.7)
with constants C_j independent of n. Furthermore, there is an event
G_n subset G_n^coef of probability tending to one as n -> infinity,
on which
|U_n(x)| + n^{-1} |grad U_n(x)|_g >= n^{-B} W_n(x) for every x in S^3.
(3.4.8)
In frame language, |J_n(x)|_Euclid >= n^{-B} for every x.
PROOF.
(a) Derivative and residual bounds. Assume G_n^coef. By (3.3.8),
each |grad^j Z_{p,ell}(x)|_g <= C_j n^j e^{n phi(x)}. There are
O(n^3) waves, each with coefficient bounded by |gamma| <= sqrt(2) n
(both real and imaginary parts bounded by n). Summing:
|sum_{p,ell} Re(gamma_{p,ell} grad^j Z_{p,ell}(x))|_g
<= C_j n * n^j e^{n phi(x)} * O(n^3) = C_j n^{j+4} e^{n phi(x)}.
The background w_n has grad^j w_n <= C_j n^j W_n (the check is a
direct power count; see Section 1.3 for j=1 and differentiate).
Combining gives (3.4.6). For the residual, use (3.3.7) with order
D + 4 (so the per-wave residual is at most C n^{-(D+4)} e^{n phi}).
The sum has O(n^3) terms with |gamma| <= sqrt(2) n, so the accumulated
residual is at most C n^{-(D+4) + 1 + 3} = C n^{-D} e^{n phi}. The
background residual (Delta_g + Lambda_n) w_n = (Delta_g - Delta_{g_*})
w_n is supported in r <= R_0 and, since phi - log r > 0 strictly there,
is exponentially smaller than e^{n phi}; it therefore also contributes
<= C n^{-D} W_n. This gives (3.4.7).
(b) Two regimes for |J_n|. Define
E_n := { x in S^3 : r(x)^n > n^6 e^{n phi(x)} }, (3.4.9a)
F_n := S^3 \ E_n = { x : r(x)^n <= n^6 e^{n phi(x)} }. (3.4.9b)
Regime E_n is where the background w_n alone already provides the
noncancellation bound (deterministic). Regime F_n is where the random
waves must do the work.
(c) Regime E_n: the background alone gives |J_n| bounded below.
Introduce the ambient vector field R = x_1 partial_{x_2} -
x_2 partial_{x_1}. On the sphere, R is tangent, with uniformly
bounded g-length. Writing x_1 + i x_2 = r e^{i theta}, one has
w_n = r^n cos(n theta), R w_n = -n r^n sin(n theta). (3.4.10)
Consequently
|w_n(x)| + n^{-1} |grad w_n(x)|_g >= c_0 r(x)^n, (3.4.11)
where c_0 > 0 is a universal constant (from the trace of the rotation
field: max(|cos(n theta)|, |sin(n theta)|) >= 1/sqrt(2)). On
G_n^coef, the random contribution to the jet at x is bounded by
|sum Re(gamma grad^j Z)| <= C n^{j+4} e^{n phi}, j = 0, 1. (3.4.12)
Therefore on E_n, where r^n > n^6 e^{n phi}, we have
|U_n| + n^{-1} |grad U_n| >= c_0 r^n / 2,
i.e., |J_n(x)| >= c_0 r^n / (2 W_n) >= c_0/4
> 0. This is far stronger than n^{-B}; so E_n causes no difficulty.
(d) Regime F_n: where the waves are needed.
Claim: F_n is contained in { r <= t_1 } for all large n. If r >= t_1,
then phi(x) = phi_0(x) = log t + r^2 - t^2. The function
phi_0(x) - log r(x) = log t - t^2 + r^2 - log r has a strictly
NEGATIVE maximum on [t_1, 1] (Lemma 2.8(d) on [t_1,1], with
t_1 > t). Call this maximum -sigma, sigma > 0. Then for r >= t_1:
r^n / e^{n phi(x)} = exp( n (log r - phi(x)) ) >= exp( sigma n ),
contradicting r^n <= n^6 e^{n phi(x)} once n is large enough that
exp(sigma n) > n^6. This proves the claim.
Fix x in F_n. Choose the center p of S_n nearest to x. By
maximality, dist_g(p, x) <= n^{-1}. Choose also the "shifted"
center p' of S_n nearest to
x + n^{-1/2} nu(x),
where nu(x) := a(x)/|a(x)|_g if a(x) != 0, and where nu(x) is any
unit vector chosen orthogonal to the two vectors b_0(x), b_1(x) if
a(x) = 0. Then |p - p'|_g >= n^{-1/2} - 2 n^{-1} >= n^{-1/2}/2 for
large n, and p' is a valid center by the same separation argument.
The cutoffs xi at both p and p' are equal to 1 at x for large n.
Rotation trick. For each wave Z_{p,ell} selected below, factor out
|Z_{p,ell}(x)| e^{-i arg Z_{p,ell}(x)}. That is: define
gamma'_{p,ell} := gamma_{p,ell} e^{i arg Z_{p,ell}(x)}. Since
gamma_{p,ell} is a complex standard Gaussian and multiplication by a
unit complex number is an orthogonal change of its two real
coordinates, gamma'_{p,ell} is again a complex standard Gaussian.
After this change, at the fixed point x we may write Re(gamma Z) =
Re(gamma') |Z|.
The five selected real Gaussian coordinates are
Re gamma'_{p,0}, Im gamma'_{p,0}, Im gamma'_{p,1}, Im gamma'_{p,2},
Re gamma'_{p',0}. (3.4.13)
These are 5 independent real standard Gaussians: the first two come
from one complex Gaussian (independent real components), the next two
come from two DIFFERENT complex Gaussians (each independent of the
others), and the fifth comes from a further complex Gaussian. The
coefficients from all other (q, m) are also independent Gaussians,
and we condition on them (they will only shift the mean of J_n(x),
which we handle at the last step).
Amplitude-factored columns. For each of the five coordinates, the
column of J_n(x) contributed (after dividing by e^{n phi(x)} and by
the amplitude factor beta := |Z_{p,ell}(x)| / e^{n phi(x)} for the
corresponding wave) is:
C1 = (1, d Re Phi_{p,0}(x) + n^{-1} Re(A_{p,0}^{-1} dA_{p,0})(x))
[wave Z_{p,0}, Re]
C2 = (0, d Im Phi_{p,0}(x) + n^{-1} Im(A_{p,0}^{-1} dA_{p,0})(x))
[wave Z_{p,0}, Im]
C3 = (0, d Im Phi_{p,1}(x) + n^{-1} Im(A_{p,1}^{-1} dA_{p,1})(x))
[wave Z_{p,1}, Im]
C4 = (0, d Im Phi_{p,2}(x) + n^{-1} Im(A_{p,2}^{-1} dA_{p,2})(x))
[wave Z_{p,2}, Im]
C5 = (1, d Re Phi_{p',0}(x) + n^{-1} Re(A_{p',0}^{-1} dA_{p',0})(x))
[wave Z_{p',0}, Re]
The amplitude factors beta are bounded above by a fixed constant and
below by c n^{-6} (via (3.3.6) and |Z|_{W_n} lower bound from the
construction). In the FACTORED scale (i.e., before multiplying by
beta), the leading terms are:
C1 = (1, df(x) + O(n^{-1})) (p at distance n^{-1})
C2 = (0, b_0(x) + O(n^{-1}))
C3 = (0, b_1(x) + O(n^{-1}))
C4 = (0, b_2(x) + O(n^{-1}))
C5 = (1, df(x) + c_{p',0} n^{-1/2} h_x(nu(x), \cdot) + O(n^{-1})).
Justifications: the (1, df + O(n^{-1})) form of C1 comes from
V_{p,0}(x) = 1 + O(n^{-1}) (Taylor at p, distance n^{-1}) and
d Re Phi_{p,0}(x) = df(x) + O(n^{-1}) = df(x) + O(n^{-1}) (varying x
from p by O(n^{-1}) with phi derivative bounded). The (0, b_j + O(n^{-1}))
form of C2, C3, C4 comes from V_{p,j}(x) = 1 + O(n^{-1}), with the
IMAGINARY part having zero value and derivative d Im Phi_{p,j} = b_j +
O(n^{-1}) (the imaginary part of the Taylor coefficients at p is b_j).
The shifted form C5 comes from the phase expansion of section 3.3
around p': d Re Phi_{p',0}(x) = df(x) - c_{p',0} h_{p'}(x - p' , \cdot)
+ O(|x - p'|^2). Since x = p' + (x - p') and x - p' = x - p -
(p' - p), with x - p = O(n^{-1}) and p' - p = n^{-1/2} nu(x) + O(n^{-1}):
d Re Phi_{p',0}(x) - df(x)
= c_{p',0} h_x( p' - p , \cdot ) + O(|x - p'|^2) + O(|x - p'| * n^{-1})
= c_{p',0} n^{-1/2} h_x( nu(x) , \cdot ) + O(n^{-1}),
using p' - p = n^{-1/2} nu(x) + O(n^{-1}), |p' - p| ~ n^{-1/2},
|x - p'| = O(n^{-1/2}) (because x is at distance n^{-1} from p, and
p' is at distance n^{-1} from x + n^{-1/2} nu; so |x - p'| = O(n^{-1/2})
+ O(n^{-1}) = O(n^{-1/2})).
Determinant of a 4 by 4 subblock. Take the four columns
(C1, C2, C3, C5). Subtract C1 from C5:
C5 - C1 = (0, c_{p',0} n^{-1/2} h_x(nu, \cdot) + O(n^{-1})).
Multiply by n^{1/2}:
n^{1/2} (C5 - C1) = (0, c_{p',0} h_x(nu(x), \cdot) + O(n^{-1/2})).
In a local orthonormal frame at x, consider the 4 x 4 determinant
det[ C1, C2, C3, n^{1/2}(C5 - C1) ]
= det[ C1, C2, C3, (0, c_{p',0} nu(x) + O(n^{-1/2})) ] .
Expand along the value row. C1 has value 1 in that row; C2, C3, and
the last column have value 0 in that row. For columns C1, C2, C3, D
(where D = n^{1/2}(C5 - C1) has zero value component), the leading
term is the (1,1) cofactor times the value 1:
det = 1 * det_3x3[ b_0(x), b_1(x), c_{p',0} nu(x) ] + O(n^{-1/2})
= c_{p',0} det[ b_0(x), b_1(x), nu(x) ] + O(n^{-1/2}).
Since nu(x) is a unit vector perpendicular to both b_0(x) and b_1(x),
and b_0, b_1 are uniformly independent in nu^\perp with |b_j| = L >= 1
and |sin(angle between b_0, b_1)| >= eta > 0 from the phase data,
det[ b_0(x), b_1(x), nu(x) ] = |nu| * (b_0, b_1 restriction to
nu^\perp determinant) in magnitude
>= |nu| * |b_0| * |b_1| * eta
>= eta > 0.
Therefore det[C1, C2, C3, n^{1/2}(C5 - C1)] >= eta/2 for all large n.
Since C5 - C1 is a linear combination of C1 and C5, and the substitution
C5 -> C5 - C1 does not change column span (it is a unimodular column
operation on C1 and C5), we have
|det[C1, C2, C3, C5]|
= n^{-1/2} |det[C1, C2, C3, n^{1/2}(C5 - C1)]|
>= c_1 n^{-1/2}. (3.4.14)
Hence the 4 x 4 matrix [C1|C2|C3|C5] in the factorized scale has all
of its singular values at least c_2 n^{-1/2} (the determinant is
c_1 n^{-1/2} and the row norms are bounded by a fixed constant times
1; the smallest singular value is at least det / (row norm) ^3).
In the ORIGINAL scale (not factorized), each column is multiplied by
the amplitude factor beta, which is >= c n^{-6}. So the 4 x 4 matrix
in the original scale has smallest singular value
s_min >= c_3 n^{-6} * c_2 n^{-1/2} = c_4 n^{-6.5}. (3.4.15)
From small singular values to a small-ball probability. Write
M for this 4 x 4 matrix (in the original W_n-normalized scale), and
note J_n(x) - m_0 = M * (five selected real standard gaussians), where
m_0 is the conditional mean from the deterministic part w_n and the
other gaussian coefficients (which we conditioned on). Since the five
selected gaussians are independent standard real, and M is invertible:
P( |J_n(x) - m_0| < r ) <= (density sup of M * std_gauss_4)
* volume( B_r^4 )
density sup = 1 / ( (2 pi)^2 * |det M| ) = 1 / ( (2 pi)^2 *
(product of singular values) )
With s_min >= c_4 n^{-6.5} and s_max <= C n^{-6} * C (i.e., each
column norm <= C n^{-6} * C * (bounded) = O(n^{-6})),
|det M| = s_1 s_2 s_3 s_4 >= (c_4 n^{-6.5})^4 = c_4^4 n^{-26}.
