Riemannian three-manifolds of positive scalar curvature are wide along at
most one direction. The product of a round two-sphere with a line has
scalar curvature 2 and infinite diameter, so a positive lower bound on
scalar curvature does not by itself bound the diameter. A width, and not
a diameter, is the natural measure of size.

For a metric space (X, d), the Urysohn k-width w_k(X) is the infimum of
the eps >= 0 such that X admits a continuous map to a k-dimensional
simplicial complex with every fiber of diameter < eps. The empty fiber
counts as diameter zero. P. S. Urysohn introduced the k-width in 1925 in
the study of curves in metric spaces, and the k = 1 case is now called
the Urysohn 1-width.

P. S. Urysohn, Sur les multiplicites Cantoriennes, Fund. Math. 7 (1925),
30-137.
https://doi.org/10.4064/fm-7-1-30-137

M. Gromov conjectured that a uniformly positive lower bound on scalar
curvature bounds w_1, and established bounds in a torus-stabilized
framework. O. Chodosh and C. Li obtained the graph-diameter conclusion
in low dimensions by a slice-and-dice construction, which in dimension
three reduces to the following: a compact three-manifold of positive
scalar curvature admits a continuous map to a one-dimensional graph
whose every fiber has uniformly bounded diameter.

M. Gromov, Four lectures on scalar curvature, in Perspectives in Scalar
Curvature, Vol. 1, World Scientific, 2023, 1-514.
sources/gromov-2023-four-lectures.pdf

M. Gromov, Product inequalities for T-stabilized scalar curvature, SIGMA
20 (2024), Paper No. 038.
sources/gromov-2024-stabilized.pdf

O. Chodosh and C. Li, Generalized soap bubbles and the topology of
manifolds with positive scalar curvature, Ann. of Math. (2) 199 (2024),
707-740.
sources/chodosh-li-2024.pdf

The result is a spectral form of this graph bound, valid
without any compactness, orientability, or bounded-geometry assumption.
Let (M^3, g) be a connected complete smooth Riemannian three-manifold
without boundary. Write Delta_g = div_g grad_g with the sign convention
that on R^n this is the sum of second partial derivatives, and let
Scal_g denote scalar curvature. Suppose that for some lambda > 0 and
every compactly supported smooth real 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 is a one-dimensional simplicial complex K with its usual
polyhedral topology, and a continuous map f : M -> K, such that every
fiber f^{-1}(y), including all of its connected components and the
distances among them, has diameter at most 500/sqrt(lambda) in the
original metric g.

The notation -4 Delta_g + Scal_g >= lambda always abbreviates (*). A
pointwise lower bound Scal_g >= lambda implies (*) because the gradient
term is nonnegative, but (*) is strictly weaker: it may hold while
scalar curvature is negative on part of M. The constant 500 is a
convenience; the estimates in the argument give a slightly smaller
bound and no attempt at sharpness is made. Fibers are not required to
be connected.

The spectral hypothesis is natural for the surface method. The operator
-4 Delta_g + Scal_g is the second-variation operator of area for
minimal surfaces in a three-manifold, as observed by R. Schoen and
S.-T. Yau in 1979, and D. Fischer-Colbrie and R. Schoen later turned
this observation into a positive-solution method for Schrodinger
operators. A positive supersolution of -4 Delta_g + Scal_g - lambda is
what the proof needs, and no pointwise bound on Scal_g is required.

R. Schoen and S.-T. Yau, On the structure of manifolds with positive
scalar curvature, Manuscripta Math. 28 (1979), 159-183.
https://doi.org/10.1007/BF01647970

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), 199-211.
sources/fischer-colbrie-schoen-1980.pdf

The proof runs in stages. After rescaling so that lambda = 1, (*) gives
a smooth real function t on M with

    D(g, t) := Scal_g + 2 Delta_g t - |grad_g t|_g^2 >= 1 .

The function t is the positive weight. Weighted area is area weighted
by e^{-t}, but the fiber diameters of the conclusion are still measured
in g: the weight changes which surfaces are cheap to cut along, and
does not distort the metric that measures the theorem's bound.
Weighted stable minimal surfaces with a pressure mu satisfy an
Euler-Lagrange equation H - nu t = mu, and a second-variation identity
whose two-dimensional part is again D. Positivity of D bounds the
intrinsic diameter of every such surface by 2 pi and forces it to be a
round sphere or a real projective plane.

M. Gromov and H. B. Lawson, Jr., Positive scalar curvature and the
Dirac operator on complete Riemannian manifolds, Publ. Math. Inst.
Hautes Etudes Sci. 58 (1983), 83-196.
sources/gromov-lawson-1983.pdf

R. Schoen and S.-T. Yau, The existence of a black hole due to
condensation of matter, Comm. Math. Phys. 90 (1983), 575-579.
https://doi.org/10.1007/BF01216187

M is then cut along a maximal disjoint family of such small stable
surfaces, chosen so that no finite subfamily disconnects M. Each
resulting piece is doubled along its boundary, and the doubled metric
and weight are smoothed so that the closed-surface estimate applies
across the seam at the cost of a factor of two in the curvature
inequality. In the smoothed double, a weighted perimeter minimization
with a cotangent pressure in an annulus produces separating interfaces
of diameter at most 4 pi. Because the interfaces separate, the graph
of complementary regions is a tree, so a single interface component
separates the base point from a whole distance slab. Every slab
component has diameter at most 220 in the smoothed metric, and the
overlapping slabs cover each piece by sets of diameter at most 440 in
the original metric g. A partition of unity subordinate to that cover
builds a graph map, and collar homotopies gluing the graph maps across
the cut surfaces extend it to all of M. The final fiber bound is
440 + 2(2 pi + 2) < 500.

