==== TABLE OF CONTENTS ====
Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//main.tex
Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//references.tex
Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/01-introduction.tex
Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/02-weighted-surfaces.tex
Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/03-cutting.tex
Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/04-doubling-bubbles.tex
Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/05-slabs.tex
Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/06-graph-gluing.tex
===========================

==== BEGIN Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//main.tex ====
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\hypersetup{pdftitle={Spectral scalar curvature and Urysohn width in dimension three},pdfauthor={OpenAI}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
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\newcommand{\Z}{\mathbb Z}
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\DeclareMathOperator{\Ric}{Ric}
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\title{Spectral scalar curvature and Urysohn width\\in dimension three}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\input{sections/01-introduction}
\input{sections/02-weighted-surfaces}
\input{sections/03-cutting}
\input{sections/04-doubling-bubbles}
\input{sections/05-slabs}
\input{sections/06-graph-gluing}
\input{references}
\end{document}
==== END Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//main.tex ====

==== BEGIN Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//references.tex ====
\begin{thebibliography}{99}

\bibitem{ChodoshLi2024}
O.~Chodosh and C.~Li,
\emph{Generalized soap bubbles and the topology of manifolds with positive
scalar curvature},
Ann. of Math. (2) \textbf{199} (2024), no.~2, 707--740.
\href{https://doi.org/10.4007/annals.2024.199.2.3}{doi:10.4007/annals.2024.199.2.3}.

\bibitem{FedererFleming1960}
H.~Federer and W.~H.~Fleming,
\emph{Normal and integral currents},
Ann. of Math. (2) \textbf{72} (1960), no.~3, 458--520.
\href{https://doi.org/10.2307/1970227}{doi:10.2307/1970227}.

\bibitem{FischerColbrieSchoen1980}
D.~Fischer-Colbrie and R.~Schoen,
\emph{The structure of complete stable minimal surfaces in $3$-manifolds
of non-negative scalar curvature},
Comm. Pure Appl. Math. \textbf{33} (1980), no.~2, 199--211.
\href{https://doi.org/10.1002/cpa.3160330206}{doi:10.1002/cpa.3160330206}.

\bibitem{GilbargTrudinger2001}
D.~Gilbarg and N.~S.~Trudinger,
\emph{Elliptic Partial Differential Equations of Second Order},
Classics in Mathematics, Springer-Verlag, Berlin, 2001,
reprint of the 1998 edition.
\href{https://doi.org/10.1007/978-3-642-61798-0}{doi:10.1007/978-3-642-61798-0}.

\bibitem{GromovFourLectures2023}
M.~Gromov,
\emph{Four lectures on scalar curvature},
in \emph{Perspectives in Scalar Curvature}, Vol.~1,
World Scientific, Hackensack, NJ, 2023, 1--514.
Section and statement locators refer to
\href{https://arxiv.org/abs/1908.10612v6}{arXiv:1908.10612v6},
July 8, 2021.

\bibitem{GromovStabilized2024}
M.~Gromov,
\emph{Product inequalities for $\mathbb T^{\rtimes}$-stabilized scalar
curvature},
SIGMA Symmetry Integrability Geom. Methods Appl.
\textbf{20} (2024), Paper No.~038, 25~pp.
\href{https://doi.org/10.3842/SIGMA.2024.038}{doi:10.3842/SIGMA.2024.038}.

\bibitem{GromovLawson1983}
M.~Gromov and H.~B.~Lawson, Jr.,
\emph{Positive scalar curvature and the Dirac operator on complete
Riemannian manifolds},
Publ. Math. Inst. Hautes \`Etudes Sci. \textbf{58} (1983), 83--196.
\href{https://doi.org/10.1007/BF02953774}{doi:10.1007/BF02953774}.

\bibitem{HirschKazarasKhuriZhang2024}
S.~Hirsch, D.~Kazaras, M.~Khuri, and Y.~Zhang,
\emph{Spectral torical band inequalities and generalizations of the
Schoen--Yau black hole existence theorem},
Int. Math. Res. Not. IMRN (2024), no.~4, 3139--3175.
\href{https://doi.org/10.1093/imrn/rnad129}{doi:10.1093/imrn/rnad129}.

\bibitem{Kemper2020}
M.~Kemper,
\emph{Gromov Hyperbolic Manifolds, Weighted Isoperimetry and Bubbles},
Ph.D. thesis, Westf\"alische Wilhelms-Universit\"at M\"unster, 2020.

\bibitem{LiokumovichMaximo2023}
Y.~Liokumovich and D.~Maximo,
\emph{Waist inequality for $3$-manifolds with positive scalar curvature},
in \emph{Perspectives in Scalar Curvature}, Vol.~2,
World Scientific, Hackensack, NJ, 2023, 799--831.
\href{https://doi.org/10.1142/9789811273230_0022}{doi:10.1142/9789811273230\_0022}.

\bibitem{LiokumovichWang2026}
Y.~Liokumovich and Z.~Wang,
\emph{On the waist and width inequality in complete $3$-manifolds with
positive scalar curvature},
Camb. J. Math. \textbf{14} (2026), no.~2, 349--375.
\href{https://arxiv.org/abs/2308.04044}{arXiv:2308.04044}.

\bibitem{Maggi2012}
F.~Maggi,
\emph{Sets of Finite Perimeter and Geometric Variational Problems:
An Introduction to Geometric Measure Theory},
Cambridge Studies in Advanced Mathematics, vol.~135,
Cambridge University Press, Cambridge, 2012.

\bibitem{SchoenYau1979}
R.~Schoen and S.-T.~Yau,
\emph{On the structure of manifolds with positive scalar curvature},
Manuscripta Math. \textbf{28} (1979), no.~1--3, 159--183.
\href{https://doi.org/10.1007/BF01647970}{doi:10.1007/BF01647970}.

\bibitem{SchoenYau1983}
R.~Schoen and S.-T.~Yau,
\emph{The existence of a black hole due to condensation of matter},
Comm. Math. Phys. \textbf{90} (1983), no.~4, 575--579.
\href{https://doi.org/10.1007/BF01216187}{doi:10.1007/BF01216187}.

\bibitem{Simon2014}
L.~Simon,
\emph{Introduction to Geometric Measure Theory},
Tsinghua lectures, 2014.
\url{https://web.stanford.edu/class/math285/ts-gmt.pdf}.

\end{thebibliography}
==== END Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//references.tex ====

==== BEGIN Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/01-introduction.tex ====
\begin{abstract}
Every connected complete smooth Riemannian three-manifold without
boundary satisfying $-4\Delta+\Scal\ge\lambda>0$ as a quadratic-form
inequality admits a continuous map to a graph whose entire fibers have
diameter at most $500/\sqrt\lambda$ in the original metric.
No orientability, spin, compactness, or bounded-geometry assumption
is required.
\end{abstract}

\section{Introduction}\label{sec:introduction}

A positive scalar-curvature lower bound need not bound the diameter of a
three-manifold: the product of a round two-sphere with a line is a basic
example. A more suitable measure of size allows the manifold to extend
along a graph while requiring the inverse image of every point of the
graph to remain small. For a metric space $(X,d)$, its \emph{Urysohn
$1$-width} is
\[
 \operatorname{UW}_1(X,d)
 =\inf_{K,f}\ \sup_{y\in K}\diam_d f^{-1}(y),
\]
where $K$ ranges over simplicial complexes of dimension at most one, with
their usual polyhedral topology, and $f:X\to K$ ranges over continuous
maps. We give the empty fiber diameter zero. The diameter here is that of
the entire fiber, including distances between its different connected
components.

We use the sign convention $\Delta=\Div\nabla$ for the Laplacian.
The following theorem gives a width bound under a spectral, rather than
a pointwise, scalar-curvature hypothesis.

\begin{theorem}\label{thm:main}
Let $(M^3,g)$ be a connected complete smooth Riemannian three-manifold
without boundary. Suppose that $\lambda>0$ and
\begin{equation}\label{eq:main-spectral}
 \int_M\bigl(4|\nabla_g\phi|_g^2+\Scal_g\phi^2\bigr)\,d\vol_g
 \ge \lambda\int_M\phi^2\,d\vol_g
 \qquad\bigl(\phi\in C_c^\infty(M;\R)\bigr).
\end{equation}
Then there exist a simplicial complex $K$ of dimension at most one and a
continuous map $f:M\to K$ such that
\[
 \diam_g f^{-1}(y)\le \frac{500}{\sqrt\lambda}
 \qquad(y\in K).
\]
The complex has its usual polyhedral topology. No orientability, spin,
compactness, or bounded-geometry assumption is imposed on $M$.
\end{theorem}

The notation $-4\Delta_g+\Scal_g\ge\lambda$ will always mean
\eqref{eq:main-spectral}. In particular, a pointwise bound
$\Scal_g\ge\lambda$ implies the hypothesis, but is not required.
The theorem concerns the given metric $g$; passing to a conformally
related metric of positive scalar curvature would not by itself give
the required distance estimate. The constant $500$ is convenient and
is not optimized.

