# Stability of the Positive Mass and Torus Rigidty Theorems Under Integral
Curvature Bounds

Brian Allen  Lehman College, CUNY https://sites.google.com/view/brian-allen ,
Edward Bryden  Universiteit Antwerpen https://etbryden.com/ and  Demetre
Kazaras  Duke University https://sites.google.com/view/dkazaras-homepage

###### Abstract.

Work of D. Stern [Stern-19] and Bray-Kazaras-Khuri-Stern [BKKS19] provide
differential-geometric identities which relate the scalar curvature of
Riemannian $3$-manifolds to global invariants in terms of harmonic functions.
These quantitative formulas are useful for stability results [Stern-19,
KKL-2021] and show promise for more applications of this type. In this paper,
we analyze harmonic maps to flat model spaces in order to address conjectures
concerning the geometric stability of the positive mass theorem and the Geroch
conjecture. By imposing integral Ricci curvature and isoperimetric bounds, we
leverage the previously mentioned formulas to establish strong control on
these harmonic maps. When the mass of an asymptotically flat manifold is
sufficiently small or when a Riemannian torus has almost non-negative scalar
curvature, we upgrade the maps to diffeomorphisms and give quantitative Hölder
closeness to the model spaces.

##  1\. Introduction

The geometric control which follows from integral bounds on Ricci curvature is
well understood. This theory has been developed by Anderson [Anderson-Ricci],
Anderson-Cheeger [Anderson-Cheeger], Colding [Colding-shape, Colding-volume],
Cheeger-Colding [Cheeger-Colding-1], Dai-Wang-Zhang [Dai_Wei_Zhang-2018],
Petersen [Petersen-97, Petersen-2006], Petersen-Sprouse [Petersen-Sprouse-
integral], Petersen-Wei [Petersen_Wei-2001, Petersen-Wei-integral1, Petersen-
Wei-integral2], Gao [Gao-integral1], and Yang [DYang-integral1, DYang-
integral2, DYang-integral3]. Readers unfamiliar with this convergence theory
are encouraged to consult Petersen’s fantastic survey article [Petersen-
Survey]. Speaking generally, methods in this area are based upon harmonic
coordinate systems, which directly relate the metric to its Ricci curvature by
an elliptic PDE. Through analysis of this equation, one is able to show that
the class of manifolds with integral bounds on Ricci (or Riemann) curvature
and injectivity radius bounds (or volume growth of balls) enjoys a system of
harmonic coordinate charts in which the metric is controlled in Sobolev and
Hölder spaces. However, the arguments which obtain this control often proceed
by contradiction, and the resulting metric control is generally not explicitly
quantitative in terms of given geometric bounds.

In this paper we take a new approach to the harmonic coordinate method
described above. On asymptotically flat $3$-manifolds and certain compact
$3$-manifolds, we study three distinct harmonic functions which encode global
geometric and topological information, but generally do not define a global
system of coordinates. These harmonic functions are related to the scalar
curvature of $3$-tori by Stern [Stern-19] and the mass of asymptotically flat
$3$-manifolds by Bray-Kazaras-Khuri-Stern [BKKS19]. By using these
breakthrough formulas in tandem with the geometric controls following from
integral Ricci curvature and isoperimetric bounds, we strongly control the
regularity and behavior of the harmonic functions. Next, when the appropriate
quantities are chosen to be small enough, we are able to show that these
harmonic functions produce a global coordinate system. This global coordinate
system is the source of the topological stability as well as quantitative
Hölder stability for both the positive mass theorem and scalar torus rigidity
theorem. For the positive mass theorem, this stability is explicitly
computable in terms of integral Riemann curvature and isoperimetric bounds. In
the case of the scalar torus rigidity theorem, the stability remains
implicitly quantitative due to the use of a contradiction argument in the
spirit of Petersen.

The positive mass theorem states that a complete, asymptotically flat manifold
with non-negative scalar curvature has positive ADM mass. This was first
proved by Schoen-Yau [SchoenYauPMT] using minimal surfaces and later by Witten
[Witten-PMT] using spinors and the Lichnerowicz formula. The geometric
stability of the positive mass theorem was first stated by Huisken-Illmanen
[Huisken-Ilmanen] in terms of Gromov-Hausdorff convergence and more recently
by Lee-Sormani [LeeSormani1] in terms of Sormani-Wenger intrinsic flat
convergence. Recently, Bray-Kazaras-Khuri-Stern [BKKS19] gave a new proof of
the positive mass theorem in dimension $3$ with a formula which relates the
mass and scalar curvature to harmonic functions defined on the asymptotically
flat manifold. This formula was then used by Kazaras-Khuri-Lee [KKL-2021] to
show Gromov-Hausdorff convergence for a sequence of asymptotically flat
manifolds with pointwise lower bounds on Ricci curvature whose mass is tending
to zero. In this paper we also use this formula to obtain quantitative
$C^{0,\gamma}$-stability, for some $\gamma\in(0,1)$, under integral Ricci
curvature bounds and isoperimetric constants for metric balls. It is
interesting to compare our methods with those of Finster-Bray [Finster-
Bray-99], Finster-Kath [Finster-Kath-02], and Finster [Finster-09], which are
based on a spinorial mass formula due to Witten [Witten-PMT]. In their work,
an isoperimetric condition is used to obtain a relation between the mass and
the $W^{1,2}$\- and $L^{\infty}$-norms of the Riemann curvature. Stability of
the positive mass theorem in the asymptotically flat setting has also been
studied by Allen [Allen18], Bryden [Bryden20], Bryden-Khuri-Sormani [BKS],
Huang-Lee-Sormani [HLS], Huang-Lee-Perales [Huang-Lee-Perales], Lee-Sormani
[LeeSormani1], Sormani-Stavrov [Sormani-Stavrov-1], and the references
therein.

The main tool we use to study the geometric stability of the positive mass
theorem is the following formula relating the ADM mass to scalar curvature
through an asymptotically linear harmonic function.

######  Theorem 1.1 (Theorem 1.2 of [BKKS19]).

Let $(M,g)$ be an orientable complete asymptotically flat manifold with no
spherical classes in $H_{2}(M;\mathbb{Z})$, and let $u$ be an asymptotically
linear harmonic function (see Section 4). Then, the ADM mass satisfies

(1) $m(g)\geq\frac{1}{16\pi}\int_{M}\frac{|\nabla^{2}u|^{2}}{|\nabla
u|}+R_{g}|\nabla u|dV_{g},$

where $R_{g}$ denotes the scalar curvature of $(M,g)$.

The first main result leverages controlled asymptotic falloff with good
control on the local analysis of Sobolev functions in order to show that ADM
mass strongly controls geometry. In order to effectively utilize the mass
formula (1), we will need Sobolev and Poincaré inequalities on our manifold.
Crucially, these inequalities are known to be equivalent to isoperimetric
constants (See Li [li_2012]). Let us now recall the isoperimetric constant
central to our result.

######  Definition 1.2 (Definition 9.2 in [li_2012]).

Let us define $IN_{\alpha}(M^{n},g)$ for $1\leq\alpha\leq\tfrac{n}{n-1}$ as
follows:

(2)
$IN_{\alpha}(M^{n},g)=\inf\left\\{\frac{\operatorname{Area}_{g}(S)}{\min\\{\operatorname{Vol}_{g}(\Omega_{1}),\operatorname{Vol}_{g}(\Omega_{2})\\}^{\frac{1}{\alpha}}}:M=\Omega_{1}\cup
S\cup\Omega_{2},\partial\Omega_{1}=S=\partial\Omega_{2}\right\\}.$

$IN_{\alpha}(M^{n},g)$ is called the Neumann $\alpha-$isoperimetric constant
of $M$.

With the above in hand, we can make precise what it means for a class of
asymptotically flat Riemannian manifolds to have controlled asymptotic falloff
and good control on the local analysis of Sobolev functions, by defining the
following class of asymptotically flat Riemannian manifolds.

######  Definition 1.3.

Fix $b,\bar{m},\Lambda,\kappa>0$, $\tau>\tfrac{1}{2}$, $p>1$,
$\alpha\in[1,\tfrac{3}{2}]$. An oriented connected complete asymptotically
flat $3$-dimensional Riemannian manifold $(M,g)$ is said to be _homologically
simple, $(b,\tau,\bar{m})$ asymptotically flat, $(\Lambda,\alpha)$ Neumann-
isoperimetrically bounded, and $\kappa$ curvature-$L^{p}$ bounded_ if

  1. (1)

there are no spherical classes in $H_{2}(M;\mathbb{Z})$ and $\partial
M=\emptyset$,

  2. (2)

there exists a coordinate chart
$\Phi:M\setminus\Omega\rightarrow\mathbb{R}^{3}\setminus B_{1}(0)$ such that
on $\mathbb{R}^{3}\setminus B_{1}(0)$ we have

$|D^{(k)}\left(g_{ij}-\delta_{ij}\right)|(x)\leq b|x|^{-\tau-k},$

where $k=0,1,$ and $2$,

  3. (3)

$R_{g}\geq 0,$ and $m_{\mathrm{ADM}}(g)\leq\bar{m}$,

  4. (4)

for all metric balls, $B_{r}(x)\subset M$, we have

$IN_{\alpha}\left(B_{r}(x)\right)\geq\Lambda,$

  5. (5)

$\|\mathrm{Rc}_{g}\|_{L^{p}}\leq\kappa.$

We will denote the family of such $3$-dimensional asymptotically flat
Riemannian manifolds by
$\mathcal{M}\left(b,\tau,\bar{m},\Lambda,\alpha,\kappa,p\right)$. In most of
what follows, we will impose $\alpha=\tfrac{3}{2}$ and $p=3$. In this case, we
adopt the shorthand
$\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)=\mathcal{M}\left(b,\tau,\bar{m},\Lambda,\tfrac{3}{2},\kappa,3\right)$.

Figure 1. The above short and stout bump is Neumann
$\Lambda$-isoperimetrically bounded if the ratio of its radius (in green) to
its height (in blue) is bounded above and below. By taking the radius and the
height to be arbitrarily small, one obtains Neumann
$\Lambda$-isoperimetrically bounded spaces with arbitrarily small injectivity
radius.

######  Remark 1.4.

Ricci curvature has a unique character in dimension $3$. In particular, all
components of a $3$-manifold’s Riemann tensor can be algebraically expressed
in terms of its Ricci curvature. This means that condition $(5)$ in Definition
1.3 is actually equivalent to a bound on $\|\mathrm{Rm}_{g}\|$.

The first main result of the paper gives quantitative stability of the
positive mass theorem.

######  Theorem 1.5.

Let $b,\bar{m},\Lambda,\kappa>0,\tau>\tfrac{1}{2}$, and
$\gamma\in(0,\tfrac{1}{2})$ be given parameters. For any $\varepsilon>0$ there
exists $\delta=\delta(b,\tau,\bar{m},\Lambda,\kappa,\varepsilon,\gamma)>0$
such that the following holds: if $(M,g)$ lies in
$\mathcal{M}\left(b,\tau,\bar{m},\Lambda,\kappa\right)$ and
$m_{\mathrm{ADM}}(g)\leq\delta$, then $M$ is diffeomorphic to $\mathbb{R}^{3}$
and

(3)
$\displaystyle\|g-g_{\mathbb{E}}\|_{C^{0,\gamma}(\mathbb{R}^{3})}<\varepsilon$

where $g_{\mathbb{E}}$ denotes the flat metric on $\mathbb{R}^{3}$.

######  Remark 1.6.

The Hölder norm in (3) can be computed using distances measured with respect
to either $g$ or $g_{\mathbb{E}}$ – since these metrics are $C^{0}$-close,
this choice is immaterial. Also, an interesting consequence of Theorem 1.5 is
the following topological stability: if a manifold
$(M,g)\in\mathcal{M}\left(b,\tau,\bar{m},\Lambda,\kappa\right)$ has
sufficiently small mass, then $M$ is diffeomorphic to $\mathbb{R}^{3}$.