So density sup <= c_5 n^{26}. Volume of 4-ball of radius r = 2 n^{-B}:
C r^4 = 16 C n^{-4B}. Hence
P( |J_n(x) - m_0| < 2 n^{-B} )
<= C n^{26 - 4B}. (3.4.16)
Unconditional form. The conditional mean m_0 is independent of the
five selected gaussians. Fix any value of m_0 (which amounts to
conditioning on the complementary gaussians and the background w_n).
Conditional on m_0, J_n(x) - m_0 = M g for the five selected real
standard gaussians g; only the image M g in R^4 matters. The
probability that |m_0 + M g| < n^{-B} is bounded by
(pointwise supremum of the density of M g)
* | B_4(0, n^{-B}) | ,
where B_4(0, r) is the 4-ball of radius r. The density of M g at
any point of R^4 is at most (2 pi)^{-2} |det M|^{-1} <= c_5 n^{26}
(from (3.4.15) and the lower bound on |det M|). The 4-ball volume is
C n^{-4B}. So the conditional probability is at most C n^{26 - 4B}.
Averaging over m_0 cannot increase this bound, so unconditionally
P( |J_n(x)| < n^{-B} ) <= C n^{26 - 4B} =: p_n. (3.4.17)
(e) Union over a mesh-net and Lipschitz to all points.
On G_n^coef, the pointwise jet bound (3.4.6) with j=2 plus the bound
|grad W_n| <= C n W_n (from phi's bounded derivatives and r^n bound)
give
|grad J_n(x)| <= C n^6. (3.4.18)
Indeed J_n = V/W where V = (U, n^{-1} grad U) and W = W_n.
d J_n = (dV)/W - V * (dW)/W^2,
|dV| <= C n^5 W (from (3.4.6) at j=1; j=0 for value)
|V| * |dW| / W <= (C n^4 W) * (C n W) / W = C n^5 W,
so |dJ_n| <= C n^5 <= C n^6. (3.4.18) holds.
Take a net N_n of S^3 with mesh n^{-B-10}; it has at most
C (n^{-B-10})^{-3} = C n^{3(B+10)} = C n^{3B+30} points (S^3 is
3-dimensional; the constant in the mesh-net size is universal). On
G_n^coef, at each point x_j of N_n we have by (3.4.17)
P( |J_n(x_j)| < n^{-B}/2 ) <= C n^{26-4B}. Union bound over the net:
P( exists x_j in N_n with |J_n(x_j)| < n^{-B}/2 )
<= C n^{3B+30} * C n^{26-4B} = C n^{56 - B}. (3.4.19)
With B = 110, this bound is C n^{-54}.
Now let x be ANY point of S^3 (not necessarily in N_n), and let
x_j in N_n be within n^{-B-10}. By (3.4.18),
|J_n(x) - J_n(x_j)| <= C n^6 * n^{-B-10} = C n^{-B-4}.
Choose n large enough that C n^{-4} < 1/2. If x_j is a net point and
|J_n(x_j)| >= n^{-B}/2, then
|J_n(x)| >= |J_n(x_j)| - |J_n(x) - J_n(x_j)|
>= n^{-B}/2 - C n^{-B-4}
>= n^{-B}/4 (for all large n).
Let G_n be the intersection of G_n^coef with the complement of the
net-failure event. Then P(G_n) -> 1. On G_n, we have
|J_n(x)| >= n^{-B}/4 for every x in S^3, where B (= 110) is the
runtime exponent used in the mesh, the small-ball radius, and the
union bound. Since n^{-110}/4 >= n^{-112} for all n >= 2, the same
conclusion holds with the clean exponent
|J_n(x)| >= n^{-112} for every x in S^3 on the event G_n.
This is what the lemma statement claims with its stated B := 112.
The gap of 2 between the runtime 110 and the stated 112 simply
absorbs the harmless factor of 4 arriving from the net interpolation.
Only the fact that the exponent is a FIXED inverse power of n
(independent of n, and the same for all large n) matters downstream.
This closes Lemma 3.4.
REMARK 3.5. The exponent 110 is arbitrary; the sharp exponent of the
argument would be around B = 60. The margin of safety is used only to
absorb finitely many poly(n) constants coming from the net size and
the small-ball density, so a larger B costs nothing downstream.
REMARK 3.6. Conditioning on the other coefficients at step (d) is
allowed because they only shift the mean m_0; the small-ball bound
(3.4.17) depends only on the five SELECTED gaussians and is therefore
uniform in the value of m_0.
End of section 3.4.
3.5 THE SIGN SCORE: EXPECTED SIZE AND A GOOD REALIZATION
The good realization of Lemma 3.4 gives a real quasimode U_n with
small weighted residual and a pointwise lower bound on (value,
first derivative) jointly. It is still random, and we need to
pick out ONE outcome of it that ALSO has a large sign score. This
section does that, in three steps: compute the variance and the
directional decorrelation of the random part; show the expected
sign score is comparable to the natural scale n int B; and use
the deterministic upper bound on the score plus the good event to
extract one realization with all three properties at once.
Throughout this section, phi is a FIXED smooth envelope produced
by Proposition 2.10 for a target volume V_0 that we will fix in
Proposition 3.5. Nothing in section 3.5 varies the envelope; all
constants are uniform in n.
3.5.1 Setup.
Fix a nonempty open subset Omega of K_1 whose closure lies in the
interior of K_1, and let sigma_1 > 0 be its distance to the
boundary of K_1. On Omega, define
B(x) := |a_phi(x)|_g = |grad phi(x)|_g. (3.5.1)
This is smooth and, by (ENV-C) ("no critical points in Omega"), is
uniformly bounded below by a positive constant B_min and above by
a fixed constant B_max. Its INTEGRAL over Omega can be arbitrarily
large by Proposition 2.10, even though its pointwise value is
bounded; the enlargement comes from the corrugations of Part 2.
Let delta_0 := n^{-1/2}. Decompose Omega (up to a negligibly small
set) into a finite disjoint family of coordinate cubes { Q } of
side length L_Q in (delta_0/2, delta_0). Because the coordinate
chart has bounded distortion and the cubes are half-open, the lost
set has measure at most (1/4)|Omega| and every Q sits in an
orthonormal coordinate patch. For each Q, set d_Q := L_Q / 4 and,
for each of the three coordinate directions d_1, d_2, d_3,
Q^-_i := { x in Q : x + d_Q d_i in Q }, |Q^-_i| = (3/4) |Q|.
(3.5.2)
Define
F_n(f) := (n/c_0) sum_Q B_Q sum_{i=1}^3
int_{Q^-_i} [ 1_{f(x) f(x + d_Q d_i) < 0} ] dx, (3.5.3)
where c_0 is the volume-growth constant from Lemma 3.1(d) and
B_Q := min_{x in Q} B(x). This is the sign score from Lemma 3.1 at
the intermediate scale delta_0. We recall its three key properties,
all proven in section 3.1:
(i) F_n(f) <= C_area H_g^2({f = 0}) for every continuous f, (3.5.4)
with C_area > 0 depending only on Omega and the C^1-size of
the metric neighborhood;
(ii) F_n(c f) = F_n(f) for every nonzero real c and every strictly
positive continuous multiplier (so F_n is invariant under
rescaling and under multiplying an eigenfunction by a
nowhere-vanishing factor), and F_n is lower semicontinuous
for uniform convergence; (3.5.5)
(iii) F_n(f) <= (3 n / c_0) (1 + O(delta_0 / L_min)) int_Omega B
for every continuous f, (3.5.6)
where L_min = delta_0/2 is the smallest side length.
(Proof of (i)-(iii) is in Lemma 3.1, with the minor geometric
constants absorbing the delta_0 / L_min corrections; we use only
that the dependence on n is exactly a factor n and the region-
integral is at most 2 int_Omega B.)
3.5.2 The random part and its covariance.
Write U_n = w_n + e^{n phi} X where
X(x) := e^{-n phi(x)} (U_n(x) - w_n(x))
= Re sum_{p in S_n} sum_{ell=0}^2 gamma_{p,ell} V_{p,ell}(x),
V_{p,ell}(x) := e^{-n phi(x)} Z_{p,ell}(x). (3.5.7)
Each gamma_{p,ell} is a complex standard Gaussian with Re and Im
independent real standard Gaussians; the family (p, ell, Re, Im) is
total independence. For any such family, a short calculation on
second moments (use E Re(gamma a) Re(gamma b) = Re(a conj b) for
complex standard Gaussian gamma and complex a, b, and the same for
independent gamma's giving zero) yields
Cov(X(x), X(y)) = Re sum_{p,ell} V_{p,ell}(x) conj(V_{p,ell}(y)), (3.5.8)
Var X(x) = sum_{p,ell} |V_{p,ell}(x)|^2.
Because V_{p,ell} = e^{-n phi} Z_{p,ell}, Lemma 3.3 gives (for a
packet centred at p with x within its support)
|V_{p,ell}(x)| <= C e^{-c n |x - p|^2} (near x = p). (3.5.9)
Away from p, V_{p,ell} decays as a Gaussian, faster than any power
of n; the number of packets is at most C n^3.
LEMMA 3.5.1 (Variance lower bound and directional decorrelation).
There are constants c_var > 0 and q_0 > 0, depending only on
g, phi, and the finite coordinate and cutoff data of section 3.3
(in particular, NOT on n), such that for every x in Omega,
Var X(x) >= c_var, (3.5.10)
and for every cube Q, every direction i in {1, 2, 3},
(1/|Q^-_i|) int_{Q^-_i} P( X(x) and X(x + d_Q d_i) have
opposite signs ) dx >= q_0. (3.5.11)
PROOF. (a) Variance. Let p = p(x) be the nearest centre of S_n to
x, so |p - x|_g <= n^{-1}. Lemma 3.3 gives, for ell = 0, 1, 2,
|Z_{p,ell}(x)| >= c e^{n phi(x)} hence |V_{p,ell}(x)| >= c.
Summing over ell = 0, 1, 2 gives
Var X(x) >= 3 c^2 =: c_var.
(c_var > 0 depends on c, which is the lower bound in (3.3.6)/the packet
amplitude, fixed once and for all.)
(b) A pair of centered Gaussians. For two real jointly centered
Gaussians with positive variances and correlation r in [-1, 1], the
probability of opposite signs is arccos(r) / pi. Elementary bounds
arccos(r) >= (pi/2) - r for r in [0, 1],
arccos(r) >= (sqrt(2)/2) sqrt(1 - r) for r in [-1, 1],
give specifically
P_opp(r) >= (1/(2 pi)) sqrt(1 - r) >= (1/(2 pi)) (1 - r). (3.5.12)
So it suffices to bound the average of (1 - r(x)) over the cube Q^-_i
from below, where
r(x) := Cov(X(x), X(x + d_Q d_i)) / sqrt(Var X(x) Var X(x+d_Q d_i)).
(c) Decomposition of r(x). Let p = p(x) be the nearest centre, and
write V_{p,ell} = |V_{p,ell}| e^{i theta_{p,ell}} with
theta_{p,ell}(x) := n Im S_{p,ell}(x) + arg A_{p,ell}(x). (3.5.13)
At the centre, Lemma 3.3 gives Im S_{p,ell}(p) = 0, and the transverse
phase covector is b_{p,ell} := Im dS_{p,ell}(p) with
| b_{p,ell} - df(x) |_g <= C n^{-1/2}. (3.5.14)
For x and x + d_Q d_i both within the packet's supporting ball,
theta_{p,ell}(x + d_Q d_i) - theta_{p,ell}(x)
= n d_Q b_{p,ell}(d_i) + O(n d_Q^2)
= n d_Q b_{p,ell}(d_i) + O(n^{-1/2}), (3.5.15)
since d_Q <= delta_0 = n^{-1/2}.
Only packets within distance n^{-2/5} of x (at most C n^3 of them)
contribute more than an exponentially small amount. For each such
(p, ell), the covariance entry is
Re( V_{p,ell}(x) conj(V_{p,ell}(x + d_Q d_i)) )
= |V_{p,ell}(x)| |V_{p,ell}(x+d_Q d_i)|
cos( theta_{p,ell}(x) - theta_{p,ell}(x+d_Q d_i) ).
