# The Half-Volume Spectrum of a Manifold

Liam Mazurowski  and  Xin Zhou  Cornell University, Department of Mathematics,
Ithaca, New York 14850<EMAIL_ADDRESS>Cornell University, Department of
Mathematics, Ithaca, New York 14850<EMAIL_ADDRESS>
###### Abstract.

We define the half-volume spectrum
$\\{{\tilde{\omega}_{p}\\}_{p\in\mathbb{N}}}$ of a closed manifold
$(M^{n+1},g)$. This is analogous to the usual volume spectrum of $M$, except
that we restrict to $p$-sweepouts whose slices each enclose half the volume of
$M$. We prove that the Weyl law continues to hold for the half-volume
spectrum. We define an analogous half-volume spectrum $\tilde{c}(p)$ in the
phase transition setting. Moreover, for $3\leq n+1\leq 7$, we use the Allen-
Cahn min-max theory to show that each $\tilde{c}(p)$ is achieved by a constant
mean curvature surface enclosing half the volume of $M$ plus a (possibly
empty) collection of minimal surfaces with even multiplicities.

##  1\. Introduction

The spectrum of the Laplacian is an important invariant of a closed Riemannian
manifold $(M^{n+1},g)$. A number $\lambda$ is called an eigenvalue of the
Laplacian provided there is a function $u\colon M\to\mathbb{R}$ such that
$\Delta u+\lambda u=0$. It is well-known that the eigenvalues form a discrete
sequence $0=\lambda_{0}<\lambda_{1}\leq\lambda_{2}\leq\ldots$ and
$\lambda_{p}\to\infty$ as $p\to\infty$. In fact, the eigenvalues of Laplacian
are characterized by the min-max formula

$\lambda_{p}=\inf_{(p+1)\text{-planes }P\subset W^{1,2}(M)}\left[\sup_{u\in
P\setminus\\{0\\}}\frac{\int_{M}|\nabla u|^{2}}{\int_{M}u^{2}}\right],$

and they satisfy the Weyl law

$\lambda_{p}\sim{4\pi^{2}}\operatorname{Vol}(B)^{-\frac{2}{n+1}}\operatorname{Vol}(M)^{-\frac{2}{n+1}}p^{\frac{2}{n+1}}$

as $p\to\infty$. Here $B$ is the unit ball in $\mathbb{R}^{n+1}$.

In [11], Gromov proposed a non-linear analog of the spectrum of the Laplacian.
Roughly speaking, he defines a $p$-sweepout of $M$ to be a family $X$ of
hypersurfaces with the following property: given any $p$ points in $M$, there
is a hypersurface $\Sigma$ belonging to the family $X$ which passes through
all $p$ of these points. Then he defines the $p$-widths

$\omega_{p}=\inf_{p\text{-sweepouts }X}\left[\sup_{\Sigma\in
X}\operatorname{Area}(\Sigma)\right].$

See Section 2 for precise definitions. The sequence
$\\{\omega_{p}\\}_{p\in\mathbb{N}}$ is called the volume spectrum of $M$.

Gromov [12] and Guth [14] proved that the volume spectrum satisfies sublinear
growth bounds. Namely, there are constants $C_{1}$ and $C_{2}$ depending on
$M$ such that

$C_{1}p^{\frac{1}{n+1}}\leq\omega_{p}\leq C_{2}p^{\frac{1}{n+1}}.$

Later, Liokumovich, Marques, and Neves [17] showed that the volume spectrum
satisfies a Weyl law. That is, there is a universal constant $a_{n}$ depending
only on the dimension such that

$\omega_{p}\sim a_{n}\operatorname{Vol}(M)^{\frac{n}{n+1}}p^{\frac{1}{n+1}}$

as $p\to\infty$; see Chodosh-Mantoulidis [6] for the calculation of $a_{n}$
when $n=2$. The Weyl law for the volume spectrum has been instrumental in the
proof of many results on the existence of minimal surfaces in Riemannian
manifolds.

In the early 1980s, Almgren [2], Pitts [20], and Schoen-Simon [21] developed a
min-max theory for the area functional on closed Riemannian manifolds. Their
combined work implies that every closed Riemannian manifold of dimension
$3\leq n+1\leq 7$, contains a closed, smooth, embedded minimal surface. Around
the same time, Yau [24] conjectured that every closed manifold should contain
infinitely many minimal surfaces. Marques and Neves devised a program to prove
Yau’s conjecture by using the Almgren-Pitts min-max theory to find a minimal
surface with area $\omega_{p}$ for each $p\in\mathbb{N}$.

This program has now been successfully carried out. Fix a closed Riemannian
manifold $(M^{n+1},g)$ with $3\leq n+1\leq 7$. Irie, Marques, and Neves [16]
showed that, for a generic metric $g$, the union of all minimal surfaces in
$M$ is dense in $M$. In particular, this proved Yau’s conjecture for generic
metrics. Later Marques, Neves, and Song [19] refined this result to show that,
for a generic metric $g$, there is a sequence of minimal surfaces in $M$ which
becomes equidistributed in $M$. The Weyl law for the volume spectrum was a key
ingredient in the proof of both of these results. Following the second named
author’s proof of the Multiplicity One Conjecture [25], Marques and Neves [18]
showed that, for a generic metric $g$, there is a sequence of minimal surfaces
$\Sigma_{p}$ with index $p$ and $\operatorname{Area}(\Sigma_{p})=\omega_{p}$.
Song [22] proved Yau’s conjecture for arbitrary metrics $g$.

The Almgren-Pitts min-max theory relies heavily on tools from geometric
measure theory. There is a parallel min-max theory for finding minimal
surfaces based on the theory of phase transitions. This theory relies on the
Allen-Cahn PDE and the varifold regularity theory of Wickramasekera. Gaspar
and Guaraco [9] defined a phase transition spectrum
$\\{c(p)\\}_{p\in\mathbb{N}}$ associated to a Riemannian manifold via the
Allen-Cahn PDE. They showed that each $c(p)$ is achieved by a collection of
minimal surfaces with multiplicities. Chodosh and Mantoulidis [5] proved the
Multiplicity One Conjecture in the phase transition setting in ambient
dimension three. Thus, for generic metrics on $M^{3}$, they obtained the
existence of a sequence of minimal surfaces $\Sigma_{p}$ with index $p$ and
$\operatorname{Area}(\Sigma_{p})=c(p)$.

Gaspar and Guaraco [10] showed that the phase transition spectrum also
satisfies a Weyl law. That this, there is a constant $\tau_{n}$ depending only
on the dimension such that

$c(p)\sim\tau_{n}\operatorname{Vol}(M)^{\frac{n}{n+1}}p^{\frac{1}{n+1}}.$

Dey [7] proved that actually $\omega_{p}=c(p)$ for all $p\in\mathbb{N}$ and
thus the Almgren-Pitts volume spectrum and the phase transition volume
spectrum coincide. In particular, the constants $a_{n}$ and $\tau_{n}$
appearing in the two Weyl laws are equal.

In this paper, we define a “half-volume” spectrum associated to a Riemannian
manifold. In the Almgren-Pitts setting, we restrict to $p$-sweepouts by
families of hypersurfaces that each enclose half the volume of $M$. Then we
define

$\tilde{\omega}_{p}=\inf_{\text{half-volume $p$-sweepouts
}X}\left[\sup_{\Sigma\in X}\operatorname{Area}(\Sigma)\right].$

The sequence $\\{\tilde{\omega}_{p}\\}_{p\in\mathbb{N}}$ is called the half-
volume spectrum of $M$. In the phase transition setting, we define an
analogous half-volume spectrum $\tilde{c}(p)$ by looking at critical points of
the Allen-Cahn energy subject to the volume constraint $\int_{M}u=0$. In both
cases, we show that the Weyl law continues to hold. This gives the following
theorems.

######  Theorem 1.

The Almgren-Pitts half-volume spectrum satisfies

$\tilde{\omega}_{p}\sim
a_{n}\operatorname{Vol}(M)^{\frac{n}{n+1}}p^{\frac{1}{n+1}},\quad\text{as
}p\to\infty.$

######  Theorem 2.

The phase transition half-volume spectrum satisfies

$\tilde{c}(p)\sim\tau_{n}\operatorname{Vol}(M)^{\frac{n}{n+1}}p^{\frac{1}{n+1}},\quad\text{as
}p\to\infty.$

In the Allen-Cahn setting, we are able to use the results of Bellettini and
Wickramasekera [3] to find varifolds achieving each $\tilde{c}(p)$. In the
following theorem, a hypersurface $\Sigma$ is called almost-embedded if near
each point in $M$ either $\Sigma$ is embedded or $\Sigma$ decomposes into an
ordered union of embedded sheets.

######  Theorem 3.

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\leq n+1\leq 7$. Fix
a number $p\in\mathbb{N}$. There are

  * (i)

a Caccioppoli set $\Omega\subset M$ with
$\operatorname{Vol}(\Omega)=\frac{1}{2}\operatorname{Vol}(M)$ whose boundary
is smooth and almost-embedded with constant mean curvature,

  * (ii)

a (possibly empty) collection of smooth, disjoint minimal surfaces
$\Sigma_{1},\ldots,\Sigma_{k}\subset M\setminus\Omega$,

  * (iii)

and positive integers $\theta_{0}\in\mathbb{Z}$ and
$\theta_{1},\ldots,\theta_{k}\in 2\mathbb{Z}$

such that
$\tilde{c}(p)=\theta_{0}\operatorname{Area}(\partial\Omega)+\theta_{1}\operatorname{Area}(\Sigma_{1})+\ldots+\theta_{k}\operatorname{Area}(\Sigma_{k}).$
Moreover, $\theta_{0}=1$ unless $\partial\Omega$ is also a minimal surface.

Note that Theorem 3 produces a constant mean curvature surface that encloses
half the volume of $M$. Previously, the second author and Zhu [26] developed a
min-max theory in the Almgren-Pitts setting capable of finding surfaces of
constant mean curvature $c$. However, there is no control over the volume
enclosed by the surface. Likewise, Bellettini and Wickramasekera [3] developed
a min-max theory in the Allen-Cahn setting capable of finding surfaces of
constant mean curvature $c$. Again there is no control over the volume
enclosed by the surface. Thus there is a trade off. Theorem 3 produces
constant mean curvature surfaces enclosing half the volume of $M$, but at the
expense of losing control over the exact value of the mean curvature.

We conclude the introduction with some open problems. First, we conjecture
that $\tilde{\omega}_{p}=\tilde{c}(p)$ for all $p\in\mathbb{N}$. Second, we
conjecture that, for a generic metric $g$, the phase transition half-volume
spectrum is achieved by multiplicity one constant mean curvature surfaces
enclosing half the volume of $M$. In other words, in Theorem 3 the collection
$\Sigma_{1},\ldots,\Sigma_{k}$ is empty and $\theta_{0}=1$ for every $p$. In
particular, we conjecture that generically there are infinitely many constant
mean curvature surfaces enclosing half the volume of $M$. Finally, it is
interesting to know whether one can find surfaces achieving
$\tilde{\omega}_{p}$ by applying the Almgren-Pitts min-max theory with a
volume constraint. This seems to be a difficult task. Already it is not
obvious how to define a suitable pull-tight on the space of half-volume
cycles.

### Acknowledgement

X.Z. was supported by NSF grant DMS-2243149, and an Alfred P. Sloan Research
Fellowship.

##  2\. The Almgren-Pitts Half-Volume Spectrum

In this section, we investigate the topology of the space of half-volume
cycles in a given manifold, and then define the Almgren-Pitts half-volume
spectrum. Let $(M^{n+1},g)$ be a closed Riemannian manifold. We will use the
following notation.