Next, we turn to the second main application of our methods. The Geroch
conjecture asserts that a Riemannian $3$-torus with non-negative scalar
curvature must be flat. This rigidity theorem was first proven by Schoen-Yau
[Schoen-Yau-min-surf] and soon after in higher dimensions by Gromov-Lawson
[Gromov-Lawson-torus] using different methods. The stability problem
associated to the Geroch conjecture was first considered by Gromov [GroD] and
made more precise by Sormani [Sormani-scalar], who observed the necessity of
an extra geometric constraint known as a minA condition. A minA condition is a
lower bound on the area of all closed minimal surfaces within the manifold,
and is used to eliminate pathological bubbling examples first observed by
Basilio-Sormani [Basilio-Sormani]. See the survey paper by Sormani
[ScalarSurvey-Sormani] for a recent discussion of this conjecture as well as
many other conjectures involving scalar curvature.

Various special cases of these conjectures have been studied by Allen [Allen-
Conformal-Torus], Allen-Bryden [AllenBryden21], Allen-Hernandez-Vazquez-
Parise-Payne-Wang [AHMPPW1], Cabrera Pacheco-Ketterer-Perales [PKP19], Chu-Lee
[Chu-Man-Chun22], and Lee-Naber-Neumeyer [LNN]. In this paper, we establish
quantitative stability and Hölder convergence in the context of integral
bounds on the Ricci curvature. We are able to avoid bubbling by requiring a
uniform lower bound on the Neumann isoperimetric constant of metric balls. The
present work suggests that a lower bound on the Neumann isoperimetric constant
is an analytically effective – yet still geometric – replacement to a minA
condition when considering the stability of $3$-dimensional rigidity theorems
related to scalar curvature.

The main ingredient we use to establish quantitative stability of the Geroch
conjecture is a formula relating the scalar curvature of a closed $3$-manifold
to $\mathcal{S}^{1}$-valued harmonic maps developed by D. Stern.

######  Theorem 1.7 (Theorem 1.1 of [Stern-19]).

Let $(M,g)$ be a closed oriented Riemannian $3$-manifold, and suppose
$u:M\rightarrow\mathcal{S}^{1}$ is a non-constant harmonic map. Then

(4)
$\begin{split}4\pi\int_{\mathcal{S}^{1}}\chi\left(\Sigma_{\theta}\right)d\theta&\geq\int_{M}\frac{|\nabla
du|^{2}}{|du|}+R_{g}|du|dV_{g}.\end{split}$

where $\chi(\Sigma_{\theta})$ is the Euler characteristic of a regular level
set $\Sigma_{\theta}=u^{-1}(\theta)$ and $du$ denotes the $1$-form
$u^{*}d\theta$.

######  Remark 1.8.

For closed manifolds $M$, recall that there is a bijection between
$H^{1}(M;\mathbb{Z})$ and homotopy classes of maps from $M$ to a circle. For
each such homotopy class of maps, there is an harmonic representative, unique
up to rotations of the circle. Consequently, the first integral cohomology
group of a manifold provides the harmonic maps appearing in Theorem 1.7.

We can now define the class of manifolds which we will consider for our second
main result.

######  Definition 1.9.

Given parameters $\Lambda,V,\kappa>0$, we denote by
$\mathcal{N}(\Lambda,V,\kappa)$ the collection of closed connected oriented
Riemannian $3$-manifolds $(M,g)$ which satisfy the following:

  1. (1)

$(M,g)$ is $(\Lambda,\frac{3}{2})$-Neumann-isoperimetrically bounded,

  2. (2)

$\frac{1}{V}\leq\operatorname{Vol}_{g}(M)\leq V$,

  3. (3)

$\|\mathrm{Rc}_{g}\|_{L^{3}}\leq\kappa$,

  4. (4)

the second integral homology $H_{2}(M;\mathbb{Z})$ contains no spherical
classes,

  5. (5)

there are $3$ cohomology classes $\alpha_{1},\alpha_{2},\alpha_{3}\in
H^{1}(M;\mathbb{Z})$ so that the cup product
$\alpha_{1}\cup\alpha_{2}\cup\alpha_{3}$ generates $H^{3}(M;\mathbb{Z})$.

######  Remark 1.10.

Let us briefly remark on the homological conditions $(4)$ and $(5)$ imposed on
the underlying manifolds in the class $\mathcal{M}(\Lambda,V,\kappa)$. First,
a class $\alpha\in H_{2}(M;\mathbb{Z})$ is said to be _spherical_ if there is
an embedding $\mathcal{S}^{2}\hookrightarrow M$ such that $\alpha$ is the
image of the fundamental class $[\mathcal{S}^{2}]$. The torus $T^{3}$ is the
prototypical example satisfying $(4)$ and $(5)$, but manifolds of the form
$T^{3}\\#N^{3}$ where $N^{3}$ has the topology of a closed hyperbolic manifold
or a spherical space-form also satisfy these conditions. On the other hand,
manifolds with $S^{2}\times S^{1}$ summands will fail to satisfy $(4)$.

Next, we apply our methods to investigate the stability of the Geroch
conjecture. Below, we adopt the notation $R_{g}^{-}=-\min(0,R_{g})$ for the
negative part of scalar curvature.

######  Theorem 1.11.

Let $\Lambda,V,\kappa>0$ and $\gamma\in(0,\tfrac{1}{2})$ be given. For any
$\varepsilon>0$, there exists a
$\delta=\delta(\Lambda,V,\kappa,\varepsilon)>0$ so that the following holds:
If $(M^{3},g)\in\mathcal{N}(\Lambda,V,\kappa)$ and
$\|R_{g}^{-}\|_{L^{1}}\leq\delta$, then $M$ is diffeomorphic to the torus
$\mathbb{T}^{3}$ and

(5) $\|g-g_{F}\|_{C^{0,\gamma}}<\varepsilon$

for some flat metric $g_{F}$.

The following sequential stability result is a straight-forward consequence of
Theorem 1.11.

######  Corollary 1.12.

Let $\Lambda,V,\kappa>0$ be given and suppose
$\\{(M^{3}_{i},g_{i})\\}_{i=1}^{\infty}$ is a sequence in
$\mathcal{N}(\Lambda,V,\kappa)$. If
$\displaystyle\lim_{i\to\infty}\|R_{g_{i}}^{-}\|_{L^{1}}=0$, then
$(M_{i},g_{i})$ subsequentially converge to a flat torus in the $C^{0,\gamma}$
topology for some $\gamma\in(0,1)$.

To conclude this section, we give a brief outline of the paper. We begin in
Section 2 by introducing the fundamental definitions and theorems which
describe volume controls for manifolds with integral bounds on Ricci
curvature. In Section 3, we define isoperimetric and Sobolev constants for
manifolds and prove various relationships between them. At the end of this
section we show that uniform control on the volume growth of balls and a
$(1,p)$ Poincaré inequality are enough to guarantee a uniformly controlled
constant in Morrey’s inequality.

Section 4 is devoted to the analysis of asymptotically linear harmonic
functions on asymptotically flat manifolds. We review the asymptotic features
of such functions established in [KKL-2021] and develop related decay results
which are needed in Sections 5 and 6. Next, Section 5 develops $L^{2}$ control
on higher order derivatives of harmonic functions. These estimates are derived
by integrating by parts and commuting covariant derivatives. This is an
essential step where the integral Ricci bounds and asymptotic estimates show
up as necessary assumptions to obtain uniform $W^{3,2}$ regularity of harmonic
functions. Finally, in Section 6, we combine the mass formula, Sobolev
inequalities, the asymptotic estimates, and the $W^{3,2}$ regularity of
harmonic functions in order to prove Theorem 1.5.

This accomplished, we turn to the proof of Theorem 1.11 in section 7. We
follow a fairly similar argument as in the proof of Theorem 1.5. The main
difference is that we cannot leverage the asymptotic control which comes from
the asymptotic region of asymptotically flat manifolds. Instead, we are able
to establish control on the $L^{2}$ norm of harmonic $1-$forms on tori and
related manifolds which is used to produce the desired Hölder stability.

##  2\. Background

In this section we review some results and notions central to our strategy. We
begin by recalling the following isoperimetric constant.

######  Definition 2.1 (Definition 9.1 in [li_2012]).

Given an $n$-dimensional Riemannian manifold $(M,g)$ and a number
$1\leq\alpha\leq\tfrac{n}{n-1}$, the _Dirichlet $\alpha-$isoperimetric
constant of $(M,g)$_ is denoted by $ID_{\alpha}(M,g)$ and defined as

(6)
$ID_{\alpha}(M^{n},g)=\inf\left\\{\frac{|\partial\Omega|}{|\Omega|^{\frac{1}{\alpha}}}:\Omega\subset
M,\text{ }\partial\Omega\cap\partial M=\emptyset\right\\}.$

Happily, a lower bound on $ID_{\alpha}(M,g)$ for $\alpha>1$ gives a lower
bound on the volume growth of metric balls.

######  Lemma 2.2.

Let $(M,g)$ be an n-dimensional Riemannian manifold with $ID_{\alpha}(M,g)>0$
for some $\alpha\in(1,\tfrac{n}{n-1}]$. Then, we have

(7) $\displaystyle\frac{|\partial
B_{r}(x)|}{|B_{r}(x)|^{\frac{1}{\alpha}}}\geq ID_{\alpha}(M,g),$

and, for any $r\leq\operatorname{Diam}(M,g)$,

(8) $\displaystyle|B_{r}(x)|\geq
ID_{\alpha}(M,g)^{\frac{\alpha}{\alpha-1}}r^{\frac{\alpha}{\alpha-1}}.$

In particular, if $\alpha=\tfrac{n}{n-1}$ and $r\leq\operatorname{Diam}(M,g)$,
then

(9) $\displaystyle|B_{r}(x)|\geq ID_{\frac{n}{n-1}}(M,g)^{n}r^{n}.$

###### Proof.

The following proof is taken from [Heinonen-2001] chapter 3, page 25. By
taking the derivative of the volume of geodesic balls, we find for almost
every $r>0$

(10) $\displaystyle\frac{d}{dr}|B_{r}(x)|=|\partial B_{r}(x)|\geq
ID_{\alpha}|B_{r}(x)|^{\frac{1}{\alpha}},$

and hence by integrating the ODE inequality we find

(11) $\displaystyle|B_{r}(x)|\geq
ID_{\alpha}^{\frac{\alpha}{\alpha-1}}r^{\frac{\alpha}{\alpha-1}}.$

∎

Whereas lower bounds on $ID_{\alpha}(M)$ give lower bounds on the volume
growth of metric balls, lower bounds on Ricci curvature give upper bounds on
the volume growth of metric balls. This is the content of the following two
important results due to Petersen-Wei.

######  Lemma 2.3 (Lemma 2.3 in [Petersen-Wei-1997]).

Given an $n$-dimensional Riemannian manifold $(M,g)$, let $h(x)$ denote the
smallest eigenvalue of the Ricci endomorphism at $x\in M$, and $k(\lambda,p)$
be given by

(12)
$k(\lambda,p)=\int_{M}\textrm{max}\left\\{0,-h(x)+(n-1)\lambda\right\\}^{p}dV_{g}.$

If $\lambda\leq 0$, $r<R$, and $p>\tfrac{n}{2}$, then there is a constant
$C(n,p,\lambda,R)$, such that

(13)
$\left(\frac{|B_{r}(x)|}{v(n,\lambda,R)}\right)^{\frac{1}{2p}}-\left(\frac{|B_{r}(x)|}{v(n,\lambda,r)}\right)^{\frac{1}{2p}}\leq
C\left(n,p,\lambda,R\right)\left(k(\lambda,p)\right)^{\frac{1}{2p}},$

where $v(n,\lambda,t)$ is the volume of a ball of radius $t$ in the simply
connected $n$-dimensional space-form with curvature $\lambda$.

As noted in [Petersen-Wei-1997], the following is an immediate consequence.