\subsection{Background and scope}

Stable minimal surfaces provide a classical link between scalar
curvature and the geometry of three-manifolds. Schoen and Yau used the
stability inequality and the Gauss equation to obtain curvature and
topological restrictions on such surfaces \cite{SchoenYau1979}.
Fischer-Colbrie and Schoen developed the associated positive-solution
method for Schr\"odinger operators \cite{FischerColbrieSchoen1980}.
The stable-surface radius estimate of Schoen and Yau
\cite{SchoenYau1983} and the distance-level estimates of Gromov and
Lawson \cite[Section~10]{GromovLawson1983} show how local surface
stability can control the large-scale geometry of the ambient
three-manifold. The latter distance-level argument uses a topological
condition on the first homology; it is not a diameter bound for an
arbitrary distance sphere in an arbitrary three-manifold.

Gromov's torus-stabilized scalar-curvature framework gives the closest
precedent for the spectral formulation here. Torus stabilization means
considering metrics on $M\times\mathbb T^N$ of the form
$g+\sum_{i=1}^N\varphi_i^2\,d\theta_i^2$, with smooth positive
functions $\varphi_i$ on $M$, while measuring width in the base metric
$g$. In his \emph{Four lectures on scalar curvature}, Gromov gives a
three-dimensional graph-width theorem in this framework
\cite[Section~3.10.1, Theorem~(G)]{GromovFourLectures2023}.
That statement is formulated for orientable complete three-manifolds,
with a mean-convex boundary allowed in the torus extension. The relation between torus
stabilization and the coefficient $4$ in
$-4\Delta+\Scal$ is made explicit in
\cite[Sections~1.A--1.B]{GromovStabilized2024}. Thus the spectral
viewpoint and the scalar-curvature-to-width principle are established
parts of this theory. Our purpose is to give a direct weighted proof
of Theorem~\ref{thm:main}, including its nonorientable and noncompact
cases, without a global bound on the auxiliary weight or its derivatives.

The prescribed-mean-curvature method used below also belongs to this
development. Gromov's $\mu$-bubbles minimize area with an added volume
term, whose density prescribes the mean curvature. Chodosh and Li
developed weighted versions and a slice-and-dice construction that first
simplifies the topology and then produces pieces of bounded diameter
\cite[Sections~3--6]{ChodoshLi2024}. The two-dimensional differential
inequality underlying our closed-surface diameter estimate is
equivalent to theirs \cite[Lemma~16]{ChodoshLi2024}. Under a pointwise
positive scalar-curvature lower bound, Liokumovich and Maximo constructed
singular foliations of compact three-manifolds with simultaneous area,
diameter, and genus control \cite{LiokumovichMaximo2023}.
Liokumovich and Wang extended area and diameter control to complete
noncompact three-manifolds \cite{LiokumovichWang2026}. These are stronger
geometric conclusions in their pointwise setting, and they also imply
graph-width bounds. Their level-set statements concern connected
components of fibers of proper Morse functions; passing to their Reeb
graphs distinguishes such components. The graph target in
Theorem~\ref{thm:main} likewise permits a uniform bound on whole fibers.

There are also sharp spectral band inequalities. For example,
Hirsch, Kazaras, Khuri, and Zhang obtain quantitative bounds for bands
from positivity of $-\Delta+c\Scal$, including $c=1/4$, under the
relevant orientability and topological hypotheses
\cite{HirschKazarasKhuriZhang2024}. Such an end-to-end distance estimate
is a different conclusion from a graph-width bound. We use the
weighted surface argument directly, keeping the spectral condition,
the topology of separating surfaces, and the final fiber estimate
within one proof.

\subsection{The proof and its useful components}

After scaling, assume $\lambda=1$. A positive supersolution of
$(-4\Delta_g+\Scal_g)v\ge v$ gives a smooth function $t=-2\log v$
such that
\begin{equation}\label{eq:intro-weighted-scalar}
 D(g,t):=\Scal_g+2\Delta_g t-|dt|_g^2\ge1.
\end{equation}
The function $t$ supplies the positive area density $e^{-t}$, without
changing the metric in which the final width is measured. All
minimization and smoothing arguments involving this density take
place in compact regions.

The first step is a weighted form of the stable-surface estimate.
If a closed connected two-sided surface is stable for weighted area
minus the weighted volume integral of a smooth pressure $\mu$, its
equation is $H-\nu t=\mu$, where $H$ is its mean curvature and $\nu$
is the chosen unit normal. A positive function for its stability operator converts
the ambient inequality into a two-dimensional inequality of the same
form as \eqref{eq:intro-weighted-scalar}. A conformal geodesic index
argument then gives an intrinsic diameter bound in the \emph{original}
surface metric whenever
$D(g,t)+\mu^2-2|d\mu|_g$ has a positive lower bound. The local version,
proved on relatively compact surface domains in
Section~\ref{sec:weighted-surfaces}, also prevents stable minimizing
sheets from escaping a sufficiently large fixed neighborhood.

We next cut along a locally finite family of closed stable weighted
minimal surfaces. The family is chosen so that no finite subfamily
disconnects $M$, and is maximal with this property. Every component
$P$ of the cut manifold then has the following separation property:
each compact connected two-sided surface meeting $\partial P$
transversely, with its boundary exactly on $\partial P$, separates
$P$. Section~\ref{sec:cutting} proves this by a localized minimizing
argument. On the orientation cover, antisymmetric integral currents
retain coorientations downstairs. A local area-nonincreasing
retraction at a cut boundary permits full-neighborhood comparisons,
so ordinary interior regularity applies even at boundary contacts.

The separation property allows small surfaces to control large
distance slabs. To construct those surfaces without a boundary
regularity problem, Section~\ref{sec:doubling-bubbles} doubles $P$ and
smooths the metric and density compatibly with reflection. The useful
smoothing variable is the logarithm of the weighted area density of
the normal slices. Weighted minimality makes its first normal
derivative vanish at the doubling locus, and the curvature inequality
survives smoothing. A reflection-invariant weighted $\mu$-bubble then
restricts to a finite collection of compact separating surfaces in $P$,
each with uniformly bounded diameter. The graph with complementary
regions as vertices and interface components as edges is a tree,
because each interface separates. The bubble places the basepoint and
a fixed distant slab component in different regions, so one interface
meets every path between them. Section~\ref{sec:slabs} uses this common
interface to bound the slab component's diameter.

Finally, overlapping slabs give a cover of multiplicity at most two,
and a subordinate partition of unity maps $P$ to a graph.
Section~\ref{sec:graph-gluing} joins these maps across the original
cuts, following the bounded-interface gluing principle of
\cite[Section~3.10.1, Lemma~(A)]{GromovFourLectures2023}.
Each cut surface is a sphere or a projective plane, so its map
to a finite graph is nullhomotopic. The homotopies are kept inside
finite subgraphs whose vertices correspond to cover sets meeting
the relevant collar. This extra condition ensures that all collar
pieces mapping to a given old graph point lie near one common cover
set. New edges join only the two collars belonging to a single cut.
These observations control entire fibers after gluing, rather than
only their connected components.

The weighted surface estimate, the compatible smoothing of doubles,
and the collar construction each isolate a useful part of the
argument. Their proofs require local elliptic and geometric measure
theory, but no uniform bounds on the geometry at infinity.
==== END Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/01-introduction.tex ====

==== BEGIN Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/02-weighted-surfaces.tex ====
\section{Weighted curvature and stable surfaces}
\label{sec:weighted-surfaces}

The spectral hypothesis produces a positive density for which closed
stable weighted minimal surfaces have uniformly bounded diameter.  We
prove both the closed-surface estimate and a local version that does not require the surface to be
complete.  The latter will be used to extract compact surfaces in
Section~\ref{sec:cutting}.

We use the sign convention $\Delta=\Div\nabla$ and take round spheres to
have positive scalar curvature.  A closed surface is compact without
boundary.  All integrals use Riemannian densities, so none of the formulas
requires an orientation.  For a Riemannian metric
$h$ and a smooth real function $T$ on a manifold of any dimension, set
\begin{equation}\label{eq:weighted-scalar}
 D(h,T)=\Scal_h+2\Delta_hT-|dT|_h^2.
\end{equation}
We work with the normalization $\lambda=1$ until the final rescaling in
Section~\ref{sec:graph-gluing}.

