SCALAR CURVATURE AND URYSOHN WIDTH IN DIMENSION THREE
A reader-friendly version of the paper
"Spectral scalar curvature and Urysohn width in dimension three"
0. HOW TO READ THIS FILE
This file has two jobs.
First, it explains the main theorem in ordinary language. A high-school
student who knows what distance, a graph, and a continuous map are can
read Sections 1 through 3 and understand the statement and the strategy.
Second, it gives the full proof. Sections 6 through 14 use standard
tools from differential geometry and geometric measure theory. Those
sections are included so that the paper is complete, not because every
reader must read every line on the first pass. A good first pass is:
read Sections 1-3, then read the statement of each lemma in Sections
6-14, then go back for details.
The original paper was written for experts. This version tries to be
clearer, shorter, and more honest about what each step is doing. The
mathematics is the same. The constant 500 is not optimized.
1. THE QUESTION
Imagine a long tube. It can be very long, so its diameter is huge.
But it is thin in two directions. If we squash the tube onto a line,
every point of the line comes from a small cross-section of the tube.
So the tube is "large" in one direction and "small" in the other two.
The Urysohn width measures exactly this idea. We want to map a space
onto a graph so that every fiber -- the set of points mapping to one
point of the graph -- is small.
For a metric space (X,d), define the Urysohn 1-width by
UW_1(X,d) = inf over (K,f) of sup over y in K of diam_d f^{-1}(y),
where K runs over one-dimensional simplicial complexes (graphs), f runs
over continuous maps X -> K, and diam_d of a fiber means the largest
distance in X between any two points of that fiber. The empty fiber has
diameter zero.
A graph is the simplest possible target that can have one large
direction. The theorem says that a positive lower bound for a certain
spectral version of scalar curvature forces the Urysohn 1-width to be
finite, with an explicit bound.
Why "spectral"? The hypothesis is not that scalar curvature is large
at every point. It is that a certain energy inequality holds for every
compactly supported smooth test function. This is weaker and more
flexible than a pointwise bound. It is the natural condition produced
by the positive-weight construction in Section 4.
2. THE ANSWER
THEOREM (main theorem).
Let (M^3,g) be a connected complete smooth Riemannian three-manifold
without boundary. Suppose there is a number lambda > 0 such that
integral_M ( 4 |grad_g phi|_g^2 + Scal_g phi^2 ) dvol_g
>= lambda integral_M phi^2 dvol_g
for every compactly supported smooth function phi on M.
Then there exist a one-dimensional simplicial complex K and a continuous
map f : M -> K such that every fiber has diameter at most
500 / sqrt(lambda)
in the original metric g.
No orientability, spin structure, compactness, or bounded geometry is
required.
The notation -4 Delta_g + Scal_g >= lambda always means the integral
inequality above. If Scal_g >= lambda at every point, then the integral
inequality holds, because the gradient term is nonnegative. But the
integral inequality can hold even when scalar curvature is negative
somewhere. The theorem uses only the integral inequality.
The theorem is about the given metric g. It is not enough to conformally
change g to a metric of positive scalar curvature: the fiber diameters
are measured in g, and the proof must control distances in g itself.
The constant 500 is convenient. The proof actually gives a bound below
500, but no attempt is made to sharpen it.
3. WORDS YOU NEED
This section is a dictionary. Skip it if the words are familiar.
RIEMANNIAN MANIFOLD. A space that looks locally like ordinary Euclidean
space and has a way to measure lengths and angles. A three-manifold
looks locally like R^3. A surface looks locally like R^2.
METRIC g. The rule for measuring lengths. The distance between two
points is the infimum of lengths of paths joining them.
COMPLETE. Every Cauchy sequence converges. Equivalently, every path
that tries to leave every compact set has infinite length. Completeness
lets us take limits and use minimizing geodesics.
SCALAR CURVATURE Scal_g. A single number at each point that measures,
in an averaged sense, how much the manifold is curved. In dimension
three it is half the full curvature information. Positive scalar
curvature is a weak kind of positive curvature.
LAPLACIAN Delta_g. The divergence of the gradient. It measures how a
function differs from its average nearby. We use the sign convention
Delta_g = Div_g grad_g,
so that on R^n the Laplacian is the sum of second partial derivatives.
GRADIENT grad_g phi. The vector field dual to the differential dphi.
Its squared length is |grad_g phi|_g^2.
COMPACTLY SUPPORTED. A function that is zero outside some compact set.
The test functions phi in the hypothesis vanish near infinity.
FIBER. For a map f : X -> K and a point y in K, the fiber is
f^{-1}(y) = { x in X : f(x) = y }.
The fiber can have several connected components. The diameter of the
fiber is the largest distance between any two of its points, including
points in different components. This is important: the theorem bounds
whole fibers, not just connected components.
GRAPH. A one-dimensional simplicial complex. It has vertices and edges,
and its topology is the usual polyhedral topology.
WEIGHTED AREA. For a smooth function T on the ambient manifold, the
weighted area of a surface Sigma is
integral_Sigma e^{-T} dArea.
The factor e^{-T} is a positive density. Weighted area is ordinary area
in the conformal metric e^{-T} g, but we measure final distances in g.
STABLE SURFACE. A surface is stable if the second variation of area is
nonnegative for every compactly supported normal variation. A soap film
that does not lower its area when wiggled is stable. Stability is a
differential inequality, not a global minimality statement.
PRESSURE. A smooth function mu used to force a surface into a region.
We minimize weighted area minus the weighted volume integral of mu.
The Euler-Lagrange equation is
H - nu T = mu,
where H is mean curvature and nu is the chosen unit normal. The function
mu acts like a pressure difference across the surface.
INTEGRAL CURRENT. A generalized oriented surface used in geometric
measure theory. It has a mass, a boundary, and weak compactness
properties. We use currents only to obtain a smooth minimizer in the
interior and across the cut boundary. Once the minimizer is smooth, the
rest of the argument is ordinary surface geometry.
4. WHY THE HYPOTHESIS IS NATURAL
The operator -4 Delta_g + Scal_g appears when one studies the second
variation of area for a minimal surface in a three-manifold. If a
minimal surface is stable, the Gauss equation and the stability
inequality force a certain two-dimensional curvature expression to be
nonnegative. That expression is exactly the two-dimensional version of
D(g,t) = Scal_g + 2 Delta_g t - |grad_g t|_g^2.
The spectral hypothesis produces a positive function v with
(-4 Delta_g + Scal_g) v >= lambda v.
Setting t = -2 log v gives
D(g,t) >= lambda.
After scaling the metric, we may assume lambda = 1. The function t is a
weight: it tells us how much to discount area near different parts of M.
The weighted area density is e^{-t}. The key point is that t does not
change the metric in which we measure the final fiber diameters. It only
changes which surfaces are cheap to cut along.
This is the same mechanism that appears in Gromov's torus-stabilized
scalar curvature and in the Chodosh-Li slice-and-dice method. The new
feature here is a direct weighted proof that also handles nonorientable
and noncompact manifolds without a global bound on the weight or its
derivatives.
5. PROOF ROADMAP
After scaling, assume lambda = 1.
STEP A. POSITIVE WEIGHT.
The spectral hypothesis gives a smooth function t with D(g,t) >= 1.
Weighted area uses density e^{-t}. This is Lemma 6.1.
STEP B. STABLE SURFACES ARE SMALL.
If a closed connected two-sided surface is stable for weighted area with
pressure mu, and
D(g,t) + mu^2 - 2 |grad mu|_g >= c > 0
along the surface, then its diameter in the induced metric is at most
2 pi / sqrt(c). It is also a sphere or a real projective plane.
This is Proposition 7.4.
STEP C. CUT ALONG SURFACES.
Choose a maximal locally finite family of pairwise disjoint closed
connected stable weighted minimal surfaces such that no finite subfamily
disconnects M. Cut M along them. In each resulting component P, every
compact connected two-sided surface that meets the boundary neatly
separates P. The boundary components are small spheres or projective
planes. This is Proposition 8.1.