######  Corollary 2.4.

With the same notation and assumptions of Lemma 2.3, for all $0\leq r\leq R$,
we have

(14)
$|B_{r}(x)|\leq\left(1+C(n,p,\lambda,R)k(\lambda,p)^{\frac{1}{2p}}\right)^{2p}v\left(n,\lambda,r\right).$

##  3\. Sobolev, Poincaré, and Morrey inequalities

In this section we remind the reader of the definitions of various Sobolev and
isoperimetric constants, review well known relationships between them, and
prove some new relationships which are needed in later sections.

######  Notation 3.1.

Below and throughout, if $f$ is a function defined on a domain $B$ in a
Riemannian manifold $(M,g)$, we write $f_{B}$ for the average value
$\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B}fdV_{g}:=\tfrac{1}{|B|}\int_{B}fdV_{g}$.
Secondly, without further mention, we will use $C(a_{1},\dots,a_{N})$ to
denote a constant which depends only on parameters $a_{1},\dots,a_{N}$. If we
require the use of multiple such constants in the course of a given proof, we
will use subscripts $C_{k}(a_{1},\dots,a_{N})$, $k=1,2,\dots$, to
differentiate them.

Since the Neumann and Dirichlet isoperimetric constants are crucial to our
forthcoming analysis, we begin with a result relating them to each other. This
lemma follows from a remark in [Dai_Wei_Zhang-2018].

######  Lemma 3.2.

Let $(M,g)$ be an $n$-dimensional Riemannian manifold. If domains
$\Omega_{1}\subset\Omega_{2}\subset M$ satisfy
$\partial\Omega_{1}\cap\partial\Omega_{2}=\emptyset$ and
$|\Omega_{1}|\leq\frac{1}{2}|\Omega_{2}|$, then

(15) $\displaystyle IN_{\alpha}(\Omega_{2},g)\leq ID_{\alpha}(\Omega_{1},g).$

Furthermore, if $\Omega_{1}^{\prime}\subset\Omega_{2}^{\prime}$ so that
$\partial\Omega_{1}^{\prime}\cap\partial\Omega_{2}^{\prime}=\emptyset$ we find

(16) $\displaystyle ID_{\alpha}(\Omega_{2}^{\prime},g)\leq
ID_{\alpha}(\Omega_{1}^{\prime},g).$

###### Proof.

Consider $U\subset\Omega_{1}$ so that $\partial
U\cap\partial\Omega_{1}=\emptyset$. Then we see that $U$ is a valid competitor
for $ID_{\alpha}(\Omega_{1},g)$ where

(17) $\displaystyle|U|\leq|\Omega_{1}|\leq\frac{1}{2}|\Omega_{2}|,$

and

(18) $\displaystyle|\Omega_{2}\setminus
U|=|\Omega_{2}|-|U|\geq\frac{1}{2}|\Omega_{2}|.$

Now we notice that $U$ is a valid competitor for $IN_{\alpha}(\Omega_{2},g)$
and hence

(19) $\displaystyle IN_{\alpha}(\Omega_{2},g)\leq\frac{|\partial
U|}{\min\\{|U|,|\Omega_{2}\setminus
U|\\}^{\frac{1}{\alpha}}}\leq\frac{|\partial U|}{|U|^{\frac{1}{\alpha}}}.$

Since this is true for any $U$ so that $U\subset\Omega_{1}$ and $\partial
U\cap\partial\Omega_{1}=\emptyset$ we can take the infimum over the right hand
side of (19) to find the first result.

The second result follows from the fact that any competitor for
$ID_{\alpha}(\Omega_{1}^{\prime},g)$ is a competitor for
$ID_{\alpha}(\Omega_{2}^{\prime},g)$. ∎

###  3.1. Dirichlet and Neumann Sobolev Constants

We start by defining the Dirichlet and Neumann Sobolev constants.

######  Definition 3.3 (Definition 9.3 in [li_2012]).

Let us define $SD_{\alpha}(M^{n},g)$ for $1\leq\alpha\leq\frac{n}{n-1}$ as
follows:

(20) $SD_{\alpha}(M^{n},g)=\inf\left\\{\frac{\|\nabla
f\|_{L^{1}}}{\|f\|_{L^{\alpha}}}:f\in W_{0}^{1,1}(M^{n},g)\right\\}.$

$SD_{\alpha}(M^{n},g)$ is called the Dirichlet $\alpha-$Sobolev constant of
$M$.

######  Definition 3.4 (Definition 9.4 in [li_2012]).

Given an $n$-dimensional Riemannian manifold $(M,g)$, let us define
$SN_{\alpha}(M^{n},g)$ for $1\leq\alpha\leq\frac{n}{n-1}$ as follows:

(21) $SN_{\alpha}(M,g)=\inf\left\\{\frac{\|\nabla
f\|_{L^{1}}}{\inf_{k\in\mathbb{R}}\|f-k\|_{L^{\alpha}}}:f\in
W^{1,1}(M,g)\right\\}$

$SN_{\alpha}(M,g)$ is called the Neumann $\alpha$-Sobolev constant of $(M,g)$.

Let us now recall and observe some relationships between the above quantities.
It is shown in [li_2012, Theorem 9.5] that

(22) $\displaystyle SD_{\alpha}(M,g)=ID_{\alpha}(M,g),$

and in [li_2012, Theorem 9.6] we see that

(23) $\displaystyle\min\\{1,2^{\frac{\alpha-1}{\alpha}}\\}IN_{\alpha}(M,g)\leq
SN_{\alpha}(M,g)\leq\max\\{1,2^{\frac{\alpha-1}{\alpha}}\\}IN_{\alpha}(M,g).$

The quantity $SN_{\alpha}$ is a bit mysterious in its present form, mainly
because of the term involving $k$. However, the following result, mentioned in
[li_2012], helps us understand it a little better. We will reproduce the
statement and proof for the convenience of the reader.

######  Lemma 3.5.

Let $(M,g)$ be an $n$-dimensional Riemannian manifold. Suppose $|M|<\infty$
and $f\in L^{q}$, $q\geq 1$. Then, there exists a unique $k_{q}(f)$ such that

(24) $\inf_{k\in\mathbb{R}}\|f-k\|_{L^{q}}=\|f-k_{q}(f)\|_{L^{q}}.$

Further, $k_{q}(f)$ is given by the equation

(25) $\int_{M}\mathrm{sgn}\left(f-k_{q}(f)\right)|f-k_{q}(f)|^{q-1}dV_{g}=0.$

In particular, when $q=2$ we have that

(26)
$k_{2}(f)=\frac{1}{|M|}\int_{M}fdV_{g}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{M}fdV_{g}.$

###### Proof.

Observe that for $q>1$ the function $F(k)=\int_{M}|f-k|^{q}dV_{g}$ is strictly
convex and differentiable. Furthermore, observe that
$\displaystyle\lim_{|k|\rightarrow\infty}\int|f-k|^{q}=\infty$. Therefore, the
function has a unique minimal point, which is also its unique critical point.
We calculate that

(27)
$F^{\prime}(k)=\frac{d}{dk}\int_{M}|f-k|^{q}dV_{g}=q\int_{M}\text{sign}(f-k)|f-k|^{q-1}dV_{g}.$

In the case that $q=1$, one can observe that for $k_{1},k_{2}\in\mathbb{R}$,
$t\in[0,1]$ that

(28) $\displaystyle F(k_{1}t+k_{2}(1-t))\leq tF(k_{1})+(1-t)F(k_{2}),$

and hence $F$ is convex, differentiable,
$\displaystyle\lim_{|k|\rightarrow\infty}F(k)=\infty$, and with only one
critical point

(29)
$F^{\prime}(k)=\frac{d}{dk}\int_{M}|f-k|dV_{g}=\int_{M}\text{sign}(f-k)dV_{g}=|\\{f\geq
k\\}|-|\\{f\leq k\\}|.$

∎

###  3.2. Relationship to Classical Poincaré and Sobolev Inequalities

Our goal in this subsection is to show how the Neumann and Dirichlet Sobolev
constants are related to the classical Poincaré and Sobolev inequalities. We
begin by recalling a result from [li_2012].

######  Proposition 3.6 (Found in [li_2012]).

Let $(M,g)$ be an $n$-dimensional Riemannian manifold with non-empty boundary,
and suppose that $SD_{\alpha}(M,g)>0$. Then, for any $1\leq p<\infty$ we have
for $f\in W^{1,p}_{0}(M)$ that

(30) $\|f\|_{\frac{\alpha p}{p-(p-1)\alpha}}\leq
SD^{-1}_{\alpha}(M,g)\frac{p}{p-\alpha(p-1)}\|\nabla f\|_{L^{p}}.$

###### Proof.

For the moment, suppose we are given $f\in W_{0}^{1,p}\bigcap L^{q}$, for
$q=\frac{p}{p-\alpha(p-1)}$. Then, we have that $g=f^{q}$

(31) $\begin{split}\|f\|^{\frac{p}{p-\alpha(p-1)}}_{\frac{\alpha
p}{p-\alpha(p-1)}}&=\left(\int_{M}|g|^{\alpha}dV_{g}\right)^{\frac{1}{\alpha}}\leq
SD^{-1}_{\alpha}\left(M\right)\int_{M}|\nabla g|dV_{g}\\\
&=qSD^{-1}_{\alpha}(M)\int_{M}|g|^{q-1}|\nabla f|dV_{g}\\\ &\leq
qSD^{-1}_{\alpha}(M)\|f\|^{\frac{\alpha
p}{p-\alpha(p-1)}\frac{p-1}{p}}_{\frac{\alpha p}{p-\alpha(p-1)}}\|\nabla
f\|_{L^{p}}.\end{split}$

Rearranging the above gives

(32) $\|f\|_{\frac{\alpha p}{p-\alpha(p-1)}}\leq qSD^{-1}_{\alpha}(M)\|\nabla
f\|_{L^{p}}.$

In order to remove the assumption that $f$ is in $L^{q}$, we note that
$W_{0}^{1,p}\bigcap L^{q}$ is dense in $W_{0}^{1,p}$. ∎

We now use this result to prove a Sobolev inequality.

######  Theorem 3.7.

Let $M$ be asymptotically flat and suppose that
$SD_{\alpha}(B_{r}(x),g)\geq\Lambda>0$, $\forall x\in M,r>0$. Then, for any
$1\leq p<\infty$ we have for $f\in W^{1,p}(M)$ that

(33) $\|f\|_{\frac{\alpha
p}{p-(p-1)\alpha}}\leq\Lambda^{-1}\frac{p}{p-\alpha(p-1)}\|f\|_{W^{1,p}}$

###### Proof.

Consider $f\in W^{1,p}(M)$ and let $\phi\in C_{c}^{\infty}(M)$ so that
$0\leq\phi\leq 1$ and $|\nabla\phi|\leq 1$. Then we have that $\phi f\in
W_{0}^{1,p}(B_{r}(x))$ for some $x\in M,r>0$ and so by Proposition 3.6 we find

(34) $\displaystyle\|\phi f\|_{\frac{\alpha p}{p-(p-1)\alpha}}$
$\displaystyle\leq
SD^{-1}_{\alpha}(B_{r}(x))\frac{p}{p-\alpha(p-1)}\|\nabla(\phi
f)\|_{L^{p}(B_{r}(x)}$ (35)
$\displaystyle=SD^{-1}_{\alpha}(B_{r}(x))\frac{p}{p-\alpha(p-1)}\left(\|\phi\nabla
f\|_{L^{p}(B_{r}(x)}+\|f\nabla\phi\|_{L^{p}(B_{r}(x)}\right)$ (36)
$\displaystyle\leq\Lambda\frac{p}{p-\alpha(p-1)}\left(\|\nabla
f\|_{L^{p}(B_{r}(x))}+\|f\|_{L^{p}(B_{r}(x))}\right)$ (37)
$\displaystyle\leq\Lambda\frac{p}{p-\alpha(p-1)}\left(\|\nabla
f\|_{L^{p}(M)}+\|f\|_{L^{p}(M)}\right).$

Now by choosing a sequence of functions $\phi_{n}\in C_{c}^{\infty}(M)$ that
converge locally uniformly to $1$ so that $0\leq\phi_{n}\leq 1$,
$|\nabla\phi_{n}|\leq 1$, and supp$(\phi_{n})\subset B_{n}(x)$ for some $x\in
M$ we can apply the previous inequality to obtain the desired result

(38) $\displaystyle\|f\|_{\frac{\alpha
p}{p-(p-1)\alpha}}=\lim_{n\rightarrow\infty}\|\phi_{n}f\|_{\frac{\alpha
p}{p-(p-1)\alpha}}$
$\displaystyle\leq\Lambda\frac{p}{p-\alpha(p-1)}\left(\|\nabla
f\|_{L^{p}(M)}+\|f\|_{L^{p}(M)}\right).$

∎

We have the following proposition, which can be found, in essence, in
[li_2012, Corollary 9.9].