\subsection{From the quadratic form to a positive density}

We use the standard ground-state construction from a nonnegative
quadratic form; see Fischer-Colbrie--Schoen~\cite[Theorem~1]{FischerColbrieSchoen1980}.
We recall the exhaustion argument in the form needed here.

\begin{lemma}[Positive weight]\label{lem:positive-weight}
Let $(M,g)$ be a connected smooth Riemannian manifold without boundary.
Suppose that
\[
 \int_M\bigl(4|d\phi|_g^2+\Scal_g\phi^2\bigr)\,d\vol_g
 \ge \int_M\phi^2\,d\vol_g
 \qquad(\phi\in C_c^\infty(M;\R)).
\]
There is a smooth function $t:M\to\R$ such that $D(g,t)\ge1$.
\end{lemma}

\begin{proof}
It suffices to find a positive smooth function $v$ satisfying
\[
 (-4\Delta_g+\Scal_g)v\ge v,
\]
since $t=-2\log v$ then gives
\begin{equation}\label{eq:log-weight}
 D(g,t)=\Scal_g-4\frac{\Delta_gv}{v}\ge1.
\end{equation}
If $M$ is compact, take a positive first eigenfunction of
$-4\Delta_g+\Scal_g$.  Its eigenvalue is at least one by the hypothesis.

If $M$ is noncompact, fix $o\in M$ and choose an increasing exhaustion
$\Omega_j$ by connected relatively compact smooth domains containing
$o$.  Let $\lambda_j$ and $v_j$ be the first Dirichlet eigenvalue and a
positive first eigenfunction on $\Omega_j$, normalized by $v_j(o)=1$.
The quadratic-form hypothesis, extended to $H_0^1(\Omega_j)$ by density,
gives $\lambda_j\ge1$.  Domain monotonicity makes $\lambda_j$ decreasing,
and a fixed nonzero test function in $\Omega_1$ bounds it from above.
Thus $\lambda_j\to\lambda_\infty\ge1$.  On each compact subset of $M$,
the local Harnack inequality, applied along finitely many chains of
coordinate balls joining it to $o$, bounds $v_j$ above and away from zero
for all sufficiently large $j$.  Interior elliptic estimates
\cite[Theorems~6.2 and~8.20]{GilbargTrudinger2001} and a
diagonal subsequence now give smooth convergence on compact subsets to
a positive function $v$ with
\[
 (-4\Delta_g+\Scal_g)v=\lambda_\infty v.
\]
Equation~\eqref{eq:log-weight} proves the assertion.
\end{proof}

Fix this function $t$ on $M$.  Its density $e^{-t}$ determines the weighted
area used to cut $M$.  We first carry out the surface calculation for
arbitrary ambient data $(h,T)$, since a modified metric and weight will
also occur later.

\subsection{The stability inequality}

Let $\Sigma$ be a smooth two-sided surface, possibly with boundary, in a
Riemannian three-manifold $(X,h)$, and choose its unit normal $\nu$.
All surface variations and test functions will be compactly supported
in $\operatorname{int}\Sigma$.  Write $k=h|_\Sigma$ and use
the conventions
\[
 B(V,W)=h(\nabla_V\nu,W),\qquad H=\operatorname{tr}_k B.
\]
For a smooth ambient function $T$, weighted area means
$\int_\Sigma e^{-T}\,d\Area_k$.  If $\Sigma$ is the boundary of a set
$E$, with outward normal $\nu$, the functional
\begin{equation}\label{eq:weighted-pressure-functional}
 \int_{\partial E}e^{-T}\,d\Area_h
 -\int_E\mu e^{-T}\,d\vol_h
\end{equation}
has Euler--Lagrange equation
\begin{equation}\label{eq:weighted-mean-curvature}
 H-\nu T=\mu,
\end{equation}
where $\mu$ is a smooth function on $X$.  For a surface not globally
bounding a set, the same equation and second variation are defined
locally using its chosen normal.  Stability always means unconstrained
stability: all normal variations compactly supported in the surface
interior are allowed, with no volume constraint.

The weighted surface Laplacian and the potential in the second variation
are
\begin{equation}\label{eq:weighted-jacobi}
 \begin{split}
 \Delta_{\Sigma,T}u
   &=\Delta_ku-\langle d(T|_\Sigma),du\rangle_k,\\
 Q&=\Ric_h(\nu,\nu)+\Hess_hT(\nu,\nu)+|B|_k^2+\nu\mu.
 \end{split}
\end{equation}
In particular, stability is the inequality
\begin{equation}\label{eq:stability-form}
 \int_\Sigma\bigl(|du|_k^2-Qu^2\bigr)e^{-T}\,d\Area_k\ge0
 \qquad(u\in C_c^\infty(\operatorname{int}\Sigma)).
\end{equation}
To check the signs, a normal variation of speed $u$ satisfies
\[
 H'=-\Delta_ku-(\Ric_h(\nu,\nu)+|B|_k^2)u,
 \qquad \nu'=-\nabla_ku.
\]
Differentiating $H-\nu T-\mu$ therefore gives
$-\Delta_{\Sigma,T}u-Qu$.  The first variation of
\eqref{eq:weighted-pressure-functional} has factor $H-\nu T-\mu$,
so its second variation at a stationary surface is exactly
\eqref{eq:stability-form}.

\begin{lemma}[Curvature transferred to a stable surface]
\label{lem:weighted-stability}
Suppose that a smooth two-sided surface $\Sigma$ satisfies
\eqref{eq:weighted-mean-curvature} and is stable in the sense of
\eqref{eq:stability-form}.  Let $\Omega$ be either the whole surface,
if it is closed and connected, or a connected smooth surface domain
with compact closure in $\operatorname{int}\Sigma$.  There is a smooth
positive function $w$ in $\Omega$ such that, with $p=\log w$ and
$U=T|_\Sigma-p$,
\begin{equation}\label{eq:surface-curvature-transfer}
 D(k,U)\ge
 D(h,T)+\mu^2+|B|_k^2+2\nu\mu+|dp|_k^2
 \qquad\text{in }\Omega.
\end{equation}
In the domain case, no completeness hypothesis is imposed on $\Sigma$.
\end{lemma}

\begin{proof}
Take a positive first eigenfunction of the self-adjoint operator
$-\Delta_{\Sigma,T}-Q$ in the measure $e^{-T}\,d\Area_k$, with
Dirichlet boundary condition in the domain case.  Stability makes its
first eigenvalue nonnegative.  Thus
\begin{equation}\label{eq:log-jacobi}
 -\Delta_{\Sigma,T}w\ge Qw,
 \qquad
 -\Delta_{\Sigma,T}p\ge Q+|dp|_k^2.
\end{equation}
The function $w$ is positive in $\Omega$; its possible vanishing on
$\partial\Omega$ will not enter the argument.

The Gauss equation and the decomposition of the ambient Laplacian read
\[
 \Scal_k=\Scal_h-2\Ric_h(\nu,\nu)+H^2-|B|_k^2,
 \qquad
 \Delta_hT=\Delta_k(T|_\Sigma)+H\nu T+\Hess_hT(\nu,\nu).
\]
Together with $|dT|_h^2=|d(T|_\Sigma)|_k^2+(\nu T)^2$, these give
\begin{equation}\label{eq:surface-gauss-weighted}
 \begin{split}
 D(k,T|_\Sigma)={}&D(h,T)
 -2\bigl(\Ric_h(\nu,\nu)+\Hess_hT(\nu,\nu)\bigr)\\
 &+(H-\nu T)^2-|B|_k^2.
 \end{split}
\end{equation}
Changing $T|_\Sigma$ to $T|_\Sigma-p$ gives the exact identity
\[
 D(k,U)=D(k,T|_\Sigma)-2\Delta_{\Sigma,T}p-|dp|_k^2.
\]
Now substitute \eqref{eq:log-jacobi},
\eqref{eq:weighted-jacobi}, and $H-\nu T=\mu$ into
\eqref{eq:surface-gauss-weighted}.  The ambient normal Ricci and Hessian
terms cancel, leaving \eqref{eq:surface-curvature-transfer}.
\end{proof}

In particular, the lower bound
\begin{equation}\label{eq:pressure-curvature-condition}
 D(h,T)+\mu^2-2|d\mu|_h\ge c>0
\end{equation}
along a stable surface gives $D(k,U)\ge c$ on the closed surface or
the chosen relatively compact domain in Lemma~\ref{lem:weighted-stability}.
We next convert this curvature inequality into a length estimate in the
original induced metric $k$, not merely in a conformal metric.