No compactness, orientability, spin structure, or injectivity-radius
bound is used. Completeness enters only through properness of the
Riemannian distance, and every instance of that use is identified as
it occurs.


The positive weight is a standard construction. Write L = -4 Delta_g +
Scal_g. Then (*) says L >= lambda as a quadratic form on compactly
supported H^1 functions on M.

LEMMA. Suppose (*) holds on (M, g) with lambda > 0. Then there is a
smooth real function t on M with D(g, t) >= lambda.

PROOF. The identity
  2 Delta_g t - |grad_g t|^2 = -4 Delta_g v / v
for t = -2 log v turns the claim into: there is a smooth positive v on M
with L v >= lambda v. The compact and noncompact cases are treated
separately.

If M is compact, L is a second-order self-adjoint elliptic operator with
smooth coefficients. Let v > 0 be its first eigenfunction. Its
eigenvalue mu_1 is the infimum of the Rayleigh quotient of L over all
nonzero compactly supported smooth functions; (*) says that this
infimum is at least lambda. So L v = mu_1 v >= lambda v.

For the noncompact case, fix o in M and a smooth proper rho : M -> [0,
infty) with rho(o) = 0, and let Omega_j = rho^{-1}([0, j)). For each j
let v_j be the positive first Dirichlet eigenfunction of L on Omega_j,
normalized by sup v_j = 1. The Rayleigh quotient over test functions
supported in Omega_j is at least lambda, so L v_j = mu_j v_j with
mu_j >= lambda. Since Omega_j is increasing and any Dirichlet test
function on Omega_j remains admissible on Omega_{j+1}, the sequence
mu_j is nonincreasing in j. Thus lambda <= mu_j <= mu_1 for every j.

Elliptic interior estimates applied to L v_j = mu_j v_j on a compact W,
with mu_j in the fixed interval [lambda, mu_1] and max |v_j| = 1, bound
v_j in C^{2, alpha} uniformly in j. Harnack's inequality applied to the
same equation gives min_W v_j >= c_W max_W v_j = c_W > 0 for every
compact W, with c_W independent of j.

Extract a diagonal subsequence v_j -> v in C^infty_loc. Then v : M ->
[0, 1] is smooth with L v >= lambda v (weak limit of L v_j >= lambda
v_j). Strong maximum principle: if v(x) = 0 at some x, the minimum
principle forces v to be zero everywhere on M, contradicting the
uniform lower bound on any W. So v > 0 everywhere. QED.

The function t is not unique, but only its existence and smoothness
enter what follows. No bound on t or on its gradient is used: every
perimeter and area minimization happens inside a compact set
where t is automatically bounded, and the fiber estimates at the end
are made in the original metric g without integrating t.



The second ingredient is a surface estimate, due to Schoen and Yau and
refined by Chodosh and Li. A stable weighted minimal surface, in
ambient curvature that is positive on average, has intrinsic diameter
bounded uniformly and topology restricted to a sphere or a projective
plane. The calculation is entirely local.

Let Sigma be a smooth two-sided surface, possibly with boundary, in a
smooth Riemannian three-manifold (X, h). Choose a unit normal nu, write
k = h|Sigma, B(V,W) = h(nabla_V nu, W) and H = tr_k B. For a smooth
ambient function T, the weighted area of a surface is

    Area_T(Sigma) = integral_Sigma e^{-T} dArea_k .

If Sigma is the boundary of a set E with outward normal nu, and mu is a
smooth ambient function, the pressure functional is

    J(E) = Area_T(Sigma) - integral_E mu e^{-T} dvol_h .

Its first variation is int_Sigma (H - nu T - mu) u e^{-T} dArea, so the
Euler-Lagrange equation is

    H - nu T = mu .

The second variation along a normal variation of speed u is the
quadratic form

    int_Sigma ( |du|_k^2 - Q u^2 ) e^{-T} dArea_k >= 0 ,

where

    Q = Ric_h(nu, nu) + Hess_h T(nu, nu) + |B|_k^2 + nu mu .

Here nu T and nu mu are directional derivatives of T and mu along nu,
Delta_k is the surface Laplacian of k, and the inequality, unconstrained
by any volume condition, is the definition of stability. A surface that
does not lower the area-type functional under a wiggle is stable.

Define also Delta_{Sigma,T} u = Delta_k u - < d(T|Sigma), du >_k.

LEMMA (surface transfer). Suppose Sigma satisfies H - nu T = mu and is
stable. Let Omega be Sigma if Sigma is closed and connected, and
otherwise a connected relatively compact smooth domain in int Sigma.
Let w be the positive first eigenfunction of -Delta_{Sigma,T} - Q on
Omega, with Dirichlet boundary condition if dOmega is nonempty. Set
p = log w and U = T|Sigma - p. Then, on Omega,

    D(k, U) >= D(h, T) + mu^2 + |B|_k^2 + 2 nu mu + |dp|_k^2 .

PROOF. Stability is precisely the statement that the first eigenvalue
of -Delta_{Sigma,T} - Q is nonnegative, so

    -Delta_{Sigma,T} w >= Q w ,
    -Delta_{Sigma,T} p = -Delta_{Sigma,T} w / w - |dp|_k^2 >= Q + |dp|_k^2 .