STEP D. DOUBLE EACH PIECE.
Double P along its boundary and smooth the metric and weight so that
they are symmetric under reflection. The smoothed data satisfy
D(h,T) >= 1/2, (1/4) g <= h <= 4 g.
This lets us use the closed-surface estimate inside P without a boundary
regularity problem. This is Lemma 9.1.
STEP E. BUILD SEPARATING INTERFACES.
For a base point o in P and a large radius ell, minimize weighted
perimeter minus a pressure volume term in an annulus around o. The
minimizer is a finite union of small separating surfaces F_i between o
and the distant slab. Each F_i lies in a controlled annulus and has
diameter at most 4 pi. This is Proposition 10.1.
STEP F. BOUND SLABS.
The surfaces F_i separate P, so the graph of complementary regions is a
tree. Some single F_i separates o from any given slab component C.
Every minimizing path from o to a point x in C crosses F_i. This gives
diam(C) < 220. This is Proposition 11.1.
STEP G. COVER BY SLABS.
The sets
{ j - 3/4 < r < j + 3/4 }, j = 0, 1, 2, ...
form a cover of multiplicity at most two by sets of diameter at most 440
in the original manifold M. Here r is distance from o in the smoothed
metric.
STEP H. MAP TO A GRAPH.
Choose a partition of unity subordinate to the slab cover. Make a graph
with one vertex for each partition function. Join two vertices by an
edge when their supports overlap. The formula
F(x) = sum_v phi_v(x) v
defines a continuous map to the graph. If F(x) = y and v is a vertex of
the simplex containing y, then phi_v(x) > 0, so x lies in the cover set
U_v. Thus a whole fiber is controlled by one small set. This is
Section 12.
STEP I. GLUE ACROSS THE CUTS.
The two boundary copies of a cut surface may map to different graphs.
On a collar of each cut surface, homotope the boundary map to a point.
The homotopy stays in a finite subgraph whose vertices still satisfy the
anchor condition. Add a new edge path joining the two chosen points.
The resulting maps agree on the two boundary copies, so they descend to
M. This is Section 13.
STEP J. BOUND THE WHOLE FIBER.
If y lies in the old graph, every point of f^{-1}(y) is within L = 2 pi + 2
of the same cover set q(U_v). Two such points are within 440 + 2L < 500
of each other. If y lies on a newly added edge, the fiber is contained
in two collars and has diameter at most L. This proves the theorem for
lambda = 1. Rescaling g to lambda g gives the general bound 500/sqrt(lambda).
This is Section 14.
6. THE POSITIVE WEIGHT
From now on, until the final rescaling, assume lambda = 1. The first
step is to turn the spectral inequality into a positive density on M.
Define
D(g,t) = Scal_g + 2 Delta_g t - |grad_g t|_g^2
for a smooth function t on M.
LEMMA 6.1 (positive weight).
Let (M,g) be a connected smooth Riemannian manifold without boundary.
Suppose that
integral_M ( 4 |grad_g phi|_g^2 + Scal_g phi^2 ) dvol_g
>= integral_M phi^2 dvol_g
for every compactly supported smooth function phi. Then there is a
smooth function t : M -> R such that
D(g,t) >= 1
everywhere on M.
PROOF.
It is enough to find a smooth positive function v with
(-4 Delta_g + Scal_g) v >= v.
Then t = -2 log v gives
D(g,t) = Scal_g - 4 (Delta_g v)/v >= 1,
because
Delta_g t = -2 (Delta_g v)/v + 2 |grad_g v|_g^2/v^2
|grad_g t|_g^2 = 4 |grad_g v|_g^2/v^2.
If M is compact, take a positive first eigenfunction v of the operator
-4 Delta_g + Scal_g. Its eigenvalue is at least one by the hypothesis.
If M is noncompact, fix a point o in M. Choose an increasing exhaustion
Omega_1 subset Omega_2 subset ... , union Omega_j = M,
by connected relatively compact smooth domains containing o. Let
lambda_j be the first Dirichlet eigenvalue of -4 Delta_g + Scal_g on
Omega_j, and let v_j be a positive first eigenfunction normalized by
v_j(o) = 1.
The quadratic-form hypothesis, extended to the Sobolev space H_0^1(Omega_j)
by density, gives lambda_j >= 1. Domain monotonicity makes lambda_j
decreasing. A fixed nonzero test function supported in Omega_1 bounds
lambda_j from above. Therefore lambda_j decreases to some
lambda_infinity >= 1.
On each compact subset of M, the local Harnack inequality, applied along
finitely many chains of coordinate balls joining that subset to o,
bounds v_j above and away from zero for all sufficiently large j.
Interior elliptic estimates give smooth convergence on compact subsets,
after passing to a diagonal subsequence, to a positive smooth function v
with
(-4 Delta_g + Scal_g) v = lambda_infinity v.
Thus D(g,t) >= 1. This proves the lemma.
Fix such a function t for the rest of the proof. Weighted area means
area with density e^{-t}. The weighted area of a surface Sigma is
Area_t(Sigma) = integral_Sigma e^{-t} dArea.
All minimization below takes place in compact regions, so no global
bound on t or its derivatives is needed.
7. STABLE WEIGHTED SURFACES
7.1. Weighted surfaces and stability
Let Sigma be a smooth two-sided surface, possibly with boundary, in a
Riemannian three-manifold (X,h). Choose a unit normal nu. Write
k = h restricted to Sigma. The second fundamental form is
B(V,W) = h(nabla_V nu, W),
and the mean curvature is H = trace_k B.
For a smooth function T on X, weighted area is
integral_Sigma e^{-T} dArea_k.
If Sigma bounds a region E, with outward normal nu, consider the
functional
integral_{partial E} e^{-T} dArea_h
- integral_E mu e^{-T} dvol_h,
where mu is a smooth function on X called the pressure. Its
Euler-Lagrange equation is
H - nu T = mu.
Here nu T means the normal derivative of T.
Stability means that the second variation of this functional is
nonnegative for every compactly supported normal variation in the
interior of Sigma. Define the weighted surface Laplacian
Delta_{Sigma,T} u = Delta_k u - < d(T|Sigma), du >_k
and the potential
Q = Ric_h(nu,nu) + Hess_h T(nu,nu) + |B|_k^2 + nu mu.
Then stability is the inequality
integral_Sigma ( |du|_k^2 - Q u^2 ) e^{-T} dArea_k >= 0
for every smooth u compactly supported in the interior of Sigma.
We use the sign convention Delta = Div grad, and round spheres have
positive scalar curvature. A closed surface means compact without
boundary.
7.2. Curvature transferred to the surface
LEMMA 7.1 (weighted stability).
Suppose a smooth two-sided surface Sigma satisfies
H - nu T = mu
and is stable in the sense above. Let Omega be either the whole surface,
if it is closed and connected, or a connected smooth surface domain with
compact closure in the interior of Sigma. Then there is a smooth
positive function w on Omega such that, with p = log w and
U = T|Sigma - p,
D(k,U) >= D(h,T) + mu^2 + |B|_k^2 + 2 nu mu + |dp|_k^2
on Omega.
PROOF.
Take a positive first eigenfunction w of the self-adjoint operator
-Delta_{Sigma,T} - Q on Omega, using the measure e^{-T} dArea_k. In the
domain case use Dirichlet boundary conditions. Stability makes the first
eigenvalue nonnegative. Therefore
-Delta_{Sigma,T} w >= Q w,
and, with p = log w,
-Delta_{Sigma,T} p >= Q + |dp|_k^2.
The function w is positive in Omega. Its possible vanishing on the
boundary of Omega will not enter the calculation.
The Gauss equation and the decomposition of the ambient Laplacian give
Scal_k = Scal_h - 2 Ric_h(nu,nu) + H^2 - |B|_k^2,
Delta_h T = Delta_k(T|Sigma) + H nu T + Hess_h T(nu,nu),
and
|dT|_h^2 = |d(T|Sigma)|_k^2 + (nu T)^2.
