SCALAR CURVATURE AND URYSOHN WIDTH IN DIMENSION THREE
A complete, beginner-friendly account of the theorem
"Spectral scalar curvature and Urysohn width in dimension three."
PART I. THE IDEA
1. WHAT THE THEOREM SAYS
Imagine a very long tube. Its diameter (the distance between two far
ends) is huge. But if you squash the tube onto a line -- its axis --
every point of the line comes from a short cross-section. So the tube
is "large" in one direction and "small" in the other two.
The Urysohn width is the mathematical version of this idea. It asks
how small we can make every fiber if we are allowed to map a space to
a graph.
DEFINITION (Urysohn 1-width). For a metric space (X, d),
UW_1(X, d) = inf sup diam_d( f^{-1}(y) ),
(K,f) y in K
where K runs over one-dimensional simplicial complexes (graphs), f runs
over continuous maps X -> K, and diam_d of a fiber is the largest
distance in X between any two points of that fiber. An empty fiber has
diameter zero.
A graph is the simplest target that can have one large direction. The
number UW_1 measures exactly that.
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, for every compactly supported smooth function phi
on M,
integral_M ( 4 |grad_g phi|_g^2 + Scal_g phi^2 ) dvol_g
>= lambda * integral_M phi^2 dvol_g . (*)
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 shorthand -4 Delta_g + Scal_g >= lambda always means (*). A
pointwise lower bound Scal_g >= lambda implies (*), because the gradient
term is nonnegative. But (*) can hold even where scalar curvature is
negative somewhere. The theorem uses only (*).
The theorem concerns the given metric g. It is not enough to
conformally rescale g to have positive scalar curvature: the fiber
diameters are measured in g, and the proof must control distances in g
itself.
The number 500 is convenient and is not optimized. The actual proof
gives a bound strictly below 500.
2. WHY A GRAPH TARGET, AND WHY SCALAR CURVATURE
Why not just bound the diameter? A positive scalar-curvature lower
bound does not bound the diameter in dimension three. The product of
a round two-sphere with a line has positive scalar curvature and is
infinitely long. So any correct theorem must allow one large direction.
A graph target gives exactly that room: the map can unfold the long
direction along edges, while cross-sections stay small.
Why this particular energy inequality? The operator -4 Delta_g +
Scal_g governs the second variation of area for minimal surfaces in a
three-manifold. A lower bound for this operator is therefore the
natural hypothesis for a surface-based argument.
Why scalar curvature at all? Positive scalar curvature in dimension
three is known to rule out certain topology (Schoen-Yau,
Fischer-Colbrie-Schoen, Gromov-Lawson; and in the torus-stabilized
framework, Gromov 2023/2024). The theorem turns that topological
restriction into a metric statement: the manifold can be wide only
along one direction.
3. THE ONE KEY TRICK
The energy inequality (*) is not used directly. Instead it is used to
build a positive function v satisfying
(-4 Delta_g + Scal_g) v >= v ,
and then one sets t = -2 log v. A short calculation (Lemma 6.1) gives
D(g, t) >= 1 ,
where
D(g, t) = Scal_g + 2 Delta_g t - |grad_g t|_g^2 .
The function t is a weight. It defines a weighted area
Area_t(Sigma) = integral_Sigma e^{-t} dArea_g
on surfaces. But the final fiber diameters are still measured in the
UNWEIGHTED metric g. The weight only changes which surfaces are cheap
to cut along; it does not distort the notion of distance that appears
in the conclusion.
The rest of the proof is standard surface geometry, applied to the
weighted area functional. The point is that the two-dimensional
analogue of D(g, t) appears in the stability inequality for a weighted
minimal surface, and a positive lower bound on D gives an explicit
diameter bound (Lemma 7.2). Stable surfaces are small spheres or
projective planes, at most 2 pi across.
4. DICTIONARY
Skim this. Come back to it when a term appears later.
RIEMANNIAN MANIFOLD. Locally looks like R^n, with a way to measure
lengths and angles. A 3-manifold looks locally like R^3; a surface
like R^2.
METRIC g. The length rule. Distance is the infimum of the lengths of
paths joining two points.
COMPLETE. Every Cauchy sequence converges. Equivalently, no path of
finite length escapes to infinity. Completeness lets us take limits
and use minimizing geodesics.
SCALAR CURVATURE Scal_g. One number per point, curvature averaged at
that point. In dimension three, positive scalar curvature is a weak
kind of positive curvature.
LAPLACIAN Delta_g = div_g grad_g. The standard second-order operator,
chosen with this sign so that on R^n it is the sum of second partial
derivatives.
GRADIENT. The vector field dual to the differential. Its squared
length is |grad_g phi|_g^2.