The Gauss equation says Scal_k = Scal_h - 2 Ric_h(nu,nu) + H^2 - |B|_k^2.
The decomposition of the ambient Laplacian into surface and normal
parts says Delta_h T = Delta_k(T|Sigma) + H nu T + Hess_h T(nu,nu).
The norm split says |dT|_h^2 = |d(T|Sigma)|_k^2 + (nu T)^2.

From these,
    D(k, T|Sigma) - D(h, T)
      = Scal_k - Scal_h - 2 H nu T - 2 Hess_h T(nu,nu) + (nu T)^2
      = -2 (Ric_h(nu,nu) + Hess_h T(nu,nu)) + (H - nu T)^2 - |B|_k^2
      = -2 (Ric_h(nu,nu) + Hess_h T(nu,nu)) + mu^2 - |B|_k^2 .

Also, 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 .

Substituting the inequality for p and the definition of Q:

    D(k, U) >= D(k, T|Sigma) + 2 Q + |dp|_k^2
             = D(h, T) + mu^2 + |B|_k^2 + 2 nu mu + |dp|_k^2 .

QED.

COROLLARY. If along a stable surface the stronger bound

    D(h, T) + mu^2 - 2 |grad mu|_h >= c > 0

holds, then D(k, U) >= c on Omega.

PROOF. Since |nu| = 1, the directional derivative nu mu satisfies
2 nu mu >= -2 |grad mu|_h. So D(h,T) + mu^2 + 2 nu mu >= D(h,T) + mu^2
- 2 |grad mu|_h >= c, and |B|_k^2 and |dp|_k^2 are nonnegative. QED.

The hypothesis of the corollary is available with c = 1 on the initial
(M, g, t) and mu = 0, because D(g, t) >= 1. It is also available with
c = 1/4 on the smoothed double, because D(h, T) >= 1/2 there and the
pressure mu is chosen with 2 |d mu|_h <= (1/4)(1 + mu^2).

LEMMA (conformal index). Let (S, k) be a smooth surface, U a smooth
function on S, and suppose

    D(k, U) >= c > 0

on a neighborhood of a finite geodesic segment gamma for the conformal
metric k_tilde = e^{-2U} k. If gamma has nonnegative index form for
normal variations compactly supported in its interior, then

    length_k(gamma) <= 2 pi / sqrt(c) .

PROOF. The Gauss curvature of k_tilde is K_tilde = e^{2U}(K_k +
Delta_k U). Since Scal_k = 2 K_k,

    D(k, U) = 2 (K_k + Delta_k U) - |dU|_k^2 >= c ,

so K_k + Delta_k U >= (c + |dU|_k^2)/2 and

    K_tilde >= (1/2) e^{2U} (c + |dU|_k^2) .

Parametrize gamma by k-arclength s in [0, L], and let s_tilde be the
arclength in k_tilde, so d(s_tilde) = e^{-U} ds. Choose a unit normal
along the segment that is parallel for k_tilde; this is possible along
an interval even when S is nonorientable. The index inequality, in the
conformal metric and with test function phi e^{-U/2}, gives

    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). Set phi = e^{-U/2}
psi. Using |U'| <= |dU|_k and completing the square,

    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 .

So 2 int_0^L (psi')^2 ds >= (c/2) int_0^L psi^2 ds for every smooth
psi compactly supported in (0, L). Substituting psi = sin(pi s / L)
gives 2 pi^2 / L^2 >= c/2, which is L <= 2 pi / sqrt(c). QED.

The conformal metric is a device for the argument. The metric in which
the length is actually bounded is k itself, and the test variations
are supported away from the segment endpoints so the local
version of the lemma holds.

PROPOSITION (closed stable surfaces). Let (X, h) be a smooth Riemannian
three-manifold, let T and mu be smooth ambient functions, and let Sigma
be a smooth closed connected two-sided surface in X with H - nu T = mu
and stable. Suppose D(h, T) + mu^2 - 2 |grad mu|_h >= c > 0 along
Sigma. Then, with k = h|Sigma,

    diam_k(Sigma) <= 2 pi / sqrt(c) .

Moreover, Sigma is diffeomorphic to S^2 or to RP^2.

PROOF. The transfer lemma 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; such a geodesic
has nonnegative index form, so the conformal index lemma bounds its
k-length by 2 pi / sqrt(c). The k-distance between the endpoints is no
greater. This proves the diameter bound.

For the topology, integrate D(k, U) >= c over Sigma with respect to
dArea_k. The conformal Laplacian term integrates to zero, so

    4 pi chi(Sigma) - integral_Sigma |dU|_k^2 dArea_k >= c Area_k(Sigma) .

Since the integral of |dU|^2 is nonnegative, chi(Sigma) > 0. A closed
connected surface has Euler characteristic at most two and positive
Euler characteristic only for S^2 and RP^2. QED.

LEMMA (local exit). Let (S, k) be a smooth surface, U a smooth function
with D(k, U) >= c > 0, and Omega a connected relatively compact smooth
domain with nonempty boundary. Then for every x in Omega, the k-distance
from x to dOmega is at most 2 pi / sqrt(c).

PROOF. A connected relatively compact smooth domain of a smooth surface
is a geodesic space, so there is a minimizing geodesic from x to dOmega
in the intrinsic metric of Omega. This geodesic is locally minimizing,
hence stable as a geodesic curve, so the conformal index lemma applies
to it. Its k-length is at most 2 pi / sqrt(c). QED.