Combining these gives
D(k,T|Sigma)
= D(h,T)
- 2 ( Ric_h(nu,nu) + Hess_h T(nu,nu) )
+ (H - nu T)^2
- |B|_k^2.
Changing T|Sigma to U = T|Sigma - p gives the exact identity
D(k,U) = D(k,T|Sigma) - 2 Delta_{Sigma,T} p - |dp|_k^2.
Now substitute the inequality for p, the definition of Q, and
H - nu T = mu. The ambient normal Ricci and Hessian terms cancel,
leaving the stated inequality. This proves the lemma.
In particular, if
D(h,T) + mu^2 - 2 |grad mu|_h >= c > 0
along a stable surface, then D(k,U) >= c on the closed surface or on
the chosen relatively compact domain, because
mu^2 + 2 nu mu >= mu^2 - 2 |grad mu|_h.
7.3. The conformal index estimate
The next lemma converts a two-dimensional curvature inequality into a
bound on length in the original metric.
LEMMA 7.2 (conformal index estimate).
Let (S,k) be a smooth surface, let U be a smooth function on S, and
suppose that
D(k,U) >= c > 0.
Let gamma be a finite geodesic segment for the conformal metric
k_tilde = e^{-2U} k whose index form is nonnegative for normal variations
compactly supported in its interior. Then
length_k(gamma) <= 2 pi / sqrt(c).
The same conclusion holds when the curvature hypothesis is assumed only
on a neighborhood of the interior of the segment.
PROOF.
The Gauss curvature of k_tilde is
K_tilde = e^{2U} ( K_k + Delta_k U )
>= (1/2) e^{2U} ( c + |dU|_k^2 ),
because Scal_k = 2 K_k.
Parametrize gamma by k-arclength s in [0,L]. Choose a parallel
k_tilde-unit normal along its interior. Such a choice exists along an
interval even when S is nonorientable. Since d(s_tilde) = e^{-U} ds,
the index inequality and the curvature bound imply
0 <= integral_0^L e^U
( (phi')^2 - (1/2)(c + |dU|_k^2) phi^2 ) ds
for every smooth phi compactly supported in (0,L). Here U and dU are
evaluated along gamma.
Set phi = e^{-U/2} psi. Since |U'| <= |dU|_k, completing the square
gives
e^U ( (phi')^2 - (1/2)(c + |dU|_k^2) phi^2 )
= 2 (psi')^2 - (c/2) psi^2
- ( psi' + (1/2) U' psi )^2
- (1/2)( |dU|_k^2 - (U')^2 ) psi^2
<= 2 (psi')^2 - (c/2) psi^2.
Therefore
2 integral_0^L (psi')^2 ds >= (c/2) integral_0^L psi^2 ds
for every smooth psi compactly supported in (0,L). Approximating
sin(pi s / L) in the Sobolev space H_0^1(0,L) gives
2 pi^2 / L^2 >= c/2,
which is L <= 2 pi / sqrt(c). All test variations are supported away
from the endpoints, so the local version also follows. This proves the
lemma.
7.4. Closed stable surfaces are small
PROPOSITION 7.4 (closed stable surfaces).
Let Sigma be a smooth closed connected two-sided surface in a Riemannian
three-manifold (X,h). Let T and mu be smooth ambient functions. Suppose
that Sigma satisfies
H - nu T = mu,
is stable for weighted area with pressure mu, and satisfies
D(h,T) + mu^2 - 2 |grad mu|_h >= c > 0
for a constant c > 0. Then, in the induced metric k = h|Sigma,
diam_k(Sigma) <= 2 pi / sqrt(c).
Moreover, Sigma is diffeomorphic to S^2 or to RP^2.
PROOF.
Lemma 7.1 gives a smooth function U on Sigma with D(k,U) >= c. The
conformal metric e^{-2U} k is complete because Sigma is compact. Join
any two points by a minimizing geodesic for that metric. Its index form
is nonnegative, so Lemma 7.2 bounds its k-length by 2 pi / sqrt(c). The
k-distance between the endpoints is no greater.
Integrating D(k,U) >= c and using Gauss-Bonnet gives
4 pi chi(Sigma)
= integral_Sigma 2 K_k dArea_k
>= c Area_k(Sigma) + integral_Sigma |dU|_k^2 dArea_k
> 0.
This formula is valid also for nonorientable surfaces, either directly
with densities or by passing to the orientation double cover. The
classification of closed connected surfaces now gives precisely S^2 and
RP^2. This proves the proposition.
7.5. A local exit estimate
The closed-surface estimate is not enough when a stable sheet is not
known to be compact. We need a local version that prevents such a sheet
from escaping a fixed neighborhood.
LEMMA 7.5 (local exit estimate).
Let Sigma be a smooth two-sided stable surface satisfying
H - nu T = mu in a Riemannian three-manifold (X,h). Suppose that
D(h,T) + mu^2 - 2 |grad mu|_h >= c > 0
on Sigma. Let Omega_0 and Omega_1 be connected smooth domains with
compact closures in the interior of Sigma such that the closure of Omega_0
is contained in Omega_1 and the boundary of Omega_0 is nonempty. For
every x in Omega_0 there is a path from x to the boundary of Omega_0,
contained in the closure of Omega_0, whose length in k = h|Sigma is at
most 2 pi / sqrt(c).
PROOF.
Apply Lemma 7.1 in Omega_1. The resulting U is smooth on a neighborhood
of the closure of Omega_0 and satisfies D(k,U) >= c there. In the
compact Riemannian manifold with boundary given by the closure of Omega_0,
choose a path of least e^{-2U}k-length from x to the boundary of Omega_0,
and stop it at its first boundary point. Its interior is a geodesic in
Omega_0 and its index form is nonnegative for compactly supported interior
variations. Lemma 7.2 gives the stated k-length estimate. This argument
uses compactness only of the two chosen domains, not completeness of
Sigma. This proves the lemma.
For the original data (M,g,t), weighted minimality means mu = 0. Since
D(g,t) >= 1, Proposition 7.4 bounds every closed connected two-sided
stable weighted minimal surface by 2 pi and identifies its topology. The
same constant in Lemma 7.5 prevents an open stable sheet from extending
arbitrarily far inside a region where it has compact intersections with
ambient balls.
8. CUTTING UNTIL EVERY SURFACE SEPARATES
The surface estimate controls the size of a closed stable weighted
minimal surface. It does not make an arbitrary separating construction
connected. So we first remove the topological obstruction: after cutting
along a locally finite family of small stable surfaces, every compact
connected two-sided hypersurface neatly embedded in a resulting piece
will separate that piece.
A hypersurface in a manifold with boundary is NEATLY EMBEDDED if its
boundary is exactly its intersection with the ambient boundary and the
intersection is transverse. A closed surface means a compact surface
without boundary. All hypersurfaces below are smooth.
8.1. The maximal family
PROPOSITION 8.1 (separating cut).
There is a locally finite family S of pairwise disjoint closed connected
embedded two-sided stable weighted minimal surfaces in M such that the
following statements hold. Let N be the manifold obtained by cutting M
along S, and let q : N -> M be the natural gluing map.
1. The map q is proper and is a quotient map. Each connected component P
of N, with the intrinsic length metric of the pulled-back metric g, is
proper.
2. Each boundary component of P is a sphere or a real projective plane,
is stable weighted minimal, and has intrinsic g-diameter at most 2 pi.
Each surface of S gives two boundary copies in N.
3. Every compact connected neatly embedded two-sided hypersurface in P
separates P.
PROOF.
Order by inclusion the families of pairwise disjoint closed connected
embedded two-sided stable weighted minimal surfaces for which no finite
subfamily disconnects M. The empty family is admissible. The union of
a chain is admissible because each finite subfamily lies in one member of
the chain. Zorn's lemma gives a maximal family S.