\subsection{The conformal length estimate}

The conformal geodesic argument is part of the stable-surface radius
method of Schoen--Yau~\cite{SchoenYau1983}; the differential inequality
used here also appears in Chodosh--Li~\cite[Lemma~16]{ChodoshLi2024}.
We give the index calculation explicitly to track length in the original
metric.

\begin{lemma}[Conformal index estimate]\label{lem:conformal-length}
Let $(S,k)$ be a smooth surface, let $U\in C^\infty(S)$, and suppose
that $D(k,U)\ge c>0$.  Let $\gamma$ be a finite geodesic segment for
$\widetilde k=e^{-2U}k$ whose index form is nonnegative for normal
variations compactly supported in its interior.  Then
\[
 \operatorname{length}_k(\gamma)\le\frac{2\pi}{\sqrt c}.
\]
The same conclusion holds when the curvature hypothesis is assumed only
on a neighborhood of the segment's interior.
\end{lemma}

\begin{proof}
The Gauss curvature of $\widetilde k$ is
\begin{equation}\label{eq:conformal-gauss}
 K_{\widetilde k}=e^{2U}(K_k+\Delta_kU)
 \ge\tfrac12 e^{2U}(c+|dU|_k^2),
\end{equation}
because $\Scal_k=2K_k$.  Parametrize $\gamma$ by $k$-arclength
$s\in[0,L]$.  Choose a parallel $\widetilde k$-unit normal along its
interior; such a choice exists along an interval even when $S$ is
nonorientable.  Since $d\widetilde s=e^{-U}ds$, the index inequality
and \eqref{eq:conformal-gauss} imply
\begin{equation}\label{eq:conformal-index}
 0\le\int_0^L e^U
 \left((\phi')^2-\tfrac12(c+|dU|_k^2)\phi^2\right)\,ds
 \qquad(\phi\in C_c^\infty(0,L)).
\end{equation}
Here and below $U$ and its differential are evaluated along $\gamma$.
Set $\phi=e^{-U/2}\psi$.  As $|U'|\le|dU|_k$, completing the square
gives
\begin{align*}
 e^U\left((\phi')^2-\tfrac12(c+|dU|_k^2)\phi^2\right)
 &=2(\psi')^2-\tfrac c2\psi^2
   -\left(\psi'+\tfrac12U'\psi\right)^2\\
 &\quad-\tfrac12\bigl(|dU|_k^2-(U')^2\bigr)\psi^2\\
 &\le2(\psi')^2-\tfrac c2\psi^2.
\end{align*}
Consequently
\[
 2\int_0^L(\psi')^2\,ds\ge\frac c2\int_0^L\psi^2\,ds
 \qquad(\psi\in C_c^\infty(0,L)).
\]
Approximating $\sin(\pi s/L)$ in $H_0^1(0,L)$ gives
$2\pi^2/L^2\ge c/2$, which is the asserted bound.  All test variations
are supported away from the endpoints, proving also the last assertion.
\end{proof}

\begin{proposition}[Closed stable surfaces]\label{prop:stable-surface}
Let $\Sigma$ be a smooth closed connected two-sided surface in a
Riemannian three-manifold $(X,h)$.  Let $T,\mu$ be smooth ambient
functions.  Suppose that $\Sigma$ satisfies $H-\nu T=\mu$, is stable
for weighted area with pressure $\mu$, and satisfies
\eqref{eq:pressure-curvature-condition} for a constant $c>0$.
Then, in the induced metric $k=h|_\Sigma$,
\[
 \diam_k\Sigma\le\frac{2\pi}{\sqrt c}.
\]
Moreover, $\Sigma$ is diffeomorphic to $S^2$ or $\R P^2$.
\end{proposition}

\begin{proof}
Lemma~\ref{lem:weighted-stability} gives a smooth function $U$ on
$\Sigma$ with $D(k,U)\ge c$.  The conformal metric $e^{-2U}k$ is
complete because $\Sigma$ is compact.  Join any two points by a
minimizing geodesic for that metric.  Its index form is nonnegative,
so Lemma~\ref{lem:conformal-length} bounds its $k$-length by
$2\pi/\sqrt c$.  The $k$-distance between the endpoints is no greater.

Integrating $D(k,U)\ge c$ and using Gauss--Bonnet gives
\[
 4\pi\chi(\Sigma)
 =\int_\Sigma 2K_k\,d\Area_k
 \ge c\Area_k(\Sigma)+\int_\Sigma|dU|_k^2\,d\Area_k>0.
\]
This formula is valid also for nonorientable surfaces, either directly
with densities or by passing to the orientation double cover.  The
classification of closed connected surfaces now gives precisely
$S^2$ and $\R P^2$.
\end{proof}

\begin{lemma}[Local exit estimate]\label{lem:stable-exit}
Let $\Sigma$ be a smooth two-sided stable surface satisfying
$H-\nu T=\mu$ in a Riemannian three-manifold $(X,h)$.  Suppose that
\eqref{eq:pressure-curvature-condition} holds on $\Sigma$.
Let $\Omega_0,\Omega_1$ be connected smooth domains with compact closures
in $\operatorname{int}\Sigma$ such that $\overline{\Omega_0}\subset\Omega_1$ and
$\partial\Omega_0\ne\varnothing$.  For every $x\in\Omega_0$ there
is a path from $x$ to $\partial\Omega_0$, contained in
$\overline{\Omega_0}$, whose length in $k=h|_\Sigma$ is at most
$2\pi/\sqrt c$.
\end{lemma}

\begin{proof}
Apply Lemma~\ref{lem:weighted-stability} in $\Omega_1$.  The resulting
$U$ is smooth on a neighborhood of $\overline{\Omega_0}$ and
satisfies $D(k,U)\ge c$ there.  In the compact Riemannian manifold
with boundary $\overline{\Omega_0}$, choose a path of least
$e^{-2U}k$-length from $x$ to $\partial\Omega_0$, 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~\ref{lem:conformal-length} gives the stated
$k$-length estimate.  This argument uses compactness only of the two
chosen domains, not completeness of $\Sigma$.
\end{proof}

For the original data $(M,g,t)$, weighted minimality means $\mu=0$.
Since $D(g,t)\ge1$, Proposition~\ref{prop:stable-surface} bounds every
closed connected two-sided stable weighted minimal surface by $2\pi$
and identifies its topology.  The same constant in
Lemma~\ref{lem:stable-exit} prevents an open stable sheet from extending
arbitrarily far inside a region where it has compact intersections with
ambient balls.
==== END Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/02-weighted-surfaces.tex ====

==== BEGIN Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/03-cutting.tex ====
\section{Cutting to obtain separation}\label{sec:cutting}

The surface estimate controls the size of a closed stable weighted minimal
surface, but does not make an arbitrary separating construction connected.
We therefore first remove the topological obstruction: after cutting along
a locally finite family of small stable surfaces, every compact connected
two-sided hypersurface neatly embedded in a resulting piece will separate
it. Throughout this
section, $M$ is complete and $D(g,t)\geq1$.

A hypersurface in a manifold with boundary is \emph{neatly embedded} if
its boundary is exactly its intersection with the ambient boundary and
the intersection is transverse. A closed surface means a compact surface
without boundary. All hypersurfaces considered below are smooth.

\begin{proposition}\label{prop:separating-cut}
There is a locally finite family $\mathcal S$ of pairwise disjoint closed
connected embedded two-sided stable weighted minimal surfaces in $M$
such that the following statements hold. Let $N$ be the manifold obtained
by cutting $M$ along $\mathcal S$, and let $q:N\to M$ be the natural
gluing map.
\begin{enumerate}
\item 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.
\item 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 $\mathcal S$ gives two boundary copies in $N$.
\item Every compact connected neatly embedded two-sided hypersurface
in $P$ separates $P$.
\end{enumerate}
\end{proposition}

\subsection{A maximal locally finite 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, and 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 $\mathcal S$.

We first verify local finiteness. By Proposition~\ref{prop:stable-surface},
every member has intrinsic, and hence ambient, diameter at most $2\pi$.
Consequently, all members meeting a fixed compact set $A\subset M$ lie
in the compact closed $2\pi$-neighborhood of $A$. Here completeness is
used through the properness of the Riemannian distance on $M$.

For any finite subfamily $S_1,\ldots,S_k$, the classes
$[S_i]\in H_2(M;\Z/2)$ are linearly independent. Indeed, the connected
complement of the subfamily contains a path joining the two sides of
$S_i$; closing it by a short transverse arc gives a loop meeting $S_i$
once and missing the other surfaces. Modulo two, intersection with this
loop pairs to one with $[S_i]$ and to zero with the other classes. This
uses only the intersection pairing of compact cycles, so neither
orientability nor compactness of $M$ is required. A compact smooth
submanifold neighborhood of the closed $2\pi$-neighborhood of $A$ has
finite-dimensional $H_2(-;\Z/2)$. Its image in $H_2(M;\Z/2)$ therefore
bounds the size of every finite subfamily meeting $A$. Thus only finitely
many members meet $A$.