  * •

Let $\mathfrak{h}=\frac{1}{2}\operatorname{Vol}M$.

  * •

Let $\mathcal{C}(M)$ denote the collection of all Caccioppoli sets in $M$.

  * •

Let $\mathcal{C}_{\mathfrak{h}}(M)$, $\mathcal{C}_{\geq\mathfrak{h}}(M)$, and
$\mathcal{C}_{\leq\mathfrak{h}}(M)$ denote the space of Caccioppoli sets with
volume equal to $\mathfrak{h}$, greater than or equal to $\mathfrak{h}$, and
less than or equal to $\mathfrak{h}$, respectively.

  * •

Let $\mathcal{Z}(M,\mathbb{Z}_{2})$ denote the set of all $n$-dimensional flat
chains mod 2 in $M$.

  * •

Let $\mathcal{B}(M,\mathbb{Z}_{2})$ denote the set of all
$T\in\mathcal{Z}(M,\mathbb{Z}_{2})$ such that $T=\partial\Omega$ for some
$\Omega\in\mathcal{C}(M)$. This is the connected component of the empty set in
$\mathcal{Z}(M,\mathbb{Z}_{2})$ in the flat topology.

  * •

Let $\mathcal{H}(M,\mathbb{Z}_{2})$ be the set of all
$T\in\mathcal{Z}(M,\mathbb{Z}_{2})$ such that $T=\partial\Omega$ for some
$\Omega\in\mathcal{C}_{\mathfrak{h}}(M)$. This is the space of “half-volume
cycles.”

  * •

We use $\mathcal{F}$ to denote the flat topology, $\mathbf{F}$ to denote the
$\mathbf{F}$-topology, and $\mathbf{M}$ to denote the mass topology. All
spaces are assumed to be equipped with the flat topology except where
otherwise noted.

  * •

We will abuse notation and write $\operatorname{Vol}(\Omega)$ and
$\operatorname{Area}(T)$ instead of $\mathbf{M}(\Omega)$ and $\mathbf{M}(T)$
for $\Omega\in\mathcal{C}(M)$ and $T\in\mathcal{Z}(M,\mathbb{Z}_{2})$,
respectively.

We will show that $\mathcal{H}(M,\mathbb{Z}_{2})$ is weakly homotopy
equivalent to ${\mathbb{R}\text{P}}^{\infty}$. The first step is to show that
the double cover $\mathcal{C}_{\mathfrak{h}}(M)$ is contractible.

######  Proposition 4.

The space $\mathcal{C}_{\leq\mathfrak{h}}(M)$ deformation retracts to
$\mathcal{C}_{\mathfrak{h}}(M)$.

###### Proof.

The union of two Caccioppoli sets is a Caccioppoli set. Choose a Morse
function $f\colon M\to\mathbb{R}$. For $s\in[0,\mathfrak{h}]$, let $B_{s}$ be
the sublevel set of $f$ with volume equal to $s$. Note that each $B_{s}$ is a
Caccioppoli set. For each $\Omega\in\mathcal{C}_{\leq\mathfrak{h}}(M)$ and
$t\in[0,1]$, choose a number $s(\Omega,t)$ so that

$\operatorname{Vol}(\Omega\cup
B_{s(\Omega,t)})=\operatorname{Vol}(\Omega)+t(\mathfrak{h}-\operatorname{Vol}(\Omega)).$

Note that there is not necessarily a unique choice for $s(\Omega,t)$, and the
mapping $(\Omega,t)\mapsto s(\Omega,t)$ may not be continuous. Nevertheless,
we claim that the map
$\phi\colon\mathcal{C}_{\leq\mathfrak{h}}(M)\times[0,1]\to\mathcal{C}_{\leq\mathfrak{h}}(M)$
defined by

$\phi(\Omega,t)=\Omega\cup B_{s(\Omega,t)}$

is continuous in the flat topology. Given this claim, $\phi$ is the required
deformation retraction. Indeed, $\phi(\Omega,t)=\Omega$ for all
$\Omega\in\mathcal{C}_{\mathfrak{h}}(M)$ and all $t\in[0,1]$, and moreover,
$\phi(\Omega,1)\in\mathcal{C}_{\mathfrak{h}}(M)$ for all
$\Omega\in\mathcal{C}_{\leq\mathfrak{h}}(M)$.

To see that $\phi$ is continuous, let $\varepsilon,\eta>0$ be small positive
numbers. Assume that $\Omega,\Theta\in\mathcal{C}_{\leq\mathfrak{h}}(M)$
satisfy $\operatorname{Vol}(\Omega\operatorname{\Delta}\Theta)<\varepsilon$
and that $t,r\in[0,1]$ satisfy $|t-r|<\eta$. We need to check that
$\operatorname{Vol}\big{(}\phi(\Omega,t)\operatorname{\Delta}\phi(\Theta,r)\big{)}$
is small. Let $U=B_{s(\Omega,t)}\setminus\Omega$ and let
$V=B_{s(\Theta,r)}\setminus\Theta$. Now observe that

$\displaystyle\operatorname{Vol}\big{(}\phi(\Omega,t)\operatorname{\Delta}\phi(\Theta,r)\big{)}$
$\displaystyle=\operatorname{Vol}\big{(}[\Omega\cup
U]\operatorname{\Delta}[\Theta\cup V]\big{)}$
$\displaystyle=\operatorname{Vol}(\Omega)+\operatorname{Vol}(U)+\operatorname{Vol}(\Theta)+\operatorname{Vol}(V)$
$\displaystyle\quad-2\operatorname{Vol}(\Omega\cap\Theta)-2\operatorname{Vol}(\Omega\cap
V)-2\operatorname{Vol}(U\cap\Theta)-2\operatorname{Vol}(U\cap V).$

Note that
$\operatorname{Vol}(\Omega)+\operatorname{Vol}(\Theta)-2\operatorname{Vol}(\Omega\cap\Theta)=\operatorname{Vol}(\Omega\operatorname{\Delta}\Theta)<\varepsilon$.
We also have $\operatorname{Vol}(\Omega\cap
V)\leq\operatorname{Vol}(\Omega\setminus\Theta)<\varepsilon$ and
$\operatorname{Vol}(U\cap\Theta)\leq\operatorname{Vol}(\Theta\setminus\Omega)<\varepsilon$.

Therefore, it remains to show that
$\operatorname{Vol}(U)+\operatorname{Vol}(V)-2\operatorname{Vol}(U\cap V)$ is
small. Observe that $\operatorname{Vol}(U)-\operatorname{Vol}(U\cap
V)=\operatorname{Vol}(U\setminus V)$ and that
$\operatorname{Vol}(V)-\operatorname{Vol}(U\cap
V)=\operatorname{Vol}(V\setminus U)$. Without loss of generality, we can
assume that $s(\Theta,r)\geq s(\Omega,t)$. In this case, we have

$\displaystyle\operatorname{Vol}(U\setminus V)$
$\displaystyle=\operatorname{Vol}\big{(}[B_{s(\Omega,t)}\setminus\Omega]\setminus[B_{s(\Theta,r)}\setminus\Theta]\big{)}$
$\displaystyle\leq\operatorname{Vol}\big{(}[B_{s(\Theta,r)}\setminus\Omega]\setminus[B_{s(\Theta,r)}\setminus\Theta]\big{)}\leq\operatorname{Vol}(\Theta\setminus\Omega)<\varepsilon.$

We also have

$\displaystyle\operatorname{Vol}(V\setminus U)$
$\displaystyle=\operatorname{Vol}\big{(}[B_{s(\Theta,r)}\setminus\Theta]\setminus[B_{s(\Omega,t)}\setminus\Omega]\big{)}$
$\displaystyle=\operatorname{Vol}\big{(}[B_{s(\Theta,r)}\setminus
B_{s(\Omega,t)}]\setminus\Theta\big{)}+\operatorname{Vol}\big{(}[B_{s(\Omega,t)}\setminus\Theta]\setminus[B_{s(\Omega,t)}\setminus\Omega]\big{)}$
$\displaystyle\leq\operatorname{Vol}\big{(}[B_{s(\Theta,r)}\setminus
B_{s(\Omega,t)}]\setminus\Theta\big{)}+\operatorname{Vol}(\Omega\setminus\Theta)$
$\displaystyle<\operatorname{Vol}\big{(}[B_{s(\Theta,r)}\setminus
B_{s(\Omega,t)}]\setminus\Theta\big{)}+\varepsilon.$

Thus to prove the claim, it remains to show that
$\operatorname{Vol}\big{(}[B_{s(\Theta,r)}\setminus
B_{s(\Omega,t)}]\setminus\Theta\big{)}$ is small. Here we must use the
assumption that $t$ is close to $r$. Notice that

$\displaystyle
r(\mathfrak{h}-\operatorname{Vol}(\Theta))=\operatorname{Vol}(V)=\operatorname{Vol}\big{(}[B_{s(\Theta,r)}\setminus
B_{s(\Omega,t)}]\setminus\Theta\big{)}+\operatorname{Vol}(B_{s(\Omega,t)}\setminus\Theta),$
$\displaystyle
t(\mathfrak{h}-\operatorname{Vol}(\Omega))=\operatorname{Vol}(U)=\operatorname{Vol}(B_{s(\Omega,t)}\setminus\Omega).$

Now observe that

$\displaystyle|r(\mathfrak{h}-\operatorname{Vol}(\Theta))-t(\mathfrak{h}-\operatorname{Vol}(\Omega))|$
$\displaystyle\leq|t-r|\mathfrak{h}+t|\operatorname{Vol}(\Omega)-\operatorname{Vol}(\Theta)|+|t-r|\operatorname{Vol}(\Theta)$
$\displaystyle\leq 2\eta\mathfrak{h}+\varepsilon.$

Also we have

$\displaystyle|\operatorname{Vol}(B_{s(\Omega,t)}\setminus\Theta)-\operatorname{Vol}(B_{s(\Omega,t)}\setminus\Omega)|$
$\displaystyle\leq\operatorname{Vol}\big{(}[B_{s(\Omega,t)}\setminus\Theta]\operatorname{\Delta}[B_{s(\Omega,t)}\setminus\Omega]\big{)}$
$\displaystyle\leq\operatorname{Vol}(\Theta\setminus\Omega)+\operatorname{Vol}(\Omega\setminus\Theta)<\varepsilon.$

This implies that

$\operatorname{Vol}\big{(}[B_{s(\Theta,r)}\setminus
B_{s(\Omega,t)}]\setminus\Theta\big{)}\leq 2\eta\mathfrak{h}+2\varepsilon,$

which completes the proof of the claim. ∎

######  Proposition 5.

The space $\mathcal{C}(M)$ deformation retracts to
$\mathcal{C}_{\mathfrak{h}}(M)$, and the space $\mathcal{B}(M,\mathbb{Z}_{2})$
deformation retracts to $\mathcal{H}(M,\mathbb{Z}_{2})$.