COMPACTLY SUPPORTED. Zero outside a compact set. The test functions
phi in (*) vanish near infinity.
FIBER. For f : X -> K and y in K, f^{-1}(y) is the set of points
mapping to y. It may have several components; its diameter includes
distances between different components. The theorem bounds whole
fibers, not just components.
GRAPH. A one-dimensional simplicial complex, with the usual polyhedral
topology.
WEIGHTED AREA. integral_Sigma e^{-T} dArea. A positive density e^{-T}
multiplies the ordinary area element. Equivalent to ordinary area in
the conformal metric e^{-T} g, but final distances are in g.
STABLE SURFACE. The second variation of the area-type functional is
nonnegative for every compactly supported normal variation. A soap film
that does not lower its area when wiggled is stable.
PRESSURE. A smooth function mu used to force a surface into a region.
The functional is weighted area minus the weighted volume integral of
mu. Its Euler-Lagrange equation is H - nu T = mu, where H is mean
curvature and nu is the chosen unit normal.
INTEGRAL CURRENT. A generalized oriented surface used in geometric
measure theory. It has a mass (area), a boundary, and good compactness
properties. We use currents only to obtain a smooth minimizer; the
rest of the argument is ordinary surface geometry.
5. THE ROADMAP
After scaling g by lambda (Section 15), assume lambda = 1.
STEP 1. POSITIVE WEIGHT (Section 6). The energy inequality gives a
smooth t with D(g, t) >= 1. Weighted area uses e^{-t}; distances stay
in g.
STEP 2. SMALL STABLE SURFACES (Section 7). A closed two-sided surface
that is stable for weighted area with pressure mu, and that satisfies
D(g, t) + mu^2 - 2 |grad mu|_g >= c > 0
along itself, has diameter at most 2 pi / sqrt(c) in its intrinsic
metric and is a sphere or a projective plane. A local version controls
noncompact stable sheets.
STEP 3. CUT (Section 8). Cut M along a maximal locally finite family
of small stable surfaces S so that no finite subfamily disconnects M.
In each resulting piece P, every compact connected two-sided
hypersurface that meets the boundary neatly separates P. Boundary
components of P are small spheres or projective planes.
STEP 4. DOUBLE (Section 9). Double P along its boundary, and smooth
metric and weight so that they are invariant under reflection, with
D(h, T) >= 1/2, (1/4) g <= h <= 4 g .
This avoids a boundary regularity problem.
STEP 5. SEPARATING INTERFACES (Section 10). Minimize weighted
perimeter minus a cotangent pressure in an annular region. The
minimizer is a finite family of small separating surfaces F_i between
the base point and a distant slab. Each F_i lies in a controlled
annulus and has diameter at most 4 pi.
STEP 6. BOUND SLABS (Section 11). The complementary regions of the F_i
form a tree. Some single F_i separates the base point from the entire
slab component C. Every minimizing path from o to a point of C meets
that F_i. Hence diam(C) < 220. The slabs
{ 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 M.
STEP 7. GRAPH MAP (Section 12). A partition of unity on that cover
produces a graph K_P and a continuous map F_P : P -> K_P such that each
fiber is controlled by a single cover set.
STEP 8. GLUE ACROSS CUTS (Section 13). On the two copies of each cut
surface, boundary maps are homotoped to single points and then joined
by a new edge path. The pieces descend to a single map f : M -> K.
STEP 9. FIBER DIAMETER (Section 14). A fiber of f meets one anchored
cover set up to a controlled collar error L = 2 pi + 2, so
diam f^{-1}(y) <= 440 + 2 L < 500 .
PART II. THE FULL PROOF
6. A POSITIVE WEIGHT
From now until the rescaling in Section 15, assume lambda = 1.
Define D(g, t) = Scal_g + 2 Delta_g t - |grad_g t|_g^2.
LEMMA 6.1 (positive weight). Let (M, g) be a connected smooth
Riemannian manifold without boundary. Suppose that, for every compactly
supported smooth phi,
integral_M ( 4 |grad_g phi|_g^2 + Scal_g phi^2 ) dvol_g
>= integral_M phi^2 dvol_g .
Then there is a smooth function t : M -> R with D(g, t) >= 1.
PROOF. It is enough to find a smooth positive v with
(-4 Delta_g + Scal_g) v >= v ,
since t = -2 log v then gives
D(g, t) = Scal_g - 4 (Delta_g v)/v >= 1 .
Indeed, Delta_g t = -2 (Delta_g v)/v + 2 |grad_g v|_g^2 / v^2 and
|grad_g t|_g^2 = 4 |grad_g v|_g^2 / v^2, and these two combine to
Scal_g - 4 (Delta_g v)/v.