Applying the closed-surface proposition to (M, g, t) with mu = 0 and
c = 1 gives: every closed stable weighted minimal surface in M has
intrinsic diameter at most 2 pi and is a sphere or a projective plane.
The local exit estimate then prevents a noncompact stable sheet with
D(k, U) >= 1 from escaping a fixed neighborhood: any open stable piece
of length exceeding 2 pi admits a shorter path to its boundary than its
own inradius, a contradiction.



The remaining work is topological. It turns the family of small stable
surfaces into a partition of M into pieces topologically simple enough
for the slab argument.

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. All hypersurfaces below are smooth.

PROPOSITION (separating cut). There is a locally finite family S of
pairwise disjoint closed connected embedded two-sided stable weighted
minimal surfaces in M, with no finite subfamily disconnecting 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. 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, and
the union of a chain is admissible because each finite subfamily lies in
one member of the chain. By Zorn's lemma there is a maximal such
family S.

Local finiteness. By the closed-surface proposition with mu = 0 and
c = 1, every member of S has intrinsic and ambient diameter at most
2 pi. Consequently every member that meets a fixed compact A is
contained in the closed 2 pi-neighborhood of A. For a 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 exactly once and missing all other S_j.
Modulo two, intersection with this loop pairs to one with [S_i] and to
zero with every other class. 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 of S
meet A, and S is locally finite.

Properness and quotient. In a tubular chart at a cut surface, q glues
two closed half-charts along their boundary; away from the cuts q is a
diffeomorphism. These charts show q is a quotient map, and is proper
locally over relatively compact neighborhoods. A finite cover of a
compact subset of M by such neighborhoods proves global properness.
The pulled-back metric induces the manifold topology on N, also for the
intrinsic length metric on each component P. A closed bounded subset of
P maps into a bounded subset of M and is closed in the compact inverse
image of a sufficiently large closed ball; so it is compact. Hence P
is proper.

Boundary surfaces. Each boundary component of P is one of the cut
surfaces of S, hence by the closed-surface proposition has intrinsic
diameter at most 2 pi and is homeomorphic to S^2 or RP^2. Weighted
minimality is inherited from the cut surface.

Separation. It remains to prove the last assertion. The proof reduces
to a closed-surface version and a neat-surface version. Fix a connected
component P of N, and let F be a compact connected two-sided neatly
embedded hypersurface in P. If F is closed and lies in int P, then F
separates P; this is the closed-surface lemma. Given that, the
general neat case is reduced to the closed case by a collar argument:
if a compact connected two-sided neat hypersurface F in P did not
separate, its bicollar and a path between its sides would produce a
smooth map P -> S^1 of nonzero degree that is constant outside a
compact set. Boundary components of P have finite fundamental group
(S^2 or RP^2), so the restriction of this map to a collar of dP has a
real lift; cutting this lift off at the boundary and exponentiating
produces a compactly supported homotopy to a map that is constant on a
neighborhood of dP, without changing its class on the crossing loop.
A regular value then produces a compact closed hypersurface in int P
which still does not separate, contradicting the closed-surface lemma.
The two lemmas are proved next.



LEMMA (closed case). Every closed connected embedded two-sided
hypersurface F in int P separates P.

PROOF. Suppose F does not separate. Fix a coorientation of F. A loop
crossing F once has nonzero mod-2 intersection. If it reverses the
ambient orientation, traverse it twice. A general-position perturbation
in dimension three then gives a smooth embedded, transverse,
orientation-preserving loop gamma in int 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 let tau be the deck involution. The
coorientation of F and the ambient orientation orient its full lift as
an integral 2-cycle T_F with

    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 with T_F(beta) nonzero. Concretely, the normal disk
bundle of the 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, and its evaluation on
T_F is the signed crossing number up to the orientation chosen for the
disk. The tube and supp beta lie in int P. Preserving this nonzero
evaluation will detect a nonseparating component of the minimizing
current, once the surface estimate has made the components meeting the
form support compact.

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 W exists by properness of P. Let J be a compact set in P
containing closure(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) and satisfying

    tau_# T = -T ,    T(beta) = T_F(beta) .

These are absolute cycles, not relative cycles. To interpret them at
dP, smoothly extend P and g_bar across dP, 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 in the constraints are closed
under weak convergence.

Practical references for the tool. H. Federer and W. H. Fleming,
Normal and integral currents, Ann. of Math. 72 (1960), 458-520.
https://doi.org/10.2307/1970227

L. Simon, Introduction to Geometric Measure Theory, Tsinghua lectures
(2014).
sources/simon-2014-gmt.pdf

Local mass minimization. In a sufficiently small ball over W inside
int P, T_0 is locally mass minimizing. Indeed, a compactly supported
integral-cycle change in that ball evaluates to zero on beta, since
beta is exact in the ball, so the change preserves both constraints;
making the opposite change in the disjoint deck translate preserves
the constraints again; antisymmetry of T_0 makes the two mass changes
equal; and a strict decrease would produce a strict decrease for
twice the amount upstairs, contradicting minimality of T_0. To see the
same minimizing property across dP, use a contractible flow box U'
upstairs with coordinates (z, s), whose boundary leaf is s = 0 and
whose allowed side is s >= 0, arranged so that the allowed side lies
over W and closure(U') and tau closure(U') are disjoint. Take a
smaller box U in U'. Suppose a compactly supported integral 2-cycle S
in U gave T' = T_0 + S with strictly smaller mass. Fold along s = 0
by the locally Lipschitz map R(z, s) = (z, max{0, s}) near closure(U),
extended by the identity before leaving U' and equal to the identity
on the allowed side. Since the exterior mass sits in U where the
folding is exact projection, the area formula gives mass_g_bar(R_# T')
<= mass_g_bar(T'). Then S' = R_# T' - T_0 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, using exactness in the two
contractible boxes and cutting off primitives beyond the cycle
supports. So T_0 + S' - tau_# S' again satisfies the constraints, and
the two mass changes are equal by antisymmetry, giving a strict
decrease and a contradiction. This establishes local mass minimization
across dP.

