================================================================
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
jetties; 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 jetties, 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 - (K + J + 1 - m)/2 <= 2 + m - (K - m)/2 - (J+1)/2
= 2 + m - D - 2m - 3 - (J+1)/2 + O(1/n)
<= -D - 1 (for J >= 2D + 2m + 4). Actually direct:
since K > 2D + 2m, (K - m)/2 > D + m/2, so
2 + m - (K - m)/2 - (J+1)/2 < 2 + m - (D + m/2) - (D+m+2)/2
= 2 + m - D - m/2 - (D+m)/2 - 1 = 1 + m/2 - 3D/2 - 3m/2
= 1 - 3D/2 - m, which is <= -D - 1 when D + m >= 2 (always
since D, m >= 0). Good.
(R): (1 - j) + m - (K + J - j - m)/2 <= -D - 1 for every
0 <= j <= J: since (K + J - m) - j >= K + J - m - J = K - m
and RHS - D - 1 requires (1-j) + m - (K+J-j-m)/2 >=
(1-j) - (J-j)/2 + m - (K-m)/2. The first group
(1-j) - (J-j)/2 = 1 - j - J/2 + j/2 = 1 - J/2 - j/2,
minimized at j = J, giving value 1 - J. So R <= 1 - J +
m - (K-m)/2 <= -D - 1 as long as J >= D + m + 2 -
(K-m)/2. With J = D + m + 1 and (K-m)/2 >= D + m + 3,
R <= 1 - J + m - D - m - 3 = -J - D - 2 = -2D - m - 3,
<= -D - 2. Good.
(G): -J + m - (K-m)/2 <= -(D + m + 1) + m - (D + m + 3)
= -2D - m - 4 <= -D - 1. 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.