Compact case. Take a positive first eigenfunction of -4 Delta_g +
Scal_g. Its eigenvalue is at least one by the hypothesis.
Noncompact case. Fix o in M and choose an exhaustion
Omega_1 subset Omega_2 subset ...
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 v_j a positive first eigenfunction with v_j(o) = 1. The
hypothesis, extended to H_0^1(Omega_j) by density, gives lambda_j >= 1.
Domain monotonicity makes lambda_j decreasing, and a fixed nonzero test
function supported in Omega_1 bounds it above. Thus 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 then give smooth
convergence on compact subsets, after a diagonal subsequence, to a
positive smooth v with
(-4 Delta_g + Scal_g) v = lambda_infinity v .
Hence D(g, t) >= 1. QED.
Fix this t for the rest of the proof. Weighted area means area with
density e^{-t}. All minimization below happens 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_{dE} 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 ,
where nu T means the normal derivative of T.
Stability. The second variation of this functional is nonnegative for
every compactly supported normal variation in the interior of Sigma.
Let
Delta_{Sigma,T} u = Delta_k u - <d(T|Sigma), du>_k
and
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.
Sign conventions. We use Delta = Div grad, and round spheres have
positive scalar curvature. A closed surface means compact without
boundary. All formulas use Riemannian densities, so none requires an
orientation.
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 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, in the measure e^{-T} dArea_k,
with Dirichlet boundary conditions in the domain case. Stability makes
its first eigenvalue nonnegative, so
-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) ,
|dT|_h^2 = |d(T|Sigma)|_k^2 + (nu T)^2 .
Combining these gives the exact identity
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. QED.
In particular, if
D(h, T) + mu^2 - 2 |grad mu|_h >= c > 0
along a stable surface, then D(k, U) >= c, because
mu^2 + 2 nu mu >= mu^2 - 2 |grad mu|_h.
7.3. The conformal index estimate.
LEMMA 7.2 (conformal index estimate). Let (S, k) be a smooth surface,
U a smooth function on S, and suppose 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 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 (this exists along an interval
even when S is nonorientable). Since d(s_tilde) = e^{-U} ds, the index
inequality and the curvature bound give
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 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. QED.
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 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 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 it. 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 holds also for nonorientable surfaces (directly with
densities, or by passing to the orientation double cover). The
classification of closed connected surfaces now gives exactly S^2 and
RP^2. QED.
7.5. A local exit estimate.
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 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 closure(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
closure(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 closure(Omega_0) and satisfies D(k, U) >= c there. In
the compact Riemannian manifold with boundary closure(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 k-length estimate. Only
compactness of the two chosen domains is used, not completeness of
Sigma. QED.
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 forces it to be
a sphere or a projective plane. The same constant in Lemma 7.5
prevents an open stable sheet from escaping a fixed neighborhood.
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 a 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
separates that piece.
Definitions. 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
compact without boundary. 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 hold. Let N be the
manifold obtained by cutting M along S, and q : N -> M the natural
gluing map.
1. 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. Local finiteness of the family. 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.
By Proposition 7.4, every member has intrinsic and hence ambient
diameter at most 2 pi. So all members meeting a fixed compact set A in
M lie in the compact closed 2 pi-neighborhood of A. Completeness is
used here through 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. So only
finitely many members meet A. This proves local finiteness.
Cutting. In a tubular chart at a cut surface, q glues two closed
half-charts along their boundary; away from the cuts it is a
diffeomorphism. These charts show 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. So it is compact. This proves properness of P. The
boundary assertions follow from Proposition 7.4.
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. Every closed connected embedded two-sided hypersurface in
the interior of P separates P.
PROOF. Suppose 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
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 (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.
Choose a relatively open set W in P with compact closure containing 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 exactly area weighted by e^{-t}. Minimize g_bar-mass
among integral 2-cycles T supported in pi^{-1}(J) satisfying
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 support constraint. The form beta
extends by zero. The competitor T_F shows the class is nonempty.
Federer-Fleming compactness and closure, then lower semicontinuity of
mass, give a minimizer T_0: support confined to a fixed compact set,
boundaries vanish, and both conditions in the constraints 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 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
closure(U') and tau closure(U') 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 closure(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 Jacobian bound give
M_g_bar(R_# T') <= M_g_bar(T') .
No Jacobian bound is needed in the interpolation region, 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. So
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 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 contained in W; the projected support
is closed relative to W because the cover is finite, and its components
are closed there.
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 B_P(x, 10). Choose another connected
relatively compact smooth surface domain Omega_1 with closure(Omega_0)
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 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 sigma the exchange
involution (a disjoint pair of copies when the boundary is empty).
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 - 2 H' ,
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 near 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 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. 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. QED.