Since the amplitude |V_{p,ell}| varies slowly across the shift d_Q,
|V_{p,ell}(x)| |V_{p,ell}(x + d_Q d_i)|
= |V_{p,ell}(x)|^2 (1 + O(n^{-1/2})), (3.5.16)
and similarly for the variance factor. Consequently, up to O(n^{-1/2})
errors (which are absorbed in the final q_0),
r(x) = sum_ell w_ell(x) cos( n d_Q b_{p,ell}(d_i) ) + O(n^{-1/2}),
w_ell(x) := |V_{p,ell}(x)|^2 / sum_{ell'} |V_{p,ell'}(x)|^2. (3.5.17)
The weights w_ell(x) are in [0, 1] and sum to 1.
(d) Uniform span of the transverse covectors. Fix x in Omega. The
three covectors b_{p,0}, b_{p,1}, b_{p,2} live in (a(x))^perp
(section 3.2 transverse construction), span (a(x))^perp with positive
minimum determinant eta_0 > 0, and have |b_{p,ell}|_g >= 1 (because
the definition |b|_g^2 = 2 + |a|^2 makes |b|_g >= sqrt 2). Hence
there exists i in {1, 2, 3} such that
max_{ell=0,1,2} | b_{p,ell}(d_i) |_g >= eta_1 > 0. (3.5.18)
(Derivation: put u := a(x)/|a(x)|. The three numbers |<u, d_i>|^2
sum to |u|^2 = 1, so min_i |<u, d_i>|^2 <= 1/3. For that i, the
projection of d_i onto a(x)^perp has size >= sqrt(2/3). The three
covectors b_{p,ell} restricted to a(x)^perp span that two-dimensional
subspace with determinant >= eta_0 > 0 and row norms >= sqrt 2; hence
for that i, at least one ell has |b_{p,ell}(d_i)| >= eta_1 > 0, with
eta_1 depending only on eta_0 and the finite data. We use this i in
part (e).)
(e) Phase oscillation. Fix i. Choose the ell = ell(i, x) that
attains the maximum in (3.5.18) applied to d = d_i. Then
Phi_ell(x) := n d_Q b_{p,ell}(d_i) (3.5.19)
satisfies |Phi_ell(x)| >= (n^{1/2}/4) * eta_1 > 0.
More importantly, the derivative dPhi_ell/dx_i along the i-direction
is bounded below:
|d/dx_i Phi_ell(x)| >= n d_Q |d/dx_i [b_{p,ell}(x)(d_i)]|
>= n^{1/2} eta_2 / 4, (3.5.20)
for some fixed eta_2 > 0, because b_{p,ell}(x) is a smooth function
of x whose derivative in x_i is uniformly bounded (bounded data
of phi and the packet amplitudes).
It follows that Phi_ell(x) traverses at least
(1 / (4 pi)) * (n^{1/2} eta_2/4) * L_Q >= eta_2 n / (64 pi)
(3.5.21)
full periods of cos as x_i varies over a side of length L_Q ~ n^{-1/2}.
The elementary computation
(1/T) int_0^T (1 - cos(a s)) ds >= 1/2 - 1/(aT) for aT >= 2,
applied with T = L_Q, a = n^{1/2} eta_2/4 and integrating out all
coordinates except x_i, gives
(1/|Q^-_i|) int_{Q^-_i} (1 - cos(Phi_ell(x))) dx >= c_2 > 0. (3.5.22)
For definiteness c_2 = 1/3 suffices for all large n.
Since all weights w_ell are nonnegative and the other ell' != ell
terms contribute nonnegatively to (1 - cos(Phi_ell')) (which also
have bounded oscillating phase when relevant), we conclude
(1/|Q^-_i|) int_{Q^-_i} (1 - r(x)) dx >= c_2 - O(n^{-1/2})
>= c_2/2 (3.5.23)
for all large n. Combining with (3.5.12):
(1/|Q^-_i|) int_{Q^-_i} P_opp(x, i) dx
>= (1/(2 pi)) * (c_2/2) =: q_0 > 0. (3.5.24)
Set q_0 = c_2/(4 pi). This proves (3.5.11).
End of Lemma 3.5.1.
3.5.3 Expected size and upper bound.
LEMMA 3.5.2 (Sign-score expectation and deterministic upper).
There are constants c_E, c_sign > 0, independent of n and depending
only on g, phi, and the finite data of section 3.3, with
E F_n(U_n) >= c_E n int_Omega B(x) dx, (3.5.25)
F_n(f) <= c_sign n int_Omega B(x) dx for every continuous f.
(3.5.26)
PROOF. Upper bound. This is (3.5.6) with c_sign = 6/c_0 (absorbing
the (1 + O(delta_0 / L_min)) correction into a larger c_sign).
Lower bound. Expand (3.5.3) and use Tonelli's theorem:
E F_n(U_n) = (n/c_0) sum_Q B_Q sum_i int_{Q^-_i} P_opp(x, i) dx.
By Lemma 3.5.1, the inner integral is at least q_0 |Q^-_i|. Using
|Q^-_i| = (3/4)|Q| and summing over i = 1, 2, 3:
E F_n(U_n) >= (n/c_0) q_0 sum_Q B_Q (9/4) |Q|.
Since B_Q >= (1/2) B(x) for every x in Q (compatible with the cube
side length if we shrink side lengths further at negligible cost), and
the tiles Q cover Omega up to measure (1/4)|Omega|,
(1/2) sum_Q int_Q B(x) dx >= (1/2) (3/4) int_Omega B(x) dx
= (3/8) int_Omega B(x) dx. (3.5.27)
Hence
E F_n(U_n) >= (9/4) (n/c_0) q_0 * (3/8) int_Omega B(x) dx
= c_E n int_Omega B(x) dx, c_E := (27/32) q_0 / c_0.
(3.5.28)
(Here we have replaced B_Q >= (1/2) B by B_Q >= B - Cdiam(Q)|grad B|,
absorbing the error into the constant c_E; equivalently, refine the
cube side lengths for the pointwise comparison, which is allowed
because B is fixed and smooth, before choosing n.)
Deterministic mean m(x) := e^{-n phi(x)} w_n(x) = (r(x)/e^{phi(x)})^n
cos(n theta(x)). On Omega, phi >= log r + sigma_2 by (ENV-D), so
|m(x)| <= e^{-sigma_2 n} is exponentially small in n. A small mean
shifts the opposite-sign probability by O(|m| / sqrt(Var X)) =
O(e^{-sigma_2 n}), which is exponentially small; it therefore does
not affect the constant c_E.
End of Lemma 3.5.2.
3.5.4 A good realization.
PROPOSITION 3.5 (A good realization with large sign score).
Fix integers m, D >= 0 and a target volume V_0 > 0. There is a
smooth envelope phi (from Part 2) with int_Omega B(x) dx >= V_0, and
an integer threshold n_0, such that for every n >= n_0 there is a
smooth real function u = U_n on S^3 for which the following three
properties hold.
(a) Support. u = w_n on {r >= t_2}; in particular u is compactly
supported in K_1 up to the smooth continuation outside.
(b) Regularity. For every fixed j, kappa,
|grad^j u|_g
<= C_{j,kappa} n^{j+kappa} W_n on all of S^3, (3.5.29)
|grad^j (Delta_g + Lambda_n) u|_g
<= C_j n^{-D} W_n on all of S^3, 0 <= j <= m. (3.5.30)
(The point is: exponents are independent of the D of the residual
accuracy -- increasing D forces larger polynomials but never
larger powers of n in (3.5.29).)
(c) Non-cancellation. |(u(x), n^{-1} grad u(x))|_g >= n^{-112} W_n(x)
for every x in S^3. (3.5.31)
(d) Large sign score. With F_n as in (3.5.3) and c_s := c_E / 4,
F_n(u) >= c_s n int_Omega B(x) dx. (3.5.32)
PROOF. Fix phi with int_Omega B >= V_0 (Prop. 2.10 with a large
enough K of amplification cycles; K is finite because B is bounded
below on Omega and the cycles multiply the integral by (1 + eta)^K).
Choose n_0 so large that for all n >= n_0:
* Lemma 3.4 applies with the given m, D (so the event G_n and the
bounds (3.4.6)-(3.4.8) are available).
* P(G_n) >= 1 - epsilon_n with epsilon_n -> 0.
* c_sign epsilon_n <= c_E / 4, using (3.5.26) and (3.5.25).
Recall the set G_n from Lemma 3.4 (the simultaneous good event of
non-cancellation and small residual). Estimate:
E[ F_n(U_n) 1_{G_n} ] = E F_n(U_n) - E[ F_n(U_n) 1_{G_n^c} ]
>= c_E n int_Omega B
- c_sign n int_Omega B * P(G_n^c)
>= (c_E - c_sign * (c_E/(4 c_sign))) *
n int_Omega B
= (3/4) c_E n int_Omega B.
(Here the second line bounds E[F_n 1_{G_n^c}] by c_sign n int_Omega B *
P(G_n^c) via the deterministic upper bound (3.5.26) and the trivial
1_{G_n^c} <= 1.) Hence some realization of U_n in G_n has
F_n(U_n) >= (3/4) c_E n int_Omega B >= (c_E/2) n int_Omega B.
Take that realization as u. Properties (a), (b) come from Lemma 3.4
on G_n (with the extra constant absorbed). Property (c) is the
non-cancellation bound (3.4.8) on G_n. Property (d) holds with
c_s := c_E/4 (the constant in the statement), since (3/4) c_E is
trivially at least c_E/4.
End of Proposition 3.5.
3.5.5 What comes next.
Proposition 3.5 gives one fixed realization u with residual k^{-D}.
That is NOT yet an exact eigenfunction: (Delta_g + Lambda_n) u is
nonzero but small. Part 4 converts u into an exact eigenfunction
of a nearby metric g_hat, without changing u or g_hat in the region
r >= t_3. Because the sign score F_n in Proposition 3.5 uses only
data in Omega (deep inside r <= t_1), and the correction keeps the
function positive multiple of u on K_1, the score F_n(u) transfers
verbatim to the exact eigenfunction t = positive * u.
Part 5 then makes the eigenvalue simple and assembles nested metric
neighborhoods into one fixed metric g_infty.
End of section 3.5.
PART 4
CONVERTING THE QUASIMODE INTO AN EXACT EIGENFUNCTION
The quasi-eigenfunction U = U_n of Proposition 3.5 satisfies
(Delta_g + Lambda_n) U = R, |grad^j R| <= C_j n^{-D_0} W,
(4.0.1)
with R as small as we like (D_0 arbitrarily large), and a lower bound
|(U(x), n^{-1} grad U(x))|_g >= n^{-B} W(x) (B = 112 > 0),
(4.0.2)
at every point. It is not yet an eigenfunction. This part converts
U into an exact eigenfunction t of a nearby smooth Riemannian metric
g_hat, with the following properties:
(E.1) g_hat is C^k-close to g_* for any prescribed k (n^{-D_1} with
D_1 arbitrarily large, independent of U).
(E.2) g_hat = g_* and t = w_n on r >= t_3 (outside the region
where U differs from the background).
(E.3) t is a nonzero smooth real eigenfunction: -Delta_{g_hat} t =
Lambda_n t.
(E.4) t equals a positive scalar multiple of U on K_1 (so it has
the same nodal set and node structure there, hence the same
sign score F_n(t) = F_n(U) by (3.5.5)).
The conversion uses four steps, one per subsection:
(4.1) a scalar conductivity-density correction (b, s);
(4.2) a semilinear equation for a positive change of variable v;
(4.3) an annular correction that fixes the determinant defect;
(4.4) a standard lemma identifying a conductivity equation as the
Laplace-Beltrami equation of an explicit metric.
Then (4.5) assembles the pieces into the final statement (Theorem
4.6, the exactification theorem).
Throughout Part 4, g is a fixed smooth metric on S^3 that lies in
the fixed C^1-neighborhood U of g_* from Lemma 3.1. All constants may
depend on g, phi, B, and the finite data of Parts 2-3, but not on n.
The parameter Lambda := Lambda_n = n(n+2) is always exactly n(n+2).
We use Delta_g = Div_g grad_g with the POSITIVE sign convention
(Delta_g = +sum partial_i^2 in flat coordinates; equivalently
-Delta_g is the standard positive-semidefinite Laplace-Beltrami
operator if the reader prefers the other convention, in which case
the eigen equation reads -Delta_{g_hat} t = Lambda_n t). We stick
with Delta_g = +Div_g grad_g throughout Part 4, so that the eigen
equation is Delta_g t + Lambda_n t = 0. The choice is purely
cosmetic; only the relative signs matter.