######  Proposition 3.8.

Let $(M,g)$ be an $n$-dimensional Riemannian manifold, $1<p<n$,
$1<\alpha\leq\frac{n}{n-1}$, $f\in W^{1,p}(M)$, and let $k(f)$ be the constant
such that

(39)
$\int_{M}\mathrm{sign}\left(f-k(f)\right)\left|f-k(f)\right|^{\frac{(\alpha-1)p}{p-\alpha(p-1)}}dV_{g}=0.$

Then, we have

(40) $\begin{split}&\|f-k(f)\|_{L^{\frac{\alpha
p}{p-(p-1)\alpha}}}\leq\frac{p}{p-(p-1)\alpha}SN^{-1}_{\alpha}(M)\|\nabla
f\|_{L^{p}}\\\ &\|f\|_{L^{\frac{\alpha p}{p-\alpha(p-1)}}}\leq
C\left(p,\alpha,SN_{\alpha}(M),|M|\right)\|f\|_{W^{1,p}}\end{split}$

###### Proof.

Let $f\in W^{1,p}$ and assume for the moment that $f\in L^{\alpha q}$, for
some $q$ to be determined. Let $k(f)$ be the unique number such that

(41)
$\int_{M}\mathrm{sign}\left(f-k(f)\right)\left|f-k(f)\right|^{q(\alpha-1)}dV_{g}=0.$

Let us define a function $\psi$ as follows

(42) $\psi=\mathrm{sign}\left(f-k(f)\right)\left|f-k(f)\right|^{q}.$

Then, by our choice of $k(f)$, we see that

(43) $\int_{M}\mathrm{sign}\left(\psi\right)|\psi|^{\alpha-1}dV_{g}=0,$

and so

(44)
$\inf_{k\in\mathbb{R}}\int_{M}|\psi-k|^{\alpha}dV_{g}=\int_{M}|\psi|^{\alpha}dV_{g}.$

Let us now apply the Sobolev-Neumann inequality to $\psi$ to obtain

(45) $\left(\int_{M}|\psi|^{\alpha}dV_{g}\right)^{\frac{1}{\alpha}}\leq
SN^{-1}_{\alpha}\int_{M}\left|\nabla\psi\right|dV_{g}.$

Plugging in our formula for $\psi$,

(46) $\left(\int_{M}|f-k(f)|^{q\alpha}dV_{g}\right)^{\frac{1}{\alpha}}\leq
qSN^{-1}_{\alpha}\int_{M}|f-k(f)|^{q-1}\left|\nabla f\right|dV_{g}.$

Applying Hölder’s inequality to the right hand side leads to

(47) $\left(\int_{M}|f-k(f)|^{q\alpha}dV_{g}\right)^{\frac{1}{\alpha}}\leq
qSN^{-1}_{\alpha}\left(\int_{M}|f-k(f)|^{\frac{(q-1)p}{p-1}}dV_{g}\right)^{\frac{p-1}{p}}\|\nabla
f\|_{L^{p}}.$

In order to match exponents on the left and the right of the above inequality,
we choose $q$ to be the solution to

(48) $q\alpha=\frac{(q-1)p}{p-1},$

which is $q=\frac{p}{p-\alpha(p-1)}$. Note that because $p<n$ we have
$\tfrac{n}{n-1}<\frac{p}{p-1}$, so $q$ is well defined. Plugging this value in
for $q$ shows that if $f$ is in $L^{\alpha q}$, then we have

(49) $\|f-k(f)\|_{L^{\frac{\alpha p}{p-\alpha(p-1)}}}\leq
SN^{-1}_{\alpha}\frac{p}{p-\alpha(p-1)}\|\nabla f\|_{L^{p}}.$

Still assuming that $f$ is in $L^{\frac{\alpha p}{p-\alpha(p-1)}}$, we get
that

(50) $SN^{-1}_{\alpha}\frac{p}{p-\alpha(p-1)}\|\nabla
f\|_{L^{p}}+|M|^{\frac{p-\alpha(p-1)}{p\alpha}}|k(f)|\geq\|f\|_{L^{\frac{\alpha
p}{p-\alpha(p-1)}}}.$

In fact, since $\alpha>1$, we can estimate $|k(f)|$ in terms of
$\|f\|_{W^{1,p}}$. This is done as follows: We begin with the estimate

(51) $|M||k(f)|=\int_{M}|k(f)|dV_{g}\leq\int_{M}|k(f)-f|+|f|dV_{g}.$

We may now use Hölder’s inequality twice to obtain

(52) $|M||k(f)|\leq|M|^{\frac{2\alpha p-p-\alpha}{\alpha
p}}\|f-k(f)\|_{\frac{\alpha
p}{p-(p-1)\alpha}}+|M|^{\frac{p-1}{p}}\|f\|_{L^{p}}.$

At this point, we can apply (49) to obtain

(53) $|M||k(f)|\leq|M|^{\frac{2\alpha p-p-\alpha}{\alpha
p}}\frac{p}{p-(p-1)\alpha}SN^{-1}_{\alpha}(M)\|\nabla
f\|_{L^{p}}+|M|^{\frac{p-1}{p}}\|f\|_{p}.$

We can now put everything together to get

(54) $\|f\|_{\frac{\alpha p}{p-(p-1)\alpha}}\leq
C(p,\alpha,SN_{\alpha},|M|)\|f\|_{W^{1,p}}$

so long as $f$ is in $L^{\alpha q}$. Since it is well known that
$W^{1,p}\bigcap L^{\alpha q}$ is dense in $W^{1,p}$, this completes the proof.

∎

We now establish that a bound on the Neumann isoperimetric constant of balls
gives a Poincaré inequality.

######  Lemma 3.9.

Let $(M,g)$ be a $n$-dimensional Riemannian manifold. Suppose there exists a
$\Lambda>0$ such that, for any ball $B_{r}(x)$, we have
$SN_{\tfrac{n}{n-1}}(B_{r}(x),g)\geq\Lambda$. Then, for any $f$ in
$W^{1,p}(M)$ we have

(55) $\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|f-f_{B_{r}(x)}|^{n}dV_{g}\right)^{\frac{1}{n}}\leq(n-1)\Lambda^{-1}|B_{r}(x)|^{\frac{1}{n}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|\nabla
f|^{\frac{n}{2}}dV_{g}\right)^{\frac{2}{n}},$

from which it follows that

(56) $\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|f-f_{B_{r}(x)}|dV_{g}\leq(n-1)\Lambda^{-1}|B_{r}(x)|^{\frac{1}{n}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|\nabla
f|^{p}dV_{g}\right)^{\frac{1}{p}}$

for any $p\geq\frac{n}{2}$.

###### Proof.

The starting point is to make sure that the constant $k(f)$ in Equation (39)
is

(57) $f_{B_{r}(x)}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}fdV_{g}.$

Inspecting the exponent in Equation (39), while keeping in mind that
$\alpha=\tfrac{n}{n-1}$, shows that we need to pick $p$ such that

(58) $1=\frac{(\alpha-1)p}{p-\alpha(p-1)}=\frac{p}{n-p}.$

That is, we have $p=\tfrac{n}{2}$. So, using this exponent in Proposition 3.8,
we get

(59)
$\left(\int_{B_{r}(x)}|f-f_{B_{r}(x)}|^{n}dV_{g}\right)^{\frac{1}{n}}\leq(n-1)\Lambda^{-1}\left(\int_{B_{r}(x)}|\nabla
f|^{\frac{n}{2}}dV_{g}\right)^{\frac{2}{n}}.$

Now, we can rewrite this in terms of averages as follows:

(60) $\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|f-f_{B_{r}(x)}|^{n}dV_{g}\right)^{\frac{1}{n}}|B_{r}(x)|^{\frac{1}{n}}\leq(n-1)\Lambda^{-1}|B_{r}(x)|^{\frac{2}{n}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|\nabla
f|^{\frac{n}{2}}dV_{g}\right)^{\frac{n}{2}}.$

Dividing both sides by $|B_{r}(x)|^{\frac{1}{n}}$ gives the first result.

To get the second result, we can begin by applying Hölder’s inequality to get

(61)
$\begin{split}\int_{B_{r}(x)}|f-f_{B_{r}(x)}|dV_{g}&\leq|B_{r}(x)|^{\frac{n-1}{n}}\left(\int_{B_{r}(x)}|f-f_{B_{r}(x)}|^{n}dV_{g}\right)^{\frac{1}{n}}\\\
&\leq|B_{r}(x)|^{\frac{n-1}{n}}(n-1)\Lambda^{-1}\left(\int_{B_{r}(x)}|\nabla
f|^{\frac{n}{2}}dV_{g}\right)^{\frac{2}{n}}\\\
&\leq|B_{r}(x)|^{\frac{n-1}{n}}(n-1)\Lambda^{-1}|B_{r}(x)|^{\frac{2(\sigma-1)}{n\sigma}}\left(\int_{B_{r}(x)}|\nabla
f|^{\frac{n\sigma}{2}}dV_{g}\right)^{\frac{2}{n\sigma}}\\\
&=(n-1)\Lambda^{-1}|B_{r}(x)|^{1+\frac{1}{n}}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|\nabla
f|^{\frac{n\sigma}{2}}dV_{g}\right)^{\frac{2}{n\sigma}}\end{split}$

So, if we divide out by $|B_{r}(x)|$ and let $p=\frac{n\sigma}{2}$, we get the
second result. ∎

###  3.3. Morrey’s Inequality

In this subsection we would like to understand what conditions are needed to
conclude a uniform Morrey inequality from a uniform Sobolev inequality. We
start by defining a local Hölder norm for functions which will be advantageous
when implying a Morrey inequality from a Poincaré inequality.

######  Definition 3.10.

Let $(M,g)$ be a Riemannian manifold. The Hölder norm of scale $R>0$ and power
$0<\gamma<1$ is defined as follows:

(62) $\displaystyle\|u\|_{C_{R}^{0,\gamma}(M,g)}=\sup_{x\in
M}|u(x)|+\sup_{x,y\in M,d_{g}(x,y)<R}\frac{|u(x)-u(y)|}{d_{g}(x,y)^{\gamma}}.$

######  Proposition 3.11.

Let $(M,g)$ be a Riemannian manifold. For any radius $R>0$ and function $u$,
we have

(63)
$\|u\|_{C_{R}^{0,\gamma}(M,g)}\leq\|u\|_{C^{0,\gamma}(M,g)}\leq\left(1+\frac{2}{R^{\gamma}}\right)\|u\|_{C_{R}^{0,\gamma}(M,g)}.$

###### Proof.

The first inequality follows quickly from the definition of
$\|u\|_{C_{R}^{0,\gamma}(M,g)}$. For the second inequality, for any $x,y$ in
$M$ we have one of two cases. Either $d_{g}(x,y)<R$ or $d_{g}(x,y)\geq R$. In
the first case, we have

(64) $\frac{|u(x)-u(y)|}{d_{g}(x,y)^{\gamma}}\leq\sup_{x,y\in
M,d_{g}(x,y)<R}\frac{|u(x)-u(y)|}{d_{g}(x,y)^{\gamma}},$

and in the second case, we have

(65)
$\frac{|u(x)-u(y)|}{d_{g}(x,y)^{\gamma}}\leq\frac{|u(x)|+|u(y)|}{R^{\gamma}}\leq\frac{2}{R^{\gamma}}\sup_{x\in
M}|u(x)|.$

∎

Now we show that a uniform, local Poincaré inequality combined with volume
growth bounds for small balls implies a uniform Morrey inequality exists. This
is the analytical backbone of both Theorem 1.5 and Theorem 1.11.