Regularity and sheets. Codimension-one interior regularity for
mass-minimizing integral currents says: 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. See Simon 2014, Chapter 7, Section 5. The
full-neighborhood comparison just proved supplies the interior
hypothesis even at points of dP. The strong maximum principle (D.
Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations
of Second Order, Springer, 2001, Chapter 3) implies that any sheet
touching dP agrees there with the boundary leaf. Hence over W the
projected support is smooth, and each connected component is either
contained in dP or is disjoint from dP.

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential
Equations of Second Order, Springer, 2001.
https://doi.org/10.1007/978-3-642-61798-0



The detected components 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 dP. Its intersection with
the closed ball B_P(x, 10) is compact: the 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; and 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
containing closure(Omega_0). Stability holds throughout Omega_1, and
D(g, t) >= 1. The local exit estimate 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.
So Sigma is a closed surface in int 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,
choose for each component the closure of one of its sides, 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 and boundary portions in dP contribute nothing because the
support of beta lies in int P. Multiplying by the constant
multiplicities and summing would give T_0(beta) = 0, a contradiction.

Take S to be such a nonseparating component. It is closed, connected,
embedded, two-sided, stable weighted minimal, and lies in int 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\S and
push the path into int 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 closed case.

QED.



LEMMA (neat case). Every compact connected two-sided neatly embedded
hypersurface in P separates P.

PROOF. A compact connected two-sided neat hypersurface F in P has a
bicollar. Suppose F does not separate P. Then the bicollar of F and a
path joining its two sides produce a loop in int P crossing F once.
Choose the transverse profile of the bicollar constant near both ends.
The bicollar coordinate extends by the same constant outside the
bicollar, and defines a smooth map

    u : P -> S^1

with F as a regular level and nonzero degree on any loop crossing F
once. Since the profile is constant outside a compact set, u is
constant outside a compact set.

Boundary components of P are spheres or projective planes, hence have
finite fundamental group. So the restriction of u to the collar of any
boundary component lifts smoothly to a real-valued map under the
covering R -> S^1. Only finitely many boundary components meet the
compact region where u is not constant, by local finiteness of the cut
surfaces. On each collar, take the difference between this lift and a
chosen lift of the outside constant, and multiply by a smooth cutoff
that is zero near dP and one near the interior end of the collar.
Exponentiating and multiplying over the finite collection of collars
that meet the nonconstant region gives a compactly supported homotopy
of u to a smooth map u_1 : P -> S^1 that equals the same constant on
a neighborhood of all of dP. Its degree on the crossing loop is
unchanged.

Choose a regular value of u_1 that is distinct from the constant value.
Its inverse image is a compact cooriented hypersurface in int P, with
finitely many connected components. Perturb the crossing loop to be
transverse to it. The sum of its signed crossings with the components
of u_1^{-1}(theta) equals the degree of u_1 on the loop, hence is
nonzero.

If every connected component of u_1^{-1}(theta) separated P, each
component would have two sides, and a closed loop crossing that
component transversely would pass alternately from one side to the
other at successive crossings, so its signed crossings would sum to
zero. Summing over components, the total signed intersection would be
zero. But it equals the nonzero degree. Contradiction. So some
component of u_1^{-1}(theta) is a compact closed two-sided embedded
hypersurface in int P that does not separate P. This contradicts the
closed case. QED.

The Proposition is now proved. The family S and the cut manifold N give
pieces P whose boundary is small spheres or projective planes, in which
every compact two-sided neatly embedded hypersurface separates.



The next step turns a compact piece P with small spherical boundary into
a closed manifold with basically the same geometry. The tool is
doubling.

Fix a connected component P of the cut manifold N. Let Z be the double
of P along its boundary, and sigma the exchange involution. When dP is
empty, Z is two disjoint copies of P, and sigma interchanges them.

LEMMA (weighted doubling). There are smooth sigma-invariant data (h, T)
on Z such that, on each copy of P,

    D(h, T) >= 1/2,    (1/4) g <= h <= 4 g .

The data are unchanged outside disjoint, locally finite collars of the
boundary components of P.

PROOF. Choose disjoint normal collars of the compact boundary
components, with widths allowed to vary from component to 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 is a ratio of densities on the boundary surface, so no
orientation is needed to define it. 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 gives

    m'(0) = 0 .

The curvature identity in these variables reads

    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, and t(s) is
t restricted to the slice. To see the identity, 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'') .

The first identity follows from the Gauss equation for the slices; the
second from the decomposition of the ambient Laplacian into the slice
part and the normal second derivative and normal divergence H t'.