4.1 SCALAR CORRECTION
Let U = U_n and R = (Delta_g + Lambda_n) U be from Proposition 3.5.
We construct smooth scalar multipliers b, s > 0 with b = s = 1
outside supp R, both arbitrarily C^infty-close to 1, such that
Div_g(b grad U) + Lambda_n s U = 0 on all of S^3. (4.1.1)
Because b = s = 1 reduces (4.1.1) to Div_g(grad U) + Lambda_n U = R
(which is (4.0.1)), the left side at b = s = 1 is exactly R. A
correction that contributes -R makes the total exactly 0. This is
what the following computation constructs.
LEMMA 4.1 (Scalar correction). Fix a positive integer D_1 and a
nonnegative integer K <= m, where m is the derivative order in
Proposition 3.5. Suppose D_0 >= D_1 + 2 BK + K + 40 (choosing the
residual accuracy in Proposition 3.5 large enough; the required n_0
grows with D_0 but is finite). Then there are smooth b, s on S^3
with
b = s = 1 outside supp R,
|b - 1|_{C^k} + |s - 1|_{C^k} <= C_k n^{-D_1} for 0 <= k <= K,
Div_g(b grad U) + Lambda_n s U = 0 on S^3,
(4.1.2)
where C_k depends only on g, phi, B, and the finite data of Parts 2-3.
PROOF. Construction.
Let gamma_* in C_c^infty(R; [0,1]) be a fixed smooth bump function
equal to 1 on [-1/4, 1/4] and to 0 on |t| >= 1/2. Set
gamma(x) := gamma_*( n^{2B+2} U(x) / W(x) ). (4.1.3)
Because W >= c > 0 on S^3 (e^{n phi} and r^n are both bounded away
from 0, uniformly in n) the argument of gamma_* is a bounded smooth
function, and gamma is smooth on S^3. Note:
gamma = 1 where |U| <= (1/4) n^{-2B-2} W, (u small),
gamma = 0 where |U| >= (1/2) n^{-2B-2} W, (u large). (4.1.4)
Put
q := gamma R / |grad U|_g^2 (scalar), (4.1.5)
F := -U q (scalar), (4.1.6)
theta := Lambda_n^{-1} [ -(1 - gamma) (R / U) + div_g(q grad U) ],
(4.1.7)
with the CONVENTION that the term (1 - gamma) (R / U) is set to 0
identically wherever 1 - gamma = 0. Since 1 - gamma = 0 on the open
set { |U| <= (1/4) n^{-2B-2} W }, and on its complement we have
|U| >= (1/4) n^{-2B-2} W > 0, this convention makes (4.1.7) a
smooth function of x: the quotient R / U is defined and smooth on
the open set where it is used, and it is multiplied by the smooth
cutoff 1 - gamma that vanishes identically at the boundary. Set
b := 1 + F, s := 1 + theta. (4.1.8)
Identity. Let P := grad_g U. Dropping the subscript g from div and
grad for brevity, and using div(f X) = f div X + X . grad f (with
. being the g-inner product on covectors):
div(-U q P) = -q div(U P) - U (grad q) . P
= -q (U Delta_g U + |P|_g^2) - U (grad q) . P.
Write Delta_g U = R - Lambda_n U using (4.0.1), so
U Delta_g U + |P|_g^2
= U R - Lambda_n U^2 + |P|^2.
Then
div(-U q P) = -q U R + Lambda_n q U^2 - q |P|^2 - U (grad q) . P.
(4.1.9)
Similarly,
U div(q P) = U (q Delta_g U + (grad q) . P)
= q U R - Lambda_n q U^2 + U (grad q) . P.
(4.1.10)
Adding (4.1.9) and (4.1.10):
div(-U q P) + U div(q P) = -q |P|^2. (4.1.11)
By (4.1.5), q |P|^2 = gamma R. So
div(F P) = -gamma R - U div(q P). (4.1.12)
Now compute:
div(F P) + Lambda_n theta U
= -gamma R - U div(q P)
+ Lambda_n * Lambda_n^{-1} [-(1 - gamma) R/U + div(q P)] U
= -gamma R - U div(q P) - (1 - gamma) R + U div(q P)
= -gamma R - (1 - gamma) R
= -R. (4.1.13)
Combining with b = 1 + F, s = 1 + theta, and (4.0.1) read as
Div_g(grad U) + Lambda_n U = R:
Div_g(b grad U) + Lambda_n s U
= [Div_g(grad U) + Lambda_n U] + [Div_g(F grad U) + Lambda_n theta U]
= R + (-R) (by (4.1.13))
= 0.
So (4.1.1) holds exactly. The b = s = 1 contribution equals the
residual R, and the (F, theta) correction equals -R.
Smallness. Bounds on F and theta on S^3.
(a) Bound on |F| at the C^0 level. F = -U q is supported where
gamma != 0, hence where |U| <= (1/2) n^{-2B-2} W. On this set,
non-cancellation gives
|grad U| >= n^{-B} W / 2
for large n (since |U| <= n^{-2B-2} W contributes only a small
correction to the inequality). Also R is bounded by n^{-D_0} W.
Hence
|F| <= |U| * |R| / |grad U|^2
<= (n^{-2B-2} W) * (n^{-D_0} W) * (4 n^{2B} W^{-2})
<= 4 n^{-D_0 - 2}. (4.1.15)
So |F| is at most C n^{-D_0 - 2} at the C^0 level.
(b) Bound on |F|_{C^k} for 1 <= k <= K. Each derivative of F
costs at most a factor C n^{2B+6} (from differentiating gamma, which
introduces chain-rule factors n^{2B+2} * |grad U/W - U grad W/W^2| <=
C n^{2B+2+4}; also from |grad U|^{-2} and its derivatives, which cost
at most n^{4}); a rough count gives
|F|_{C^k} <= C_k n^{-D_0 - 2} * (C n^{2B+6})^k
<= C_k n^{-D_0 - 2 + k (2B + 6)}. (4.1.16)
(c) Bound on |theta|. From (4.1.7),
|theta| <= Lambda_n^{-1} [ (1 - gamma) |R| / |U|
+ |div(gamma R |grad U|^{-2} grad U)| ].
First term: (1 - gamma)|R|/|U| <= n^{-D_0} W / ((1/4) n^{-2B-2} W)
= 4 n^{-D_0 + 2B + 2}.
Second term: bound the vector field A := gamma R |grad U|^{-2} grad U.
Similar to (a),
|A| <= n^{-D_0 - 1 + B}. (4.1.17)
Its first derivatives add at most factor C n^{2B + 8}, so
|div A| <= C n^{-D_0 - 1 + B + 2B + 8} = C n^{-D_0 + 3B + 7}.
(4.1.18)
So
|theta| <= C n^{-2} (n^{-D_0 + 2B + 2} + n^{-D_0 + 3B + 7})
<= C n^{-D_0 + 3B + 5}. (4.1.19)
(d) Higher derivatives of theta. Each derivative beyond the C^0
level adds at most a factor C n^{2B + 8} in the (1 - gamma) R/U term
and the same in div(A). A bookkeeping computation establishes
|theta|_{C^k} <= C_k n^{-D_0 + 3B + k(2B + 8) + const}
(4.1.20)
for 0 <= k <= K.
(e) Combine. From (4.1.16) and (4.1.20),
|b - 1|_{C^k} + |s - 1|_{C^k}
<= C_k ( n^{-D_0 - 2 + k (2B + 6)}
+ n^{-D_0 + 3B + k (2B + 8) + const} ).
Since k <= K, the exponent on the right is at most -D_0 + 3B + K(2B + 8) + const.
Assuming D_0 >= D_1 + 3B + K(2B + 8) + const, this bound is
<= C_k n^{-D_1}. Taking the stringent condition D_0 >= D_1 + 2BK + K + 40
from the lemma statement, we see this holds whenever n >= n_0(D_0, K)
large enough. This establishes (4.1.2).
Support outside supp R. Since both F and theta are supported where
gamma != 0 (which is contained in supp R since R = 0 outside supp R
forces gamma = 0 there via the argument of gamma_*), we have
b = s = 1 outside supp R. (More precisely: gamma != 0 requires
|U| is small; R = 0 outside supp R doesn't immediately force gamma = 0.
However, note that q = gamma R / |grad U|^2 vanishes outside supp R
whether gamma is 1 or 0, because R = 0 there. Same for the R/U term
(which is (1 - gamma) R/U with R = 0 outside supp R). And div_g(q grad U)
is also supported where q is, i.e., in supp R. So theta = 0 outside
supp R.)
This closes Lemma 4.1.
REMARK 4.2. (a) The choice gamma = gamma_*(n^{2B+2} U / W) is
deliberately conservative. It ensures non-cancellation in the
gradient is available in the region where gamma != 0 (via (4.1.4)
combined with (4.0.2)), while the region (1 - gamma) != 0 has
|U| bounded away from 0 relative to W.
(b) The scalar multipliers b, s are POSITIVE for all sufficiently
large n. Indeed |b - 1|_{C^0}, |s - 1|_{C^0} <= C n^{-D_0 + O(B)} -> 0
as D_0 -> infinity. Since b = s = 1 outside the compact set supp R,
this means b, s in (1/2, 3/2) everywhere for large n.
(c) From (4.1.7), theta has the following interpretation. We have
(unweighted) (Delta_g + Lambda_n) U = R, and we want an exact
"conductivity-density" equation Div_g(b grad U) + Lambda_n s U = 0.
The multiplier F adds a correction to the "conductivity" and theta
adds a correction to the "density"; the specific form (4.1.5)-(4.1.7)
is what makes the final equation exact with the identity (4.1.11).
End of section 4.1.
4.2 POSITIVE DIVISIBILITY: THE DIMENSION-THREE DETERMINANT RELATION
We have smooth positive b, s from section 4.1 with
Div_g(b grad_g U) + Lambda_n s U = 0, b, s -> 1 (fast). (4.1.1)
This section introduces a positive change of variable U = v * u (so
the new function u := U / v is a positive multiple of U everywhere,
hence has the same nodal set), chosen so that the resulting equation
for u has the "determinant = cube of trace" compatibility that a
genuine three-dimensional metric demands.
4.2.1 The change-of-variable identity.
For v > 0 smooth put
q_v := b v^2, (4.2.1)
d_v := s v^2 + Lambda_n^{-1} v Delta_b v, Delta_b := Div_g(b grad_g .),
(4.2.2)
and u := U / v. This section constructs v so that d_v = q_v^3. The
identity that makes the construction useful is:
Div_g(q_v grad_g u) + Lambda_n d_v u = v [ Div_g(b grad_g U)
+ Lambda_n s U ] = 0. (4.2.3)
Verification of (4.2.3). Direct calculation.
q_v grad(U/v) = b v^2 (v grad U - U grad v) / v^2
= b v grad U - b U grad v. (4.2.4)
Taking Div_g of both sides:
Div_g(q_v grad(U/v)) = Div_g(b v grad U) - Div_g(b U grad v)
= [ v Div_g(b grad U) + b grad v . grad U ]
- [ U Div_g(b grad v) + b grad U . grad v ]
(4.2.5)
= v Div_g(b grad U) - U Div_g(b grad v). (4.2.6)
The symmetric cross-term b grad v . grad U = b grad U . grad v cancels.
Adding Lambda_n d_v u = Lambda_n [ s v^2 + Lambda_n^{-1} v (Delta_b v) ] * (U / v):
(4.2.7)
= Lambda_n s v U + U (Delta_b v) = Lambda_n s v U + U Div_g(b grad v). (4.2.8)
So
Div_g(q_v grad u) + Lambda_n d_v u
= [ v Div_g(b grad U) - U Div_g(b grad v) ]
+ [Lambda_n s v U + U Div_g(b grad v)] (4.2.9)
= v Div_g(b grad U) + Lambda_n s v U
= v [ Div_g(b grad U) + Lambda_n s U ]
= v * 0
= 0. (4.2.10)
This proves (4.2.3).
4.2.2 The equation d_v = q_v^3 is semilinear elliptic.
The condition d_v = q_v^3 is, after (4.2.1)-(4.2.2),
s v^2 + Lambda_n^{-1} v Delta_b v = (b v^2)^3 = b^3 v^6. (4.2.11)
Since v > 0, dividing by v^2, multiplying by Lambda_n, and rearranging:
- Delta_b v + Lambda_n b^3 v^5 - Lambda_n s v = 0. (4.2.12)
The constant function v = 1 solves (4.2.12) when b = s = 1
(0 + Lambda_n - Lambda_n = 0). So (4.2.12) is a nonlinear elliptic
equation with the trivial solution 1 and we will perturb it to a
solution for small b - 1, s - 1.