We first verify local finiteness. By Proposition 7.4, every member has
intrinsic, and hence ambient, diameter at most 2 pi. Consequently all
members meeting a fixed compact set A in M lie in the compact closed
2 pi-neighborhood of A. Here completeness is used through the properness
of the Riemannian distance on M.
For any finite subfamily S_1,...,S_k of S, the classes [S_i] in
H_2(M;Z/2) are linearly independent. Indeed, the connected complement
of the subfamily contains a path joining the two sides of S_i. Closing
it by a short transverse arc gives a loop meeting S_i once and missing
the other surfaces. Modulo two, intersection with this loop pairs to
one with [S_i] and to zero with the other classes. This uses only the
intersection pairing of compact cycles, so neither orientability nor
compactness of M is required.
A compact smooth submanifold neighborhood of the closed 2 pi-neighborhood
of A has finite-dimensional H_2(-;Z/2). Its image in H_2(M;Z/2)
therefore bounds the size of every finite subfamily meeting A. Thus only
finitely many members meet A. This proves local finiteness.
Local finiteness makes cutting well-defined. In a tubular chart at a cut
surface, q glues the two closed half-charts along their boundary; away
from the cuts it is a diffeomorphism. These charts show that q is a
quotient map and is proper locally over relatively compact neighborhoods.
A finite such cover of a compact subset of M proves global properness.
The pulled-back metric induces the manifold topology, also for its
intrinsic length metric on each component P. A closed bounded subset of
P maps into a bounded subset of M; it is closed in the compact inverse
image of a sufficiently large closed ball. It is therefore compact.
This proves properness of P. The boundary assertions follow from
Proposition 7.4.
It remains to prove separation. Fix a connected component P of N. We
first treat closed hypersurfaces in the interior. This is the only step
requiring minimization.
8.2. A closed hypersurface separates
LEMMA 8.2 (closed separation).
Every closed connected embedded two-sided hypersurface in the interior
of P separates P.
PROOF.
Suppose that F in the interior of P is such a hypersurface and does not
separate. Fix a coorientation of F. A loop crossing F once has nonzero
algebraic crossing number. If it reverses ambient orientation, traverse
it twice. A general-position perturbation in dimension three then gives
a smooth embedded, transverse, orientation-preserving loop gamma in the
interior of P with nonzero algebraic crossing number with F. The
crossing number is unchanged by this perturbation.
A COMPACT MINIMIZATION PROBLEM.
Let pi : P_hat -> P be the orientation double cover, including both
copies when P is orientable. Give P_hat its canonical orientation and
write tau for the deck involution. The coorientation of F and the
ambient orientation orient its full lift as an integral 2-cycle T_F
satisfying
tau_# T_F = -T_F.
Since gamma preserves orientation, it has two closed lifts. In a small
tubular neighborhood of one of them, choose a smooth compactly supported
closed 2-form beta such that T_F(beta) is nonzero. Concretely, the normal
disk bundle of this lifted loop is an oriented trivial 2-disk bundle. A
disk area form with a smooth bump of integral one, extended along the
loop and by zero outside the tube, is a Thom form. Its evaluation is the
signed crossing number, up to the orientation chosen for the disk. The
tube and the support of beta lie in the interior. Preserving this
nonzero evaluation will detect a nonseparating component, once the
surface estimate has made the components meeting the form support
compact.
Choose a relatively open set W in P with compact closure that contains F
and the closed 10-neighborhood of pi(supp beta) in the intrinsic metric
of P. Such a set exists by properness. Let J be a compact set in P
containing the closure of W. In the conformal metric
g_bar = e^{-t} g,
surface area is precisely area weighted by e^{-t}. Minimize g_bar-mass
among integral 2-cycles T supported in pi^{-1}(J) that satisfy
tau_# T = -T, T(beta) = T_F(beta).
These are absolute cycles, not relative cycles. To interpret them at the
boundary, smoothly extend P and g_bar across the boundary of P, take the
orientation cover of that extension, and regard the currents as currents
in the extension with the indicated support constraint. The form beta
extends by zero. The competitor T_F shows that the class is nonempty.
Federer-Fleming compactness and closure, followed by lower semicontinuity
of mass, give a minimizer T_0: support is confined to a fixed compact
set, the boundaries vanish, and both conditions above are closed under
weak convergence.
In a sufficiently small ball over W intersected with the interior of P,
T_0 is locally mass minimizing. A compactly supported integral-cycle
change there evaluates to zero on beta, since beta is exact in the ball.
Making the opposite change in the disjoint deck translate preserves the
constraints; a strict decrease would give a strict decrease of twice the
amount upstairs. We must establish the same minimizing property across
the boundary before using interior regularity.
COMPARISON ACROSS A MINIMAL BOUNDARY.
Each boundary component is minimal for g_bar, because its g_bar-area is
its original weighted area. Near any point of such a component there is
a foliation by g_bar-minimal disks, with the boundary disk as one leaf.
Here is the local construction, which does not require a globally stable
extension of the boundary. On a sufficiently small disk in that leaf,
the Dirichlet Jacobi operator, with the stability sign, has strictly
positive first eigenvalue: its zeroth-order term is bounded and the
Poincare inequality dominates it after shrinking the disk. Solve the
minimal-graph equation with small constant normal boundary height by the
implicit function theorem, using standard Dirichlet elliptic theory. The
derivative at height zero solves the Jacobi equation with boundary value
one and is strictly positive by the maximum principle. After shrinking
the disk and the height interval, these graphs give the asserted
foliation.
Let Y be its unit normal field, directed toward the allowed side of the
boundary. Since all leaves are minimal, Div_g_bar Y = 0. In an oriented
flow box the 2-form
alpha = iota_Y dvol_g_bar
is closed, annihilates Y, and is invariant under its flow. Projection
along this flow onto the boundary leaf therefore pulls its area form back
to alpha. As alpha has comass one, this projection has 2-Jacobian at
most one on every tangent 2-plane.
To apply this to currents, choose a contractible flow box U' upstairs,
with coordinates (z,s), whose boundary leaf is s = 0 and whose allowed
side is s >= 0. Require that the allowed side lie over W, and that the
closure of U' and its tau-translate be disjoint. Take a smaller box U
compactly contained in U'. Suppose a compactly supported integral
2-cycle S in U gave a strictly smaller mass for T' = T_0 + S. Use the
locally Lipschitz map
R(z,s) = (z, max{0,s})
near the closure of U, interpolating to the identity before leaving U',
and keeping R equal to the identity everywhere on the allowed side. The
interpolation can be made in the flow coordinate and can be taken to map
U' into itself. Every portion of T' on the exterior side is supported
in U, where the exact projection rule holds. On the allowed side,
including tangent planes carried by the boundary leaf, R is the identity.
The area formula and the preceding Jacobian bound give
M_g_bar(R_# T') <= M_g_bar(T').
No Jacobian bound for the interpolation region is needed, since T' has
no exterior mass there.
Put S' = R_# T' - T_0. Pushforward preserves the cycle condition, and
R_# T_0 = T_0. Thus S' is an integral cycle supported on the allowed side
in a compact subset of U'. Both S' and tau_# S' evaluate to zero on
beta: use exactness in the respective contractible boxes, with primitives
cut off beyond the cycle supports. Consequently
T_0 + S' - tau_# S'
is again supported in pi^{-1}(J) and satisfies the constraints. The
boxes are disjoint, and antisymmetry of T_0 makes the two mass changes
equal. By the mass inequality their sum is strictly negative,
contradicting minimality. This proves local mass minimization in full
neighborhoods across the boundary.
REGULAR SHEETS AND THEIR COORIENTATIONS.
We now use codimension-one interior regularity for mass-minimizing
integral currents: in a smooth Riemannian 3-manifold, a locally
mass-minimizing integral 2-current without boundary is, near each support
point, a smooth embedded minimal surface with locally constant integer
multiplicity and a smooth orientation. The full-neighborhood comparison
just proved supplies precisely the interior hypothesis even at points of
the boundary of P. The strong maximum principle implies that any sheet
touching the boundary agrees there with the boundary leaf. Hence, over
W, the projected support is smooth and each connected component is either
contained in the boundary of P or disjoint from it.