######  Theorem 3.12.

Let $(M,g)$ be an $n$-dimensional Riemannian manifold. Fix $p>1$ and $R>0$.
Assume there are constants $C_{P}$ and $C_{V}$ so that the following hold for
all $r\in(0,2R)$ and $x\in M$:

(66) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|u-u_{B_{r}(x)}|dV_{g}\leq
C_{P}\operatorname{Diam}(B_{r}(x))^{\mu}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|\nabla
u|^{p}dV_{g}\right)^{\frac{1}{p}},$

for any $u\in W^{1,p}(M,g)$ and

(67) $\displaystyle C_{V}^{-1}r^{n}\leq|B_{r}(x)|\leq C_{V}r^{n}.$

Then, for all $u\in C^{1}(M)$, points $x,y\in M$ such that $d_{g}(x,y)\leq R$,
and $\mu>\tfrac{n}{p}$, we have

(68) $\displaystyle\frac{|u(x)-u(y)|}{d_{g}(x,y)^{\frac{p\mu-n}{p}}}\leq
C(p,\mu,C_{V},C_{P})\left(\int_{M}|\nabla u|^{p}dV_{g}\right)^{\frac{1}{p}},$

and

(69) $\displaystyle\|u\|_{C_{R}^{0,\gamma}(M,g)}\leq
C(p,\mu,C_{V},C_{P})\left(\int_{M}|u|dV_{g}+\left(\int_{M}|\nabla
u|^{p}dV_{g}\right)^{\frac{1}{p}}\right),$

where $\gamma=\frac{p\mu-n}{p}$.

###### Proof.

Here we more or less follow the proof given by N. Shanmugalingam in
[Shanmugalingam-2000] where our goal is to adapt the argument to our setting
and keep careful track of the constants. Since $u\in C^{1}(M)$ we know that
every $x\in M$ is a Lebesgue point of $u$ and hence

(70) $\displaystyle\lim_{r\rightarrow
0^{+}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|u(x)-u(y)|dV_{g}(y)=0.$

Since

(71) $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}udV_{g}-u(x)\right|=\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}u(y)-u(x)dV_{g}(y)\right|\leq\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}|u(x)-u(y)|dV_{g}(y),$

we see that

(72) $\displaystyle\lim_{r\rightarrow
0^{+}}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}udV_{g}=u(x).$

Now, for $x,y\in M$ such that $d_{g}(x,y)<R$, we define nested families of
balls $\left\\{B_{i}\right\\}_{-\infty}^{\infty}$ where

(73) $\displaystyle B_{0}=B_{2d_{g}(x,y)}(x),\quad
B_{i}=B_{2^{1-i}d_{g}(x,y)}(x),\quad B_{-i}=B_{2^{1-i}d_{g}(x,y)}(y),\quad
i\in\mathbb{N}.$

Now if we let $u_{B_{i}}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{i}}udV_{g}$ then we notice that

(74) $\displaystyle|u(x)-u(y)|$ $\displaystyle=\lim_{r\rightarrow
0^{+}}\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{r}(x)}udV_{g}-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B(y,r)}udV_{g}\right|$ (75)
$\displaystyle=\lim_{i\rightarrow\infty}\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2^{1-i}d_{g}(x,y)}(x)}udV_{g}-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{2^{1-i}d_{g}(x,y)}(y)}udV_{g}\right|$ (76)
$\displaystyle=\left|\sum_{i=0}^{\infty}\left(u_{B_{i+1}}-u_{B_{i}}\right)-\sum_{i=0}^{\infty}\left(u_{B_{-i-1}}-u_{B_{-i}}\right)\right|$
(77)
$\displaystyle=\left|\sum_{-\infty}^{\infty}\left(u_{B_{i+1}}-u_{B_{i}}\right)\right|\leq\sum_{-\infty}^{\infty}|u_{B_{i+1}}-u_{B_{i}}|.$

Rewriting the terms on the right hand side of (77)

(78) $\displaystyle|u_{B_{i+1}}-u_{B_{i}}|$
$\displaystyle=\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{i+1}}udV_{g}-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{i}}udV_{g}\right|$ (79)
$\displaystyle=\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{i+1}}u-u_{B_{i}}dV_{g}\right|$ (80)
$\displaystyle\leq\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{i+1}}|u-u_{B_{i}}|dV_{g}\leq\frac{|B_{i}|}{|B_{i+1}|}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{i}}|u-u_{B_{i}}|dV_{g}.$

On the other hand, by using the volume growth assumption, the ratios of
volumes of balls satisfy

(81) $\displaystyle\frac{|B_{i}|}{|B_{i+1}|}$
$\displaystyle\leq\frac{(C_{V}2^{1-i}d_{g}(x,y))^{n}}{(C_{V}^{-1}2^{-i}d_{g}(x,y))^{n}}\leq
C_{V}^{2n}2^{n}\quad i\geq 0,$ (82) $\displaystyle\frac{|B_{i}|}{|B_{i+1}|}$
$\displaystyle\leq\frac{(C_{V}2^{2+i}d_{g}(x,y))^{n}}{(C_{V}^{-n}2^{1+i}d_{g}(x,y))^{n}}\leq
C_{V}^{2n}2^{-n}\leq C_{V}^{2n}2^{n}\quad i<0.$

Combining the above inequalities yields

(83) $\displaystyle|u_{B_{i+1}}-u_{B_{i}}|$ $\displaystyle\leq
C_{V}^{2n}2^{n}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{i}}|u-u_{B_{i}}|dV_{g}.$

Next, we apply the Poincaré inequality to find

(84) $\displaystyle|u_{B_{i+1}}-u_{B_{i}}|$
$\displaystyle\leq(2C_{V})^{2n}C_{P}\operatorname{Diam}(B_{i})^{\mu}\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{i}}|\nabla u|^{p}dV_{g}\right)^{\frac{1}{p}}$
(85)
$\displaystyle\leq\frac{(2C_{V})^{2n}C_{P}\operatorname{Diam}(B_{i})^{\mu}}{|B_{i}|^{\frac{1}{p}}}\left(\int_{B_{i}}|\nabla
u|^{p}dV_{g}\right)^{\frac{1}{p}}$ (86)
$\displaystyle\leq\frac{(2C_{V})^{2n}2^{\mu}C_{P}2^{\mu(1-|i|)}d_{g}(x,y)^{\mu}}{(C_{V}2^{1-|i|}d_{g}(x,y))^{\frac{n}{p}}}\left(\int_{B_{i}}|\nabla
u|^{p}dV_{g}\right)^{\frac{1}{p}}$ (87)
$\displaystyle\leq(2C_{V})^{2n}2^{\mu}C_{P}C_{V}^{-n/p}2^{\frac{p\mu-n}{p}(1-|i|)}d_{g}(x,y)^{\frac{p\mu-n}{p}}\left(\int_{B_{i}}|\nabla
u|^{p}dV_{g}\right)^{\frac{1}{p}}.$

Putting everything together,

(88) $\displaystyle|u(x)-u(y)|$
$\displaystyle\leq\sum_{-\infty}^{\infty}(2C_{V})^{2n}2^{\mu}C_{P}C_{V}^{-n/p}2^{\frac{p\mu-n}{p}(1-|i|)}d_{g}(x,y)^{\frac{p\mu-n}{p}}\left(\int_{B_{i}}|\nabla
u|^{p}dV_{g}\right)^{\frac{1}{p}}$ (89)
$\displaystyle\leq(2C_{V})^{2n}2^{\mu}C_{P}C_{V}^{-n/p}d_{g}(x,y)^{\frac{p\mu-n}{p}}\left(\int_{M}|\nabla
u|^{p}dV_{g}\right)^{\frac{1}{p}}\sum_{-\infty}^{\infty}2^{\frac{p\mu-n}{p}(1-|i|)}.$

Then since $p\mu>n$ we find that

(90) $\displaystyle\sum_{-\infty}^{\infty}2^{\frac{p\mu-n}{p}(1-|i|)}$
$\displaystyle=2^{\frac{p\mu-n}{p}+1}+2\sum_{i=1}^{\infty}2^{\frac{p\mu-n}{p}(1-i)}$
(91)
$\displaystyle=2^{\frac{p\mu-n}{p}+1}+2\sum_{i=0}^{\infty}\left(2^{\frac{n-p\mu}{p}}\right)^{i}=2^{\frac{p\mu-n}{p}+1}+\frac{2}{1-2^{\frac{n-p\mu}{p}}},$

and hence

(92) $\displaystyle|u(x)-u(y)|$ $\displaystyle\leq
C(p,\mu,C_{V},C_{P})d_{g}(x,y)^{\frac{p\mu-n}{p}}\left(\int_{M}|\nabla
u|^{p}dV_{g}\right)^{\frac{1}{p}}.$

Now, let $y\in B_{1}(x)$, and integrate both sides of
$|u(x)|\leq|u(y)|+|u(x)-u(y)|$ to obtain

(93)
$\displaystyle\int_{B_{1}(x)}|u(x)|dV_{g}(y)\leq\int_{B_{1}(x)}|u(y)|dV_{g}(y)+\int_{B_{1}(x)}|u(x)-u(y)|dV_{g}(y),$

or, by a slight rearrangement,

(94) $\displaystyle|u(x)|\leq\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{1}(x)}|u|dV_{g}+\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{1}(x)}|u(x)-u(y)|dV_{g}(y).$

Apply (92), (67), and the fact that $d_{g}(x,y)\leq 1$, to find that

(95) $\displaystyle|u(x)|$ $\displaystyle\leq C_{V}\int_{B_{1}(x)}|u|dV_{g}$
(96)
$\displaystyle\quad+C(p,\mu,C_{V},C_{P})\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{1}(x)}d_{g}(x,y)^{\frac{p\mu-n}{p}}\left(\int_{M}|\nabla
u|^{p}dV_{g}\right)^{\frac{1}{p}}dV_{g}(y)$ (97) $\displaystyle\leq
C_{V}\int_{M}|u|dV_{g}+C(p,\mu,C_{V},C_{P})\left(\int_{M}|\nabla
u|^{p}dV_{g}\right)^{\frac{1}{p}},$

which, together with (92), implies the desired Hölder bound. ∎

###  3.4. Application

Fix $b,\Lambda,\kappa,\bar{m}>0$ and $\tau>\tfrac{1}{2}$. Using the above
work, we can now show that members of the family
$\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$ have well behaved Sobolev spaces.
We begin by demonstrating that metric balls are well behaved.

######  Proposition 3.13.

Given $R>0$ there exists a constants $C(\Lambda)$ and $C(\kappa,R)$ such that
for any member $(M,g)$ in $\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$ we
have, for all $x\in M$ and all $r\leq R$, the following volume bounds

(98) $C(\Lambda)r^{3}\leq|B_{r}(x)|\leq C(\kappa,R)r^{3}.$

###### Proof.

We observe that the upper bound on $|B_{r}(x)|$ follows directly from the
control we have on $Rm$ applied to Corollary 2.4.