###### Proof.

Consider the deformation retraction
$\phi\colon\mathcal{C}_{\leq\mathfrak{h}}(M)\times[0,1]\to\mathcal{C}_{\leq\mathfrak{h}}(M)$
from the previous proposition. We can extend $\phi$ to an odd map
$\psi\colon\mathcal{C}(M)\times[0,1]\to\mathcal{C}(M)$ by the formula

$\psi(\Omega,t)=\begin{cases}\phi(\Omega,t),&\text{if
}\operatorname{Vol}(\Omega)\leq\mathfrak{h}\\\
M\setminus\phi(M\setminus\Omega,t),&\text{if
}\operatorname{Vol}(\Omega)\geq\mathfrak{h}.\end{cases}$

Then $\psi$ is a deformation retraction of $\mathcal{C}(M)$ onto
$\mathcal{C}_{\mathfrak{h}}(M)$. Moreover, since $\psi$ is odd, this descends
to a map
$\theta\colon\mathcal{B}(M,\mathbb{Z}_{2})\times[0,1]\to\mathcal{B}(M,\mathbb{Z}_{2})$.
This is the required deformation retraction of $\mathcal{B}(M,\mathbb{Z}_{2})$
onto $\mathcal{H}(M,\mathbb{Z}_{2})$. ∎

######  Proposition 6.

Let $K$ be the maximal area of a level set of the Morse function $f$ used in
the proof of Proposition 4. Let $\theta$ be the deformation retraction from
Proposition 5. Then
$\operatorname{Area}(\theta(T,t))\leq\operatorname{Area}(T)+K$ for all
$T\in\mathcal{B}(M,\mathbb{Z}_{2})$ and all $t\in[0,1]$.

###### Proof.

Fix some $T\in\mathcal{B}(M,\mathbb{Z}_{2})$ and some $t\in[0,1]$. Choose a
set $\Omega\in\mathcal{C}_{\leq\mathfrak{h}}(M)$ such that $\partial\Omega=T$.
Let $\phi$ be the deformation retraction from Proposition 4. Then

$\theta(T,t)=\partial\phi(\Omega,t)=\partial(\Omega\cup B_{s(\Omega,t)}).$

Note that $\partial(\Omega\cup
B_{s(\Omega,t)})\subset\partial\Omega\cup\partial B_{s(\Omega,t)}$ and
therefore
$\operatorname{Area}(\theta(T,t))\leq\operatorname{Area}(\partial\Omega)+K=\operatorname{Area}(T)+K$,
as needed ∎

The homotopy groups of the cycle spaces were originally computed by Almgren
[1]. Later, Marques and Neves [18] gave a simplified proof in the case of
codimension 1 cycles.

######  Theorem 7 (Marques-Neves).

The map $\partial\colon\mathcal{C}(M)\to\mathcal{B}(M,\mathbb{Z}_{2})$ is a
double cover. The space $\mathcal{C}(M)$ is contractible, and
$\mathcal{B}(M,\mathbb{Z}_{2})$ is weakly homotopy equivalent to
${\mathbb{R}\text{P}}^{\infty}$.

Combined with the previous propositions, this yields the following corollary.

######  Corollary 8.

The map
$\partial\colon\mathcal{C}_{\mathfrak{h}}(M)\to\mathcal{H}(M,\mathbb{Z}_{2})$
is a double cover. The space $\mathcal{C}_{\mathfrak{h}}(M)$ is contractible
and $\mathcal{H}(M,\mathbb{Z}_{2})$ is weakly homotopy equivalent to
${\mathbb{R}\text{P}}^{\infty}$. The inclusion map
$\mathcal{H}(M,\mathbb{Z}_{2})\to\mathcal{B}(M,\mathbb{Z}_{2})$ is a homotopy
equivalence.

We now recall the notion of sweepouts. Since $\mathcal{B}(M,\mathbb{Z}_{2})$
is weakly homotopy equivalent to ${\mathbb{R}\text{P}}^{\infty}$, it follows
that the cohomology ring of $\mathcal{B}(M,\mathbb{Z}_{2})$ with
$\mathbb{Z}_{2}$ coefficients is $\mathbb{Z}_{2}[\lambda]$, where the
generator $\lambda$ is of degree 1. Let $X$ be a cubical complex.

######  Definition 9.

A flat continuous map $\Phi\colon X\to\mathcal{B}(M,\mathbb{Z}_{2})$ is called
a $p$-sweepout if $\Phi^{*}\lambda^{p}\neq 0$ in $H^{p}(X,\mathbb{Z}_{2})$.

######  Definition 10.

A map $\Phi\colon X\to\mathcal{Z}(M,\mathbb{Z}_{2})$ is said to have no
concentration of mass provided

$\lim_{r\to 0}\left[\sup_{q\in M}\sup_{x\in
X}\operatorname{Area}(\Phi(x)\llcorner B(q,r))\right]=0.$

######  Definition 11.

Let $\mathcal{P}_{p}(M)$ denote the collection of all $p$-sweepouts of $M$
with no concentration of mass. Note that different $p$-sweepouts may have
different domains.

######  Definition 12.

The $p$-width of $M$ is

$\omega_{p}=\inf_{\Phi\in\mathcal{P}_{p}(M)}\left[\sup_{x\in\operatorname{dom}(\Phi)}\operatorname{Area}(\Phi(x))\right].$

######  Remark 13.

In [17], the authors state that the cohomology ring of
$\mathcal{Z}(M,\mathbb{Z}_{2})$ is isomorphic to $\mathbb{Z}_{2}[\lambda]$.
Then they define a $p$-sweepout as a map $\Phi\colon
X\to\mathcal{Z}(M,\mathbb{Z}_{2})$ such that $\Phi^{*}\lambda^{p}\neq 0$.
However, the cohomology ring of $\mathcal{Z}(M,\mathbb{Z}_{2})$ is actually
$\oplus_{i}\mathbb{Z}_{2}[\lambda_{i}]$ where the direct sum is taken over the
connected components of $\mathcal{Z}(M,\mathbb{Z}_{2})$. These connected
components are in bijection with homology classes in
$H_{n}(M,\mathbb{Z}_{2})$. Given this, there are several possible ways to
define a $p$-sweepout. The simplest, which we shall adopt, is to replace the
space $\mathcal{Z}(M,\mathbb{Z}_{2})$ with $\mathcal{B}(M,\mathbb{Z}_{2})$ as
in Definition 9 so that the cohomology ring is indeed
$\mathbb{Z}_{2}[\lambda]$. Alternatively, one could define a $p$-sweepout as a
map $\Phi\colon X\to\mathcal{Z}(M,\mathbb{Z}_{2})$ such that
$\Phi^{*}\lambda_{i}^{p}\neq 0$ for some $i$. In either case, it is
straightforward to see that one still obtains a Weyl law for the resulting
$p$-widths.

We can now introduce the central object of the paper. By Corollary 8, the
cohomology ring of $\mathcal{H}(M,\mathbb{Z}_{2})$ with $\mathbb{Z}_{2}$
coefficients is also $\mathbb{Z}_{2}[\lambda]$. Again let $X$ be a cubical
complex.

######  Definition 14.

A flat continuous map $\Phi\colon X\to\mathcal{H}(M,\mathbb{Z}_{2})$ is called
a half-volume $p$-sweepout if $\Phi^{*}\lambda^{p}\neq 0$ in
$H^{p}(X,\mathbb{Z}_{2})$.

######  Definition 15.

Let $\mathcal{Q}_{p}(M)$ denote the collection of all half-volume
$p$-sweepouts of $M$ with no concentration of mass.

######  Definition 16.

The half-volume $p$-width of $M$ is

$\tilde{\omega}_{p}=\inf_{\Phi\in\mathcal{Q}_{p}(M)}\left[\sup_{x\in\operatorname{dom}(\Phi)}\operatorname{Area}(\Phi(x))\right].$

We will call the sequence $\\{\tilde{\omega}_{p}\\}_{p\in\mathbb{N}}$ the
half-volume spectrum of $M$.

Liokumovich, Marques, and Neves [17] showed that the $p$-widths of $M$ satisfy
a Weyl law.

######  Theorem 17 (Liokumovich, Marques, Neves).

There is a universal constant $a_{n}$ such that $\omega_{p}\sim
a_{n}\operatorname{Vol}(M)^{n/(n+1)}p^{1/(n+1)}$ as $p\to\infty$.

Next, we will show that the half-volume spectrum also satisfies a Weyl law. It
is possible to prove this directly. However, this is not the approach we will
take. Rather, we will show that the Weyl law for the half-volume spectrum
follows from Theorem 17, together with the fact that every $p$-sweepout is
homotopic to a $p$-sweepout by half-volume cycles.

######  Proposition 18.

The half-volume spectrum satisfies $\omega_{p}\leq\tilde{\omega}_{p}$ for all
$p\in\mathbb{N}$.

###### Proof.

Notice that any half-volume $p$-sweepout with no concentration of mass
automatically belongs to ${\mathcal{P}_{p}}(M)$. Therefore, the proposition
follows immediately from the definitions of $\omega_{p}$ and
$\tilde{\omega}_{p}$. ∎

######  Proposition 19.

There is a constant $K$ depending only on $M$ such that
$\tilde{\omega}_{p}\leq\omega_{p}+K+1$ for all $p\in\mathbb{N}$.

###### Proof.

Choose a $p$-sweepout $\Phi\colon X\to\mathcal{B}(M,\mathbb{Z}_{2})$ in
${\mathcal{P}_{p}}(M)$ with

$\sup_{x\in X}\operatorname{Area}(\Phi(x))\leq\omega_{p}+1.$

Let
$\theta\colon\mathcal{B}(M,\mathbb{Z}_{2})\times[0,1]\to\mathcal{B}(M,\mathbb{Z}_{2})$
be the deformation retraction constructed in Proposition 5. By Proposition 6,
there is a constant $K$ such that

$\operatorname{Area}(\theta(T,t))\leq\operatorname{Area}(T)+K$

for all $T\in\mathcal{B}(M,\mathbb{Z}_{2})$ and all $t\in[0,1]$. Therefore,
the map $\Psi\colon X\to\mathcal{H}(M,\mathbb{Z}_{2})$ given by
$\Psi(x)=\theta(\Phi(x),1)$ is a half-volume $p$-sweepout with

$\sup_{x\in X}\operatorname{Area}(\Psi(x))\leq\sup_{x\in
X}\operatorname{Area}(\Phi(x))+K.$

Moreover, it is straightforward to check that $\Psi$ has no concentration of
mass. This proves that $\tilde{\omega}_{p}\leq\omega_{p}+K+1$. ∎

We are now able to prove Theorem 1.

######  Theorem 1.

The Weyl law holds for the half-volume spectrum. In other words, we have
$\tilde{\omega}_{p}\sim a_{n}\operatorname{Vol}(M)^{n/(n+1)}p^{1/(n+1)}$ as
$p\to\infty$.

###### Proof.

This follows from Proposition 18 and Proposition 19. Indeed, we have

$\omega_{p}\leq\tilde{\omega}_{p}\leq\omega_{p}+K+1.$

Theorem 17 implies that

$\lim_{p\to\infty}\frac{\omega_{p}}{a_{n}\operatorname{Vol}(M)^{n/(n+1)}p^{1/(n+1)}}=1\quad\text{and}\quad\lim_{p\to\infty}\frac{\omega_{p}+K+1}{a_{n}\operatorname{Vol}(M)^{n/(n+1)}p^{1/(n+1)}}=1,$

and it follows that

$\lim_{p\to\infty}\frac{\tilde{\omega}_{p}}{a_{n}\operatorname{Vol}(M)^{n/(n+1)}p^{1/(n+1)}}=1$

as well. ∎

##  3\. The Phase Transition Half-Volume Spectrum

There is also an analogous half-volume spectrum in the Allen-Cahn setting. Let
$W\colon\mathbb{R}\to\mathbb{R}$ be an even double-well potential. This means
that

  * (i)

$W$ is smooth and non-negative,

  * (ii)

$W(x)=W(-x)$ for all $x\in\mathbb{R}$,

  * (iii)

$W$ has non-degenerate minima $W(\pm 1)=0$,

  * (iv)

$W$ has a non-degenerate maximum $W(0)>0$,

  * (v)

$W$ is increasing on $(-1,0)$ and $(1,\infty)$ and decreasing on $(0,1)$ and
$(-\infty,-1)$,

  * (vi)

there are constants $\kappa>0$ and $\alpha\in(0,1)$ such that
$W^{\prime\prime}(x)\geq\kappa$ for all $|x|\geq\alpha$.