In a sufficiently small evenly covered neighborhood of a projected support
point, regularity gives an embedded sheet in each lift. Deck invariance
identifies their projections, so the projected support is itself an
embedded surface. Moreover, the orientations give coorientations
downstairs: tau reverses both the current orientation and the ambient
orientation, so preserves the resulting normal direction. The absolute
multiplicities on the two lifts of a connected projected component agree.
Every component in W intersected with the interior of P is therefore a
two-sided stable weighted minimal surface, initially possibly noncompact.
To see the asserted stability, extend a compactly supported normal
variation of one sheet to an ambient flow isolating that sheet and
supported in W intersected with the interior of P. Its lifts preserve
antisymmetry. They preserve evaluation on beta by the homotopy formula,
because d beta = 0 and the boundary of T_0 is zero. Minimality of T_0,
and the locally constant multiplicities, then give the weighted stability
inequality for that sheet.
THE COMPONENTS DETECTED BY THE FORM ARE CLOSED.
Let Sigma be a component of the projected support on W meeting
pi(supp beta), and choose x in that intersection. This component is
disjoint from the boundary of P. Its intersection with the closed ball
B_P(x,10) is compact. Indeed, this ball is compact and is contained in
W; the projected support is closed relative to W because the cover is
finite, and its components are closed there. The regular embedded
surface structure is the subspace topology.
If Sigma were noncompact, choose a connected relatively compact smooth
domain Omega_0 in Sigma containing this intersection in its interior.
Its nonempty boundary lies outside the closed ball B_P(x,10). Choose
another connected relatively compact smooth surface domain Omega_1 with
the closure of Omega_0 contained in Omega_1. Stability holds throughout
Omega_1, and D(g,t) >= 1. Lemma 7.5, with mu = 0 and c = 1, produces a
path from x to the boundary of Omega_0 of induced g-length at most 2 pi.
Its image in P has no greater length. This contradicts the distance of
the boundary of Omega_0 from x, which is greater than 10. Thus Sigma is
a closed surface in the interior of P.
Only finitely many projected components meet the form support, by
compactness and local regularity. Since T_0(beta) is nonzero, at least
one of these closed components is nonseparating in P. For otherwise,
for each component choose the closure of one of its sides, with the
choice agreeing with its coorientation. On the orientation cover,
Stokes' theorem gives zero for the integral of beta over its full
oriented lift. This remains valid when the side is noncompact, since
beta is compactly supported; boundary portions in the boundary of P
contribute nothing, since its support lies in the interior. Multiplying
by the constant multiplicities and summing would give T_0(beta) = 0.
Let S be such a nonseparating component. It is closed, connected,
embedded, two-sided, stable weighted minimal, and lies in the interior of
P. Its image in M is disjoint from every member of S. Adding this image
preserves the defining property of S. In fact, after deleting any finite
subfamily the ambient manifold remains connected, and the two sides of S
can still be joined while avoiding S: join them in P minus S and push the
path into the interior of P. A connected two-sided hypersurface has at
most two complementary components, so this joining path means that the
new surface does not disconnect that complement. We have contradicted
maximality, proving the lemma.
8.3. From closed to neat hypersurfaces
We finish the proof of Proposition 8.1. A compact connected two-sided
neat hypersurface has a bicollar, and its complement has at most two
components. If it does not separate, the bicollar and a path joining its
two sides produce a loop in the interior crossing it once. Suppose F is
such a nonseparating neat hypersurface. Its bicollar defines a smooth map
u : P -> S^1
which is constant outside a compact set, goes once around the circle
along the transverse collar coordinate, and has F as a regular level.
The map has nonzero degree on a loop crossing F once. One may use a
transverse profile that is constant near both ends of the bicollar, so
extension by the same constant value outside the collar is smooth.
Every boundary component of P has finite fundamental group. The
restriction of u to a collar of any such component therefore has a smooth
real lift under the covering R -> S^1. Only finitely many boundary
components meet the compact region where u is not constant, by local
finiteness. On their collars, multiply the difference between this lift
and a chosen lift of the outside constant by a smooth cutoff that is zero
near the boundary of P and one near the interior end. Exponentiating
gives a compactly supported homotopy of u to a smooth map u_1 that equals
the same constant on a neighborhood of all of the boundary of P. Its
degree on the crossing loop is unchanged.
Choose a regular value of u_1 distinct from the constant value. Its
inverse image is a compact cooriented hypersurface in the interior of P,
with finitely many connected components. Perturb the crossing loop to be
transverse to it. The sum of its signed crossings with these components
is the nonzero degree of u_1 on the loop. If every component separated P,
every such signed crossing number would be zero. Thus a component would
contradict Lemma 8.2. This proves the final assertion of Proposition 8.1.
9. DOUBLING A PIECE
Fix a component P of the cut manifold in Proposition 8.1. We want to
construct small separating surfaces between a base point and a distant
distance slab. The boundary of P prevents a direct use of the
closed-surface estimate, so we first smooth the doubled weighted
geometry.
Let Z be the double of P along its boundary, and let sigma exchange its
two copies. If the boundary of P is empty, take two disjoint copies.
LEMMA 9.1 (weighted doubling).
There are smooth sigma-invariant data (h,T) on Z such that
D(h,T) >= 1/2, (1/4) g <= h <= 4 g
on each copy of P. The data can be left unchanged outside disjoint,
locally finite collars of the boundary components.
PROOF.
Choose disjoint normal collars of the compact boundary components. Their
widths may vary with the component. In one such collar, use inward
distance s >= 0 and write
g = ds^2 + k(s),
m(s) = log( dvol_{k(s)} / dvol_{k(0)} ) - t(s).
The ratio in this formula is a ratio of densities on the boundary surface,
so no orientation is needed. With the normal d/ds, the slices have mean
curvature
H = (1/2) trace_{k(s)} k'(s),
and m' = H - t'. Weighted minimality of the boundary therefore gives
m'(0) = 0.
The useful curvature formula in these variables is
D(g,t) = D(k(s),t(s))
- | (1/2) k'(s) |_{k(s)}^2
- (m')^2
- 2 m''.
Here D(k(s),t(s)) uses the two-dimensional slice metric. To verify the
formula, use
Scal_g = Scal_{k(s)} - |(1/2)k'|^2 - H^2 - 2H',
Delta_g t = Delta_{k(s)} t + t'' + H t',
and combine the normal terms as
-(H - t')^2 - 2(H' - t'').
Reflect k and m evenly across s = 0. The reflected k is continuous and
locally Lipschitz, whereas the reflected m is C^2 by m'(0) = 0. Convolve
these data in s with an even, nonnegative smooth mollifier of radius
delta. An even cutoff, equal to one near zero and supported in a smaller
collar, blends the convolution back to the original data. Denote the
resulting smooth even data by k_delta and m_delta, and set
h_delta = ds^2 + k_delta(s),
T_delta = log( dvol_{k_delta(s)} / dvol_{k(0)} ) - m_delta(s).
For small delta, the tensor k_delta is positive definite.
We check the curvature inequality also at points approaching the seam.
On compact subcollars, k_delta, m_delta and their tangential derivatives
through order two converge uniformly to the reflected data. The first
two normal derivatives of m_delta converge uniformly as well. Thus every
term in the curvature formula, except the squared norm of k_delta'/2,
converges to its one-sided value at the seam. For a sequence of points
approaching (x,0) as delta tends to zero, the convolved k' averages
tensors approaching k'(x,0) and -k'(x,0). Convexity of the squared norm,
and convergence of the metrics used in that norm, give
limsup_{delta -> 0} | (1/2) k_delta' |_{k_delta}^2
<= | (1/2) k'(x,0) |_{k(0)}^2.
Consequently the lower limit of D(h_delta,T_delta) there is at least one.