Let us now concentrate on the lower bound on volume growth. From our
assumption on asymptotic flatness, it follows that for any $x$ in $M$ we have

(99) $\lim_{r\rightarrow\infty}|B_{r}(x)|=\infty,$

and any compact subset is eventually contained in $B_{r}(x)$ for some $r$
large enough. Thus, we may apply Proposition 3.2 together with our assumption
on the Neumann isoperimetric constant of metric balls to conclude that any
compact domain in $M$ has Dirichlet isoperimetric constant uniformly bounded
away from zero, depending on $\Lambda>0$. At this point, we may apply Lemma
2.2 to get the desired result. ∎

We can now state the form of Morrey’s inequality which is valid uniformly for
the family $\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$. It is obtained by
gathering together all the information we established thus far.

######  Theorem 3.14.

For all $(M,g)$ in $\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$, points
$x_{0}$ in $M$, and functions $f\in W^{1,p}(M,g)$ with $p>3$ and $s>0$, we
have

(100) $\|f\|_{C^{0,\frac{p-3}{p}}_{R}\left(B_{s}(x_{0})\right)}\leq
C(b,\tau,\Lambda,\kappa,R,p)\left(\|f\|_{L^{1}\left(B_{s+R}(x_{0})\right)}+\|\nabla
f\|_{L^{p}\left(B_{s+R}(x_{0})\right)}\right).$

###### Proof.

The proof will be complete when we show that every member of the family
$\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$ satisfies the conditions of
Theorem 3.12, since then we can follow the proof of Theorem 3.12 word for word
with $M=B_{s+R}(x_{0})$. Let us recall that we need to establish that metric
balls around points in $B_{s}(x_{0})$ have upper and lower volume bounds, and
uniformly satisfy a $(1,p)$ Poincaré inequality.

It follows from Proposition 3.13 that $B_{r}(x)$ satisfies the required volume
bounds for $r\leq R$ and $x$ in $B_{s}(x_{0})$. In addition, it follows from
Lemma 3.9 that $B_{r}(x)$ has the desired $(1,p)$ Poincaré inequality with
$\mu=1$ for $r\leq R$ and $x\in B_{s}(x_{0})$. Thus, we get the result by
following the proof of Theorem 3.12. ∎

##  4\. Asymptotic Estimates For the Harmonic Functions

In this section, we show how ADM mass and the condition of uniform asymptotic
flatness strongly control the behavior of asymptotically linear harmonic
functions within the asymptotically flat region. Let us first recall some
information about asymptotically linear harmonic functions.

For a complete asymptotically flat $3$-manifold $(M,g)$, let $x^{i}$ for
$i=1,2,3$ be the components of the asymptotic coordinate chart. Then we call
$u^{i}$ the $i^{\text{th}}$ asymptotically linear harmonic function if $u^{i}$
satisfies

(101) $\begin{split}&\Delta_{g}u^{i}=0\\\ &u^{i}-x^{i}\in
C^{2,\gamma}_{1-\tau}.\end{split}$

In particular, the gradients of $u^{i}$ are asymptotically constant in the
sense that

(102) $|\nabla u^{i}-\partial_{x^{i}}|\in C^{1,\gamma}_{-\tau}.$

Convention: Given asymptotically flat $(M,g)$ and $r>0$, we denote by $M_{r}$
the bounded region of $M$ lying inside the coordinate sphere of radius $r$,
denoted $\mathcal{S}_{r}$. Secondly, if $\ell$ is a linear map from
$\mathbb{R}^{3}\to\mathbb{R}$ with $||d\ell||=1$, we will call any solution to
$\Delta u=0$ with $|u-\ell(x^{1},x^{2},x^{3})|\in C^{2,\gamma}_{1-\tau}(M)$,
for some $\gamma\in(0,1)$, an asymptotically linear harmonic function, noting
that we always require the linear function to have unit norm. Also, throughout
this section, we will fix parameters $b,\Lambda,\kappa,\bar{m}>0$ and
$\tau>\tfrac{1}{2}$.

It was observed in [KKL-2021] that the asymptotics (102) are uniform in the
following sense.

######  Proposition 4.1.

Let $(M,g)$ be an orientable complete $3$-dimensional $(b,\tau,\bar{m})$
asymptotically flat manifold. Assume that $H_{2}(M;\mathbb{Z})$ contains no
spherical classes and that $R_{g}\geq 0$. For a sufficiently large $r_{0}$
(depending on $b$ and $\tau$), there is a constant $C(r_{0},b,\tau,\bar{m})$
so that any asymptotically linear functions $\\{u^{i}\\}_{i=1}^{3}$ satisfy

(103) $||\nabla u^{i}-\partial_{x^{i}}||_{C^{1,\alpha}_{-\tau}(M\setminus
M_{r_{0}})}\leq C(r_{0},b,\tau,\bar{m})$

and in particular,

(104) $\left|g\left(\nabla u^{i},\nabla u^{j}\right)-\delta^{ij}\right|\leq
C(r_{0},b,\tau,\bar{m})|x|^{-\tau}$

holds on $M\setminus M_{r_{0}}$. Moreover, there is a particular collection of
asymptotically linear functions $\\{\hat{u}^{i}\\}_{i=1}^{3}$, independent of
$r_{0}$, which satisfy

(105) $\sup_{M_{r}}|\hat{u}^{i}|\leq C(r_{0},b,\tau,\bar{m})r$

for all $r>r_{0}$.

###### Proof.

Only the final statement requires proof since the other statements are
contained in [KKL-2021, Lemma 3.2]. Fix the choice of
$\\{\hat{u}^{i}\\}_{i=1}^{3}$ by requiring that the average value of
$\hat{u}^{i}$ over the annulus $M_{r_{0}}\setminus M_{1}$ vanishes. According
to [KKL-2021, Proposition 3.1], using a Sobolev inequality and the mass
formula, we have $\sup_{M_{r_{0}}}|\hat{u}^{i}|<C_{1}(r_{0},b,\tau,\bar{m})$.
For a larger radius $r>r_{0}$, one can estimate $\hat{u}^{i}$ at a point
$x\in(M_{r}\setminus M_{r_{0}})$ by integrating the gradient estimate (103)
along a radial curve connecting $x$ to $M_{r_{0}}$. The fundamental theorem of
calculus then shows that
$|\hat{u}^{i}(x)-C_{1}(r_{0},b,\tau,\bar{m})|<C_{2}(r_{0},b,\tau,\bar{m})r$
and the result follows. ∎

Next, we leverage the uniform weighted control (103) to show that the
difference between $g$ and the flat metric on $\mathbb{R}^{3}$ is small in the
average sense within the asymptotic region.

######  Proposition 4.2.

Let $(M,g)$ be an oriented complete $3$-dimensional $(b,\tau,\bar{m})$
asymptotically flat manifold with mass $m$. Assume that $H_{2}(M;\mathbb{Z})$
contains no spherical classes and that $R_{g}\geq 0$. Given a sufficiently
large $r_{0}=r_{0}(b,\tau)>0$, $r>r_{0}$, and an $\varepsilon>0$, there is a
$\delta=\delta(r_{0},b,\tau,\varepsilon)>0$ so that the following holds: If
$m(M,g)\leq\delta$, then

(106) $\int_{M_{r}\setminus M_{r_{0}}}\left|\left<\nabla u^{i},\nabla
u^{j}\right>-\delta^{ij}\right|dV_{g}<\varepsilon,$

for any $i,j\in\\{1,2,3\\}$ where $\\{u^{i}\\}_{i=1}^{3}$ are the harmonic
coordinates of $(M,g)$.

###### Proof.

Let $r_{0}>0$ be the radius given in Proposition 4.1. By (103),

(107) $|\nabla^{2}u^{i}|+|\nabla u^{i}|\leq C_{1}(b,\tau)$

holds on $M_{r_{0}}$. In light of the $|\nabla u^{i}|$’s boundedness, the mass
formula (1) implies

(108) $\int_{M\setminus M_{r_{0}}}|\nabla^{2}u^{i}|^{2}dV\leq C_{2}(b,\tau)m.$

The next step is to apply a Sobolev inequality on asymptotically flat
manifolds due to Schoen-Yau [SchoenYauPMT, Lemma 3.1],

(109) $\displaystyle\int_{M\setminus M_{r_{0}}}(\langle\nabla u^{i},\nabla
u^{j}\rangle-\delta^{ij})^{6}dV$ $\displaystyle\leq
C_{3}(b,\tau)\left(\int_{M\setminus M_{r_{0}}}|\nabla\langle\nabla
u^{i},\nabla u^{j}\rangle|^{2}\right)^{3}$ (110) $\displaystyle\leq
C_{4}(b,\tau)\sup_{M\setminus M_{r_{0}}}|\nabla
u^{i}|^{2}\left(\int_{M\setminus M_{r_{0}}}|\nabla^{2}u^{j}|^{2}dV\right)^{3}$
(111) $\displaystyle+C_{4}(b,\tau)\sup_{M\setminus M_{r_{0}}}|\nabla
u^{j}|^{2}\left(\int_{M\setminus M_{r_{0}}}|\nabla^{2}u^{i}|^{2}dV\right)^{3}$
(112) $\displaystyle\leq C_{5}(b,\tau)m^{3}.$

Also notice that the uniform asymptotic flatness condition implies that
$|M_{r}\setminus M_{r_{0}}|\leq C_{6}(b,\tau)r^{3}$. Combining this
observation with Hölder’s inequality and (109),

(113) $\displaystyle\int_{M_{r}\setminus M_{r_{0}}}|\langle\nabla u^{i},\nabla
u^{j}\rangle-\delta^{ij}|dV$ $\displaystyle\leq|M_{r}\setminus
M_{r_{0}}|^{5/6}\left(\int_{M_{r_{0}}\setminus M_{r}}|\langle\nabla
u^{i},\nabla u^{j}\rangle-\delta^{ij}|^{6}dV\right)^{1/6}$ (114)
$\displaystyle\leq C_{7}(b,\tau)r^{5/2}\sqrt{m}$

and the result follows.

∎

######  Theorem 4.3.

Let $(M,g)$ be a $(b,\tau,\bar{m})$ asymptotically flat Riemannian manifold.
If $u$ is a asymptotically linear harmonic function, then there is a
$r_{0}(b,\tau)$ so that the following holds on $M\setminus M_{r_{0}}$

(115) $\displaystyle|\nabla u|+|\nabla^{2}u|+|\nabla^{3}u|\leq C(b,\tau).$

###### Proof.

Throughout this proof we will work in $(M,g)$’s asymptotically flat coordinate
chart and all partial derivatives will be in these coordinates. Using
Proposition 4.1, we know that we can choose an $r_{0}$ large enough so that
the first and second derivatives satisfy $|\partial u|\leq C_{1}(b,\tau)$ and
$|\partial\partial u|\leq C_{2}(b,\tau)$ on $M\setminus M_{r_{0}}$. It
suffices, therefore, to estimate the third derivatives of $u$ and relate the
partial derivatives to covariant derivatives with respect to the connection
induced by $g$. Expanding the Hessian of $u$ in asymptotically flat
coordinates,

(116)
$\displaystyle\nabla^{2}u=g^{lm}\left(\partial_{l}\partial_{m}-\Gamma_{lm}^{k}\partial_{k}\right)u.$

According to the uniform asymptotics, the terms in (116) can be bounded in
$M_{r_{0}}$ to find $|\nabla^{2}u|\leq C_{3}(b,\tau)$.