Define the constant

$\sigma=\int_{-1}^{1}\sqrt{W(s)/2}\,ds.$

Let $u\colon M\to\mathbb{R}$ be an $W^{1,2}$ function. For $\varepsilon>0$
define the Allen-Cahn energy

$E_{\varepsilon}(u)=\int_{M}\frac{\varepsilon}{2}|\nabla
u|^{2}+\frac{W(u)}{\varepsilon}.$

In [9], Gaspar and Guaraco define a phase-transition spectrum associated to
$M$ via the Allen-Cahn energy.

In order to state the definition of the spectrum, we shall need some further
background. A paracompact topological space $X$ is called a
$\mathbb{Z}_{2}$-space if it admits a free $\mathbb{Z}_{2}$-action. Given such
a space, there is always a quotient space $T=X/\mathbb{Z}_{2}$ and the natural
map $X\to T$ is a principal $\mathbb{Z}_{2}$-bundle. Any such bundle arises as
a pullback of the universal bundle
$S^{\infty}\to{\mathbb{R}\text{P}}^{\infty}$. More precisely, there is a
classifying map $f\colon T\to{\mathbb{R}\text{P}}^{\infty}$ such that $X\to T$
is the pullback of $S^{\infty}\to{\mathbb{R}\text{P}}^{\infty}$ via $f$. The
Alexander-Spanier cohomology ring of ${\mathbb{R}\text{P}}^{\infty}$ with
$\mathbb{Z}_{2}$ coefficients is $\mathbb{Z}_{2}[\mu]$ where the generator
$\mu$ is in degree one. The map $f$ is unique up to homotopy, and therefore
the cohomology classes $f^{*}\mu^{p}$ are well-defined in the Alexander-
Spanier cohomology ring $H^{*}(T,\mathbb{Z}_{2})$. The $\mathbb{Z}_{2}$-index
of $X$ is defined to be the largest $p$ such that $f^{*}\mu^{p-1}\neq 0$ in
$H^{*}(T,\mathbb{Z}_{2})$. A subspace $A$ of $X$ is called invariant if it is
closed under the $\mathbb{Z}_{2}$-action.

The $\mathbb{Z}_{2}$-index enjoys the following useful properties. See Fadell
and Rabinowitz [8] for more details.

  * (i)

(Monotonicity) If $X_{1}$ and $X_{2}$ are $\mathbb{Z}_{2}$-spaces and there is
a continuous equivariant map $X_{1}\to X_{2}$ then
$\operatorname{ind}_{\mathbb{Z}_{2}}(X_{1})\leq\operatorname{ind}_{\mathbb{Z}_{2}}(X_{2})$.

  * (ii)

(Subadditivity) If $X$ is a $\mathbb{Z}_{2}$-space and $A_{1}$ and $A_{2}$ are
closed, invariant subsets with $A_{1}\cup A_{2}=X$ then
$\operatorname{ind}_{\mathbb{Z}_{2}}(X)\leq\operatorname{ind}_{\mathbb{Z}_{2}}(A_{1})+\operatorname{ind}_{\mathbb{Z}_{2}}(A_{2})$.

  * (iii)

(Continuity) If $X$ is a $\mathbb{Z}_{2}$-space and $A$ is a closed, invariant
subset of $X$ then there is an invariant neighborhood $V$ of $A$ in $X$ such
that
$\operatorname{ind}_{\mathbb{Z}_{2}}(A)=\operatorname{ind}_{\mathbb{Z}_{2}}(\overline{V})$.

The space $W^{1,2}(M)\setminus\\{0\\}$ is paracompact since it is a metric
space. Moreover, it admits a natural $\mathbb{Z}_{2}$ action $u\mapsto-u$.
Note that $E_{\varepsilon}$ respects this action since
$E_{\varepsilon}(u)=E_{\varepsilon}(-u)$. This uses the fact that $W$ is even.
A set $A\subset W^{1,2}(M)\setminus\\{0\\}$ is called invariant provided $u\in
A$ if and only if $-u\in A$. The $\mathbb{Z}_{2}$-action on
$W^{1,2}(M)\setminus\\{0\\}$ descends to any such $A$. Define the families

$\mathcal{F}_{p}=\\{A\subset W^{1,2}(M)\setminus\\{0\\}:\ A\text{ is compact
and invariant with }\operatorname{ind}_{\mathbb{Z}_{2}}(A)\geq p+1\\}.$

Gaspar and Guaraco define the min-max values

$c(\varepsilon,p)=\frac{1}{2\sigma}\inf_{A\in\mathcal{F}_{p}}\left[\sup_{u\in
A}E_{\varepsilon}(u)\right].$

Then they set $c(p)=\liminf_{\varepsilon\to 0}c(\varepsilon,p)$. The sequence
$\\{c(p)\\}_{p\in\mathbb{N}}$ is the phase transition spectrum of $M$.

Gaspar and Guaraco [10] showed that the Weyl law also holds for the phase
transition spectrum.

######  Theorem 20 (Gaspar and Guaraco).

There is a universal constant $\tau_{n}$ such that
$c(p)\sim\tau_{n}\operatorname{Vol}(M)^{n/(n+1)}p^{1/(n+1)}$ as $p\to\infty$.

Dey [7] proved that $\omega_{p}=c(p)$ for all $p\in\mathbb{N}$. In particular,
this implies that the constant $\tau_{n}$ is equal to the constant $a_{n}$.

######  Remark 21.

Gaspar and Guaraco do not include the normalization constant
$\frac{1}{2\sigma}$ in the definition of $c(\varepsilon,p)$ and $c(p)$. We
have chosen to include it so that one has $\omega_{p}=c(p)$.

It is also possible to define a half-volume spectrum in the phase transition
setting. Define

$Y=\\{u\in W^{1,2}(M):\int_{M}u=0\\}.$

Note that $Y$ is a closed subspace of $W^{1,2}(M)$ and so $Y$ is also a
Hilbert space. We can run essentially the same construction using $Y$ in place
of $W^{1,2}(M)$. For each $p\in\mathbb{N}$, define

$\mathcal{G}_{p}=\\{A\subset Y\setminus\\{0\\}:\,A\text{ is compact and
invariant with }\operatorname{ind}_{\mathbb{Z}_{2}}(A)\geq p+1\\},$

and then set

$\tilde{c}(\varepsilon,p)=\frac{1}{2\sigma}\inf_{A\in\mathcal{G}_{p}}\left[\sup_{u\in
A}E_{\varepsilon}(u)\right].$

Taking the limit as $\varepsilon\to 0$ gives the phase-transition half volume
spectrum.

######  Definition 22.

For each $p\in\mathbb{N}$, let $\tilde{c}(p)=\liminf_{\varepsilon\to
0}\tilde{c}(\varepsilon,p)$. The phase transition half volume spectrum of $M$
is the sequence $\\{\tilde{c}(p)\\}_{p\in\mathbb{N}}$.

######  Proposition 23.

The phase transition half-volume spectrum satisfies $c(p)\leq\tilde{c}(p)$ for
all $p\in\mathbb{N}$.

###### Proof.

Note that $\mathcal{G}_{p}\subset\mathcal{F}_{p}$ for every $p\in\mathbb{N}$.
Therefore, for every $\varepsilon>0$, it holds that
$c(\varepsilon,p)\leq\tilde{c}(\varepsilon,p)$. The result then follows by
sending $\varepsilon\to 0$. ∎

######  Proposition 24.

The phase transition half-volume spectrum satisfies $\tilde{c}(p)\leq c(p+1)$
for all $p\in\mathbb{N}$.

###### Proof.

Fix an $\varepsilon>0$. Select a set $A\in\mathcal{F}_{p+1}$ with

$\sup_{u\in A}E_{\varepsilon}(u)\leq
2\sigma\left[c(\varepsilon,p+1)+\varepsilon\right].$

Define the set $B=\\{u\in A:\int_{M}u=0\\}$ and note that $B$ is closed and
invariant. We claim that $\operatorname{ind}_{\mathbb{Z}_{2}}(B)\geq p+1$ so
that $B\in\mathcal{G}_{p}$. Given this, we obtain that
$\tilde{c}(\varepsilon,p)\leq c(\varepsilon,p+1)+\varepsilon$, and the result
follows upon sending $\varepsilon\to 0$.

It remains to prove the claim. By the continuity of the index, there is a
neighborhood $V$ of $B$ in $A$ such that
$\operatorname{ind}_{\mathbb{Z}_{2}}(B)=\operatorname{ind}_{\mathbb{Z}_{2}}(\overline{V})$.
There is an $\eta>0$ such that

$\\{u\in A:-\eta<\int_{M}u<\eta\\}\subset V.$

Indeed, if not, then there is a sequence $u_{k}$ in $A\setminus V$ with
$\int_{M}u_{k}\to 0$. Since $A\setminus V$ is compact, we can find a
subsequence $u_{k_{j}}$ that converges to a limit $u$ in $A\setminus V$. But
$u$ satisfies $\int_{M}u=0$ and therefore $u\in B\subset V$ and this is a
contradiction. Therefore, such an $\eta$ exists.

Let $K=\\{u\in A:\,\left|\int_{M}u\right|\geq\frac{\eta}{2}\\}$. Then $K$ is a
closed invariant subset of $A$ and $K\cup\overline{V}=A$. Define a map $K\to
S^{0}$ by sending $u$ to $1$ if $\int_{M}u>0$ and sending $u$ to $-1$ if
$\int_{M}u<0$. This map is continuous and equivariant and so by the
monotonicity of the index we have

$\operatorname{ind}_{\mathbb{Z}_{2}}(K)\leq\operatorname{ind}_{\mathbb{Z}_{2}}(S^{0})=1.$

Hence by the subadditivity of the index, we get

$p+2\leq\operatorname{ind}_{\mathbb{Z}_{2}}(A)\leq\operatorname{ind}_{\mathbb{Z}_{2}}(K)+\operatorname{ind}_{\mathbb{Z}_{2}}(\overline{V})\leq\operatorname{ind}_{\mathbb{Z}_{2}}(\overline{V})+1.$

This implies that
$\operatorname{ind}_{\mathbb{Z}_{2}}(B)=\operatorname{ind}_{\mathbb{Z}_{2}}(\overline{V})\geq
p+1$, and so $B\in\mathcal{G}_{p}$ as needed. ∎

We can now prove Theorem 2.

######  Theorem 2.

The phase transition half-volume spectrum satisfies the Weyl law. In other
words, we have
$\tilde{c}(p)\sim\tau_{n}\operatorname{Vol}(M)^{n/(n+1)}p^{1/(n+1)}$ as
$p\to\infty$.