Reflect k and m evenly across s = 0. The reflected k is continuous and
locally Lipschitz, but the reflected m is C^2 because m'(0) = 0 makes
the two one-sided pieces fit with matching first and second derivatives.
Convolve these reflected data in the s variable with an even, nonnegative
smooth mollifier of radius delta, and multiply by an even cutoff that is
one near s = 0 and is supported in the collar, blending the convolution
back to the original data near the far end of the collar. Write

    k_delta, m_delta

for the results, 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.

The curvature inequality survives at the seam. On compact subcollars,
k_delta, m_delta and their tangential derivatives through order two
converge uniformly to the reflected data, and the first two normal
derivatives of m_delta also converge uniformly. Every term in the
curvature formula except |(1/2) k_delta'|^2 converges to its one-sided
value at the seam. For points approaching (x, 0) as delta -> 0, 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) at the seam is at
least D(g, t)(x, 0) >= 1. Away from the seam the convergence is smooth,
and the compactness of the collar region in which changes occur yields

    D(h_delta, T_delta) >= 1/2

for all sufficiently small delta. Uniform convergence also gives the
metric comparison (1/4) g <= h_delta <= 4 g. Choose the smoothing
radius separately on each collar; the collars are locally finite, so
these choices give globally defined smooth data, equal to (g, t) away
from the collars. Setting h = h_delta and T = T_delta completes the
proof. QED.

For the rest of the construction, fix o in int P and let d be the
intrinsic length metric of h restricted to P. By the metric comparison
and properness of the original length metric on P, the metric d is
proper. The function

    r(x) = d(o, x)        (x in P)

extends evenly to Z, and 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 the
consequence

    d(x, y) <= length_h(gamma)        (x, y in P)

for every path gamma in Z from x to y.

The projection back to M is a contraction up to a factor of two:

    dist_g(q(x), q(y)) <= 2 d(x, y)     (x, y in P) .

Indeed, the metric comparison h <= 4 g says that every h-path in P has
g-length at most twice its h-length; taking the infimum over such paths
and applying q, which is a local isometry away from the cuts, gives the
inequality.



The final geometric ingredient is a family of small separating
interfaces. Fix P, o, h, T, d and r. For each level ell >= 100 in the
metric d, the goal is a finite disjoint family of compact two-sided
neatly embedded hypersurfaces in P, each with d-diameter at most 4 pi,
each contained in the annulus ell - 91 < r < ell - 18, and each
separating the base point o from the slab {ell < r < ell + 2}.

Smooth radial function. Fix a smooth even function rho on Z with

    |rho - r| < 1 ,    |d rho|_h <= 2 .

Such a function exists by local mollification of the Lipschitz function
r in coordinate charts where h is within factor 2 of a constant metric.
On each chart, the local smoothing has |d f_j|_h <= 3/2 with any
prescribed positive value error; patching with a locally finite smooth
partition of unity {phi_j} has gradient error sum_j (f_j - r) d phi_j
(because sum d phi_j = 0), and taking the local value errors below
2^{-j-2}/(1 + |d phi_j|_h) makes this error smaller than 1/2 in norm
and smaller than one in value. Averaging with the reflection then
produces a sigma-invariant smooth rho satisfying these bounds.
Compact sublevel sets are inherited from r.

The pressure. Choose regular values

    a in (ell - 90, ell - 89) ,
    b in (ell - 20, ell - 19) ,

and define, on the band a < rho < b,

    mu = cot( pi (rho - a) / (b - a) ) .

Because b - a > 69 and |d rho|_h <= 2,

    2 |d mu|_h <= (4 pi / (b - a)) (1 + mu^2) <= (1/4)(1 + mu^2) .

The pressure tends to +infinity as rho -> a+ and to -infinity as
rho -> b-. These signs keep a minimizing interface away from the
constraints used to obtain it.

Existence and reflection symmetry. Let mu_* be a smooth,
reflection-invariant, compactly supported extension of mu that agrees
with mu on a neighborhood of the closed band
{a + epsilon <= rho <= b - epsilon}. Such an extension exists because
this band is compactly contained in {a < rho < b}. For a set E of
finite perimeter in Z, 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 .

For sufficiently small epsilon > 0, minimize J 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} .

Regular sublevel sets provide competitors, so the class is nonempty.
All admissible sets lie in a fixed compact set, where e^{-T} is bounded
above and below by positive constants; a minimizing sequence therefore
has uniformly bounded ordinary perimeter. Compactness for functions
of bounded variation then gives an L^1-convergent subsequence. The
constraints are closed under the convergence, the volume term
converges, and weighted perimeter is lower semicontinuous; the standard
theory of sets of finite perimeter covers both the compactness and the
semicontinuity. Thus a minimizer E exists.

F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems,
Cambridge University Press, 2012.
https://doi.org/10.1017/CBO9781139108133

The lattice inequality for perimeter reads

    J(E union E') + J(E intersect 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 its intersection with E are admissible.
Minimality and the lattice inequality show both are minimizers, so
replacing E by E union sigma E leaves J unchanged; after this
replacement, E is invariant under reflection, up to a null set.



Removing the constraints near the interface. The minimization obtained
this way is constrained by two level sets of rho. We now show that the
minimizer fills a larger inner sublevel set and avoids a larger outer
superlevel set; this is exactly the statement that the interface sits
well inside the band, away from the constraints, so that its
neighbors in Z are locally unconstrained.

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
works for all sufficiently small epsilon. Set A = {rho < a + 2
epsilon}. The field X equals the outward unit normal along dA, 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 intersect A)
                             <= int_{A \ E} Div_h(e^{-T} X) dvol_h .