To estimate the third derivatives of $u$, we apply the Böchner formula to find

(117) $\displaystyle\Delta\nabla
u=g^{lm}\left(\partial_{l}\partial_{m}-\Gamma_{lm}^{k}\partial_{k}\right)\nabla
u=Rc(\nabla u,\cdot)=:\mathbf{f},$

which we consider as a PDE for $\nabla u$. Again using the uniform
asymptotics, $|\mathbf{f}|,|g^{lm}|,|\Gamma_{lm}^{k}|\leq C_{4}(b,\tau)$ on
$M\setminus M_{r_{0}}$. Applying elliptic $L^{p}$ estimates [GT, Theorem
9.11], for every $p>1$

(118) $\displaystyle\|\partial_{i}u\|_{W^{2,p}(M\setminus
M_{r_{0}},\delta)}\leq C_{5}(b,\tau),\quad 1\leq i\leq 3.$

Since (118) holds for all $p>1$, the third partial derivatives
$|\partial\partial\nabla u|\leq C_{6}(b,\tau)$ and again since

(119)
$\displaystyle\nabla^{2}\nabla_{i}u=g^{lm}\left(\partial_{l}\partial_{m}-\Gamma_{lm}^{k}\partial_{k}\right)\nabla_{i}u,$

we find $|\nabla^{3}u|\leq C_{8}(b,\tau)$, as claimed. ∎

We conclude this section by showing that there is a strong relationship
between the regions $M_{r}$ and metric balls $B(x,r)$ for manifolds in the
class $\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$. This information is
important since the regions $M_{r}$ have well behaved boundary, but no
inherent geometric significance, and the balls $B(x,r)$ are geometrically
important, but could have wild boundaries. Before accomplishing this, we show
that a uniform volume bound on the region within a given coordinate sphere.

######  Lemma 4.4.

Let $(M,g)$ belong to $\mathcal{M}\left(b,\tau,\bar{m},\Lambda,\kappa\right)$.
Then, there exists a constant $K=K(r_{0},b,\tau,\Lambda)$ such that, for any
$r>r_{0}$, we have $|M_{r}|\leq Kr^{3}.$

###### Proof.

By the definition of $M_{r}$ and the $(b,\tau)$ asymptotic flatness of $M$ on
$M_{r_{0}}$, we know that

(120) $|\mathcal{S}_{r}|\leq C_{1}(r_{0},b,\tau)r^{2}.$

Further, $\mathcal{S}_{r}$ is a valid competitor for the isoperimetric ratio,
and so it follows from the fact that $M$ is $(\Lambda,\frac{3}{2})$ Neumann-
isoperimetrically bounded and Lemma 3.2 that

(121) $\displaystyle|M_{r}|\leq C_{2}(r_{0},b,\tau,\Lambda)r^{3}.$

∎

The following lemma allows us to simultaneously take advantage of the good
boundary behavior of the regions $M_{r}$ and the good geometric behavior of
the metric balls $B(x,r)$.

(a) $M_{r}$ is contained in $B_{d}(x)$

(b) $B_{r}(x)$ is contained $M_{\hat{r}}$

######  Lemma 4.5.

Given $b,\tau,\Lambda,r$, there is $d=d(b,\tau,\Lambda,r)$ and
$V=V(b,\tau,\Lambda,r)$ so that the following holds: If
$(M,g)\in\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$, then
$\mathrm{diam}(M_{r})\leq d$ and $|M_{r}|\leq V$.

Moreover, there is an $\hat{r}=\hat{r}(b,\tau,r)$ so that the following holds:
If $(M,g)\in\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$ and $x\in M_{1}$, then
$B_{r}(x)\subset M_{\hat{r}}$.

###### Proof.

From Lemma 4.4 it follows that $|M_{r}|$ is bounded, say by
$V=V(b,\tau,\kappa,r)$. Let $x$ and $y$ be any two points in $M_{r}$. By Lemma
2.2, we know that $|B_{r}(x)|$ and $|B_{r}(y)|$ have volume bounded below by
$C(\Lambda)r^{3}$. Let $S=\left(\frac{V+1}{C(\Lambda)}\right)^{\frac{1}{n}}$.
Then, by looking at the relevant volumes, we can conclude that

(122) $\begin{split}B_{S}(x)\bigcap\partial M_{r}\neq\emptyset,\qquad
B_{S}(y)\bigcap\partial M_{r}\neq\emptyset.\end{split}$

Since $\partial M_{r}$ is in the asymptotically flat region, it follows that
$\operatorname{Diam}(\partial M_{r})$ is uniformly bounded above, by
$C=C(b,\tau,\kappa,r)$. Putting the above together, we see that we have

(123) $\operatorname{Diam}(M_{r})\leq 2S+C=:d,$

proving the first statement of Lemma 4.5.

For the second statement, let $x\in M_{1}$ and consider the metric ball
$B_{r}(x)$. Using the uniform asymptotics, we can find a large enough
$\hat{r}=\hat{r}(b,\tau,r)$ so that any path from $\partial M_{1}$ to
$\partial M_{\hat{r}}$ has length least $r$. It follows that $B_{r}(x)\subset
M_{\hat{r}}$. ∎

##  5\. Integral Estimates For Harmonic Functions

In this section we leave the asymptotically flat setting and work with general
and fixed oriented complete $n$-dimensional Riemannian manifold $(M,g)$, and
study harmonic functions $u:M\rightarrow\mathbb{R}$, which is to say, $\Delta
u=0$. We would like to develop integral estimates for $u$. Let $\Omega\subset
M$ be bounded region with smooth boundary $\partial\Omega$ and outward
pointing unit normal vector $\nu$. Our goal in this section is to obtain
Sobolev type estimates for harmonic functions, which will ultimately lead to
Hölder control. We begin with an elementary observation which leads to $L^{2}$
control on the gradient of harmonic functions.

######  Proposition 5.1.

Let $(M,g)$ be an oriented complete $n$-dimensional Riemannian manifold, $u$ a
harmonic function, and $\Omega\subset M$ a bounded domain with smooth boundary
$\partial\Omega$ and outward pointing unit normal vector $\nu$. Then

(124) $\displaystyle\int_{\Omega}|\nabla u|^{2}dV_{g}$
$\displaystyle=\int_{\partial\Omega}u\frac{\partial u}{\partial\nu}dA_{g}.$

###### Proof.

We can integrate by parts $-u\Delta u=0$ to find

(125) $\displaystyle\int_{\Omega}|\nabla u|^{2}dV_{g}$
$\displaystyle=\int_{\partial\Omega}u\frac{\partial u}{\partial\nu}dA_{g}.$

∎

Now we obtain an $L^{2}$ estimate on the hessian of harmonic functions.

######  Theorem 5.2.

Let $(M,g)$ be an oriented complete $n$-dimensional Riemannian manifold, $u$ a
harmonic function, and $\Omega\subset M$ a bounded domain with smooth boundary
$\partial\Omega$ and outward pointing unit normal vector $\nu$. Then

(126) $\displaystyle\int_{\Omega}|\nabla^{2}u|^{2}dV_{g}$
$\displaystyle\leq\int_{\partial\Omega}|\nabla^{2}u||\nabla
u|dA_{g}+\left(\int_{\Omega}|Rc|^{p}dV_{g}\right)^{1/p}\left(\int_{\Omega}|\nabla
u|^{\frac{2p}{p-1}}dV_{g}\right)^{\frac{p-1}{p}}.$

###### Proof.

Now we calculate

(127) $\displaystyle 0$ $\displaystyle=\int_{\Omega}(\Delta u)^{2}dV_{g}$
(128)
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}\nabla_{i}\nabla_{j}u\nabla_{p}\nabla_{q}udV_{g}$
(129)
$\displaystyle=-\int_{\Omega}g^{ij}g^{pq}\nabla_{j}u\nabla_{i}\nabla_{p}\nabla_{q}udV_{g}+\int_{\partial\Omega}g^{ij}g^{pq}\nabla_{j}u\nabla_{p}\nabla_{q}u\nu_{i}dA_{g}$
(130)
$\displaystyle=-\int_{\Omega}g^{ij}g^{pq}\nabla_{j}u\nabla_{p}\nabla_{i}\nabla_{q}u-g^{ij}g^{pq}\nabla_{j}uR_{ipqk}\nabla_{k}udV_{g}+\int_{\partial\Omega}\frac{\partial
u}{\partial\nu}\Delta udA_{g}$ (131)
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}\nabla_{p}\nabla_{j}u\nabla_{i}\nabla_{q}u+Rc(\nabla
u,\nabla
u)dV_{g}-\int_{\partial\Omega}g^{ij}g^{pq}\nabla_{j}u\nabla_{i}\nabla_{q}u\nu_{p}dA_{g}$
(132)
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}\nabla_{p}\nabla_{j}u\nabla_{q}\nabla_{i}u+Rc(\nabla
u,\nabla u)dV_{g}-\int_{\partial\Omega}\nabla\nabla u(\nabla u,\nu)dA_{g}$
(133) $\displaystyle=\int_{\Omega}|\nabla^{2}u|^{2}+Rc(\nabla u,\nabla
u)dV_{g}-\int_{\partial\Omega}\nabla\nabla u(\nabla u,\nu)dA_{g},$

and so by rearranging we find

(134)
$\displaystyle\int_{\Omega}|\nabla^{2}u|^{2}dV_{g}=\int_{\partial\Omega}\nabla\nabla
u(\nabla u,\nu)dA_{g}-\int_{\Omega}Rc(\nabla u,\nabla u)dV_{g}.$

If we take the absolute value of both sides, use Cauchy-Schwarz, and then
Hölder’s inequality we find

(135) $\displaystyle\int_{\Omega}|\nabla^{2}u|^{2}dV_{g}$
$\displaystyle\leq\int_{\partial\Omega}|\nabla^{2}u||\nabla
u|dA_{g}+\int_{\Omega}|Rc||\nabla u|^{2}dV_{g}$ (136)
$\displaystyle\leq\int_{\partial\Omega}|\nabla^{2}u||\nabla
u|dA_{g}+\left(\int_{\Omega}|Rc|^{p}dV_{g}\right)^{1/p}\left(\int_{\Omega}|\nabla
u|^{\frac{2p}{p-1}}dV_{g}\right)^{\frac{p-1}{p}}.$

∎

Now we would like to estimate the integral of third derivatives of harmonic
functions in a similar way. Below and throughout, given two tensors $T_{1}$
and $T_{2}$, we will use $T_{1}*T_{2}$ as a stand in for any linear function
of $T_{1}$ and $T_{2}$.

######  Theorem 5.3.

Let $(M,g)$ be an oriented complete $n$-dimensional Riemannian manifold, $u$ a
harmonic function, and $\Omega\subset M$ a bounded domain with smooth boundary
$\partial\Omega$ and outward pointing unit normal vector $\nu$. Then

(137)
$\displaystyle\begin{split}\int_{\Omega}|\nabla^{3}u|^{2}dV_{g}&=\int_{\Omega}2Rc(\nabla
u,\Delta\nabla u)-Rc^{2}(\nabla u,\nabla u)-Rc*\nabla\nabla u*\nabla\nabla
udV_{g}\\\ &+\int_{\Omega}Rm*\nabla\nabla\nabla u*\nabla u-Rm*\nabla\nabla
u*\nabla\nabla udV_{g}\\\ &+\int_{\partial\Omega}\nabla\nabla\nabla
u(\nabla\nabla u,\nu)-\nabla\nabla u(\Delta\nabla u,\nu)dA_{g},\end{split}$

where $Rm$ denotes the Riemann curvature tensor of $(M,g)$.