###### Proof.

By Propositions 23 and 24 we have

$c(p)\leq\tilde{c}(p)\leq c(p+1).$

By the Weyl law for the phase-transition spectrum, we have

$\lim_{p\to\infty}\frac{c(p)}{\tau_{n}\operatorname{Vol}(M)^{n/(n+1)}p^{1/(n+1)}}=1\quad\text{
and
}\quad\lim_{p\to\infty}\frac{c(p+1)}{\tau_{n}\operatorname{Vol}(M)^{n/(n+1)}p^{1/(n+1)}}=1$

and therefore

$\lim_{p\to\infty}\frac{\tilde{c}(p)}{\tau_{n}\operatorname{Vol}(M)^{n/(n+1)}p^{1/(n+1)}}=1$

as well. ∎

##  4\. Surfaces Associated to the Half-Volume Spectrum

In this section, we use the Allen-Cahn min-max theory to construct surfaces
associated to the phase transition half-volume spectrum. The goal is to prove
Theorem 3. Fix a closed Riemannian manifold $(M^{n+1},g)$ with $3\leq n+1\leq
7$. Fix a number $p\in\mathbb{N}$. In this section, we require the following
additional hypothesis on the double-well potential $W$.

  * (vii)

There are constants $0<C_{1}<C_{2}$ and $\beta>1$ and $2<q<\frac{11}{5}$ such
that

$C_{1}|x|^{q}\leq W(x)\leq C_{2}|x|^{q}\quad\text{and}\quad
C_{1}|x|^{q-1}\leq|W^{\prime}(x)|\leq C_{2}|x|^{q-1}$

for all $|x|\geq\beta$.

The first step of the proof is to construct, for each small enough
$\varepsilon>0$, a critical point $u_{\varepsilon}$ of $E_{\varepsilon}$
subject to the volume constraint

$\int_{M}u_{\varepsilon}=0.$

Given such a $u_{\varepsilon}$, there is a Lagrange multiplier
$\lambda_{\varepsilon}\in\mathbb{R}$ such that $u_{\varepsilon}$ is a critical
point of

$F_{\varepsilon,\lambda_{\varepsilon}}(v)=E_{\varepsilon}(v)+\lambda_{\varepsilon}\int_{M}v$

on all of $W^{1,2}$. The construction of $u_{\varepsilon}$ is similar to that
of Gaspar and Guaraco [9] in the unconstrained case.

######  Remark 25.

There are two purposes for imposing the growth condition (vii). The first is
that it allows us to verify the Palais-Smale condition with the volume
constraint. In the unconstrained case, one has

$E_{\varepsilon}(\max(\min(u,1),-1))\leq E_{\varepsilon}(u)$

and so by a truncation argument it is enough to verify the Palais-Smale
condition along Palais-Smale sequences which are bounded in $L^{\infty}$. See
[9] for more details. However, truncation may not preserve the volume
constraint. In the volume constrained case, we instead rely on (vii) to show
that $W^{\prime}(u)\in L^{2}$ whenever $u\in W^{1,2}$. The second purpose is
to get uniform $L^{\infty}$ bounds on critical points of
$F_{\varepsilon,\lambda}$. Given a sequence $\varepsilon_{k}\to 0$ and
critical points $u_{\varepsilon_{k}}$ of
$F_{\varepsilon_{k},\lambda_{\varepsilon_{k}}}$ with
$E_{\varepsilon_{k}}(u_{\varepsilon_{k}})\leq C$, the growth condition (vii)
implies that $\|u_{\varepsilon_{k}}\|_{L^{\infty}}\leq\beta$ provided $k$ is
large enough.

We recall (see Proposition 4.4 in [13]) that the first variation of
$E_{\varepsilon}$ is given by

$DE_{\varepsilon}(u)(v)=\int_{M}\frac{\varepsilon}{2}\nabla u\cdot\nabla
v+\frac{W^{\prime}(u)}{\varepsilon}v.$

Fix a number $\varepsilon>0$. A sequence $A_{k}$ in $\mathcal{G}_{p}$ is
called a critical sequence if

$\lim_{k\to\infty}\left[\sup_{u\in
A_{k}}E_{\varepsilon}(u)\right]=2\sigma\tilde{c}(\varepsilon,p).$

A sequence $u_{k}\in A_{k}$ is called a min-max sequence provided
$\lim_{k\to\infty}E_{\varepsilon}(u_{k})=2\sigma\tilde{c}(\varepsilon,p).$

In the unconstrained case, it is not necessarily true that every min-max
sequence is bounded in $W^{1,2}$. However, one can obtain the existence of a
bounded min-max sequence via a truncation argument. See, for example, the
remarks before Proposition 4.5 in [13]. We cannot employ truncation because it
doesn’t preserve the volume constraint. Fortunately, in the volume constrained
case, every min-max sequence is automatically bounded in $W^{1,2}$.

######  Proposition 26.

Any min-max sequence $u_{k}$ is uniformly bounded in $W^{1,2}(M)$.

###### Proof.

Assume that $u\in Y$ satisfies $E_{\varepsilon}(u)\leq K$. Since $W\geq 0$, it
follows immediately that

$\int_{M}|\nabla u|^{2}\leq\frac{2K}{\varepsilon}.$

Since $u$ has average $0$, the Poincare inequality implies that
$\|u\|_{W^{1,2}}\leq CK/\varepsilon$. This proves the result. ∎

######  Proposition 27.

Assume that $u_{k}$ is a sequence uniformly bounded in $W^{1,2}$. Then
$W^{\prime}(u_{k})$ is uniformly bounded in $L^{2}$.

###### Proof.

By assumption the sequence $u_{k}$ is uniformly bounded in $W^{1,2}$. As
$3\leq n+1\leq 7$, the Sobolev embedding theorem implies that $u_{k}$ is
uniformly bounded in $L^{12/5}$. Now $|W^{\prime}(u_{k})|\leq
C|u_{k}|^{q-1}\leq C|u_{k}|^{6/5}$ whenever $|u_{k}|\geq\beta$. Therefore

$\displaystyle\int_{M}W^{\prime}(u_{k})^{2}$
$\displaystyle=\int_{|u_{k}|\leq\beta}W^{\prime}(u_{k})^{2}+\int_{|u_{k}|>\beta}W^{\prime}(u_{k})^{2}$
$\displaystyle\leq C\operatorname{Vol}(M)+C\int_{M}|u_{k}|^{12/5},$

and it follows that $W^{\prime}(u_{k})$ is uniformly bounded in $L^{2}$. ∎

The functional $E_{\varepsilon}|_{Y}$ satisfies the Palais-Smale condition.
See [13] Proposition 4.4 for the proof without a volume constraint.

######  Proposition 28.

The functional $E_{\varepsilon}|_{Y}$ satisfies the Palais-Smale condition.
More precisely, assume that $u_{k}$ is a bounded sequence in $Y$ and that
$\|DE_{\varepsilon}|_{Y}(u_{k})\|\to 0$. Then a subsequence of $u_{k}$
converges strongly to a limit $u\in Y$.

###### Proof.

Assume that $u_{k}$ is a bounded sequence in $Y$ such that
$\|DE_{\varepsilon}|_{Y}(u_{k})\|\to 0$. We need to show that some subsequence
of $u_{k}$ converges strongly to a limit $u\in Y$. Note that $Y$ is closed and
convex in $W^{1,2}(M)$ and so $Y$ is weakly closed. Thus, passing to a
subsequence, we can assume that $u_{k}$ converges weakly in $W^{1,2}$ and
strongly in $L^{12/5}$ to a point $u\in Y$.

Observe that

$\displaystyle DE_{\varepsilon}|_{Y}(u)(u_{k}-u)=\int_{M}\varepsilon\nabla
u\cdot\nabla(u_{k}-u)+\int_{M}\frac{W^{\prime}(u)}{\varepsilon}(u_{k}-u).$

The first term on the right hand side goes to 0 by the weak convergence
$u_{k}\rightharpoonup u$. Note that $W^{\prime}(u)\in L^{2}$ since $u\in
L^{12/5}$. Therefore the second term on the right hand side also goes to 0
since $u_{k}\to u$ in $L^{2}$. Thus we obtain

$DE_{\varepsilon}|_{Y}(u)(u_{k}-u)\to 0,\quad\text{as }k\to\infty.$

Also note that $DE_{\varepsilon}|_{Y}(u_{k})(u_{k}-u)\to 0$ since
$\|DE_{\varepsilon}|_{Y}(u_{k})\|\to 0$ and $u_{k}-u$ is uniformly bounded in
$W^{1,2}$. On the other hand,

$DE_{\varepsilon}|_{Y}(u_{k})(u_{k}-u)=\int_{M}\varepsilon\nabla
u_{k}\cdot\nabla(u_{k}-u)+\int_{M}\frac{W^{\prime}(u_{k})}{\varepsilon}(u_{k}-u).$

The second term on the right hand side goes to 0 as $W^{\prime}(u_{k})$ is
uniformly bounded in $L^{2}$ and $u_{k}-u\to 0$ in $L^{2}$.

Now observe that

$\displaystyle
DE_{\varepsilon}|_{Y}(u_{k})(u_{k}-u)-DE_{\varepsilon}|_{Y}(u)(u_{k}-u)$
$\displaystyle\qquad=\int_{M}\varepsilon|\nabla u_{k}-\nabla
u|^{2}+\int_{M}\frac{W^{\prime}(u_{k})}{\varepsilon}(u_{k}-u)-\int_{M}\frac{W^{\prime}(u)}{\varepsilon}(u_{k}-u).$

We have already seen that every term in this formula goes to 0 except
$\int_{M}\varepsilon|\nabla u_{k}-\nabla u|^{2}$, and therefore
$\int_{M}\varepsilon|\nabla u_{k}-\nabla u|^{2}$ goes to 0 as well. This
proves that $u_{k}\to u$ strongly in $W^{1,2}$, as needed. ∎

According to Gaspar and Guaraco [9], for each given $p$, we have $2\sigma
c(\varepsilon,p+1)<E_{\varepsilon}(0)$ provided $\varepsilon$ is small enough.
Therefore, we also have $2\sigma\tilde{c}(\varepsilon,p)<E_{\varepsilon}(0)$
provided $\varepsilon$ is small enough. Hence, for $\varepsilon$ small enough,
any min-max sequence remains bounded away from 0. By the classical theory for
functionals satisfying the Palais-Smale condition (see [23]), we get the
following existence result for critical points of $E_{\varepsilon}|_{Y}$. See
Theorem 3.3 in [9] for the corresponding result in the unconstrained case.

######  Proposition 29.

Fix $p\in\mathbb{N}$. For all small enough $\varepsilon$, there is a critical
point $u_{\varepsilon}\in Y$ of $E_{\varepsilon}|_{Y}$ with
$E_{\varepsilon}(u_{\varepsilon})=\tilde{c}(\varepsilon,p)$. There is a number
$\lambda_{\varepsilon}\in\mathbb{R}$ such that $u_{\varepsilon}$ is a critical
point of

$v\mapsto
F_{\varepsilon,\lambda_{\varepsilon}}(v)=E_{\varepsilon}(v)+\lambda_{\varepsilon}\int_{M}v$

on all of $W^{1,2}$, and $u_{\varepsilon}$ satisfies the PDE

$-\varepsilon\Delta
u_{\varepsilon}+\frac{W^{\prime}(u_{\varepsilon})}{\varepsilon}=\lambda_{\varepsilon}$

in the weak sense. Moreover, we have $\int_{M}u_{\varepsilon}=0$. The index of
$u_{\varepsilon}$ as a critical point of $E_{\varepsilon}|_{Y}$ is at most
$p$.