By the inner constraint, the difference set A \ E 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)

for small epsilon: the divergence is bounded on a fixed compact set,
while mu_* tends uniformly to +infinity as rho -> a+. Subtracting the
volume terms from the inequality shows that J(E union A) would be
strictly smaller than J(E) unless A \ E is null. Since E is a
minimizer, A \ E must be null:

    {rho < a + 2 epsilon} contained in E ,    up to a null set.

The outer end is the same comparison applied 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 dB. Both the field and the change
of phase have compact support near dB; the divergence calculation
still applies if B itself is unbounded, because the support of the
change does not see the far end. The pressure -mu_* tends uniformly to
+infinity on {b - 2 epsilon < rho <= b - epsilon}. So

    E intersect {rho > b - 2 epsilon} = empty ,    up to a null set.

With the constraints removed near the interface, S = supp(|D chi_E|_h)
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 satisfies the full local minimizing property for weighted
perimeter-plus-pressure near every point of S.

Local regularity. For smooth positive weights and smooth pressures, the
perimeter support of a locally minimizing set is smooth, and its
outward measure-theoretic normal extends smoothly to the support. One
proof is to absorb the weight into a conformal metric h_bar = e^{-T} h,
under which surface area is precisely weighted h-area and the pressure
becomes mu e^{T/2}. Local regularity for unweighted perimeter
minimizers gives the smoothness; see M. Kemper, Gromov Hyperbolic
Manifolds, Weighted Isoperimetry and Bubbles, Ph.D. thesis, University
of Munster, 2020, and the singular-set dimension argument there, which
in dimension three has no singular set at all. The same conclusion
follows from local regularity for unweighted perimeter minimizers, by
the conformal change described.

M. Kemper, Gromov Hyperbolic Manifolds, Weighted Isoperimetry and
Bubbles, Ph.D. thesis, University of Munster, 2020.
sources/kemper-2020-thesis.pdf

Thus S is a compact smooth two-sided hypersurface with finitely many
connected components, and the phase chi_E is locally constant on the
complement. 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.

Diameter bound. From the doubling inequality D(h, T) >= 1/2 and the
pressure gradient 2 |d mu|_h <= (1/4)(1 + mu^2),

    D(h, T) + mu^2 - 2 |d mu|_h >= (1/4) + (3/4) mu^2 >= 1/4

along Sigma. The closed-surface proposition with c = 1/4 now gives

    diam_{h|Sigma}(Sigma) <= 4 pi .

Reflection invariance. Reflection preserves both the full phase and the
empty phase, and hence preserves S together with its outward normal.
At a point of S on the seam, that normal is fixed by d(sigma), while
d(sigma) reverses the normal of the seam; so the normal of S is
tangential to the seam. Hence S meets the seam transversally. Its
restriction to P is consequently a finite union of disjoint compact
two-sided neatly embedded hypersurfaces. Let F_1, ..., F_m denote their
connected components.

Each F_i is contained in a single connected component Sigma of S. Two
points of F_i are joined by a path in Sigma of h-length at most 4 pi by
the closed-surface bound. Folding this path into P does not increase
its length, and applying the projection inequality gives

    diam_d F_i <= 4 pi .

The separating-cut proposition then says that every F_i separates P.

Inclusion in the annulus. Since S lies in {a + epsilon < rho < b -
epsilon} and |rho - r| < 1, every F_i is contained in

    ell - 91 < r < ell - 18 .

Hence the interfaces F_1, ..., F_m lie in the intended distance annulus
and have d-diameter at most 4 pi. Each separates the base point o from
the slab {ell < r < ell + 2}: the base point lies in the full phase,
the slab component lies in the empty phase, and each separating
interface lies between them.



LEMMA (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 a
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 the 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 works with boundary and for a neat hypersurface,
using its bicollar.

The graph is connected: points in complementary components can be moved
into int P and joined by an interior path transverse to the
hypersurfaces; a transverse compact path has finitely many crossings,
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, contradicting
separation. A connected finite graph in which every edge is a bridge is
a tree; the final assertion follows from the unique path between two
vertices. QED.

PROPOSITION (slab diameter). Fix a connected component P of the cut
manifold, with the smoothed data (h, T), the intrinsic metric d, the
base point o, and r(x) = d(o, x). For any 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 d-diameter at most 204.

Suppose ell >= 100 and the slab is nonempty. The interface construction
at level ell gives finitely many disjoint compact connected neatly
embedded 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 of the union of the F_i, and C lies in a single
region because it is connected and misses every F_i.

By the region tree lemma, some one F_i separates o from all of C.
A connected relatively compact domain of a proper length space is
geodesic after compact exhaustion, 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, hence the length after this crossing equals
r(x) - r(z_x), and r(z_x) > ell - 91 because z_x lies on F_i. So

    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)
            < 93 + 4 pi + 93 < 220 .

Together with the first case this proves the proposition. QED.

The slab cover. Use the 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 fixed j are disjoint, so U_P has
multiplicity at most two. Each set has compact closure because r is
proper. By the slab proposition and the projection inequality, for
every U in U_P,

    diam_h U <= 220,    diam_g q(U) <= 440 .

The cover itself need not be locally finite; the next step fixes this
without losing the diameter bound.



Fix a connected component P of N, with the cover U_P from the
previous paragraph. Choose a locally finite smooth partition of unity
{psi_alpha} with each psi_alpha compactly supported and 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 the functions assigned to the
same set:

    phi_U = sum over alpha with U(alpha) = U of psi_alpha .