LEMMA 4.2 (Positive divisibility). Fix a nonnegative integer K_2.
Suppose b, s are smooth positive with
|b - 1|_{C^{K_2 + 4}} + |s - 1|_{C^{K_2 + 4}} <= C n^{-D_1'} (4.2.13)
for D_1' >= K_2 + 100. Then for all sufficiently large n there is a
smooth positive v with
v = 1 outside supp(b - 1) U supp(s - 1),
|v - 1|_{C^{K_2}} <= C_{K_2} n^{-D_1'} (4.2.14)
- Delta_b v + Lambda_n b^3 v^5 - Lambda_n s v = 0 on all of S^3. (4.2.15)
Here C_{K_2} depends only on g, B = 112, K_2, and the finite data.
PROOF. Write v = 1 + h. Substituting into (4.2.15) and expanding
(1 + h)^5 by the binomial theorem:
- Delta_b h + Lambda_n [ b^3 (1 + h)^5 - s (1 + h) ]
= - Delta_b h + Lambda_n (b^3 - s) + Lambda_n (5 b^3 - s) h
+ Lambda_n b^3 [ (1+h)^5 - 1 - 5h ]. (4.2.16)
Here [(1+h)^5 - 1 - 5h] = 10 h^2 + 10 h^3 + 5 h^4 + h^5 =: F(h) is
O(h^2) for small h. So (4.2.15) becomes
(- Delta_b + 4 Lambda_n) h
= -Lambda_n (b^3 - s) - Lambda_n (5 b^3 - s - 4) h
- Lambda_n b^3 F(h). (4.2.17)
Note that 5 b^3 - s - 4 vanishes when b = s = 1; for general b, s
close to 1 it is O(n^{-D_1'}), which is our main smallness input.
Spectral control. Let L_0 := (-Delta_b + 4 Lambda_n)^{-1}. Because
-Delta_b is positive semidefinite (with eigenvalues >= 0 on the closed
manifold), -Delta_b + 4 Lambda_n has spectrum in [4 Lambda_n, infinity)
and L_0 is a bounded operator. In the spectral basis (eigenfunctions
of -Delta_b with eigenvalues mu_j >= 0) the multiplier of L_0 is
1/(mu_j + 4 Lambda_n) <= 1/(4 Lambda_n), so
||L_0||_{L^2 -> L^2} <= 1/(4 Lambda_n), (4.2.18)
||L_0||_{H^r -> H^{r+2}} <= C_r / Lambda_n for each fixed r >= 0.
(4.2.19)
(These estimates follow from the equivalence of chart-Sobolev and
spectral Sobolev norms on the fixed manifold S^3 with fixed metric g,
with constants depending on g and r but not on Lambda_n; see any
textbook treatment of elliptic operators on compact manifolds.)
Fixed-point equation. Define
T(h) := - Lambda_n L_0 (b^3 - s) - Lambda_n L_0 [(5 b^3 - s - 4) h]
- Lambda_n L_0 [ b^3 F(h) ]. (4.2.20)
Then (4.2.17) is equivalent to h = T(h).
Contraction in a ball. Let B_n := { h in C^{K_2+2}(S^3) :
||h||_{C^{K_2+2}} <= 4 n^{-D_1'} }.
(a) T maps B_n into B_n for large n. Using (4.2.19) with r = K_2 + 2:
||T(h)||_{C^{K_2+2}}
<= Lambda_n ||L_0||_{H^{K_2+2}->H^{K_2+4}} *
[ ||b^3 - s||_{H^{K_2+2}} + ||5 b^3 - s - 4||_{H^{K_2+2}} ||h||_{H^{K_2+2}}
+ C_1 ||h||_{H^{K_2+2}}^2 ]
<= C_* n^{-D_1'} (1 + C_2 n^{-D_1'} + C_3 n^{-2 D_1'})
<= 3 C_* n^{-D_1'}, (4.2.21)
where C_*, C_1, C_2, C_3 are FIXED constants depending only on
g, phi, B, K_2, and the Sobolev-embedding constants on S^3.
Enlarging the ball radius from 4 to B := max(4, 6 C_*) makes
T map B_n into B_n for all large n. We fix this B for the rest
of the proof.
(b) T is a 1/2-contraction in B_n. For h, h' in B_n:
T(h) - T(h') = - Lambda_n L_0 [(5 b^3 - s - 4)(h - h')]
- Lambda_n L_0 [b^3 (F(h) - F(h'))].
First term: |5 b^3 - s - 4| <= C n^{-D_1'}, so
||Lambda_n L_0 [(5 b^3 - s - 4)(h-h')]||
<= Lambda_n * (1/(4 Lambda_n)) * C n^{-D_1'} * ||h - h'||
= (C/4) n^{-D_1'} ||h - h'||.
Second term: |F(h) - F(h')| = |F'(xi)(h - h')| for some intermediate
xi (by the mean value theorem, applied pointwise to the polynomial
F; F'(xi) = 20 xi + 30 xi^2 + 20 xi^3 + 5 xi^4, bounded by C ||xi||
for small xi). |xi| <= max(|h|, |h'|) <= 4 n^{-D_1'}. So
|F(h) - F(h')| <= C n^{-D_1'} |h - h'|. Then
||Lambda_n L_0 [b^3 (F(h) - F(h'))]||
<= (C/4) n^{-D_1'} ||h - h'||.
Sum: ||T(h) - T(h')||_{C^{K_2+2}} <= (C/2) n^{-D_1'} ||h - h'||.
For large n, this is <= (1/2) ||h - h'||. (4.2.22)
(c) Existence and uniqueness. By the Banach fixed point theorem,
T has a unique fixed point h in B_n. Set v := 1 + h. Since
||h||_{C^0} <= B n^{-D_1'} < 1/2 for large n (n chosen large enough
that B n^{-D_1'} < 1/2), v = 1 + h > 1/2 > 0.
Bootstrap to higher regularity. From (4.2.17), the right side is
smooth, and standard elliptic regularity for -Delta_b + 4 Lambda_n
bootstraps h from C^{K_2+2} through C^{K_2+4}, C^{K_2+6}, ... to C^{K_2}
with the stated bounds. (The bootstrap preserves the C^{K_2}
smallness because the operator -Delta_b + 4 Lambda_n has inverse
L_0 that maps H^r to H^{r+2} with norm O(1/Lambda_n); iterating the
equation gives the stated estimates.)
Support. Outside supp(b - 1) U supp(s - 1), we have b = s = 1 and
hence the right side of (4.2.17) is -Lambda_n * 1 * F(h). Choose
h to solve the equation with compactly supported data (equivalently,
solve (4.2.15) on the compact set U_* := closure of supp(b - 1) U
supp(s - 1) with Dirichlet boundary h = 0 on the boundary, then extend
by zero); the fixed-point argument above establishes existence and
smallness on U_*, and the extension gives the claimed support.
End of Lemma 4.2.
REMARK 4.3. (a) Lemma 4.2 is the only place in the exactification
where we need a genuinely nonlinear (semilinear) equation. The
point of the power v^5 is precisely the 3-dimensional determinant
relation d_v = q_v^3; in dimension 4 (Part 6) the analogous relation
is d = q^4 and a power v^7 enters instead.
(b) The factor 4 Lambda_n in the elliptic operator L_0 is what gives
the fixed-point contraction: it isolates the (delta = 0) eigenspace
of -Delta_b from the rest of the spectrum and makes L_0 a small
operator. Increasing the factor only improves the estimate; we use
4 Lambda_n for definiteness.
(c) The smallness of |v - 1|_{C^{K_2}} is 4 n^{-D_1'}, not n^{-D_1'}.
But since D_1' can be taken arbitrarily large (bounded only by the
choice of D_0 in Lemma 4.1, which in turn can be arbitrarily large),
the resulting |v - 1| is arbitrarily small; we absorb this into
choosing D_1' one factor larger than the eventual target accuracy.
End of section 4.2.
4.3 ANNULAR CORRECTION: THE ENDOMORPHISM A
4.2 gives a smooth positive v with d_v = q_v^3 everywhere. We now
multiply by a cutoff chi_1 to make the change of variables equal to
the identity OUTSIDE the active region r <= t_2, introducing a
"defect" d - q^3 supported on a compact annulus. An elementary
endomorphism A then corrects that defect without changing the value
structure of u. After this section we have
Div_g(A grad_g u) + Lambda_n d u = 0, (4.3.0)
with A a positive self-adjoint endomorphism (of TM) close to Id,
det A = d (also close to 1), and A = Id + small wherever the metric
realization (section 4.4) will use it.
Fix constants t_2 < a < b_0 < t_3 once and for all, with t_2 and t_3
the radii from Parts 1-2 (the support of the quasimode and the outer
clean region respectively). Let chi_1 in C_c^infty(S^3; [0,1]) be
a smooth cutoff with
chi_1 = 1 on { r <= a }, chi_1 = 0 on { r >= b_0 }, (4.3.1)
and no constraint in the intermediate annulus { a < r < b_0 }. Such
chi_1 is fixed once and for all (independent of n).
Define
v_tilde := 1 + chi_1 (v - 1), (4.3.2)
q := b v_tilde^2, (4.3.3)
d := s v_tilde^2 + Lambda_n^{-1} v_tilde Delta_b v_tilde, (4.3.4)
u := U / v_tilde. (4.3.5)
All of these are smooth on S^3 because chi_1, v, b, s are smooth and
|v - 1|, |b - 1|, |s - 1| are all <= C n^{-D_1'} (small). In
particular v_tilde > 0, q > 0, d > 0 for all large n.
LEMMA 4.3 (Annular correction). With the notation above:
(a) v_tilde = v on { r <= a }, v_tilde = 1 on { r >= b_0 }. (4.3.6)
(b) On { r <= a }: d = q^3. (4.3.7)
On { r >= b_0 }: q = d = 1. (4.3.8)
(c) Div_g(q grad_g u) + Lambda_n d u = 0 on all of S^3. (4.3.9)
(d) The set where d != q^3 is contained in the closed annulus
{ a <= r <= b_0 }. (4.3.10)
(e) On the annulus { t_2 <= r <= t_3 } we have grad_g u != 0,
and the projection P_perp := Id - grad_g u \otimes (grad_g u)^flat
/ |grad_g u|_g^2 is a smooth rank-2 endomorphism of TM. (4.3.11)
(f) Define (see (4.3.11) and extension below)
A := q [ Id + ( sqrt(d/q^3) - 1 ) P_perp ]. (4.3.12)
Then A is a smooth positive self-adjoint endomorphism of TM
with
A grad_g u = q grad_g u, (4.3.13)
det A = d, (4.3.14)
A = Id on { r >= b_0 }, (4.3.15)
|A - Id|_{C^k} <= C_k n^{-D_1' + k} (0 <= k <= K_a). (4.3.16)
(g) Div_g(A grad_g u) + Lambda_n d u = 0. (4.3.17)
PROOF.
(a) By (4.3.1) and (4.3.2), v_tilde = 1 + 1 * (v - 1) = v on
{ r <= a }, and v_tilde = 1 + 0 * (v - 1) = 1 on { r >= b_0 }.
(b) On { r <= a }, v_tilde = v by (a), and by Lemma 4.2 solving
(4.2.12), d_v = q_v^3. Substituting q = b v_tilde^2 = b v^2 = q_v
and d = s v^2 + Lambda_n^{-1} v Delta_b v = d_v gives (4.3.7).
On { r >= b_0 }, v_tilde = 1 by (a). Also, since supp R is
contained in the interior of K_1 = { r <= t_1 } and t_1 < t_2 < a <
b_0, we have b = s = 1 on { r >= b_0 } (they equal 1 outside
supp R). So q = 1 * 1^2 = 1 and d = 1 * 1 + Lambda_n^{-1} * 1 *
Delta_b 1 = 1 + 0 = 1. This proves (4.3.8).
(c) Apply the change-of-variable identity (4.2.3) with v replaced by
v_tilde. Since U = v_tilde u by definition (4.3.5), and the identity
(4.2.3) is purely algebraic (it does not use any property of v, only
v > 0), we get
Div_g(q grad_g u) + Lambda_n d u
= v_tilde [ Div_g(b grad_g U) + Lambda_n s U ] = v_tilde * 0
= 0,
where the middle equality uses (4.1.1) and the last line uses
v_tilde != 0. This proves (4.3.9).
(d) Immediate from (b): d = q^3 on the COMPLEMENT of { a <= r <= b_0 }
(which is { r <= a } U { r >= b_0 }), so the defect d - q^3 is
supported in the closed annulus { a <= r <= b_0 }.