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

The boundary assertions in Proposition~\ref{prop:separating-cut} now
follow from Proposition~\ref{prop:stable-surface}. It remains to prove
separation. Fix a connected component $P$ of $N$. We first treat closed
hypersurfaces in the interior; this is
the only step requiring minimization.

\subsection{Producing an additional stable surface}

\begin{lemma}\label{lem:closed-separation}
Every closed connected embedded two-sided hypersurface in
$\operatorname{int}P$ separates $P$.
\end{lemma}

\begin{proof}
Suppose that $F\subset\operatorname{int}P$ is such a hypersurface and
does not separate. Fix a coorientation of $F$. A loop crossing $F$ once
has nonzero algebraic crossing number. If it reverses ambient orientation,
traverse it twice. A general-position perturbation in dimension three
then gives a smooth embedded, transverse, orientation-preserving loop
$\gamma\subset\operatorname{int}P$ with nonzero algebraic crossing
number with $F$. The crossing number is unchanged by this perturbation.

\paragraph{A compact minimization problem.}
Let $\pi:\widehat P\to P$ be the orientation double cover, including
both copies when $P$ is orientable. Give $\widehat P$ 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 $T_F(\beta)\ne0$. 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 $\supp\beta$ lie in the interior.
Preserving this nonzero evaluation will detect a nonseparating component,
once the surface estimate has made the components meeting the form
support compact.

Choose a relatively open set $W\subset P$ with compact closure that
contains $F$ and the closed $10$-neighborhood of
$\pi(\supp\beta)$ in the intrinsic metric of $P$. Such a set exists
by properness. Let $J\subset P$ be a compact set containing $\overline W$.
In the conformal metric
\begin{equation}\label{eq:area-metric}
\bar g=e^{-t}g,
\end{equation}
surface area is precisely area weighted by $e^{-t}$. Minimize
$\bar g$-mass among integral $2$-cycles $T$ supported in $\pi^{-1}(J)$
that satisfy
\begin{equation}\label{eq:current-constraints}
\tau_\#T=-T,
\qquad T(\beta)=T_F(\beta).
\end{equation}
These are absolute cycles, not relative cycles. To interpret them at
the boundary, smoothly extend $P$ and $\bar g$ across $\partial P$,
take the orientation cover of that extension, and regard the currents
as currents in the extension with the indicated support constraint.
The form $\beta$ extends by zero. The competitor $T_F$ shows that the
class is nonempty. Federer--Fleming compactness and closure
\cite{FedererFleming1960}, 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 \eqref{eq:current-constraints} are closed under weak
convergence.

In a sufficiently small ball over $W\cap\operatorname{int}P$, $T_0$
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
\eqref{eq:current-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.

\paragraph{Comparison across a minimal boundary.}
Each boundary component is minimal for $\bar g$, because its
$\bar g$-area is its original weighted area. Near any point of such a
component there is a foliation by $\bar g$-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 Poincar\'e inequality dominates it after shrinking the
disk. Solve the minimal-graph equation with small constant normal boundary
height by the implicit function theorem, using the standard Dirichlet
elliptic theory \cite{GilbargTrudinger2001}. 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,
$\Div_{\bar g}Y=0$. In an oriented flow box the $2$-form
\[
\alpha=\iota_Y\,d\vol_{\bar g}
\]
is closed, annihilates $Y$, and is invariant under its flow. Projection
along this flow onto the boundary leaf therefore pulls its area form
back to $\alpha$. As $\alpha$ has comass one, this projection has
$2$-Jacobian at most one on every tangent $2$-plane.

To apply this to currents, choose a contractible flow box $U'$ upstairs,
with coordinates $(z,s)$, whose boundary leaf is $s=0$ and whose allowed
side is $s\geq0$. Require that the allowed side lie over $W$, and that
$\overline{U'}$ and $\tau\overline{U'}$ be disjoint. Take a smaller
box $U\Subset U'$. 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 $\overline U$, interpolating to the identity before leaving $U'$,
and keeping $R$ equal to the identity everywhere on the allowed side.
The interpolation can be made in the flow coordinate and can be taken
to map $U'$ into itself. Every portion of $T'$ on the exterior side is
supported in $U$, where the exact projection rule holds. On the allowed
side, including tangent planes carried by the boundary leaf, $R$ is
the identity. The area formula and the preceding Jacobian bound give
\begin{equation}\label{eq:boundary-retraction-mass}
\mathbf M_{\bar g}(R_\#T')\leq \mathbf M_{\bar g}(T').
\end{equation}
No Jacobian bound for the interpolation region is needed, since $T'$
has no exterior mass there.

Put $S'=R_\#T'-T_0$. Pushforward preserves the cycle condition, and
$R_\#T_0=T_0$. Thus $S'$ is an integral cycle supported on the allowed
side in a compact subset of $U'$. Both $S'$ and $\tau_\#S'$ evaluate
to zero on $\beta$: use exactness in the respective contractible boxes,
with primitives cut off beyond the cycle supports. Consequently
\[
T_0+S'-\tau_\#S'
\]
is again supported in $\pi^{-1}(J)$ and satisfies
\eqref{eq:current-constraints}. The boxes are disjoint, and antisymmetry
of $T_0$ makes the two mass changes equal. By
\eqref{eq:boundary-retraction-mass} their sum is strictly negative,
contradicting minimality. This proves local mass minimization in full
neighborhoods across the boundary.

\paragraph{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
\cite[Chapter~7, Section~5]{Simon2014}. The full-neighborhood
comparison just proved supplies precisely the interior hypothesis even
at points of $\partial P$. 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 $\partial P$ 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
$W\cap\operatorname{int}P$ is therefore a two-sided stable weighted
minimal surface, initially possibly noncompact. To see the asserted
stability, extend a compactly supported normal variation of one sheet
to an ambient flow isolating that sheet and supported in
$W\cap\operatorname{int}P$. Its lifts preserve antisymmetry. They
preserve evaluation on $\beta$ by the homotopy formula, because
$d\beta=0$ and $\partial T_0=0$. Minimality of $T_0$, and the locally
constant multiplicities, then give the weighted stability inequality
for that sheet.

We have obtained stable surfaces near the detecting form. The next
point is that the surface length estimate makes the relevant components
compact, even though no completeness of these intermediate sheets has
been asserted.

\paragraph{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 $\partial P$. Its intersection with
$\overline B_P(x,10)$ is compact. Indeed, this ball is compact and is
contained in $W$; the projected support is closed relative to $W$
because the cover is finite, and its components are closed there.
The regular embedded surface structure is the subspace topology.

If $\Sigma$ were noncompact, choose a connected relatively compact
smooth domain $\Omega_0\subset\Sigma$ containing this intersection
in its interior. Its nonempty boundary lies outside
$\overline B_P(x,10)$. Choose another connected relatively compact
smooth surface domain $\Omega_1$ with
$\overline{\Omega_0}\subset\Omega_1$. Stability holds throughout
$\Omega_1$, and $D(g,t)\geq1$. Lemma~\ref{lem:stable-exit}, with
$\mu=0$ and $c=1$, produces a path from $x$ to
$\partial\Omega_0$ of induced $g$-length at most $2\pi$. Its image
in $P$ has no greater length. This contradicts the distance of
$\partial\Omega_0$ 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 $T_0(\beta)\ne0$, 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 $\partial P$
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
$\operatorname{int}P$. Its image in $M$ is disjoint from every member
of $\mathcal S$. Adding this image preserves the defining property of
$\mathcal S$. 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 $P\setminus S$ 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.
\end{proof}

\subsection{From closed to neat hypersurfaces}

We finish the proof of Proposition~\ref{prop:separating-cut}. 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
\[
u:P\longrightarrow S^1
\]
which is constant outside a compact set, goes once around the circle
along the transverse collar coordinate, and has $F$ as a regular level.
The map has nonzero degree on a loop crossing $F$ once. One may use a
transverse profile that is constant near both ends of the bicollar, so
extension by the same constant value outside the collar is smooth.

Every boundary component of $P$ has finite fundamental group. The
restriction of $u$ to a collar of any such component therefore has a
smooth real lift under $\R\to S^1$. 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 $\partial P$ 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 $\partial P$.
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
$\operatorname{int}P$, 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~\ref{lem:closed-separation}. This proves the final assertion of
Proposition~\ref{prop:separating-cut}.
==== END Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/03-cutting.tex ====

==== BEGIN Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/04-doubling-bubbles.tex ====
\section{Doubling and separating interfaces}\label{sec:doubling-bubbles}

Fix a component $P$ of the cut manifold in
Proposition~\ref{prop:separating-cut}.  We will 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.  We then minimize weighted
perimeter with a pressure that forces the resulting surfaces to lie in
a prescribed annular region.