Given the existence of $u_{\varepsilon}$, the second step in the proof is to
study the convergence of $u_{\varepsilon}$ as $\varepsilon\to 0$. Fortunately
for us, Bellettini and Wickramasekera [3] have already studied the regularity
of such limits. Let us recall the setup in [3]. For each $\varepsilon>0$,
suppose $u_{\varepsilon}$ is a critical point of
$F_{\varepsilon,\lambda_{\varepsilon}}$ and that
$E_{\varepsilon}(u_{\varepsilon})$ and
$\operatorname{ind}_{F_{\varepsilon,\lambda_{\varepsilon}}}(u_{\varepsilon})$
and $\|u_{\varepsilon_{k}}\|_{L^{\infty}}$ are uniformly bounded. Choose a
sequence $\varepsilon_{j}\to 0$ and assume that
$\lambda_{\varepsilon_{j}}\to\lambda$. Passing to a subsequence if necessary,
there exist a radon measure $\mu$ on $M$ and a function $u_{\infty}\in BV(M)$
such that

$\frac{1}{2\sigma}\left(\frac{\varepsilon_{j}}{2}|\nabla
u_{\varepsilon_{j}}|^{2}+\frac{W(u_{\varepsilon_{j}})}{\varepsilon_{j}}\right)\rightharpoonup\mu$

and $u_{\varepsilon_{j}}\to u_{\infty}$ in $L^{1}$. Moreover, $u_{\infty}$
takes only the values $\pm 1$. Hutchinson and Tonegawa [15] proved that there
is an integral varifold $V$ on $M$ such that $\|V\|=\mu$. The following is the
special case of Theorem 4.1 in Bellettini and Wickramasekera [3] where the
prescription functions are assumed to be constants and the ambient dimension
is assumed to be between 3 and 7.

######  Theorem 30 (See Theorem 4.1 in [3]).

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\leq n+1\leq 7$.
Assume $(u_{\varepsilon_{j}})$ is a sequence as above, and assume that
$\lambda>0$. Let $\Omega=\operatorname{int}(\\{x\in M:\,u_{\infty}(x)=1\\})$.
Then $V=V_{0}+V_{\lambda}$ where

  * (i)

$V_{0}$ is induced by a collection of smooth, disjoint minimal surfaces
equipped with even multiplicities. Moreover,
$\operatorname{spt}(\|V_{0}\|)\subset M\setminus\Omega$.

  * (ii)

If $\Omega=\emptyset$ then $V_{\lambda}=0$. If $\Omega\neq\emptyset$ then
$V_{\lambda}=|\partial^{\star}\Omega|\neq 0$, and moreover, $V_{\lambda}$ is
induced by a smooth surface with constant mean curvature $\lambda$ whose mean
curvature vector points into $\Omega$.

The minimal surfaces may have tangential intersection with the CMC surface.
Likewise, the CMC surface may have tangential intersections with itself but it
never crosses itself.

We can now complete the proof of Theorem 3.

######  Theorem 3.

Let $(M^{n+1},g)$ be a closed Riemannian manifold with $3\leq n+1\leq 7$. Fix
a number $p\in\mathbb{N}$. There are

  * (i)

a Caccioppoli set $\Omega\subset M$ with
$\operatorname{Vol}(\Omega)=\frac{1}{2}\operatorname{Vol}(M)$ whose boundary
is smooth and almost-embedded with constant mean curvature,

  * (ii)

a (possibly empty) collection of smooth, disjoint minimal surfaces
$\Sigma_{1},\ldots,\Sigma_{k}\subset M\setminus\Omega$,

  * (iii)

and positive integers $\theta_{0}\in\mathbb{Z}$ and
$\theta_{1},\ldots,\theta_{k}\in 2\mathbb{Z}$

such that
$\tilde{c}(p)=\theta_{0}\operatorname{Area}(\partial\Omega)+\theta_{1}\operatorname{Area}(\Sigma_{1})+\ldots+\theta_{k}\operatorname{Area}(\Sigma_{k}).$
Moreover, $\theta_{0}=1$ unless $\partial\Omega$ is also a minimal surface.

###### Proof.

Choose a sequence $\varepsilon_{k}\to 0$ so that
$\tilde{c}(\varepsilon_{k},p)\to\tilde{c}(p)$. Let $u_{\varepsilon_{k}}$ be
the critical points of $F_{\varepsilon_{k},\lambda_{\varepsilon_{k}}}$
constructed in Proposition 29. Then $E_{\varepsilon_{k}}(u_{\varepsilon_{k}})$
and
$\operatorname{ind}_{F_{\varepsilon_{k},\lambda_{\varepsilon_{k}}}}(u_{\varepsilon_{k}})$
are uniformly bounded. According to Hutchinson and Tonegawa [15] Section 6.1
and Lemma 3.4 in [4], the Lagrange multipliers $\lambda_{\varepsilon_{k}}$ are
also uniformly bounded and there is a constant $K>0$ such that
$\|u_{\varepsilon_{k}}\|_{L^{\infty}}\leq K$ for all $k$. For the interested
reader, we include the details of the argument in the appendix. Applying
Theorem 30 to the sequence $u_{\varepsilon_{k}}$ now yields the result. ∎

##  5\. Appendix

The goal of the appendix is to prove the following proposition. We largely
follow the sketch in [15] section 6.1, giving details as appropriate. The
proof that the Lagrange multipliers are bounded depends on work of Chen [4].

######  Proposition 31.

Assume that the potential $W$ satisfies the growth condition (vii). Choose a
sequence $\varepsilon_{k}\to 0$ and let $u_{\varepsilon_{k}}$ be a critical
point of $F_{\varepsilon_{k},\lambda_{\varepsilon_{k}}}$. Assume that
$E_{\varepsilon_{k}}(u_{\varepsilon_{k}})$ is uniformly bounded. Then the
Lagrange multipliers $\lambda_{\varepsilon_{k}}$ are uniformly bounded and
$\|u_{\varepsilon_{k}}\|_{L^{\infty}}$ is also uniformly bounded.

###### Proof.

The first step is to check that each $u_{\varepsilon_{k}}$ is smooth. Recall
that $|W^{\prime}(u_{\varepsilon_{k}})|\leq C|u_{\varepsilon_{k}}|^{q-1}$ for
$|u_{\varepsilon_{k}}|\geq\beta$. For simplicity, we will give the argument
assuming $3\leq n+1\leq 5$. In this case, by the Sobolev embedding theorem,
$u_{\varepsilon_{k}}$ belongs to $L^{10/3}$. It follows that
$W^{\prime}(u_{\varepsilon_{k}})\in L^{10/(3q-3)}$. Hence by elliptic
regularity we have $u_{\varepsilon_{k}}\in W^{2,q_{1}}$ for $q_{1}=10/(3q-3)$.
Note that

$\frac{n+1}{q_{1}}\leq\frac{3}{2}(q-1)\leq\frac{18}{10}<2.$

Thus by the Sobolev inequalities we obtain that $u_{\varepsilon_{k}}$ is
Hölder continuous. Standard elliptic regularity then implies that
$u_{\varepsilon_{k}}$ is smooth. The cases $6\leq n+1\leq 7$ are handled
similarly. One applies the Sobolev inequalities together with elliptic
regularity several times. Each application improves the regularity of
$u_{\varepsilon_{k}}$ until eventually one obtains $u_{\varepsilon_{k}}\in
W^{2,q_{1}}$ with $2>(n+1)/q$. This gives Hölder continuity of
$u_{\varepsilon_{k}}$, and standard elliptic regularity then implies that
$u_{\varepsilon_{k}}$ is smooth. Note at this point we do not have uniform
$L^{\infty}$ estimates on $u_{\varepsilon_{k}}$.

To prove that the Lagrange multipliers are bounded we follow [4] Lemma 3.4.
Note that [4] Lemma 3.4 is proved for domains in Euclidean space. Some
addition difficulties arise in adapting the mollifier arguments used in [4] to
the case of a closed manifold. We need to prove a sequence of lemmas. In what
follows, $C$ denotes a positive constant that is allowed to change from line
to line.

######  Lemma 32.

We have

$\int_{M}(|u_{\varepsilon_{k}}|-1)^{2}\to 0,\quad\text{as }k\to\infty.$

###### Proof.

Define the sets

$\displaystyle
A_{1}=\\{||u_{\varepsilon_{k}}|-1|\leq\varepsilon_{k}^{1/4}\\},$
$\displaystyle
A_{2}=\\{\varepsilon_{k}^{1/4}\leq||u_{\varepsilon_{k}}|-1|\leq\beta-1\\},$
$\displaystyle A_{3}=\\{|u_{\varepsilon_{k}}|\geq\beta\\}.$

We will estimate the integral over the sets $A_{1}$, $A_{2}$, and $A_{3}$
separately. For $A_{1}$, we have

$\int_{A_{1}}(|u_{\varepsilon_{k}}|-1)^{2}\leq\varepsilon_{k}^{1/2}\operatorname{Vol}(M).$

Regarding $A_{2}$, property (iii) of $W$ imply that there is a constant $c>0$
independent of $k$ such that $|W(x)|\geq c\varepsilon_{k}^{1/2}$ whenever
$||x|-1|\geq\varepsilon_{k}^{1/4}$. Therefore the set $A_{2}$ has measure at
most ${\varepsilon_{k}^{1/2}}{c^{-1}}E_{\varepsilon_{k}}(u_{\varepsilon_{k}})$
and so

$\int_{A_{2}}(|u_{\varepsilon_{k}}|-1)^{2}\leq(\beta-1)^{2}c^{-1}\varepsilon_{k}^{1/2}E_{\varepsilon_{k}}(u_{\varepsilon_{k}}).$

It remains to estimate the integral over $A_{3}$. Without loss of generality
we can assume that $\beta\geq 5$. Remember that $C|W(x)|\geq|x|^{q}$ for
$|x|\geq\beta$ and so

$(|u_{\varepsilon_{k}}|-1)^{2}\leq|u_{\varepsilon_{k}}|^{2}\leq|u_{\varepsilon_{k}}|^{q}\leq
C|W(u_{\varepsilon_{k}})|$

whenever $|u_{\varepsilon_{k}}|\geq\beta$. Thus we have the estimate

$\int_{A_{3}}(|u_{\varepsilon_{k}}|-1)^{2}\leq\int_{A_{3}}C|W(u_{\varepsilon_{k}})|\leq
C\varepsilon_{k}E_{\varepsilon_{k}}(u_{\varepsilon_{k}}).$

Combining these three estimates shows that

$\int_{M}(|u_{\varepsilon_{k}}|-1)^{2}\leq
C\varepsilon_{k}^{1/2}(1+E_{\varepsilon_{k}}(u_{\varepsilon_{k}})).$

The right hand side goes to 0 as $k\to\infty$. ∎

Define

$\Phi(s)=\int_{0}^{s}\sqrt{W(s)/2}\,ds.$

Let $w_{\varepsilon_{k}}=\Phi\circ u_{\varepsilon_{k}}$. It is easy to check
that $\nabla w_{\varepsilon_{k}}$ is uniformly bounded in $L^{1}$ (see [15]).

######  Lemma 33.

There is a constant $C>0$ such that $|x-y|^{2}\leq C|\Phi(x)-\Phi(y)|$ for all
$x,y\in\mathbb{R}$.

###### Proof.

We check a number of cases. First suppose that $x,y\geq\beta$ and assume
without loss of generality that $x\geq y$. Then

$\displaystyle|\Phi(x)-\Phi(y)|$
$\displaystyle=\int_{y}^{x}\sqrt{W(s)/2}\,ds\geq\frac{1}{\sqrt{2}}\int_{y}^{x}s^{q/2}\,ds$
$\displaystyle\geq\frac{1}{\sqrt{2}}\int_{y}^{x}s\,ds=\frac{1}{2\sqrt{2}}(x^{2}-y^{2})\geq\frac{1}{2\sqrt{2}}|x-y|^{2},$

where the last inequality uses the fact that $x\geq y\geq 0$ and so $x+y\geq
x-y$. The same argument works in the case where $x,y\leq-\beta$.