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. So the even extension of r is
proper and 1-Lipschitz on each component of Z. We will also use
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 weighted perimeter-minus-volume minimization. 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
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 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
{r > ell} contained in U_-.
PROOF. Smooth approximation of the distance. 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 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 sum d phi_j = 0, the error in the gradient caused by patching is
sum (f_j - r) d phi_j. Taking the value errors smaller than
2^{-j-2}/(1 + norm(d phi_j)) makes the 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.
The pressure. 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 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} contained in E ,
E 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 both are minimizers. Replacing E by
E union sigma E, we may suppose 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 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} 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 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. So
the minimizer has the full local minimizing property for weighted
perimeter with the smooth pressure mu near every point of S.
Regularity, diameter, restriction to the half. We use the following
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 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 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. QED.
11. BOUNDED COMPONENTS OF DISTANCE SLABS
The separating interfaces from Section 10 give a cover of each cut
component by sets of bounded diameter. The key is that a single
interface component separates the base point from a whole slab.
LEMMA 11.1 (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 separating P. The graph with one
vertex per 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: points in complementary components can be moved
into the interior of P and 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,
joining the two sides of F_i without meeting F_i, contradicting
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 vertices gives
the final assertion. QED.
Fix P, h, d, o, r as in Section 9. The metric d is proper, and Lemma
9.1 gives
dist_g(q(x), q(y)) <= 2 d(x, y) (x, y in P) .
PROPOSITION 11.2 (slab diameter). 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, and 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: finitely many disjoint compact connected neat two-sided
hypersurfaces F_1, ..., F_m in {ell - 91 < r < ell - 18}, each separating
P, 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. (C lies in a single region, since 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 z_x. Each subpath is minimizing, hence
the 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. QED.
For the graph construction use the open cover
U_P = { connected components of { j - 3/4 < r < j + 3/4 } :
j = 0, 1, 2, ... } .
Open 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 set has compact closure because r is proper. By
Proposition 11.2 and the projection distance,
diam_g(q(U)) <= 440 (U in U_P) .
The cover need not be locally finite; the next section handles that
without losing the 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} with compact
supports, each support contained in some assigned U(alpha) in U_P. Such
a partition follows from a locally finite relatively compact refinement
of the cover and 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 finitely many summands in each sum: their nonempty supports all
meet the compact set closure(U), whereas the original family is locally
finite. So supp phi_U is a compact subset of U. Moreover, {supp phi_U}
remains locally finite. Discard identically zero functions.
Give each remaining function a vertex v; write U_v and phi_v. Include
the edge [v, w] 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 phi_v are positive at any point. The formula
F_P(x) = sum_v phi_v(x) v
defines a map to K_P, in barycentric coordinates on 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; set 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 is in 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 .
So a 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. 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:
the smallest subcomplex containing h_B(B). Compactness of B and local
finiteness of K_P imply finiteness of L_B; connectedness of the image
implies connectedness of the subcomplex. Its choice ensures
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. So 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
pi_1(B) is finite. Since the fundamental group of a connected graph is
free and torsion free, (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 M. 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 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 endpoint. Parametrize this path by alpha_S : [0, 1] -> A_S, taking
the first chosen endpoint at 0 and the second at 1; when the 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 the corresponding collar meets supp 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.
15. RESCALING: PROOF OF THE MAIN THEOREM
Sections 6 through 14 proved the theorem under the normalization
lambda = 1. For general lambda > 0, replace g by
g' = lambda g .
Under this scaling:
Scal_{g'} = lambda^{-1} Scal_g ,
|grad_{g'} phi|_{g'}^2 = lambda^{-1} |grad_g phi|_g^2 ,
dvol_{g'} = lambda^{3/2} dvol_g ,
dist_{g'} = sqrt(lambda) dist_g .
Therefore
integral_M ( 4 |grad_{g'} phi|_{g'}^2 + Scal_{g'} phi^2 ) dvol_{g'}
= lambda^{1/2} integral_M ( 4 |grad_g phi|_g^2 + Scal_g phi^2 ) dvol_g
>= lambda^{1/2} * lambda integral_M phi^2 dvol_g
= lambda^{3/2} integral_M phi^2 dvol_g
= integral_M phi^2 dvol_{g'} .
So g' satisfies the normalized inequality with lambda = 1.
Apply the normalized conclusion to (M, g'): there is a graph K and a
continuous f : M -> K with every fiber of g'-diameter at most 500. Since
dist_{g'} = sqrt(lambda) dist_g,
diam_g f^{-1}(y) = lambda^{-1/2} diam_{g'} f^{-1}(y)
<= 500 / sqrt(lambda) .
This is the asserted bound. The proof of the main theorem is complete.
16. WHAT TO REMEMBER
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 .
17. FURTHER READING
The main sources used above.
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.