\subsection{Smoothing the double}

Let $Z$ be the double of $P$ along $\partial P$, and let $\sigma$ exchange
its two copies.  If $\partial P$ is empty, take two disjoint copies.

\begin{lemma}[Weighted doubling]\label{lem:double-smoothing}
There are smooth $\sigma$-invariant data $(h,T)$ on $Z$ such that
\begin{equation}\label{eq:double-comparison}
 D(h,T)\ge\frac12,
 \qquad \frac14g\le h\le4g
 \quad\text{on each copy of }P.
\end{equation}
The data can be left unchanged outside disjoint, locally finite collars
of the boundary components.
\end{lemma}

\begin{proof}
Choose disjoint normal collars of the compact boundary components.  Their
widths may vary with the component.  In one such collar, use inward
distance $s\ge0$ and write
\begin{equation}\label{eq:weighted-collar-data}
 g=ds^2+k(s),\qquad
 m(s)=\log\frac{d\vol_{k(s)}}{d\vol_{k(0)}}-t(s).
\end{equation}
The ratio in this formula is a ratio of densities on the boundary
surface, so no orientation is needed.  With the normal $\partial_s$,
the slices have mean curvature
$H=\frac12\operatorname{tr}_{k(s)}k'(s)$, and
$m'=H-t'$.  Weighted minimality of the boundary therefore gives
\begin{equation}\label{eq:even-weighted-density}
 m'(0)=0.
\end{equation}
The useful curvature formula in these variables is
\begin{equation}\label{eq:collar-curvature}
 D(g,t)=D(k(s),t(s))
       -\left|\frac12k'(s)\right|_{k(s)}^2-(m')^2-2m''.
\end{equation}
Here $D(k(s),t(s))$ uses the two-dimensional slice metric.  To verify the
formula, use
\[
 \Scal_g=\Scal_{k(s)}-\left|\frac12k'\right|^2-H^2-2H',
 \qquad
 \Delta_g t=\Delta_{k(s)}t+t''+Ht',
\]
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
\eqref{eq:even-weighted-density}.  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 $k_\delta,m_\delta$, and set
\[
 h_\delta=ds^2+k_\delta(s),\qquad
 T_\delta=\log\frac{d\vol_{k_\delta(s)}}{d\vol_{k(0)}}-m_\delta(s).
\]
For small $\delta$, the tensor $k_\delta$ is positive definite.

We check the curvature inequality also at points approaching the seam.
On compact subcollars, $k_\delta,m_\delta$ and their tangential derivatives
through order two converge uniformly to the reflected data.  The first
two normal derivatives of $m_\delta$ converge uniformly as well.  Thus
every term in \eqref{eq:collar-curvature}, except the squared norm of
$k_\delta'/2$, converges to its one-sided value at the seam.  For a
sequence of points approaching $(x,0)$ as $\delta\to0$, 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
\[
 \limsup_{\delta\to0}
 \left|\frac12k_\delta'\right|_{k_\delta}^2
 \le \left|\frac12k'(x,0)\right|_{k(0)}^2.
\]
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
$D(h_\delta,T_\delta)\ge1/2$ for all sufficiently small $\delta$.
Uniform convergence also gives the metric comparison in
\eqref{eq:double-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.
\end{proof}

Write $d$ for the intrinsic length metric of $h|_P$.  It is proper by
\eqref{eq:double-comparison} and properness of the original length metric
on $P$.  Fix $o\in\operatorname{int}P$ and put
\begin{equation}\label{eq:radial-distance}
 r(x)=d(o,x)\qquad(x\in P).
\end{equation}
Extend $r$ evenly to $Z$.  The folding map $Z\to P$ preserves the length
of a path on either copy and does not increase length across the seam.
This follows directly from $h=ds^2+k(s)$, with $k$ even, by replacing
$s$ with $|s|$ along an absolutely continuous path.  In particular, the
even extension of $r$ is proper and $1$-Lipschitz on each component of
$Z$.  We will also use the consequence
\begin{equation}\label{eq:folding-distance}
 d(x,y)\le\operatorname{length}_h(\gamma)
 \quad(x,y\in P)
\end{equation}
for every path $\gamma$ in $Z$ from $x$ to $y$.

\subsection{Separating interfaces in a distance annulus}

We use a weighted perimeter-minus-volume minimization, as in
\cite[Proposition~12]{ChodoshLi2024}, to obtain the following output for
the slab argument.  An open set here is understood relative to $P$ when
it meets $\partial P$.

\begin{proposition}[Separating interfaces]\label{prop:separating-interfaces}
Let $\ell\ge100$, and suppose that $\{r>\ell\}$ is nonempty.  There are finitely
many disjoint smooth compact connected two-sided neat hypersurfaces
$F_1,\ldots,F_m$ in $P$ such that
\begin{equation}\label{eq:interface-location}
 F_i\subset\{\ell-91<r<\ell-18\},\qquad \diam_d F_i\le4\pi.
\end{equation}
Each $F_i$ separates $P$.  Moreover,
$P\setminus\bigcup_iF_i$ is the disjoint union of two open sets
$U_+,U_-$, with $o\in U_+$ and $\{r>\ell\}\subset U_-$.
\end{proposition}

\begin{proof}
First choose a smooth even function $\rho$ on $Z$ such that
\begin{equation}\label{eq:smooth-radial-distance}
 |\rho-r|<1,\qquad |d\rho|_h\le2.
\end{equation}
For completeness, smooth a Lipschitz function in relatively compact
coordinate charts in which the metric is sufficiently close to a
constant metric.  Local convolution then gives gradient norm at most
$3/2$, with any prescribed positive value error on the compact support
of a partition function.  Let $f_j$ be these local smoothings and patch
them using a countable, locally finite smooth partition of unity
$\{\phi_j\}$.  Since
$\sum_jd\phi_j=0$, the error in the gradient caused by patching is
$\sum_j(f_j-r)d\phi_j$.  Taking the value errors smaller than
$2^{-j-2}/(1+\|d\phi_j\|_\infty)$ makes its norm less than $1/2$ and
also makes the total value error less than one.  Finally average the
result with its reflection.  Since reflection is an isometry, both
bounds survive.  The function $\rho$ has compact sublevel sets because
$r$ does.

Choose regular values
\begin{equation}\label{eq:pressure-levels}
 a\in(\ell-90,\ell-89),\qquad b\in(\ell-20,\ell-19).
\end{equation}
On the band $a<\rho<b$ define
\begin{equation}\label{eq:cotangent-pressure}
 \mu=\cot\left(\frac{\pi(\rho-a)}{b-a}\right).
\end{equation}
The width $b-a>69$ and \eqref{eq:smooth-radial-distance} imply
\begin{equation}\label{eq:pressure-gradient}
 2|d\mu|_h
 \le\frac{4\pi}{b-a}(1+\mu^2)
 \le\frac14(1+\mu^2).
\end{equation}
The pressure tends to $+\infty$ at the inner end of the band and to
$-\infty$ at the outer end.  These signs will keep a minimizing
interface away from the constraints used to obtain it.

\subsubsection*{Existence and reflection symmetry}
For sufficiently small $\epsilon>0$, minimize among sets $E$ of finite
perimeter satisfying, up to sets of measure zero,
\begin{equation}\label{eq:phase-constraints}
 \{\rho<a+\epsilon\}\subset E,
 \qquad E\subset\{\rho\le b-\epsilon\}.
\end{equation}
Write
\[
 \mathcal P_T(E)=\int_Z e^{-T}\,d|D\chi_E|_h,
 \qquad
 \mathcal J(E)=\mathcal P_T(E)
       -\int_E\mu_*e^{-T}\,d\vol_h,
\]
where $\mu_*$ is smooth, compactly supported, reflection invariant, and
agrees with $\mu$ on a neighborhood of
$\{a+\epsilon\le\rho\le b-\epsilon\}$.  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; see, for example,
\cite{Maggi2012}.  Thus a minimizer exists.

The lattice inequality for perimeter gives
\begin{equation}\label{eq:perimeter-lattice}
 \mathcal J(E\cup E')+\mathcal J(E\cap E')
 \le\mathcal J(E)+\mathcal J(E').
\end{equation}
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 \eqref{eq:perimeter-lattice} show that both are minimizers.
Replacing $E$ by $E\cup\sigma E$, we may suppose that $E$ is invariant
under reflection, up to a null set.

\subsubsection*{Removing the constraints near the interface}
We show that $E$ fills a larger inner sublevel and avoids a larger
outer superlevel.  This is the step that permits unrestricted local
variations of its boundary.