Away from the seam, the convergence is smooth. Compactness of the collar
region in which changes occur now gives D(h_delta,T_delta) >= 1/2 for all
sufficiently small delta. Uniform convergence also gives the metric
comparison above. Choose the smoothing radius separately on each collar.
The collars are locally finite, so these choices give global smooth data,
equal to (g,t) elsewhere. This proves the lemma.
Write d for the intrinsic length metric of h restricted to P. It is
proper by the metric comparison and properness of the original length
metric on P. Fix o in the interior of P and put
r(x) = d(o,x) (x in P).
Extend r evenly to Z. The folding map Z -> P preserves the length of a
path on either copy and does not increase length across the seam. This
follows directly from h = ds^2 + k(s), with k even, by replacing s with
|s| along an absolutely continuous path. In particular, the even
extension of r is proper and 1-Lipschitz on each component of Z. We will
also use the consequence
d(x,y) <= length_h(gamma) (x,y in P)
for every path gamma in Z from x to y.
10. SEPARATING INTERFACES IN A DISTANCE ANNULUS
We use a weighted perimeter-minus-volume minimization, as in the
Chodosh-Li slice-and-dice method, to obtain the following output for the
slab argument. An open set here is understood relative to P when it
meets the boundary of P.
PROPOSITION 10.1 (separating interfaces).
Let ell >= 100, and suppose that the set {r > ell} is nonempty. There
are finitely many disjoint smooth compact connected two-sided neat
hypersurfaces F_1,...,F_m in P such that
F_i is contained in { ell - 91 < r < ell - 18 },
diam_d(F_i) <= 4 pi.
Each F_i separates P. Moreover, P minus the union of the F_i is the
disjoint union of two open sets U_+ and U_-, with o in U_+ and the set
{r > ell} contained in U_-.
PROOF.
First choose a smooth even function rho on Z such that
|rho - r| < 1, |d rho|_h <= 2.
For completeness, smooth a Lipschitz function in relatively compact
coordinate charts in which the metric is sufficiently close to a
constant metric. Local convolution then gives gradient norm at most 3/2,
with any prescribed positive value error on the compact support of a
partition function. Let f_j be these local smoothings and patch them
using a countable, locally finite smooth partition of unity phi_j.
Since the sum of d phi_j is zero, the error in the gradient caused by
patching is the sum of (f_j - r) d phi_j. Taking the value errors smaller
than 2^{-j-2}/(1 + norm(d phi_j)) makes its norm less than 1/2 and also
makes the total value error less than one. Finally average the result
with its reflection. Since reflection is an isometry, both bounds
survive. The function rho has compact sublevel sets because r does.
Choose regular values
a in (ell - 90, ell - 89),
b in (ell - 20, ell - 19).
On the band a < rho < b define
mu = cot( pi (rho - a) / (b - a) ).
The width b - a > 69 and |d rho|_h <= 2 imply
2 |d mu|_h
<= (4 pi / (b - a)) (1 + mu^2)
<= (1/4) (1 + mu^2).
The pressure tends to +infinity at the inner end of the band and to
-infinity at the outer end. These signs will keep a minimizing interface
away from the constraints used to obtain it.
EXISTENCE AND REFLECTION SYMMETRY.
For sufficiently small epsilon > 0, minimize among sets E of finite
perimeter satisfying, up to sets of measure zero,
{rho < a + epsilon} is contained in E,
E is contained in {rho <= b - epsilon}.
Write
P_T(E) = integral_Z e^{-T} d|D chi_E|_h,
J(E) = P_T(E) - integral_E mu_* e^{-T} dvol_h,
where mu_* is smooth, compactly supported, reflection invariant, and
agrees with mu on a neighborhood of the closed band
{ a + epsilon <= rho <= b - epsilon }.
Such an extension exists because this closed band is compactly contained
in {a < rho < b}. The perimeter is taken on all of Z, not relative to
the band.
Regular sublevel sets provide competitors. All admissible sets lie in a
fixed compact set, where the smooth weight is bounded above and below by
positive constants. A minimizing sequence therefore has uniformly
bounded ordinary perimeter; compactness for functions of bounded
variation gives an L^1-convergent subsequence. The constraints are
closed under this convergence, the volume term converges, and weighted
perimeter is lower semicontinuous. Thus a minimizer exists.
The lattice inequality for perimeter gives
J(E union E') + J(E intersection E')
<= J(E) + J(E').
Indeed the volume terms satisfy equality, while perimeter satisfies the
indicated inequality. The reflected set sigma E is another minimizer,
and both its union and intersection with E are admissible. Minimality
and the lattice inequality show that both are minimizers. Replacing E by
E union sigma E, we may suppose that E is invariant under reflection, up
to a null set.
REMOVING THE CONSTRAINTS NEAR THE INTERFACE.
We show that E fills a larger inner sublevel and avoids a larger outer
superlevel. This is the step that permits unrestricted local variations
of its boundary.
Since a is a regular value and its level set is compact, there is a fixed
compact level neighborhood on which d rho is nonzero. Choose a smooth
compactly supported field X, with |X|_h <= 1, that equals
grad rho / |d rho|_h on a smaller such neighborhood. This one field can
be used for all sufficiently small epsilon. Set A = {rho < a + 2 epsilon}.
The field X equals the outward unit normal along the boundary of A, so
the divergence theorem and the variational definition of perimeter give
P_T(E union A) - P_T(E)
<= P_T(A) - P_T(E intersection A)
<= integral_{A minus E} Div_h(e^{-T} X) dvol_h.
By the constraints, the difference set is contained, up to a null set,
in { a + epsilon <= rho < a + 2 epsilon }. On this shell,
mu_* e^{-T} > Div_h(e^{-T} X)
when epsilon is sufficiently small: the divergence is bounded on a fixed
compact set, while the pressure tends uniformly to +infinity.
Subtracting the volume terms shows that E union A would strictly improve
J unless A minus E has measure zero. Hence
{rho < a + 2 epsilon} is contained in E (up to a null set).
For the outer end apply the same comparison to the complement of E, whose
pressure is -mu_*. Use B = {rho > b - 2 epsilon} and a field equal to
-grad rho / |d rho|_h near its boundary. Both the field and the change of
phase have compact support near that boundary; thus the divergence
calculation also applies if B itself is unbounded. The pressure -mu_*
tends uniformly to +infinity on { b - 2 epsilon < rho <= b - epsilon }.
It follows that
E intersect {rho > b - 2 epsilon} is empty (up to a null set).
Let S = support of |D chi_E|_h. The two clearance statements show that S
is a compact subset of the open band
a + epsilon < rho < b - epsilon.
Every compactly supported change of E in this band is admissible. The
minimizer therefore has the full local minimizing property for weighted
perimeter with the smooth pressure mu near every point of S.
REGULARITY, DIAMETER, AND RESTRICTION TO THE HALF.
We use the following precise local regularity input: in a smooth
three-manifold, a set locally minimizing perimeter with a smooth positive
weight and a smooth pressure volume term has smooth embedded perimeter
support, and its outward measure-theoretic unit normal extends smoothly
to that support. For the weighted version, see Kemper's theorem on
weighted perimeter minimizers: the singular-set bound is ambient
dimension minus eight, and smooth data give smooth regular parts by
elliptic regularity. The weight can equivalently be absorbed in the
smooth conformal metric h_bar = e^{-T} h: in dimension three its surface
area is the weighted h-area, and its pressure is mu e^{T/2}. Only this
local regularity theorem is used here, with the full-neighborhood
minimizing property just established.
In particular S is a compact smooth two-sided hypersurface with finitely
many connected components. On its complement the phase is locally
constant: a characteristic function with zero distributional derivative
on a connected open set is constant almost everywhere.
Each connected component Sigma of S is stationary and stable for all
normal variations of the weighted perimeter-minus-volume functional. Its
outward normal satisfies
H - nu T = mu.
From the doubling comparison and the pressure gradient,
D(h,T) + mu^2 - 2 |d mu|_h
>= 1/4 + (3/4) mu^2
>= 1/4.