Each sum has finitely many summands. Indeed, the nonempty supports all
meet the compact set closure(U), whereas the original family of
supports is locally finite. So supp phi_U is a compact subset of U.
Moreover, the family {supp phi_U} remains locally finite: a
neighborhood meeting only finitely many of the original supports meets
only finitely many of the grouped supports. Discard identically zero
functions.

Give each remaining function a vertex v; write U_v and phi_v.
Include the edge [v, w] whenever phi_v and phi_w are simultaneously
positive somewhere. This gives a locally finite simplicial graph K_P:
the compact support of phi_v meets only finitely many supports of other
functions. Because U_P 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 F_P is
continuous for the usual polyhedral topology.

Use disjoint vertex sets for different components P, and set K_0 equal to
the disjoint union of the K_P. The maps F_P combine to a continuous
map F_0 : N -> K_0. If y lies 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 .

This is the whole-fiber anchor: an entire fiber of F_0 lies in a
single cover set, whose g-diameter is at most 440 by the cover
estimate.



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 the separating-cut proposition 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 the separating-cut
proposition, so pi_1(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 L_B, so it satisfies the anchor
condition for every vertex it visits.

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 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 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 .

The construction is a collar version of the bounded-width gluing
principle in Gromov's Four Lectures, Section 3.10.1, Lemma (A). Keeping
the finite image subgraphs and the partition supports explicit ensures
both whole fibers and local finiteness of the target.

M. Gromov, Four lectures on scalar curvature, in Perspectives in
Scalar Curvature, Vol. 1, World Scientific, 2023, Section 3.10.1.
sources/gromov-2023-four-lectures.pdf



The diameter of an entire fiber. Fix y in K_0, and let v be a vertex
of the unique simplex of K_0 whose relative interior contains y. The
whole fiber f^{-1}(y) lies within d-distance L of the single set
q(U_v). To see this, take x in f^{-1}(y) and lift it to N.

If F_tilde(x) equals F_0(x), which is what happens away from the
collars, the anchor property of F_0 gives x in U_v, and q(x) lies in
q(U_v). Otherwise x lies in a collar C_B. On the homotopy half of the
collar, its image lies in L_B; since L_B is a subcomplex whose
intersection with the simplex containing y is nonempty, it contains
v. On the remaining half the image can belong to K_0 only at the
endpoint a_B, so again v lies 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
bound gives dist_g(q(x), q(z)) <= L, and q(z) lies in q(U_v). So q(x)
is within g-distance L of q(U_v).

Applying this to two points of the same fiber, their anchor points lie
in the same set q(U_v), and

    diam_g f^{-1}(y) <= 440 + 2L = 440 + 2(2 pi + 2) < 500 .

If y lies in K - 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 bound gives

    diam_g f^{-1}(y) <= L .

This covers the new vertices c_S, d_S and the common value m_S. Empty
fibers require no estimate. Thus the fiber bound holds for every
y in K, and the simplicial complex K is locally finite.



Rescaling: proof of the general case. The argument so far assumed
lambda = 1. For general lambda > 0, replace g by

    g' = lambda g .

Under this scaling, the geometric quantities transform as

    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'} = lambda^{1/2} dist_g .

The energy integrand therefore transforms as

    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 spectral inequality (*) with lambda = 1.

Apply the normalized result to (M, g'): there is a locally finite
simplicial complex K of dimension at most one and a continuous map
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 for general lambda. The proof is complete.

Some remarks on the constant and the hypotheses. The bound 500 is a
convenience. The actual estimate produced is

    440 + 2(2 pi + 2) = 444 + 4 pi < 457 ,

and no attempt was made to optimize the numerical choices in the
pressure band (a factor of four in each of several places). The
important structural point is that the constant depends only on the
spectral lower bound lambda, not on any geometric quantity of M. In
particular, no compactness, injectivity radius, diameter, scalar
curvature upper bound, or orientation hypothesis enters.



The scalar-curvature hypothesis is classical. Lichnerowicz observed that
a metric with Scal_g > 0 on a closed spin manifold admits no harmonic
spinors, ruling out positive scalar curvature on many spin manifolds.
Schoen and Yau used stable minimal surfaces to prove the positive mass
theorem and to obtain the first purely geometric obstructions. Kazdan
and Warner showed that scalar curvature is far less flexible than the
full curvature tensor. The graph-width question for scalar curvature
was formulated by Gromov and carried out in low dimensions by Chodosh
and Li. Torus-stabilized scalar curvature, also due to Gromov, and the
sharp torical band inequalities of Hirsch, Kazaras, Khuri, and Zhang
supply the closest precedents; the waist and width results of
Liokumovich and Maximo, and of Liokumovich and Wang, give related
bounds under pointwise positive scalar curvature. The proof combines
the positive-solution method of Fischer-Colbrie and Schoen for the
stability operator with the slice-and-dice and mu-bubble machinery of
Chodosh and Li and the bounded-width gluing principle of Gromov. The
direct weighted treatment extends the conclusion to complete noncompact
three-manifolds with no compactness or bounded-geometry assumption.

A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris Ser. A-B
257 (1963), A7-A9.

J. L. Kazdan and F. W. Warner, Scalar curvature and conformal
deformation of Riemannian structure, J. Differential Geom. 10 (1975),
113-134.
https://doi.org/10.4310/jdg/1214432678