Since $a$ is a regular value and its level set is compact, there is a
fixed compact level neighborhood on which $d\rho\ne0$.  Choose a smooth
compactly supported field $X$, with $|X|_h\le1$, that equals
$\nabla\rho/|d\rho|_h$ 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 $\partial A$, so the divergence theorem and the variational
definition of perimeter give
\begin{align}
 \mathcal P_T(E\cup A)-\mathcal P_T(E)
 &\le\mathcal P_T(A)-\mathcal P_T(E\cap A)\notag\\
 &\le\int_{A\setminus E}\Div_h(e^{-T}X)\,d\vol_h.
 \label{eq:inner-barrier-comparison}
\end{align}
By \eqref{eq:phase-constraints}, the difference set is contained, up
to a null set, in $\{a+\epsilon\le\rho<a+2\epsilon\}$.  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 $+\infty$.  Subtracting the volume terms in
\eqref{eq:inner-barrier-comparison} shows that $E\cup A$ would strictly
improve $\mathcal J$ unless $A\setminus E$ has measure zero.  Hence
\begin{equation}\label{eq:inner-phase-clearance}
 \{\rho<a+2\epsilon\}\subset E
 \quad\text{up to a null set}.
\end{equation}

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
$-\nabla\rho/|d\rho|_h$ near its boundary.  Both the field and the
change of phase have compact support near that boundary; thus the
divergence calculation also applies if $B$ itself is unbounded.
The pressure $-\mu_*$ tends uniformly to $+\infty$ on
$\{b-2\epsilon<\rho\le b-\epsilon\}$.  It follows that
\begin{equation}\label{eq:outer-phase-clearance}
 E\cap\{\rho>b-2\epsilon\}=\varnothing
 \quad\text{up to a null set}.
\end{equation}

Let $S=\supp|D\chi_E|_h$.  Equations
\eqref{eq:inner-phase-clearance} and
\eqref{eq:outer-phase-clearance} show that $S$ is a compact subset of
the open band $a+\epsilon<\rho<b-\epsilon$.  Every compactly supported
change of $E$ in this band is admissible.  The minimizer therefore has
the full local minimizing property for weighted perimeter with the
smooth pressure $\mu$ near every point of $S$.

\subsubsection*{Regularity, diameter, and restriction to the half}
We use the following precise local regularity input: in a smooth
three-manifold, a set locally minimizing perimeter with a smooth
positive weight and a smooth pressure volume term has smooth embedded
perimeter support, and its outward measure-theoretic unit normal
extends smoothly to that support.  For the weighted version, see
\cite[Theorem~4.3 and Remarks~4.4]{Kemper2020}: 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 $\bar h=e^{-T}h$: 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 \eqref{eq:double-comparison} and \eqref{eq:pressure-gradient},
\[
 D(h,T)+\mu^2-2|d\mu|_h
 \ge\frac14+\frac34\mu^2\ge\frac14.
\]
Proposition~\ref{prop:stable-surface}, with $c=1/4$, now gives
\begin{equation}\label{eq:closed-interface-diameter}
 \diam_{h|_\Sigma}\Sigma\le4\pi.
\end{equation}

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 $F_1,\ldots,F_m$ 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 \eqref{eq:folding-distance} give $\diam_d F_i\le4\pi$.
Proposition~\ref{prop:separating-cut} says that every $F_i$ separates
$P$.

The location of $S$ in $a<\rho<b$, together with
\eqref{eq:smooth-radial-distance} and \eqref{eq:pressure-levels}, gives
\eqref{eq:interface-location}.  Restrict the full and empty phases of
$Z\setminus S$ to $P$ to obtain $U_+,U_-$.  They are disjoint relatively
open sets covering $P\setminus\bigcup_iF_i$.  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.
\end{proof}
==== END Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/04-doubling-bubbles.tex ====

==== BEGIN Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/05-slabs.tex ====
\section{Bounded components of distance slabs}
\label{sec:slabs}

The separating interfaces constructed in Section~\ref{sec:doubling-bubbles}
will now give a cover of each cut component by sets of bounded diameter.
The point is to find one interface component that every path from the
base point to a given slab component must cross.  Separate crossings of
unrelated interface components would not suffice for this purpose.

\begin{lemma}[The complementary-region tree]
\label{lem:region-tree}
Let $P$ be a connected smooth manifold, possibly with boundary.  Let
$F_1,\ldots,F_m$ be disjoint compact connected two-sided neatly embedded
hypersurfaces, each of which separates $P$.  The graph with one vertex for
each component of $P\setminus\bigcup_i F_i$ 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 $F_i$.
\end{lemma}

\begin{proof}
A bicollar of $F_i$ can be chosen disjoint from all the other
hypersurfaces.  Its two sides are connected, so each side belongs to a
unique complementary component.  This defines the endpoints of its
edge.  The complement has finitely many components: insert the
hypersurfaces one at a time, observing that removing a connected
two-sided hypersurface from a connected manifold creates at most two
components.  The same argument applies with boundary and to a neat
hypersurface, using its bicollar.

The graph is connected.  Indeed, points in complementary components
can first be moved into the interior of $P$ and then joined by an
interior path transverse to the hypersurfaces.  A transverse compact
path has finitely many crossings and hence gives a walk in the graph.
Every edge is a bridge: a graph path joining its endpoints without
that edge could be realized by paths in complementary regions and
bicollars, thereby joining the two sides of $F_i$ without meeting
$F_i$.  This contradicts separation by $F_i$.  A connected finite
graph in which every edge is a bridge is a tree.  Any edge on the
unique path between two distinct vertices gives the final assertion.
\end{proof}

Fix a connected component $P$ of the cut manifold of
Proposition~\ref{prop:separating-cut}.  As in
Section~\ref{sec:doubling-bubbles}, use the smoothed metric $h$ on its
double, let $d$ be the intrinsic length metric of $h|_P$, and fix
$o\in\operatorname{int}P$.  Write $r(x)=d(o,x)$.  The metric $d$ is
proper, and Lemma~\ref{lem:double-smoothing} gives
\begin{equation}
\label{eq:projection-distance}
 \dist_g(q(x),q(y))\le 2d(x,y) \qquad (x,y\in P).
\end{equation}
Here $\dist_g$ is distance in the original manifold $M$, not the
intrinsic distance in a cut component: project paths in $P$ and use
$g\le 4h$.

\begin{proposition}[Slab diameter bound]
\label{prop:slab-diameter}
For $\ell<u\le\ell+2$, every connected component $C$ of
\[
 \{x\in P:\ell<r(x)<u\}
\]
has $d$-diameter at most $220$.
\end{proposition}

\begin{proof}
If $\ell<100$, then $r<u<102$ on the slab.  The triangle inequality
through $o$ gives $\diam_d C\le 204$.

Suppose $\ell\ge100$ and the slab is nonempty.  Apply
Proposition~\ref{prop:separating-interfaces} at level $\ell$.  It gives
finitely many disjoint compact connected neat two-sided hypersurfaces
$F_1,\ldots,F_m$ in
\begin{equation}
\label{eq:interface-radius-window}
 \ell-91<r<\ell-18,
\end{equation}
each separating $P$ and each of $d$-diameter at most $4\pi$.
The full phase contains $o$, whereas the empty phase contains the
slab; the phase is constant on each complementary region.  Thus $o$
and $C$ lie in different complementary regions.  In this assertion,
$C$ lies in a single region because it is connected and misses every
$F_i$.

By Lemma~\ref{lem:region-tree}, some one component $F_i$ separates
$o$ from all of $C$.  Proper length spaces are geodesic, so for each
$x\in C$ choose a minimizing path from $o$ to $x$.  It meets $F_i$
at some point $z_x$.  Each subpath is minimizing, and hence its
remaining length after this crossing is
\begin{equation}
\label{eq:slab-residual-length}
 d(x,z_x)=r(x)-r(z_x)
 < (\ell+2)-(\ell-91)=93.
\end{equation}
For $x,y\in C$, the triangle inequality and the ambient
$d$-diameter bound for $F_i$ give
\[
 d(x,y)\le d(x,z_x)+d(z_x,z_y)+d(z_y,y)
 < 2\cdot93+4\pi<220.
\]
Together with the first case this proves the proposition.
\end{proof}

For the graph construction we use the particular open cover
\begin{equation}
\label{eq:slab-cover}
 \mathcal U_P=
 \left\{\text{connected components of }
   \{j-\tfrac34<r<j+\tfrac34\}:j=0,1,2,\ldots\right\}.
\end{equation}
These are open sets because a manifold with boundary is locally
path connected.  The intervals cover $[0,\infty)$, and no point
belongs to more than two of them.  Components for a fixed $j$ are
disjoint, so $\mathcal U_P$ has multiplicity at most two.  Each of
its sets has compact closure, since $r$ is proper.  By
Proposition~\ref{prop:slab-diameter} and
\eqref{eq:projection-distance},
\begin{equation}
\label{eq:cover-diameter}
 \diam_g q(U)\le440 \qquad(U\in\mathcal U_P).
\end{equation}
The cover itself need not be locally finite.  The next section makes
the partition of unity, and therefore the target graph, locally
finite without losing this diameter bound.
==== END Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/05-slabs.tex ====

==== BEGIN Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/06-graph-gluing.tex ====
\section{Graph maps and gluing across the cuts}
\label{sec:graph-gluing}

We first map each cut component to a locally finite graph using the
slab cover.  The two boundary copies of a cut surface need not have
the same image.  We will reconcile them by homotopies on small
collars and a new edge path for that surface.  Throughout this modification,
each vertex will retain its original cover set as an anchor for
the entire fiber.
This is a collar version of the bounded-width gluing argument in
\cite[Section~3.10.1, Lemma~(A)]{GromovFourLectures2023}.  We keep the
finite image subgraphs and the partition supports explicit to control
both whole fibers and local finiteness of the target.