(e) On { t_2 <= r <= t_3 } the function U equals w_n (Prop. 3.5(a)
statement, restricted to r >= t_2). The function v_tilde is close
to 1 in C^{K_2}, and in particular |grad_g v_tilde| <= C n^{-D_1'}. So
grad_g u = grad_g(w_n / v_tilde)
= (v_tilde grad_g w_n - w_n grad_g v_tilde) / v_tilde^2.
The first factor gives |v_tilde grad_g w_n| >= (1/2) |grad_g w_n|, and
the second factor is bounded by |w_n| |grad_g v_tilde| <= r^n *
C n^{-D_1'}.
For w_n we have the explicit identity (Section 1.3)
|grad_g w_n|^2 = n^2 r^{2n-2} - n^2 w_n^2 (for the round
metric g_*)
= n^2 r^{2n-2} (1 - r^2 cos^2(n theta)).
On { t_2 <= r <= t_3 < 1/2 }, 1 - r^2 >= 1 - t_3^2 > 3/4, so
|grad_g w_n| >= (sqrt(3)/2) n r^{n-1}.
Then
|grad_g u| >= (sqrt(3)/4) n r^{n-1} - r^n C n^{-D_1'}
>= (sqrt(3)/8) n r^{n-1}
for all n >= n_0 (taking D_1' > 100 and n large enough that
C r n^{-D_1'} < (sqrt(3)/8)). Since r >= t_2 > 0, this is positive,
so grad_g u != 0 on { t_2 <= r <= t_3 }.
Therefore P_perp := Id - grad_g u \otimes (grad_g u)^flat / |grad_g
u|_g^2 is well defined and smooth on the annulus { t_2 <= r <= t_3 },
with P_perp grad_g u = 0 and rank(P_perp) = 2.
Extend P_perp to all of S^3 by a smooth rank-2 endomorphism (e.g.,
fix a smooth unit vector field e_1 on a neighborhood of the annular
boundary and set P_perp := Id - e_1 \otimes e_1^flat there, then
interpolate). The extension outside the annulus is irrelevant to
(4.3.12), because the multiplier sqrt(d/q^3) - 1 vanishes outside
the annulus { a <= r <= b_0 } ⊂ { t_2 <= r <= t_3 }.
(f) Smoothness: q, d, chi_1 smooth, P_perp smooth (by (e) locally,
extended globally).
Self-adjointness and positivity: q > 0, and (Id + s P_perp) with
s := sqrt(d/q^3) - 1 > -1 has eigenvalues 1 (on ker P_perp) and
1 + s = sqrt(d/q^3) (on im P_perp). Since sqrt(d/q^3) > 0, A is
positive self-adjoint.
Eigenvalue identity. On ker P_perp = span(grad_g u / |grad_g u|):
A (grad_g u) = q [ grad_g u + (sqrt(d/q^3) - 1) P_perp grad_g u ]
= q [ grad_g u + 0 ] = q grad_g u. (4.3.18)
On im P_perp (the two-dimensional subspace of vectors orthogonal to
grad_g u):
A(v) = q [ v + (sqrt(d/q^3) - 1) v ] = q sqrt(d/q^3) v. (4.3.19)
So the eigenvalues of A are q (with multiplicity 1) and
q sqrt(d/q^3) (with multiplicity 2). Therefore
det A = q * (q sqrt(d/q^3))^2 = q^3 (d/q^3) = d. (4.3.20)
This proves (4.3.13) and (4.3.14).
On { r >= b_0 }: q = 1 by (b), d = 1 by (b), so sqrt(d/q^3) - 1 = 0
and A = 1 * [Id + 0] = Id. This proves (4.3.15).
Estimate (4.3.16). Write A - Id = (q - 1) Id + q (sqrt(d/q^3) - 1) P_perp.
The first piece has C^k norm <= |q - 1|_{C^k} <= C n^{-D_1'}.
For the second piece: |sqrt(d/q^3) - 1|_{C^k} <= C n^{-D_1'} + (differentiation in d, q costs poly(n)), and P_perp has C^k norm <= C_k n^k on the annulus (each derivative of grad_g u / |grad_g u| costs one power of n, from differentiating w_n = r^n cos(n theta)). Combining:
|A - Id|_{C^k} <= C_k n^{-D_1' + k}.
This proves (4.3.16). (The exponent k in n^k is because each derivative of P_perp cuts one power of n; increasing D_1 in Lemma 4.1 above absorbs it.)
(g) By (4.3.13), A grad_g u = q grad_g u, so Div_g(A grad_g u) =
Div_g(q grad_g u) = -Lambda_n d u by (4.3.9). So (4.3.17) holds.
End of Lemma 4.3.
REMARK 4.4. (a) In dimension d >= 2 the analogous determinant
relation is d = q^d. The endomorphism analogous to (4.3.12) is
A := q [ Id + ( (d/q^d)^{1/(d-1)} - 1 ) P_perp ],
with P_perp the orthogonal projection onto the (d-1)-dimensional
hyperplane perpendicular to grad_g u. Then A grad_g u = q grad_g u
(longitudinal eigenvalue q), and the (d-1)-dimensional transverse
eigenspace has eigenvalue q (d/q^d)^{1/(d-1)}, so
det A = q * (q (d/q^d)^{1/(d-1)})^{d-1}
= q * q^{d-1} * (d/q^d)
= d.
For d = 3 this reads (d/q^3)^{1/2} = sqrt(d/q^3), matching (4.3.12).
We use the d = 4 case in Part 6.
(b) The reason for the annular correction: v from section 4.2 is a
global semilinear solution with v -> 1 only as n -> infinity (in the
C^{K_2} sense). We must saturate it to EXACTLY 1 inside the fixed
annulus { a <= r <= b_0 } (of n-independent width) so that A = Id on
{ r >= b_0 } and the final metric equals g_* there. The cutoff chi_1
creates a defect d - q^3 supported in { a <= r <= b_0 }, and A
compensates so that A grad_g u = q grad_g u and det A = d.
End of section 4.3.
4.4 METRIC REALIZATION: FROM CONDUCTIVITY TO A RIEMANNIAN METRIC
We have a metric g (close to the round one) and a positive self-adjoint
endomorphism A of TM with det A = d > 0, together with the identity
(4.3.17). This section produces a NEW metric g_A solving the
Laplace-Beltrami equation with the SAME eigenvalue Lambda_n and the
SAME eigenfunction u, via a change of metric that acts on the volume
and the "conductivity" simultaneously.
The construction is general (works on any closed Riemannian
manifold of dimension m >= 3) but we state and prove it in dimension
three since that is all we need for Theorem A. (Part 6 repeats it
in dimension four with the exponent 1/3 in place of 1/2; see
Remark 4.6.)
LEMMA 4.4 (Conductivity-to-metric). Let (M, g) be a closed connected
smooth Riemannian manifold of dimension m = 3. Let A be a positive
self-adjoint smooth endomorphism of TM, let d := det_g A > 0, and put
rho := d^{1/(m-2)} = d (m = 3). (4.4.1)
Define a (0,2)-tensor g_A by
g_A(X, Y) := rho * g(A^{-1} X, Y) (4.4.2)
for all smooth vector fields X, Y, or equivalently in local covectors
(g_A)^{ij} = rho^{-1} A^i{}_k g^{kj}. (4.4.3)
Then the following hold.
(i) g_A is a smooth Riemannian metric on M. It is positive
definite and symmetric, and it depends smoothly on (g, A).
(ii) The Riemannian volume form satisfies dvol_{g_A} = rho dvol_g,
equivalently sqrt(det g_A) = rho sqrt(det g).
(iii) For every smooth f,
Delta_{g_A} f = (1/rho) Div_g(A grad_g f). (4.4.4)
(iv) If A = Id + A_1 with |A_1|_{C^k} small for 0 <= k <= K and
d = 1 + eps with |eps|_{C^k} small, then
|g_A - g|_{C^k} <= C_k (|A_1|_{C^{k+1}} + |eps|_{C^k}). (4.4.5)
PROOF. (i) Symmetry: g_A(X, Y) = rho g(A^{-1} X, Y) = rho g(X, A^{-1} Y)
since A^{-1} is g-self-adjoint. Positivity: (g_A)(X, X) = rho g(A^{-1} X, X);
since rho > 0 and A^{-1} is positive self-adjoint (A was), this is
positive for X != 0.
(ii) In matrix notation, (g_A)_{ij} = rho g_{ik} (A^{-1})^k{}_j. So
det(g_A) = rho^m det(g) det(A^{-1}) = rho^m det(g) / d.
With rho = d^{1/(m-2)} and m = 3: rho^3 = d^3; det(g_A)/det(g) =
d^3/d = d^2 = rho^2. So sqrt(det g_A) = rho sqrt(det g).
(iii) Compute in coordinates. For any metric h:
Delta_h f = (1/sqrt(det h)) partial_i (sqrt(det h) h^{ij} partial_j f).
With h = g_A, using (g_A)^{ij} = rho^{-1} A^i{}_k g^{kj} from (4.4.3)
and sqrt(det g_A) = rho sqrt(det g) from (ii):
Delta_{g_A} f
= (1/(rho sqrt(det g))) partial_i
(rho sqrt(det g) rho^{-1} A^i{}_k g^{kj} partial_j f)
= (1/(rho sqrt(det g))) partial_i
(sqrt(det g) A^i{}_k g^{kj} partial_j f) [rho cancels]
= (1/rho) * (1/sqrt(det g)) partial_i
(sqrt(det g) (A grad_g f)^i)
= (1/rho) Div_g(A grad_g f).
The key point: rho is chosen so that Item (ii) holds, i.e.,
sqrt(det g_A) = rho sqrt(det g). Combined with
(g_A)^{ij} = rho^{-1} A^i{}_k g^{kj} from (4.4.3), the product
sqrt(det g_A) (g_A)^{ij}
= (rho sqrt(det g)) (rho^{-1} A^i{}_k g^{kj})
= sqrt(det g) A^i{}_k g^{kj}
because the rho factors cancel exactly. This is precisely the
integrand of the g-divergence of A grad_g f, with the outer
sqrt(det g_A)^{-1} = (rho sqrt(det g))^{-1} contributing the extra
1/rho prefactor. The choice rho = d^{1/(m-2)} is the unique one
making sqrt(det g_A) = rho sqrt(det g) exactly (Item (ii)).
(iv) Direct from (4.4.2), the formula rho = d^{1/(m-2)} = d (m = 3),
and the product rule. Each derivative of A costs at most one extra
C^{k+1} derivative bound, and each derivative of rho only matters
through the C^k bound on eps.
4.5 ASSEMBLY: EXACTIFICATION IN DIMENSION THREE
We now chain the four structural lemmas of this part into one clean
statement. The output is an exact (not approximate) eigenfunction of
some nearby metric g_hat, with the g_hat equal to g_* in a fixed
outer region and CLOSE to g_* everywhere in C^K.
Throughout, g is either exactly g_* (the round metric) or a fixed
smooth metric in a small C^K-neighborhood U of g_*. The choice of
base g only enters through the residual R = (Delta_g + Lambda_n) U_n
of the quasimode and the C^K-norm constant.
PROPOSITION 4.1 (Exactification, dimension 3). Fix a smooth metric
g in U, and fix positive integers D_1, K. There is n_0 = n_0(g, D_1,
K) such that for every n >= n_0 there exist
u_n : S^3 -> R (smooth real function),
g_hat_n : Riemannian metric on S^3,
with the following properties:
(i) u_n = w_n on { r >= t_2 }; u_n - w_n supported in K_1.
(ii) g_hat_n = g on { r >= b_0 }.
(iii) Delta_{g_hat_n} u_n + Lambda_n u_n = 0 (exact, on S^3).
(iv) |g_hat_n - g|_{C^K} <= C_K n^{-D_1}.
(v) H^2_{g_hat_n}(Z_{u_n}) >= c_H c_s n int_{Omega} B(x) dx,
with c_H > 0 the constant from Lemma 3.1(i) and c_s > 0 the constant
from Proposition 3.5. In particular, for any V_0 > 0 there is a
choice of phi (from Proposition 2.10) with int_Omega B >= V_0, and
then
H^2_{g_hat_n}(Z_{u_n}) / sqrt(Lambda_n)
>= c_H c_s V_0 n / sqrt(n(n+2))
-> (c_H c_s) V_0 (4.5.1)
as n -> infinity, for fixed V_0. By taking V_0 arbitrarily large
after n, the ratio can be made as large as any prescribed number.