Proposition 7.4, with c = 1/4, now gives
diam_{h|Sigma}(Sigma) <= 4 pi.
Reflection preserves the full and empty phases and hence S with its
outward normal. At a point of S on the seam, that normal is fixed by
d sigma, so it is tangent to the seam. Thus S meets the seam
transversally. Its restriction to P is consequently a finite union of
disjoint compact two-sided neat hypersurfaces. Let F_1,...,F_m be their
connected components. Two points of one F_i lie in one connected
component Sigma of S. A path in Sigma of length at most 4 pi, followed by
folding, and the folding-distance inequality give
diam_d(F_i) <= 4 pi.
Proposition 8.1 says that every F_i separates P.
The location of S in a < rho < b, together with |rho - r| < 1 and the
choice of levels, gives
F_i is contained in { ell - 91 < r < ell - 18 }.
Restrict the full and empty phases of Z minus S to P to obtain U_+ and
U_-. They are disjoint relatively open sets covering P minus the union
of the F_i. Since rho(o) < 1 < a, the base point is in the full phase.
If r(x) > ell, then rho(x) > ell - 1 > b, so x is in the empty phase.
This proves the final assertion.
11. BOUNDED COMPONENTS OF DISTANCE SLABS
The separating interfaces constructed in Section 10 will now give a
cover of each cut component by sets of bounded diameter. The point is to
find one interface component that every path from the base point to a
given slab component must cross. Separate crossings of unrelated
interface components would not suffice for this purpose.
LEMMA 11.1 (the complementary-region tree).
Let P be a connected smooth manifold, possibly with boundary. Let
F_1,...,F_m be disjoint compact connected two-sided neatly embedded
hypersurfaces, each of which separates P. The graph with one vertex for
each component of P minus the union of the F_i and one edge across each
F_i is a finite tree. In particular, two connected subsets of the
complement that lie in different components are separated by some single
F_i.
PROOF.
A bicollar of F_i can be chosen disjoint from all the other
hypersurfaces. Its two sides are connected, so each side belongs to a
unique complementary component. This defines the endpoints of its edge.
The complement has finitely many components: insert the hypersurfaces
one at a time, observing that removing a connected two-sided hypersurface
from a connected manifold creates at most two components. The same
argument applies with boundary and to a neat hypersurface, using its
bicollar.
The graph is connected. Indeed, points in complementary components can
first be moved into the interior of P and then joined by an interior path
transverse to the hypersurfaces. A transverse compact path has finitely
many crossings and hence gives a walk in the graph.
Every edge is a bridge: a graph path joining its endpoints without that
edge could be realized by paths in complementary regions and bicollars,
thereby joining the two sides of F_i without meeting F_i. This
contradicts separation by F_i. A connected finite graph in which every
edge is a bridge is a tree. Any edge on the unique path between two
distinct vertices gives the final assertion. This proves the lemma.
Fix a connected component P of the cut manifold of Proposition 8.1. As
in Section 9, use the smoothed metric h on its double, let d be the
intrinsic length metric of h restricted to P, and fix o in the interior
of P. Write r(x) = d(o,x). The metric d is proper, and Lemma 9.1 gives
dist_g(q(x),q(y)) <= 2 d(x,y) (x,y in P).
Here dist_g is distance in the original manifold M, not the intrinsic
distance in a cut component: project paths in P and use g <= 4h.
PROPOSITION 11.2 (slab diameter bound).
For ell < u <= ell + 2, every connected component C of
{ x in P : ell < r(x) < u }
has d-diameter at most 220.
PROOF.
If ell < 100, then r < u < 102 on the slab. The triangle inequality
through o gives diam_d(C) <= 204.
Suppose ell >= 100 and the slab is nonempty. Apply Proposition 10.1 at
level ell. It gives finitely many disjoint compact connected neat
two-sided hypersurfaces F_1,...,F_m in
ell - 91 < r < ell - 18,
each separating P and each of d-diameter at most 4 pi.
The full phase contains o, whereas the empty phase contains the slab; the
phase is constant on each complementary region. Thus o and C lie in
different complementary regions. In this assertion, C lies in a single
region because it is connected and misses every F_i.
By Lemma 11.1, some one component F_i separates o from all of C. Proper
length spaces are geodesic, so for each x in C choose a minimizing path
from o to x. It meets F_i at some point z_x. Each subpath is
minimizing, and hence its remaining length after this crossing is
d(x,z_x) = r(x) - r(z_x)
< (ell + 2) - (ell - 91)
= 93.
For x,y in C, the triangle inequality and the ambient d-diameter bound
for F_i give
d(x,y) <= d(x,z_x) + d(z_x,z_y) + d(z_y,y)
< 2*93 + 4 pi
< 220.
Together with the first case this proves the proposition.
For the graph construction we use the particular open cover
U_P = { connected components of
{ j - 3/4 < r < j + 3/4 } : j = 0, 1, 2, ... }.
These are open sets because a manifold with boundary is locally path
connected. The intervals cover [0,infinity), and no point belongs to
more than two of them. Components for a fixed j are disjoint, so U_P has
multiplicity at most two. Each of its sets has compact closure, since r
is proper. By Proposition 11.2 and the projection distance,
diam_g(q(U)) <= 440 (U in U_P).
The cover itself need not be locally finite. The next section makes the
partition of unity, and therefore the target graph, locally finite
without losing this diameter bound.
12. A GRAPH MAP FROM SLABS
Fix a component P of N, with the cover U_P from Section 11. Choose a
locally finite smooth partition of unity {psi_alpha} whose supports are
compact and with each support contained in some assigned U(alpha) in U_P.
Such a partition follows by taking a locally finite relatively compact
refinement of the cover and then a subordinate partition of unity. Group
functions assigned to the same set:
phi_U = sum over alpha with U(alpha) = U of psi_alpha.
There are only finitely many summands in each of these sums. Indeed,
their nonempty supports all meet the compact set given by the closure of
U, whereas the original family of supports is locally finite.
Consequently the support of phi_U is a compact subset of U. Moreover,
the family of supports of the phi_U remains locally finite: a
neighborhood meeting only finitely many of the original supports meets
only finitely many grouped supports. Discard identically zero functions.
Give each remaining function a vertex v, and write U_v and phi_v for the
corresponding set and function. Include the edge [v,w] precisely when
phi_v and phi_w are simultaneously positive somewhere. This defines a
simplicial graph K_P. It is locally finite, because the compact support
of phi_v meets only finitely many supports of other functions. Since the
original cover has multiplicity at most two, at most two of the functions
are positive at any point. The formula
F_P(x) = sum_v phi_v(x) v
therefore defines a map to K_P, interpreted in barycentric coordinates on
its vertices and edges. Near each point only finitely many coordinates
occur, so this map is continuous for the usual polyhedral topology, not
merely for a topology on an infinite coordinate product.
Use disjoint vertex sets for different components P, and put K_0 equal to
the disjoint union of the K_P. The maps F_P give a continuous map
F_0 : N -> K_0. If y belongs to the relative interior of a simplex of K_0
and v is any vertex of that simplex, then
F_0(x) = y implies phi_v(x) > 0 implies x in U_v.
In particular, the whole fiber is controlled by a single set in the cover
diameter bound.
13. COLLAR HOMOTOPIES AND GLUING ACROSS THE CUTS
Choose disjoint closed collars C_B of all boundary components B of N,
with the family of collars locally finite. Take them from sufficiently
small tubular neighborhoods of the cut surfaces in M. Each collar is
compact, and its normal g-length may be taken at most one. If B^+ and
B^- are the two copies of a cut surface S, the intrinsic diameter
estimate diam_g(S) <= 2 pi from Proposition 8.1 gives
diam_g( q(C_{B^+} union C_{B^-}) ) <= L, L := 2 pi + 2.
Indeed, project any two collar points normally to S, join the projections
in S, and add the two normal segments.
Write each collar as B x [0,1], with B x {0} the actual boundary and
B x {1} its inner end. Let
h_B : B -> K_0, h_B(x) = F_0(x,1).