Now suppose that $x\geq\beta$ and $y\leq-\beta$. Let
$a=\int_{-\beta}^{\beta}\sqrt{W(s)/2}\,ds>0$. Then

$\displaystyle|\Phi(x)-\Phi(y)|$
$\displaystyle=\int_{\beta}^{x}\sqrt{W(s)/2}\,ds+a+\int_{y}^{-\beta}\sqrt{W(s)/2}\,ds$
$\displaystyle\geq\frac{1}{\sqrt{2}}\int_{\beta}^{x}s\,ds+a+\frac{1}{\sqrt{2}}\int_{y}^{-\beta}(-s)\,ds$
$\displaystyle\geq\frac{1}{2\sqrt{2}}(x^{2}-\beta^{2})+a+\frac{1}{2\sqrt{2}}(y^{2}-\beta^{2}).$

We have

$\displaystyle|x-y|^{2}\leq 2(x^{2}+y^{2})\leq
4\beta^{2}+2(x^{2}-\beta^{2})+2(y^{2}-\beta^{2})\leq C|\Phi(x)-\Phi(y)|.$

The same argument works if $x\leq-\beta$ and $y\geq\beta$.

It remains to handle the case when $-\beta\leq x,y\leq\beta$. Assume for
contradiction there are two sequences $x_{j},y_{j}\in[-\beta,\beta]$ such that

(1) $|x_{j}-y_{j}|^{2}>j|\Phi(x_{j})-\Phi(y_{j})|.$

Passing to a subsequence if necessary, we can assume that $x_{j}\to
x\in[-\beta,\beta]$ and $y_{j}\to y\in[-\beta,\beta]$. It is clear from (1)
that $x=y$. Passing to the limit in

$|x_{j}-y_{j}|\geq j\frac{|\Phi(x_{j})-\Phi(y_{j})|}{|x_{j}-y_{j}|}$

we obtain that $\Phi^{\prime}(x)=0$. Thus $W(x)=0$ and so $x=\pm 1$. Without
loss of generality assume that $x=1$. Note that there is a constant $C>0$ such
that $W(s)\geq C|s-1|^{2}$ for $s$ close to 1. Thus

$\displaystyle|\Phi(x_{j})-\Phi(y_{j})|=\left|\int_{y_{j}}^{x_{j}}\sqrt{W(s)/2}\,ds\right|\geq
C\left|\int_{y_{j}}^{x_{j}}|s-1|\,ds\right|.$

If $x_{j}\geq y_{j}\geq 1$ then

$\displaystyle\left|\int_{y_{j}}^{x_{j}}|s-1|\,ds\right|$
$\displaystyle=\int_{y_{j}}^{x_{j}}s-1\,ds=\frac{1}{2}\left[(x_{j}-1)^{2}-(y_{j}-1)^{2}\right]$
$\displaystyle=\frac{1}{2}\left[(x_{j}+y_{j}-2)(x_{j}-y_{j})\right]\geq\frac{1}{2}(x_{j}-y_{j})^{2},$

which combined with the previous equation yields a contradiction. If
$x_{j}\geq 1\geq y_{j}$ then

$\displaystyle\left|\int_{y_{j}}^{x_{j}}|s-1|\,ds\right|$
$\displaystyle=\int_{y_{j}}^{1}1-s\,ds+\int_{1}^{x_{j}}s-1\,ds$
$\displaystyle=\frac{1}{2}\left[(1-y_{j})^{2}+(x_{j}-1)^{2}\right]\geq\frac{1}{4}(x_{j}-y_{j})^{2}$

since either $x_{j}-1\geq\frac{1}{2}(x_{j}-y_{j})$ or
$1-y_{j}\geq\frac{1}{2}(x_{j}-y_{j})$. Again this gives a contradiction. The
remaining possibilities likewise lead to contradiction and the lemma is
proved. ∎

For each $\eta\in(0,1)$, let $u_{\varepsilon_{k},\eta}$ be a mollified version
of $u_{\varepsilon_{k}}$. More precisely, choose an isometric embedding of $M$
into $\mathbb{R}^{m}$. Let $N$ be a small tubular neighborhood of $M$ where
the nearest point projection $\Pi\colon N\to M$ is well-defined and a
submersion. Let $B_{r}(x)$ denote the open ball of radius $r$ centered at
$x\in\mathbb{R}^{m}$. Also let $B_{1}$ denote the open unit ball centered at
the origin in $\mathbb{R}^{m}$. Choose a non-negative smooth function
$\rho\colon B_{1}\to\mathbb{R}$ which is compactly supported in $B_{1}$ and
satisfies

$\int_{B_{1}}\rho(y)\,d\mathcal{L}^{m}(y)=1.$

Extend $u_{\varepsilon_{k}}$ to a function $v_{\varepsilon_{k}}$ on $N$ by
setting $v_{\varepsilon_{k}}=u_{\varepsilon_{k}}\circ\Pi$. Then let

$u_{\varepsilon_{k},\eta}(x)=\int_{B_{1}}\rho(y)v_{\varepsilon_{k}}(x-\eta
y)\,d\mathcal{L}^{m}(y),\quad x\in M$

be the mollified version of $u_{\varepsilon_{k}}$.

######  Lemma 34.

For each fixed $\eta$, there is a uniform bound
$\|u_{\varepsilon_{k},\eta}\|_{L^{\infty}(M)}\leq C(\eta)$.

###### Proof.

Observe that

$\displaystyle|u_{\varepsilon_{k},\eta}(x)|$
$\displaystyle\leq\int_{B_{1}}\rho(y)|v_{\varepsilon_{k}}(x-\eta
y)|\,d\mathcal{L}^{m}(y)$ $\displaystyle\leq
1+\int_{B_{1}}\rho(y)\big{|}|v_{\varepsilon_{k}}(x-\eta
y)|-1\big{|}\,d\mathcal{L}^{m}(y)$ $\displaystyle\leq
1+C\eta^{-(n+1)}\int_{B_{\eta}(x)}\big{|}|v_{\varepsilon_{k}}(z)|-1\big{|}\,d\mathcal{L}^{m}(z).$

By the co-area formula we have

$\displaystyle\int_{B_{\eta}(x)}\big{|}|v_{\varepsilon_{k}}(z)|-1\big{|}\,d\mathcal{L}^{m}(z)$
$\displaystyle\qquad\leq
C\int_{B_{\eta}(x)}\big{|}|v_{\varepsilon_{k}}(z)|-1\big{|}\,J\Pi(z)\,d\mathcal{L}^{m}(z)$
$\displaystyle\qquad\leq
C\int_{\Pi(B_{\eta}(x))}\int_{p\in\Pi^{-1}(q)}\big{|}|v_{\varepsilon_{k}}(p)|-1\big{|}\,d\mathcal{H}^{m-n-1}(p)\,d\mathcal{H}^{n+1}(q)$
$\displaystyle\qquad\leq
C\int_{\Pi(B_{\eta}(x))}\big{|}|u_{\varepsilon_{k}}(q)|-1\big{|}\,d\mathcal{H}^{n+1}(q).$

Inserting this into the previous equation and using Lemma 32 gives the result.
∎

######  Lemma 35.

For each fixed $\eta$, there is a uniform bound $\|\nabla
u_{\varepsilon_{k},\eta}\|_{L^{\infty}(M)}\leq C(\eta)$.

###### Proof.

Fix an index $i\in\\{1,\ldots,m\\}$. Note the formula for
$u_{\varepsilon_{k},\eta}(x)$ also makes sense for $x\in N$ so we can regard
$u_{\varepsilon_{k},\eta}$ as a function defined on $N$. For $x\in M$ we have

$\displaystyle\partial_{i}u_{\varepsilon_{k},\eta}(x)$
$\displaystyle=\int_{B_{1}}\partial_{i}\rho(y)v_{\varepsilon_{k}}(x-\eta
y)\,d\mathcal{L}^{m}(y).$

Thus we have

$\displaystyle|\partial_{i}u_{\varepsilon_{k},\eta}(x)|$
$\displaystyle\leq\int_{B_{1}}|\partial_{i}\rho(y)||v_{\varepsilon_{k}}(x-\eta
y)|\,d\mathcal{L}^{m}(y)$ $\displaystyle\leq
C+\int_{B_{1}}|\partial_{i}\rho(y)|(|v_{\varepsilon_{k}}(x-\eta
y)|-1)\,d\mathcal{L}^{m}(y)$ $\displaystyle\leq
C+C\eta^{-(n+1)}\int_{B_{\eta(x)}}\big{|}|v_{\varepsilon_{k}}(z)|-1\big{|}\,d\mathcal{L}^{m}(z).$

Using the coarea formula as in the proof of the previous lemma now gives the
result. ∎

######  Lemma 36.

There is a uniform bound
$\|u_{\varepsilon_{k},\eta}-u_{\varepsilon_{k}}\|^{2}_{L^{2}(M)}\leq C\eta$
for $k\geq K(\eta)$.

###### Proof.

Observe that

$\displaystyle\int_{M}|u_{\varepsilon_{k},\eta}-u_{\varepsilon_{k}}|^{2}$
$\displaystyle=\int_{M}\left|\int_{B_{1}}\rho(y)v_{\varepsilon_{k}}(x-\eta
y)\,d\mathcal{L}^{m}(y)-u_{\varepsilon_{k}}(x)\right|^{2}\,d\mathcal{H}^{n+1}(x)$
$\displaystyle\leq\int_{M}\int_{B_{1}}\rho(y)|v_{\varepsilon_{k}}(x-\eta
y)-u_{\varepsilon_{k}}(x)|^{2}\,d\mathcal{L}^{m}(y)\,d\mathcal{H}^{n+1}(x)$
$\displaystyle\leq C\int_{M}\int_{B_{1}}\rho(y)|f(x-\eta
y)-f(x)|\,d\mathcal{L}^{m}(y)\,d\mathcal{H}^{n+1}(x),$

where $f=\Phi\circ u_{\varepsilon_{k}}\circ\Pi=w_{\varepsilon_{k}}\circ\Pi$
and we used Lemma 33 to get the last inequality. By Fubini’s theorem we get

$\displaystyle\int_{M}\int_{B_{1}}\rho(y)|f(x-\eta
y)-f(x)|\,d\mathcal{L}^{m}(y)\,d\mathcal{H}^{n+1}(x)$
$\displaystyle\qquad\leq\eta\int_{M}\int_{B_{1}}\int_{0}^{1}\rho(y)|\nabla
f(x-t\eta y)|\,\mathcal{\,}dt\,L^{m}(y)\,d\mathcal{H}^{n+1}(x)$
$\displaystyle\qquad=\eta\int_{B_{1}}\rho(y)\int_{0}^{1}\int_{M}|\nabla
f(x-t\eta y)|\,d\mathcal{H}^{n+1}(x)\,dt\,d\mathcal{L}^{m}(y).$

Now for fixed $y$ and $t$, we have

$\int_{M}|\nabla f(x-t\eta y)|\,d\mathcal{H}^{n+1}(x)=\int_{M-t\eta y}|\nabla
f(z)|\,d\mathcal{H}^{n+1}(z).$

Provided $\eta$ is small enough, the map $\Pi\colon M-t\eta y\to M$ is a
diffeomorphism and so by the change of variables forumla

$\displaystyle\int_{M-t\eta y}|\nabla f(z)|\,d\mathcal{H}^{n+1}(z)$
$\displaystyle\leq\int_{M-t\eta y}|\nabla
w_{\varepsilon_{k}}(\Pi(z))|\,d\mathcal{H}^{n+1}(z)$ $\displaystyle\leq
C\int_{M-t\eta y}|\nabla
w_{\varepsilon_{k}}(\Pi(z))|J\Pi(z)\,d\mathcal{H}^{n+1}(z)$
$\displaystyle=C\int_{M}|\nabla
w_{\varepsilon_{k}}(q)|\,d\mathcal{H}^{n+1}(q).$

Thus we obtain

$\displaystyle\eta\int_{B_{1}}\rho(y)\int_{0}^{1}\int_{M}|\nabla f(x-t\eta
y)|\,d\mathcal{H}^{n+1}(x)\,dt\,d\mathcal{L}^{m}(y)$ $\displaystyle\qquad\leq
C\eta\int_{B_{1}}\rho(y)\|w_{\varepsilon_{k}}\|_{L^{1}(M)}\,d\mathcal{L}^{m}(y)\leq
C\eta.$

Putting everything together we get

$\int_{M}|u_{\varepsilon_{k},\eta}-u_{\varepsilon_{k}}|^{2}\leq C\eta,$

and the result follows. ∎

######  Lemma 37.

Let $\overline{u}_{\varepsilon_{k},\eta}$ be the average of
$u_{\varepsilon_{k},\eta}$. There is a uniform bound
$|\overline{u}_{\varepsilon_{k},\eta}|\leq C\eta^{\frac{1}{2}}$ for $k\geq
K(\eta)$.