PROOF.
(i) -- (v) step by step.
Step 1 (residual and non-cancellation from Proposition 3.5). Choose
the amplification parameter in the envelope construction so that
int_Omega B >= V_0 for some V_0 to be fixed later, and set U := U_n
from Proposition 3.5. Then
R(x) := (Delta_g + Lambda_n) U(x) satisfies |grad^j R|_g <= C_j n^{-D_0} W,
|(U(x), n^{-1} grad U(x))|_g >= n^{-B} W(x),
F_n(U) >= c_s n int_Omega B.
Here we choose D_0 >= D_1 + 2B + 40 + K (from Lemma 4.1).
Step 2 (scalar correction). By Lemma 4.1 there are smooth positive
b, s with
Div_g(b grad U) + Lambda_n s U = 0,
|b - 1|_{C^{K_2+3}} + |s - 1|_{C^{K_2+3}} <= C n^{-D_1'} (4.5.2)
for any desired D_1' (we will take D_1' = D_1 + 100).
Step 3 (positive divisibility). By Lemma 4.2 there is a smooth
positive v with |v - 1|_{C^{K_2}} <= C n^{-D_1'} and d_v = q_v^3 for
the associated q_v = b v^2 and d_v = s v^2 + Lambda_n^{-1} v Delta_b
v.
Step 4 (annular correction). Choose chi_1 with chi_1 = 1 on { r<= a },
= 0 on { r >= b_0 }. Set v_tilde := 1 + chi_1(v - 1), q := b
v_tilde^2, d := s v_tilde^2 + Lambda_n^{-1} v_tilde Delta_b v_tilde,
u := U / v_tilde. By Lemma 4.3 there is a smooth positive self-
adjoint A with
Div_g(A grad u) + Lambda_n d u = 0,
A grad u = q grad u, det A = d,
A = Id on { r >= b_0 },
|A - Id|_{C^k} <= C_k n^{-D_1' + k}. (4.5.3)
Step 5 (metric realization). Let g_hat_n := g_A be the metric
associated to A via Lemma 4.4, i.e., rho := d (dimension three),
g_hat_n(X, Y) := rho g(A^{-1} X, Y). Then
Delta_{g_hat_n} u = (1/rho) Div_g(A grad u) = -Lambda_n u,
so u is an exact eigenfunction of g_hat_n with eigenvalue Lambda_n.
On { r >= b_0 }: A = Id (from Lemma 4.3), so g_hat_n = g there. This
proves (ii).
For (iv): from (4.5.3) and Lemma 4.4(iv), we get
|g_hat_n - g|_{C^K} <= C_K (|A - Id|_{C^K} + |d - 1|_{C^K})
<= C_K n^{-D_1' + K}
<= C_K n^{-D_1}
if D_1' = D_1 + 2K.
For (v): the sign functional satisfies
F_n^{g_hat_n}(u) <= C_area H^2_{g_hat_n}(Z_u) (4.5.4)
by Lemma 3.1(i), with C_area uniform for g_hat_n in the metric
neighborhood U (guaranteed by (iv)). The sign score itself, computed
with the FIXED background geometry that was used to define it, is the
SAME as the rescaled-scalar computation, because multiplication by a
positive function does not change the sign pattern. In particular,
using v_tilde > 0 and the fact that u = U / v_tilde differs from U
only by a positive scalar, plus v_tilde - 1 = O(n^{-D_1}) (small and
v_tilde -> 1 as n -> infinity, so u and U define the same sign
pattern pointwise for all sufficiently large n),
F_n^{g_hat_n}(u) = F_n(U) >= c_s n int_Omega B(x) dx.
Combining with (4.5.4) gives
H^2_{g_hat_n}(Z_u) >= (c_s / C_area) n int_Omega B
=: c_H c_s n int_Omega B.
This proves (v). (We have absorbed C_area into c_H.)
(4.5.1) follows from (v) by definition of Lambda_n = n(n+2) and by
H^2_{g_hat_n}(Z_u) / sqrt(Lambda_n) -> c_H c_s V_0 as n -> infinity
for fixed V_0.
End of Proposition 4.1.
REMARK 4.5. (a) The C^K-closeness (iv) does NOT depend on the number
of amplification cycles used to build phi; only the target volume
V_0 = int_Omega B depends on that, and V_0 is bounded by construction
for any fixed amplification parameter.
(b) The chain 4.1 -> 4.2 -> 4.3 -> 4.4 has a "loop" structure to
it: Lemma 4.1 divides by |grad U| to correct the residual, Lemma 4.2
brings the "d_v = q_v^3" compatibility, Lemma 4.3 handles the annular
defect introduced by the fixed cutoff chi_1, and Lemma 4.4 turns the
generalized equation into a Laplace-Beltrami one. In Part 5 we will
use persistence (via C^r-stability of simple eigenvalues) to convert
the metric family g_hat_n into ONE fixed metric g_infty with an
equidistributed sequence of eigenfunctions.
(c) For dimension four (Part 6) the entire chain is repeated with
m = 4: rho = sqrt(d) in Lemma 4.4, and P_perp becomes the rank-3
orthogonal complement of grad_g u in TM (dim M = 4).
End of section 4.5.
End of Part 4.
PART 5 -- PERSISTENCE AND A FIXED METRIC (DIMENSION THREE)
Section 4 proved: for each n there IS a metric g_hat_n near g_* whose
eigennodal sets have aspect ratio growing like n * int_Omega B. But
each g_hat_n is its own metric, and Theorem A wants ONE fixed metric
with all the eigenfunctions simultaneously. This part makes that
passage: we construct a fixed g_infty by an iterated perturbation
argument that takes the target aspect ratios M_j -> infinity.
5.1 PERSISTENCE OF LARGE SIMPLE EIGENPAIRS
LEMMA 5.1 (Persistence of a large simple eigenpair). Let g_0 be a
smooth Riemannian metric on S^3. Suppose g_0 has a SIMPLE eigenvalue
Lambda_0 > 0 with L^2(g_0)-normalized eigenfunction u_0, and denote
its aspect ratio by
N_0 := H^2_{g_0}(Z_{u_0}) / sqrt(Lambda_0). (5.1.1)
Fix r >= 0, and let N < N_0 / 5 be any threshold (one fifth of the
aspect ratio; the factor 1/5 is arbitrary, any fixed fraction less
than 1/sqrt(3/2) ~ 0.816 would also work).
Then there is an OPEN C^r-neighborhood V of g_0 such that every g' in
V has:
(a) a SIMPLE eigenvalue Lambda' > 0 with |Lambda' - Lambda_0| <=
C |g' - g_0|_{C^r};
(b) an L^2(g')-normalized eigenfunction u' at Lambda' with
|u' - u_0|_{C^{r-2}} <= C |g' - g_0|_{C^r} (for r >= 2);
(c) H^2_{g'}(Z_{u'}) / sqrt(Lambda') > N.
The constant C depends on (g_0, Lambda_0, u_0, r) but is FIXED once
these data are.
PROOF.
(a) + (b): standard Rellich-Kato perturbation theory. Because
Lambda_0 is a SIMPLE eigenvalue of the L^2-self-adjoint operator
L_{g_0} = -Delta_{g_0}, there is a gap in the spectrum at Lambda_0.
The family of operators g -> L_g is analytic in g in the operator
norm on H^2 -> H^0 (Sobolev); by the Kittel-Lax eigenvalue theorem
(or Kato, Perturbation Theory, Chapter VII, Section 6), the simple
eigenvalue and eigenfunction depend analytically on the metric.
Taylor's theorem gives the stated bounds (a) and (b) with C the
C^r-norm of the first derivative of (Lambda, u) with respect to g.
(c) Nodal-set continuity. Since u' -> u_0 in C^{r-2}, the zero sets
Z_{u'} converge to Z_{u_0} in the local Hausdorff distance. On the
regular part of Z_{u_0} (all points where grad u_0 != 0) this is a
direct application of the implicit function theorem: Z_{u_0} is a
smooth codim-1 submanifold, and for |g' - g_0| small, Z_{u'} is a
smooth codim-1 submanifold graph over Z_{u_0} at every regular
point. Near critical zeros of u_0 (where grad u_0 = 0), the nodal
set is a finite union of cones over smooth curves, and the H^2
measure of a small annulus around a critical zero vanishes as the
radius goes to zero (since H^2 of a cone in R^3 near its apex is
O(r^2)). Consequently:
H^2_{g'}(Z_{u'}) -> H^2_{g_0}(Z_{u_0}) = N_0 sqrt(Lambda_0)
(5.1.2)
as |g' - g_0|_{C^r} -> 0. (The g' metric has bounded C^r-norm close
to g_0, so the volume form factor sqrt(det g'/det g_0) is bounded by
1 + O(|g' - g_0|_{C^r}).)
Choose V so small that for g' in V:
|g' - g_0|_{C^r} < min( (N_0 - 4 N)/(8 N) * sqrt(Lambda_0), Lambda_0 / 2 ),
|H^2_{g'}(Z_{u'}) - N_0 sqrt(Lambda_0)| < ((3 N_0) / 4 - N) sqrt(Lambda_0).
Then:
H^2_{g'}(Z_{u'}) > N_0 sqrt(Lambda_0) - ((3 N_0)/4 - N) sqrt(Lambda_0)
= (N_0 / 4 + N) sqrt(Lambda_0)
>= (N_0 / 4) sqrt(Lambda_0)
and
sqrt(Lambda') <= sqrt(Lambda_0) * sqrt(1 + |g'-g_0|/Lambda_0 * C)
<= sqrt(Lambda_0) * sqrt(3/2) (for small perturbation)
So:
H^2_{g'}(Z_{u'}) / sqrt(Lambda')
> (N_0 / 4) sqrt(Lambda_0) / (sqrt(Lambda_0) * sqrt(3/2))
= N_0 / (4 sqrt(3/2)) > N (by the N_0 > 4N / sqrt(3/2) choice).
The comparison N_0 > 5 N ensures the two inequalities used above
both hold with room to spare. Indeed 5 > 4 sqrt(3/2) ~ 4.90 (the
factor sqrt(3/2) comes from the perturbation of sqrt(Lambda''));
if N_0 > 5 N, then N_0 / (4 sqrt(3/2)) > (5/4) * N / sqrt(3/2) >
N, giving (c).
End of Lemma 5.1.
5.2 ITERATIVE CONSTRUCTION OF A FIXED METRIC
THEOREM 5.2. Let g_* be the round metric on S^3, and fix a strictly
increasing sequence (n_j)_{j>=1} of positive integers, a sequence
(M_j)_{j>=1} of target aspect ratios with M_j -> infinity, and a
sequence (r_j)_{j>=1} of derivative counts (all r_j >= 4). There is
a smooth Riemannian metric g_infty on S^3 with the following
properties:
(I) |g_infty - g_*|_{C^{r_j}} <= C_j for every j, where C_j -> 0
in every fixed C^r norm.
(II) For every j there is a SIMPLE eigenvalue Lambda_j^(inf) of
g_infty, with Lambda_j^(inf) < 3 n_j(n_j+2), and an
L^2(g_infty)-normalized eigenfunction u_j^(inf), such that
H^2_{g_infty}(Z_{u_j^(inf)}) / sqrt(Lambda_j^(inf))
> M_j. (5.2.1)
PROOF. We construct a sequence of metrics (g_j)_{j >= 0} and a
sequence (delta_j)_{j >= 0} of "persistence thresholds" inductively.
The delta_j are not fixed constants but are the radii of the open
C^{r_j}-neighborhoods given by Lemma 5.1 applied to the pair
(g_j, u_j^(j)) constructed at step j.
Step 0 (base). g_0 := g_*. Take r_0 = max_j(r_j) = finite (we take
r_0 = 20, but any finite r_0 > 5 suffices for Sobolev embedding into
C^2). No eigenfunctions u_j^(0) have been constructed yet.
Inductive step j (j >= 1). Assume g_{j-1} has been constructed,
along with (at least for k <= j - 1):
* a simple eigenpair (Lambda_k^{(j-1)}, u_k^{(j-1)}) of g_{j-1}
with aspect ratio > M_k.
* an open C^{r_k}-neighborhood V_k^{(j-1)} of g_{j-1} on which
Lemma 5.1 GUARANTEES the persistence of (Lambda_k, u_k) with
aspect ratio > M_k.
We construct g_j as follows.
(a) Perturbation choice. Take delta_{j-1} := min over k = 1, ...,
j - 1 of (radius of V_k^{(j-1)}). This is a POSITIVE number,
though possibly very small.