\subsection{A locally finite partition and its graph}

Fix a component $P$ of $N$, with the cover $\mathcal U_P$ from
\eqref{eq:slab-cover}.  Choose a locally finite smooth partition of
unity $\{\psi_\alpha\}$ whose supports are compact and with each
support contained in some assigned $U(\alpha)\in\mathcal U_P$.
Such a partition follows by taking a locally finite relatively
compact refinement of the cover and then a subordinate partition
of unity.  Group functions assigned to the same set:
\begin{equation}
\label{eq:grouped-partition}
 \phi_U=\sum_{\alpha:U(\alpha)=U}\psi_\alpha.
\end{equation}
There are only finitely many summands in each of these sums.  Indeed,
their nonempty supports all meet the compact set $\overline U$,
whereas the original family of supports is locally finite.
Consequently $\supp\phi_U$ 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 grouped supports.  Discard identically zero functions.

Give each remaining function a vertex $v$, and write $U_v$ and
$\phi_v$ for the corresponding set and function.  Include the edge
$[v,w]$ precisely when $\phi_v$ and $\phi_w$ are simultaneously
positive somewhere.  This defines a simplicial graph $K_P$.  It is
locally finite, because the compact support of $\phi_v$ meets only
finitely many supports of other functions.  Since the original
cover has multiplicity at most two, at most two of the functions
are positive at any point.  The formula
\begin{equation}
\label{eq:initial-graph-map}
 F_P(x)=\sum_v\phi_v(x)v
\end{equation}
therefore defines a map to $K_P$, interpreted in barycentric
coordinates on its vertices and edges.  Near each point only
finitely many coordinates occur, so this map is continuous for
the usual polyhedral topology, not merely for a topology on an
infinite coordinate product.

Use disjoint vertex sets for different components $P$, and put
$K_0=\coprod_P K_P$.  The maps $F_P$ give a continuous map
$F_0:N\to K_0$.  If $y$ belongs to the relative interior of a
simplex of $K_0$ and $v$ is any vertex of that simplex, then
\begin{equation}
\label{eq:initial-fiber-anchor}
 F_0(x)=y\quad\Longrightarrow\quad
 \phi_v(x)>0\quad\Longrightarrow\quad x\in U_v.
\end{equation}
In particular, the whole fiber is controlled by a single set
in \eqref{eq:cover-diameter}.

\subsection{Collar homotopies with controlled anchors}

Choose disjoint closed collars $\mathcal C_B$ 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 $B^+,B^-$ are the two
copies of a cut surface $S$, the intrinsic diameter estimate
$\diam_g S\le2\pi$ from
Proposition~\ref{prop:separating-cut} gives
\begin{equation}
\label{eq:paired-collar-diameter}
 \diam_g q(\mathcal C_{B^+}\cup\mathcal C_{B^-})
 \le L,\qquad L:=2\pi+2.
\end{equation}
Indeed, project any two collar points normally to $S$, join the
projections in $S$, and add the two normal segments.

Write each collar as $B\times[0,1]$, with $B\times\{0\}$ the
actual boundary and $B\times\{1\}$ its inner end.  Let
\[
 h_B:B\longrightarrow K_0,\qquad h_B(x)=F_0(x,1).
\]
Its image lies in a finite connected subgraph $L_B$ of the
appropriate $K_P$: take the smallest subcomplex containing
$h_B(B)$.  Compactness of $B$ and local finiteness of $K_P$ imply
finiteness of this subcomplex, and connectedness of the image
implies connectedness of the subcomplex.  More importantly, its
choice ensures that
\begin{equation}
\label{eq:homotopy-anchor}
 v\in L_B\quad\Longrightarrow\quad
 (B\times\{1\})\cap\{\phi_v>0\}\ne\varnothing.
\end{equation}
For a vertex in the image this is immediate.  Any other vertex
of $L_B$ is an endpoint of an edge whose interior meets the
image, and both endpoint coordinates are positive at such a
point.  Thus \eqref{eq:homotopy-anchor} 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~\ref{prop:separating-cut}, 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
\begin{equation}
\label{eq:boundary-nullhomotopy}
 H_B:B\times[0,1]\longrightarrow L_B,
 \qquad H_B(x,0)=h_B(x),\quad H_B(x,1)=a_B.
\end{equation}
In particular, the homotopy stays inside a subgraph all of whose
vertices satisfy \eqref{eq:homotopy-anchor}.

\subsection{Adding edge paths and descending to the manifold}

For each cut surface $S$ add an edge path $A_S$ from $a_{B^+}$ to
$a_{B^-}$, with interior disjoint from $K_0$ and from every
other added path.  More explicitly, introduce two distinct new
vertices $c_S,d_S$ and the three edges
\[
 [a_{B^+},c_S],\qquad[c_S,d_S],\qquad[d_S,a_{B^-}].
\]
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 $\alpha_S:[0,1]\to A_S$, 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 \eqref{eq:homotopy-anchor} 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 $\widetilde F:N\to K$ to agree with $F_0$ outside
the collars.  On the homotopy half of a collar, adjacent
to its inner end and given by $1/2\le s\le1$, put
\[
 \widetilde F(x,s)=H_B(x,2-2s).
\]
For the remaining halves of the paired collars of $S$, put
\begin{equation}
\label{eq:arc-collar-map}
 \begin{aligned}
 \widetilde F(x,s)&=\alpha_S(\tfrac12-s)
       &&\text{on }\mathcal C_{B^+},\quad 0\le s\le\tfrac12,\\
 \widetilde F(x,s)&=\alpha_S(\tfrac12+s)
       &&\text{on }\mathcal C_{B^-},\quad 0\le s\le\tfrac12.
 \end{aligned}
\end{equation}
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 $q:N\to M$ is the quotient that identifies
these copies, $\widetilde F$ descends to a continuous map
\begin{equation}
\label{eq:final-graph-map}
 f:M\longrightarrow K,\qquad f\circ q=\widetilde F.
\end{equation}

\subsection{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
\eqref{eq:initial-fiber-anchor}.  Otherwise it lies in some
collar $\mathcal C_B$.  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
$v\in L_B$.  By \eqref{eq:homotopy-anchor}, choose
$z\in B\times\{1\}$ with $\phi_v(z)>0$, hence
$z\in U_v$.  Equation~\eqref{eq:paired-collar-diameter}
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 \eqref{eq:cover-diameter} that
\begin{equation}
\label{eq:whole-fiber-bound}
 \diam_g f^{-1}(y)\le440+2L
 =440+2(2\pi+2)<500.
\end{equation}
If instead $y\in K\setminus K_0$, it belongs to the
open part $A_S\setminus K_0$ of exactly one added path.  Its entire
preimage is contained in the images of the two collars
of $S$, so \eqref{eq:paired-collar-diameter} gives the
stronger bound $L$.  This also covers the new vertices
and the common value $m_S$.  Empty fibers require no
estimate.  Thus \eqref{eq:whole-fiber-bound} controls
whole fibers, without any connectedness assumption.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
For the normalized spectral inequality with $\lambda=1$,
the preceding construction supplies the required graph
and map with constant $500$.  For general $\lambda>0$,
replace $g$ by $g'=\lambda g$.  Scalar curvature and
the Laplacian are multiplied by $\lambda^{-1}$, so the
spectral inequality for $g'$ has lower bound one.
Moreover $\dist_{g'}=\sqrt\lambda\,\dist_g$.
Applying the normalized conclusion and rescaling
\eqref{eq:whole-fiber-bound} gives
\[
 \diam_g f^{-1}(y)\le500\lambda^{-1/2}
 \qquad(y\in K),
\]
as claimed.
\end{proof}
==== END Spectral-scalar-curvature-and-Urysohn-width-in-dimension-three//sections/06-graph-gluing.tex ====