###### Proof.

Observe that

$\displaystyle\overline{u}_{\varepsilon_{k},\eta}$
$\displaystyle=\frac{1}{\operatorname{Vol}(M)}\int_{M}u_{{\varepsilon_{k}},\eta}\,=\frac{1}{\operatorname{Vol}(M)}\int_{M}(u_{\varepsilon_{k},\eta}-u_{\varepsilon_{k}}).$

Now the result follows from Lemma 36 and Hölder’s inequality. ∎

Now fix an $\eta$ to be specified later. Given a large integer $k$, let $\psi$
be the solution to

$-\Delta\psi=u_{\varepsilon_{k},\eta}-\overline{u}_{\varepsilon_{k},\eta},\quad\int_{M}\psi=0.$

By Lemmas 34, 35, and 37 the right hand side of the above PDE is uniformly
bounded in $C^{1}$. Therefore by elliptic regularity, $\psi$ is bounded in
$C^{2}$ by a constant that depends on $\eta$ but not on $k$. Note that
$u_{\varepsilon_{k}}$ satisfies the PDE

$-{\varepsilon_{k}}\Delta
u_{\varepsilon_{k}}+\frac{W^{\prime}(u_{\varepsilon_{k}})}{\varepsilon_{k}}=\lambda_{\varepsilon_{k}}.$

Multiplying by $\nabla\psi\cdot\nabla u_{\varepsilon_{k}}$ and integrating
yields

(2) $\lambda_{\varepsilon_{k}}\int_{M}\nabla\psi\cdot\nabla
u_{\varepsilon_{k}}=\int_{M}\nabla\psi\cdot\nabla
u_{\varepsilon_{k}}\left(-{\varepsilon_{k}}\Delta
u_{\varepsilon_{k}}+\frac{W^{\prime}(u_{\varepsilon_{k}})}{\varepsilon_{k}}\right).$

Observe that

$\int_{M}\nabla\psi\cdot\nabla
u_{\varepsilon_{k}}\frac{W^{\prime}(u_{\varepsilon_{k}})}{\varepsilon_{k}}=\int_{M}\nabla\psi\cdot\nabla\left(\frac{W(u_{\varepsilon_{k}})}{\varepsilon_{k}}\right)=-\int_{M}\frac{W(u_{\varepsilon_{k}})}{\varepsilon_{k}}\Delta\psi.$

Also, by the integration by parts formula for the Hessian, we have

$\displaystyle\varepsilon_{k}\int_{M}D^{2}\psi(\nabla
u_{\varepsilon_{k}},\nabla
u_{\varepsilon_{k}})=-\varepsilon_{k}\int_{M}\nabla\psi\cdot\nabla
u_{\varepsilon_{k}}\Delta
u_{\varepsilon_{k}}+\frac{\varepsilon_{k}}{2}\int_{M}|\nabla
u_{\varepsilon_{k}}|^{2}\Delta\psi.$

Thus we have the following formula for the right hand side of (2):

$\displaystyle\int_{M}\nabla\psi\cdot\nabla
u_{\varepsilon_{k}}\left(-{\varepsilon_{k}}\Delta
u_{\varepsilon_{k}}+\frac{W^{\prime}(u_{\varepsilon_{k}})}{\varepsilon_{k}}\right)$
$\displaystyle\qquad=\varepsilon_{k}\int_{M}D^{2}\psi(\nabla
u_{\varepsilon_{k}},\nabla
u_{\varepsilon_{k}})-\int_{M}\left(\frac{\varepsilon_{k}}{2}|\nabla
u_{\varepsilon_{k}}|^{2}+\frac{W(u_{\varepsilon_{k}})}{\varepsilon_{k}}\right)\Delta\psi.$

Since $\|\psi\|_{C^{2}}\leq C(\eta)$, this gives a bound

$\left|\int_{M}\nabla\psi\cdot\nabla
u_{\varepsilon_{k}}\left(-{\varepsilon_{k}}\Delta
u_{\varepsilon_{k}}+\frac{W^{\prime}(u_{\varepsilon_{k}})}{\varepsilon_{k}}\right)\right|\leq
C(\eta)E_{\varepsilon_{k}}(u_{\varepsilon_{k}}).$

We now turn attention to the left hand side of (2). Integrating by parts, we
have

$\lambda_{\varepsilon_{k}}\int_{M}\nabla\psi\cdot\nabla
u_{\varepsilon_{k}}=-\lambda_{\varepsilon_{k}}\int_{M}u_{\varepsilon_{k}}\Delta\psi=\lambda_{\varepsilon_{k}}\int_{M}u_{\varepsilon_{k}}(u_{\varepsilon_{k},\eta}-\overline{u}_{\varepsilon_{k},\eta}).$

Now observe that

$\displaystyle\int_{M}u_{\varepsilon_{k}}(u_{\varepsilon_{k},\eta}-\overline{u}_{\varepsilon_{k},\eta})=\int_{M}u_{\varepsilon_{k}}(u_{\varepsilon_{k},\eta}-u_{\varepsilon_{k}})+\int_{M}(u_{\varepsilon_{k}}^{2}-1)-\overline{u}_{\varepsilon_{k},\eta}\int_{M}u_{\varepsilon_{k}}+\operatorname{Vol}(M).$

By Lemmas 32 and 37 and Hölder’s inequality, we can select $\eta$ small enough
that

$\left|\int_{M}u_{\varepsilon_{k}}(u_{\varepsilon_{k},\eta}-u_{\varepsilon_{k}})\right|\leq\frac{1}{4}\operatorname{Vol}(M),\quad\left|\overline{u}_{\varepsilon_{k},\eta}\int_{M}u_{\varepsilon_{k}}\right|\leq\frac{1}{4}\operatorname{Vol}(M).$

By Lemma 32 and Hölder’s inequality, we have

$\left|\int_{M}(u_{\varepsilon_{k}}^{2}-1)\right|\leq\int_{M}||u_{\varepsilon_{k}}|-1|(|u_{\varepsilon_{k}}|+1)\leq\frac{1}{4}\operatorname{Vol}(M)$

for $k$ large enough. It follows that

$\int_{M}\nabla\psi\cdot\nabla
u_{\varepsilon_{k}}\geq\frac{1}{4}\operatorname{Vol}(M)$

for large enough $k$. Using equation (2) then gives an upper bound on
$\lambda_{\varepsilon_{k}}$ which is independent of $k$.

To complete the proof of the proposition, it remains to show that
$\|u_{\varepsilon_{k}}\|_{L^{\infty}}$ is uniformly bounded. Let
$M_{\varepsilon_{k}}=M/\varepsilon_{k}$. Define the rescaled functions
$f_{\varepsilon_{k}}\colon M_{\varepsilon_{k}}\to\mathbb{R}$ by
$f_{\varepsilon_{k}}(x)=u_{\varepsilon_{k}}(\varepsilon_{k}x)$. Then
$f_{\varepsilon_{k}}$ solves

$-\Delta
f_{\varepsilon_{k}}+W^{\prime}(f_{\varepsilon_{k}})=\varepsilon_{k}\lambda_{\varepsilon_{k}}.$

Fix some $p\geq 1$. Multiplying the equation by
$|f_{\varepsilon_{k}}|^{p-1}f_{\varepsilon_{k}}$ and integrating gives

(3) $\int_{M_{\varepsilon_{k}}}p|f_{\varepsilon_{k}}|^{p-1}|\nabla
f_{\varepsilon_{k}}|^{2}+\int_{M_{\varepsilon_{k}}}W^{\prime}(f_{\varepsilon_{k}})|f_{\varepsilon_{k}}|^{p-1}f_{\varepsilon_{k}}=\varepsilon_{k}\lambda_{\varepsilon_{k}}\int_{M_{\varepsilon_{k}}}|f_{\varepsilon_{k}}|^{p-1}f_{\varepsilon_{k}}.$

Now for $|x|\geq\beta$ we have $W^{\prime}(x)|x|^{p-1}x\geq C|x|^{p+q-1}$.
Since $\varepsilon_{k}\lambda_{\varepsilon_{k}}\to 0$, it follows from (3)
that

$\int_{\\{f_{\varepsilon_{k}}\geq\beta\\}}|f_{\varepsilon_{k}}|^{p+q-1}\leq\frac{1}{2}\int_{M_{\varepsilon_{k}}}|f_{\varepsilon_{k}}|^{p}$

assuming $k$ is large enough. This implies that

$\int_{M_{\varepsilon_{k}}}|f_{\varepsilon_{k}}|^{p+q-1}\leq
C\beta^{p+q-1}+\frac{1}{2}\int_{M_{\varepsilon_{k}}}|f_{\varepsilon_{k}}|^{p}.$

By induction, for any positive integer $r$, this gives

$\displaystyle\int_{M_{\varepsilon_{k}}}|f_{\varepsilon_{k}}|^{2+r(q-1)}$
$\displaystyle\leq\frac{C\beta^{2}}{2^{r}}\sum_{j=1}^{r}(2\beta^{q-1})^{j}+2^{-r}\int_{M_{\varepsilon_{k}}}|f_{\varepsilon_{k}}|^{2}$
$\displaystyle=\frac{C\beta^{2}}{2^{r}}\left(\frac{2\beta^{q-1}(2^{r}\beta^{r(q-1)}-1)}{2\beta^{q-1}-1}\right)+2^{-r}\int_{M_{\varepsilon_{k}}}|f_{\varepsilon_{k}}|^{2}$
$\displaystyle\leq
C\beta^{2+r(q-1)}+2^{-r}\int_{M_{\varepsilon_{k}}}|f_{\varepsilon_{k}}|^{2}.$

Raising both sides to the power $(2+r(q-1))^{-1}$ and then sending
$r\to\infty$ gives the bound
$\|f_{\varepsilon_{k}}\|_{L^{\infty}(M_{\varepsilon_{k}})}\leq\beta$ provided
$k$ is large enough. This implies that
$\|u_{\varepsilon_{k}}\|_{L^{\infty}(M)}\leq\beta$ for all large $k$, as
needed. ∎

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