Its image lies in a finite connected subgraph L_B of the appropriate K_P:
take the smallest subcomplex containing h_B(B). Compactness of B and
local finiteness of K_P imply finiteness of this subcomplex, and
connectedness of the image implies connectedness of the subcomplex.
More importantly, its choice ensures that
v in L_B implies (B x {1}) intersect {phi_v > 0} is nonempty.
For a vertex in the image this is immediate. Any other vertex of L_B is
an endpoint of an edge whose interior meets the image, and both endpoint
coordinates are positive at such a point. Thus the anchor condition
concerns the same cover sets U_v as the original partition, even if the
ensuing homotopy visits parts of L_B outside h_B(B).
Each B is a sphere or a real projective plane by Proposition 8.1, so the
fundamental group of B is finite. Since the fundamental group of a
connected graph is free and torsion free, the induced map (h_B)_* is
trivial. The map h_B therefore lifts to the universal covering tree of
L_B. Contracting this lift to a lift of a chosen vertex a_B in L_B and
projecting back gives a homotopy
H_B : B x [0,1] -> L_B,
H_B(x,0) = h_B(x), H_B(x,1) = a_B.
In particular, the homotopy stays inside a subgraph all of whose vertices
satisfy the anchor condition.
ADDING EDGE PATHS AND DESCENDING TO THE MANIFOLD.
For each cut surface S add an edge path A_S from a_{B^+} to a_{B^-},
with interior disjoint from K_0 and from every other added path. More
explicitly, introduce two distinct new vertices c_S and d_S and the three
edges
[a_{B^+}, c_S], [c_S, d_S], [d_S, a_{B^-}].
All new vertices are distinct for different S. This produces an ordinary
simplicial graph even if the two endpoints agree or other paths have the
same endpoints. Parametrize this path by alpha_S : [0,1] -> A_S, taking
the first chosen endpoint at zero and the second at one; when these
endpoints agree, the parametrization is injective on [0,1) and only its
endpoints are identified. Let m_S = alpha_S(1/2).
Call the enlarged graph K. It is locally finite. The only point
requiring verification is the number of added paths at an old vertex v.
An endpoint attached there is some a_B = v, and the anchor condition
implies that the corresponding collar meets the support of phi_v. A
compact set meets only finitely many members of a locally finite family
of collars, proving the assertion.
Define F_tilde : N -> K to agree with F_0 outside the collars. On the
homotopy half of a collar, adjacent to its inner end and given by
1/2 <= s <= 1, put
F_tilde(x,s) = H_B(x, 2 - 2s).
For the remaining halves of the paired collars of S, put
F_tilde(x,s) = alpha_S(1/2 - s) on C_{B^+}, 0 <= s <= 1/2,
F_tilde(x,s) = alpha_S(1/2 + s) on C_{B^-}, 0 <= s <= 1/2.
The formulas agree at s = 1/2 and with F_0 at s = 1. Local finiteness of
the collars proves continuity on all of N. Both boundary copies of S are
mapped constantly to m_S. Since q : N -> M is the quotient that
identifies these copies, F_tilde descends to a continuous map
f : M -> K, f o q = F_tilde.
14. THE DIAMETER OF AN ENTIRE FIBER
Fix y in K_0, and let v be a vertex of the unique simplex whose relative
interior contains y. Every point of f^{-1}(y) lies at distance at most L
from the one set q(U_v). To prove this, lift the point to N. If its
value is unchanged from F_0, use the initial fiber anchor. Otherwise it
lies in some collar C_B. On the homotopy half, its image lies in L_B;
because L_B is a subcomplex and contains y, it contains v. On the
remaining half the image can belong to K_0 only at the endpoint a_B, so
again v is in L_B. By the anchor condition, choose z in B x {1} with
phi_v(z) > 0, hence z in U_v. The paired-collar diameter estimate gives
the asserted distance to q(z).
For any two points of this fiber, their anchor points therefore belong to
the same set q(U_v). It follows from the cover diameter bound that
diam_g f^{-1}(y) <= 440 + 2L
= 440 + 2(2 pi + 2)
< 500.
If instead y lies in K minus K_0, it belongs to the open part
A_S minus K_0 of exactly one added path. Its entire preimage is contained
in the images of the two collars of S, so the paired-collar diameter
estimate gives the stronger bound L. This also covers the new vertices
and the common value m_S. Empty fibers require no estimate. Thus the
whole-fiber bound holds, without any connectedness assumption.
PROOF OF THE MAIN THEOREM.
For the normalized spectral inequality with lambda = 1, the preceding
construction supplies the required graph and map with constant 500. For
general lambda > 0, replace g by g' = lambda g. Scalar curvature and the
Laplacian are multiplied by lambda^{-1}, so the spectral inequality for
g' has lower bound one. Moreover
dist_{g'} = sqrt(lambda) dist_g.
Applying the normalized conclusion and rescaling gives
diam_g f^{-1}(y) <= 500 / sqrt(lambda) (y in K),
as claimed. This completes the proof.
15. WHAT TO REMEMBER
The proof has five reusable ideas.
1. A spectral lower bound for -4 Delta + Scal produces a positive weight
t with D(g,t) >= 1. Weighted area uses e^{-t}, but distances are still
measured in g.
2. A stable weighted surface with a pressure mu is small whenever
D(g,t) + mu^2 - 2 |grad mu|_g is positive. The conformal index
estimate gives diameter 2 pi / sqrt(c) and forces the surface to be a
sphere or a projective plane.
3. Cutting along a maximal family of such surfaces makes every remaining
compact two-sided hypersurface separate. This is where the
orientation double cover and integral currents enter.
4. Doubling and a carefully chosen pressure produce separating interfaces
of diameter at most 4 pi in a controlled annulus. The complementary
regions form a tree, so one interface separates the base point from a
whole slab. This bounds slab components by 220 and cover sets by 440
in the original metric.
5. A partition of unity turns the slab cover into a graph map. The
anchor property controls whole fibers, and the collar construction
glues the maps across the original cuts. The final bound is
440 + 2(2 pi + 2) < 500.
16. FURTHER READING
The main references used above are:
O. Chodosh and C. Li, Generalized soap bubbles and the topology of
manifolds with positive scalar curvature, Ann. of Math. 199 (2024).
H. Federer and W. H. Fleming, Normal and integral currents, Ann. of Math.
72 (1960).
D. Fischer-Colbrie and R. Schoen, The structure of complete stable
minimal surfaces in 3-manifolds of non-negative scalar curvature, Comm.
Pure Appl. Math. 33 (1980).
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations
of Second Order, Springer, 2001.
M. Gromov, Four lectures on scalar curvature, in Perspectives in Scalar
Curvature, Vol. 1, World Scientific, 2023.
M. Gromov, Product inequalities for T-stabilized scalar curvature, SIGMA
20 (2024).
S. Hirsch, D. Kazaras, M. Khuri, and Y. Zhang, Spectral torical band
inequalities and generalizations of the Schoen-Yau black hole existence
theorem, Int. Math. Res. Not. IMRN (2024).
M. Kemper, Gromov Hyperbolic Manifolds, Weighted Isoperimetry and
Bubbles, Ph.D. thesis, University of Munster, 2020.
Y. Liokumovich and D. Maximo, Waist inequality for 3-manifolds with
positive scalar curvature, in Perspectives in Scalar Curvature, Vol. 2,
World Scientific, 2023.
Y. Liokumovich and Z. Wang, On the waist and width inequality in complete
3-manifolds with positive scalar curvature, Camb. J. Math. 14 (2026).
F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems,
Cambridge University Press, 2012.
R. Schoen and S.-T. Yau, On the structure of manifolds with positive
scalar curvature, Manuscripta Math. 28 (1979).
R. Schoen and S.-T. Yau, The existence of a black hole due to
condensation of matter, Comm. Math. Phys. 90 (1983).
L. Simon, Introduction to Geometric Measure Theory, Tsinghua lectures,
2014.