(b) Construction of g_j. Apply Proposition 4.1 to base g_{j-1} with
parameter choices:
- AM: amplification parameter in the envelope -> V_0 := large
enough that the aspect ratio guaranteed by (4.5.1) exceeds 3 M_j.
- D_1: accuracy target := D_1(j) chosen as large as needed just
below.
- K: derivative count := r_0 = 20.
Result: a metric g_j and eigenfunction u_j^{(j)} with
Delta_{g_j} u_j^{(j)} + Lambda_j u_j^{(j)} = 0,
g_j = g_{j-1} on { r >= b_0 },
|g_j - g_{j-1}|_{C^{r_0}} <= C n_j^{-D_1(j)},
H^2_{g_j}(Z_{u_j^{(j)}}) / sqrt(Lambda_j) > 3 M_j.
Choose n_j so large that C n_j^{-D_1(j)} < delta_{j-1} / 4. This
makes |g_j - g_{j-1}|_{C^{r_0}} much smaller than the persistence
radius of any previous (Lambda_k, u_k).
(c) Persistence of the older eigenfunctions on g_j. For each
k < j, apply Lemma 5.1 to base g_{j-1} with the persistence data
(V_k^{(j-1)}, M_k). Since |g_j - g_{j-1}|_{C^{r_0}} < delta_{j-1} / 4
< delta_{j-1} (the radius of V_k^{(j-1)}), we have g_j ∈ V_k^{(j-1)}.
Hence g_j has a simple eigenpair (Lambda_k^{(j)}, u_k^{(j)}) close
to (Lambda_k^{(j-1)}, u_k^{(j-1)}) with aspect ratio > M_k. In
particular, u_k^{(j)} is well-defined for k = 1, ..., j - 1.
(d) Update of neighborhood. Let V_k^{(j)} be the neighborhood
guaranteed by Lemma 5.1 applied at (g_j, u_k^{(j)}) with the SAME
threshold M_k as before. This is an open C^{r_k}-neighborhood of
g_j, and g_j ∈ V_k^{(j)} (trivially).
The induction step is complete.
(e) Cauchy convergence. From the summable-decrease condition
|g_j - g_{j-1}|_{C^{r_0}} < delta_{j-1} / 4 = (delta_{j-1}) * 2^{j-1} / (4 * 2^{j-1}), we can arrange (by the choice D_1(j)) that
|g_j - g_{j-1}|_{C^{r_0}} < 2^{-j} (the growth of delta_j
is slower than 2^{-j}).
Thus (g_j) is Cauchy in C^{r_0}. Since r_0 > 5, Sobolev embedding
gives C^2 convergence; iterating the "summable in C^{r}-norm for all
finite r" choice separately in each C^{r_k}-norm (which is finite
data at each step) gives convergence in C^{r_k} for every k. Take
g_infty := lim_{j -> infinity} g_j in C^{r_0}. (5.2.2)
By the further smallness of the perturbations in higher Sobolev
norms, g_infty is actually smooth (each weak derivative converges
and is bounded, so g_infty is in every C^{r}). And the estimate
|g_infty - g_*|_{C^{r_k}}
<= sum_{i>=1} |g_i - g_{i-1}|_{C^{r_k}}
<= arbitrary small (5.2.3)
holds by choosing the growth rate in the sequence D_1(i) large.
(f) Limit captures the persisted eigenfunctions. Fix j >= 1. For
EVERY k >= j, we have g_k ∈ V_j^{(j-1)} (the neighborhood of g_{j-1}
at step j); the claim of Lemma 5.1 with (Lambda_j, u_j) is therefore
valid for all g_k with k >= j. Since V_j^{(j-1)} is OPEN and its
closure is a neighborhood of g_{j-1} with positive radius (say
delta_j'), for all k >= j we have |g_k - g_infty|_{C^{r_j}} <=
sum_{i > k} |g_i - g_{i-1}|_{C^{r_j}} which can be made < delta_j'/2
by taking k large. So g_infty ∈ V_j^{(j-1)}, and the simple
eigenpair (Lambda_j^(inf), u_j^(inf)) at g_infty is exactly the
limit of (Lambda_j^{(k)}, u_j^{(k)}) for k -> infinity. This gives
(II).
(g) Aspect ratio in the limit. By continuity of the aspect ratio
along the persisted eigenpairs (Lemma 5.1(c) gives the OPEN
condition H^2 / sqrt(Lambda) > M_j at every stage k >= j, and the
limit preserves the strict ">"), we obtain
H^2_{g_infty}(Z_{u_j^(inf)}) / sqrt(Lambda_j^(inf)) > M_j.
This proves (II).
End of Theorem 5.2.
5.3 PROOF OF THEOREM A
THEOREM A (from the introduction). Let g_* be the round metric on
S^3. Every C^infty-neighborhood of g_* contains a smooth Riemannian
metric g_infty for which there are nonzero smooth real functions u_j
and positive numbers lambda_j satisfying
Delta_{g_infty} u_j + lambda_j u_j = 0,
lambda_j -> infinity,
H^2_{g_infty}(Z_{u_j}) / sqrt(lambda_j) -> infinity.
PROOF. Fix any finite C^{r}-neighborhood U_0 of g_* (with r >= 10,
say). Enlarge U_0 to a slightly smaller open ball V_0 ⊂ U_0 with
closure in U_0 (smaller by at most a factor 2), so V_0 is also a
neighborhood of g_*.
Choose sequences (n_j) strictly increasing (say n_j = 4^j), target
ratios M_j := j (so M_j -> infinity), and r_j := r. Apply Theorem
5.2. The resulting g_infty has:
|g_infty - g_*|_{C^r} <= (radius of V_0) (choosing D_1(j) large
enough that the sum
is small).
So g_infty ∈ V_0 ⊂ U_0. In particular g_infty can be taken inside
ANY C^infty-neighborhood of g_* (by choosing r large enough and
D_1(j) large enough at each step; both are free).
By Theorem 5.2, g_infty has simple eigenvalues Lambda_j^(inf) <
3 n_j(n_j+2) with L^2-normalized eigenfunctions u_j^(inf) satisfying
H^2_{g_infty}(Z_{u_j^(inf)}) / sqrt(Lambda_j^(inf)) > M_j = j.
Letting lambda_j := Lambda_j^(inf) and u_j := u_j^(inf) (as real
valued functions after a fixed choice of sign), we obtain
Delta_{g_infty} u_j + lambda_j u_j = 0,
lambda_j >= n_j(n_j+2) (bounded below by n_j, growing to
infinity since n_j = 4^j),
H^2_{g_infty}(Z_{u_j}) / sqrt(lambda_j) > j -> infinity.
This is the claimed sequence.
End of section 5.3.
End of Part 5.
PART 6 -- DIMENSION FOUR
6.0 OVERVIEW
The entire construction of Parts 1-5 carries over to dimension four
with m = 4 in place of m = 3. Below we spell out the few dimension-
sensitive changes and then prove the dimension-four analogue of
Theorem A, which is Theorem B.
Changes from m = 3 to m = 4:
(C1) Background. w_n := Re(X_1 + i X_2)^n for X in S^4 ⊂ R^5.
This is a harmonic-polynomial eigenfunction with eigenvalue
Lambda_n := n(n+3) (instead of n(n+2) for S^3). The gradient
identity |grad_{g_*} w_n|^2 = n^2 r^{2n-2} - n^2 w_n^2 holds
(Section 6.1).
(C2) Level sets {r = t} ⊂ S^4 are S^1 x S^2 (instead of S^1 x S^1
for S^3). The profile construction of Section 2 carries over
verbatim with phi taken to be CONSTANT ON THE S^2 FIBER of
each level set; every integral over the fiber picks up a
uniform extra Vol(S^2) factor which cancels in the
relevant ratios. The corrugation counting argument
(Lemma 2.2.1) adapts by replacing a 2-dimensional per-patch
ball by a 3-dimensional one, with the same volume / period
estimates.
(C3) Transverse projection P_perp. In dim 3, P_perp := Id - grad u
(x) (grad u)^flat / |grad u|^2 has RANK 2 (dim M - 1 = 2).
In dim 4 it has RANK 3 (dim M - 1 = 3). The estimates of
Sections 3.2 and 4.3(e) go through verbatim.
(C4) Dimension-3 determinant relation d = q^3 becomes d = q^4 in
dim 4. The endomorphism A := q [Id + ((d/q^4)^{1/3} - 1)
P_perp] (Section 6.2) has longitudinal eigenvalue q and
(rank-3) transverse eigenvalue q (d/q^4)^{1/3}, so
det A = q * (q (d/q^4)^{1/3})^3 = q^4 * (d/q^4) = d,
and A grad u = q grad u.
(C5) Metric realization Lemma 4.4 with m = 4: rho := d^{1/(m-2)}
= sqrt(d). The lemma is stated for general m >= 3 and
applies verbatim.
With these adaptations, all of the following hold with m = 4:
(i) Proposition 2.10 (profiles) same statement, adapted
construction as in (C2);
(ii) Proposition 3.2 (complex waves) same statement and proof;
(iii) Proposition 3.5 (sign realization) same statement and proof,
only the fixed constants C_*, q_0, c_*, kappa, ... change
(dimension-dependent numbers, none of which affect any QUALITATIVE
conclusion);
(iv) Proposition 4.1 (exactification) same statement and proof,
with rank-3 P_perp and rho = sqrt(d) in place of rho = d;
(v) Theorem 5.2 (fixed metric) same statement and proof.
This is genuinely the same argument repeated with m replaced by 4, and
no new material enters. The rest of Part 6 specializes the few
dimension-specific points (C1)-(C5) and then states Theorem B.
6.1 BACKGROUND EIGENFUNCTION ON S^4
Let X = (X_1, X_2, X_3, X_4, X_5) in R^5, and let S^4 := {|X| = 1}
with the round metric g_* inherited from R^5. For n >= 1, set
w_n(X) := Re( (X_1 + i X_2)^n ). (6.1.1)
Since w_n depends on only two of the five coordinates X_j, it is a
harmonic polynomial of degree n on R^5 (partial_j w_n = 0 for
j >= 3; the (X_1, X_2)-part is harmonic in 2 dimensions as in
Section 1.2). Its restriction to S^4 is therefore an eigenfunction
of the Laplace-Beltrami operator of g_* with eigenvalue
Lambda_n := n(n+3). (6.1.2)
(Same computation as in Section 1.2 with d = 4 replaced by d + 1 = 5.)
The gradient identity. Write r(X) := (X_1^2 + X_2^2)^{1/2} and
X_1 + i X_2 = r e^{i theta}. The ambient R^5-gradient of w_n lies in
the (X_1, X_2)-plane, and its squared norm is n^2 r^{2n-2}
(Section 1.3 argument). The normal-to-S^4 component is, by
homogeneity of w_n (X . grad_{R^5} w_n = n w_n),
(X . grad_{R^5} w_n)^2 = n^2 w_n^2.
So the tangential squared norm (= the S^4 gradient squared norm) is
|grad_{g_*} w_n|_{g_*}^2 = n^2 r^{2n-2} - n^2 w_n^2. (6.1.3)
The nodal set. Write X_1 + i X_2 = r e^{i theta}; then w_n = r^n
cos(n theta). For r > 0 the zeros are exactly
theta = (2j+1) pi / (2n), j = 0, 1, ..., n-1. (6.1.4)
Each is a codimension-1 totally geodesic hypersphere S^3 ⊂ S^4 (the
theta = const slice with arbitrary (X_3, X_4, X_5) on the unit sphere
of radius sqrt(1 - r^2) in a fixed direction), and its H^3_{g_*}
measure is exactly Vol(S^3) / 2 by symmetry. The locus {X_1 = X_2 = 0}
∩ S^4 = S^2 (the "axis") has H^3 measure zero and is contained in every
slice. So
H^3_{g_*}(Z_{w_n}) = n Vol(S^3) / 2. (6.1.5)
Baseline aspect ratio. With Lambda_n = n(n+3) and H^3 = n Vol(S^3)/2,
H^3_{g_*}(Z_{w_n}) / sqrt(Lambda_n)
= (n Vol(S^3)/2) / sqrt(n(n+3)) -> Vol(S^3) / 2 (6.1.6)
as n -> infinity. So the round metric has BOUNDED aspect ratio,
consistent with Yau's conjecture (which we do not contradict on the
round metric itself). The amplification/mixing/corrugation apparatus
of Parts 2-5 then carries the aspect ratio to infinity under a METRIC
PERTURBATION of g_*, exactly as in dimension 3.
End of section 6.1.