###### Proof.

Now we notice that since $u$ is harmonic we can also deduce that $\nabla\Delta
u=0$ and calculate

(138) $\displaystyle 0$ $\displaystyle=\int_{\Omega}|\nabla\Delta
u|^{2}dV_{g}$ (139)
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}g^{lm}\nabla_{i}\nabla_{p}\nabla_{q}u\nabla_{j}\nabla_{l}\nabla_{m}udV_{g}$
(140)
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}g^{lm}\left(\nabla_{p}\nabla_{i}\nabla_{q}u-g^{rs}R_{ipqr}\nabla_{s}u\right)\left(\nabla_{l}\nabla_{j}\nabla_{m}u-g^{vw}R_{jlmv}\nabla_{w}u\right)dV_{g}$
(141)
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}g^{lm}\left(\nabla_{p}\nabla_{i}\nabla_{q}u\nabla_{l}\nabla_{j}\nabla_{m}u-2g^{rs}R_{ipqr}\nabla_{s}u\nabla_{l}\nabla_{j}\nabla_{m}u+g^{vw}R_{jlmv}\nabla_{w}ug^{rs}R_{ipqr}\nabla_{s}u\right)dV_{g}$
(142)
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}g^{lm}\nabla_{p}\nabla_{q}\nabla_{i}u\nabla_{l}\nabla_{m}\nabla_{j}u-2g^{ij}g^{lm}g^{rs}R_{ir}\nabla_{s}u\nabla_{l}\nabla_{m}\nabla_{j}u+Rc^{2}(\nabla
u,\nabla u)dV_{g}$ (143)
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}g^{lm}\nabla_{p}\nabla_{q}\nabla_{i}u\nabla_{l}\nabla_{m}\nabla_{j}u-2Rc(\nabla
u,\Delta\nabla u)+Rc^{2}(\nabla u,\nabla u)dV_{g}$ (144)
$\displaystyle=I+II+III.$

For $III$ we note that

(145) $\displaystyle Rc^{2}(\nabla u,\nabla
u)=g^{ij}g^{pq}g^{rs}R_{ip}R_{jr}\nabla_{q}u\nabla_{s}u,$

and for $II$ we note that

(146) $\displaystyle Rc(\nabla u,\Delta\nabla
u)=g^{ij}g^{lm}g^{rs}R_{ir}\nabla_{s}u\nabla_{l}\nabla_{m}\nabla_{j}u.$

Now we continue the calculation by focusing on $I$

(147) $\displaystyle I$
$\displaystyle=-\int_{\Omega}g^{ij}g^{pq}g^{lm}\nabla_{q}\nabla_{i}u\nabla_{p}\nabla_{l}\nabla_{m}\nabla_{j}udV_{g}+\int_{\partial\Omega}g^{ij}g^{pq}g^{lm}\nabla_{q}\nabla_{i}u\nabla_{l}\nabla_{m}\nabla_{j}u\nu_{p}dA_{g}$
(148)
$\displaystyle=-\int_{\Omega}g^{ij}g^{pq}g^{lm}\nabla_{q}\nabla_{i}u\nabla_{p}\nabla_{l}\nabla_{m}\nabla_{j}udV_{g}+\int_{\partial\Omega}\nabla\nabla
u(\Delta\nabla u,\nu)dA_{g}=A+B.$

Now we continue with the first term

(149) $\displaystyle A$
$\displaystyle=-\int_{\Omega}g^{ij}g^{pq}g^{lm}\nabla_{q}\nabla_{i}u\left(\nabla_{l}\nabla_{p}\nabla_{m}\nabla_{j}u-g^{rs}R_{plmr}\nabla_{s}\nabla_{j}u-g^{rs}R_{pljr}\nabla_{m}\nabla_{s}u\right)dV_{g}$
(150)
$\displaystyle=\int_{\Omega}-g^{ij}g^{pq}g^{lm}\nabla_{q}\nabla_{i}u\nabla_{l}\nabla_{p}\nabla_{m}\nabla_{j}u+Rc*\nabla\nabla
u*\nabla\nabla u+Rm*\nabla\nabla u*\nabla\nabla udV_{g}$ (151)
$\displaystyle=C+D+E.$

For this calculation we are using

(152) $\displaystyle Rc*\nabla\nabla u*\nabla\nabla u$
$\displaystyle=g^{ij}g^{pq}g^{rs}R_{pr}\nabla_{s}\nabla_{j}u\nabla_{q}\nabla_{i}u,$
(153) $\displaystyle Rm*\nabla\nabla u*\nabla\nabla u$
$\displaystyle=g^{ij}g^{lm}g^{rs}R_{pljr}\nabla_{m}\nabla_{s}u\nabla_{q}\nabla_{i}u.$

Now we continue with the term $C$ to find

(154) $\displaystyle C$
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}g^{lm}\nabla_{l}\nabla_{q}\nabla_{i}u\nabla_{p}\nabla_{m}\nabla_{j}udV_{g}-\int_{\partial\Omega}g^{ij}g^{pq}g^{lm}\nabla_{q}\nabla_{i}u\nabla_{p}\nabla_{m}\nabla_{j}u\nu_{l}dA_{g}$
(155)
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}g^{lm}\nabla_{l}\nabla_{q}\nabla_{i}u\nabla_{p}\nabla_{m}\nabla_{j}udV_{g}-\int_{\partial\Omega}g^{ij}g^{pq}g^{lm}\nabla_{q}\nabla_{i}u\nabla_{p}\nabla_{j}\nabla_{m}u\nu_{l}dA_{g}$
(156)
$\displaystyle=\int_{\Omega}g^{ij}g^{pq}g^{lm}\nabla_{l}\nabla_{q}\nabla_{i}u\left(\nabla_{m}\nabla_{p}\nabla_{j}u-g^{rs}R_{pmjr}\nabla_{s}u\right)dV_{g}-\int_{\partial\Omega}\nabla\nabla\nabla
u(\nabla\nabla u,\nu)dA_{g}$ (157)
$\displaystyle=\int_{\Omega}|\nabla^{3}u|^{2}-g^{ij}g^{pq}g^{lm}g^{rs}R_{pmjr}\nabla_{l}\nabla_{q}\nabla_{i}u\nabla_{s}udV_{g}-\int_{\partial\Omega}\nabla\nabla\nabla
u(\nabla\nabla u,\nu)dA_{g}$ (158)
$\displaystyle=\int_{\Omega}|\nabla^{3}u|^{2}-Rm*\nabla\nabla\nabla u*\nabla
udV_{g}-\int_{\partial\Omega}\nabla\nabla\nabla u(\nabla\nabla u,\nu)dA_{g}.$

In the calculation above we are using

(159) $\displaystyle Rm*\nabla\nabla\nabla u*\nabla
u=g^{ij}g^{pq}g^{lm}g^{rs}R_{pmjr}\nabla_{l}\nabla_{q}\nabla_{i}u\nabla_{s}u.$

Putting everything together we find

(160)
$\displaystyle\begin{split}\int_{\Omega}|\nabla^{3}u|^{2}dV_{g}&=\int_{\Omega}2Rc(\nabla
u,\Delta\nabla u)-Rc^{2}(\nabla u,\nabla u)-Rc*\nabla\nabla u*\nabla\nabla
udV_{g}\\\ &+\int_{\Omega}Rm*\nabla\nabla\nabla u*\nabla u-Rm*\nabla\nabla
u*\nabla\nabla udV_{g}\\\ &+\int_{\partial\Omega}\nabla\nabla\nabla
u(\nabla\nabla u,\nu)-\nabla\nabla u(\Delta\nabla u,\nu)dA_{g}\end{split}$

∎

The terms on the right hand side of the estimate in Theorem 5.3 involving a
square of the Hessian of $u$ are concerning since we should only expect to
have integral control on $|\nabla^{2}u|^{2}$ and we will need to use Hölder’s
inequality. To this end we prove the following proposition which will be used
to overcome this issue. The proof of the following proposition is inspired by
the calculations in [Strzelecki-06]. However, we only need the following weak
conclusion, whereas the result obtained in [Strzelecki-06] is much more
refined. Luckily, this significantly simplifies the proof.

######  Proposition 5.4.

Let $(M,g)$ be an oriented complete $n$-dimensional Riemannian manifold, $u$ a
harmonic function, and $\Omega\subset M$ a bounded domain with smooth boundary
$\partial\Omega$. If $f\in C^{3}(\Omega)$, then we have that

(161)
$\|\nabla^{2}f\|^{2}_{L^{3}}\leq\max\left(1,\|\nabla^{2}f\|^{2}_{L^{3}}\right)\leq\max\left(1,5\|\nabla
f\|_{L^{6}}\|\nabla^{3}f\|_{L^{2}}+\int_{\partial\Omega}\left|\nabla^{2}f\right|^{2}|\nabla
f|dA_{g}\right).$

###### Proof.

The proof is the result of a nice integration by parts. In particular, we have
that

(162) $\int_{\Omega}|\nabla^{2}f|^{3}dV_{g}=-\int_{\Omega}\left<\nabla
f,\nabla^{*}(|\nabla^{2}f|\nabla^{2}f)\right>dV_{g}+\int_{\partial\Omega}\left|\nabla^{2}f\right|\nabla^{2}f\left(\nabla
f,\nu\right)dA_{g}$

where $\nu$ is the unit outward normal to $\partial\Omega$. Taking absolute
values on the right hand side, the first term can be estimated as follows:

(163) $\displaystyle\left|\left<\nabla
f,\nabla^{*}(|\nabla^{2}f|\nabla^{2}f)\right>\right|\leq|\nabla
f|\left(4|\nabla^{2}f||\nabla^{3}f|+|\nabla^{2}f||\nabla^{3}f|\right)=5|\nabla
f||\nabla^{2}f||\nabla^{3}f|.$

Hence we have

(164) $\displaystyle\int_{\Omega}|\nabla^{2}f|^{3}dV_{g}\leq
5\int_{\Omega}|\nabla
f||\nabla^{2}f||\nabla^{3}f|dV_{g}+\int_{\partial\Omega}\left|\nabla^{2}f\right|^{2}|\nabla
f|dA_{g}.$

We may now use the generalized Hölder’s inequality with exponents $6$, $3$,
and $2$, respectively to get

(165) $\int_{\Omega}|\nabla^{2}f|^{3}dV_{g}\leq 5\|\nabla
f\|_{L^{6}}\|\nabla^{2}f\|_{L^{3}}\|\nabla^{3}f\|_{L^{2}}+\int_{\partial\Omega}\left|\nabla^{2}f\right|^{2}|\nabla
f|dA_{g}.$

Dividing out both sides of the above by $\|\nabla^{2}f\|_{L^{3}}$ gives us
that

(166) $\|\nabla^{2}f\|^{2}_{L^{3}}\leq 5\|\nabla
f\|_{L^{6}}\|\nabla^{3}f\|_{L^{2}}+\|\nabla^{2}f\|_{L^{3}}^{-1}\int_{\partial\Omega}\left|\nabla^{2}f\right|^{2}|\nabla
f|dA_{g}.$

Now, if $\|\nabla^{2}f\|_{L^{3}}\geq 1$ then we find that

(167) $\|\nabla^{2}f\|^{2}_{L^{3}}\leq 5\|\nabla
f\|_{L^{6}}\|\nabla^{3}f\|_{L^{2}}+\int_{\partial\Omega}\left|\nabla^{2}f\right|^{2}|\nabla
f|dA_{g},$

and if not, then $\|\nabla^{2}f\|_{L^{3}}<1$, which is a reasonable bound in
its own right. ∎

By using the previous proposition we are now able to overcome the issue of a
quadratic Hessian term in order to obtain an $L^{2}$ estimate on the third
derivative of $u$ in terms of quantities we expect to be able to estimate.

######  Theorem 5.5.

Let $(M,g)$ be an oriented complete $n$-dimensional Riemannian manifold, $u$ a
harmonic function, and $\Omega\subset M$ a bounded domain with smooth boundary
$\partial\Omega$ and outward pointing unit normal vector $\nu$. Then, for any
$p>1$,

(168) $\displaystyle\begin{split}\int_{\Omega}|\nabla^{3}u|^{2}dV_{g}&\leq
C\left(\int_{\Omega}|Rm|^{2p}+|Rc|^{2p}dV_{g}\right)^{1/p}\left(\int_{\Omega}|\nabla
u|^{\frac{2p}{p-1}}dV_{g}\right)^{\frac{p-1}{p}}\\\
&+C\max\left(2\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}},\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{2}{3}}\|\nabla
u\|_{L^{6}(\Omega)}^{2}\right)\\\
&+C\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}}\max\left(1,\int_{\partial\Omega}|\nabla^{2}u|^{2}|\nabla
u|dA_{g}\right)\\\ &+\int_{\partial\Omega}\nabla\nabla\nabla u(\nabla\nabla
u,\nu)-\nabla\nabla u(\Delta\nabla u,\nu)dA_{g}\end{split}$

###### Proof.

If we apply Cauchy-Schwarz to the result from Theorem 5.3, we find

(169)
$\displaystyle\begin{split}\int_{\Omega}|\nabla^{3}u|^{2}dV_{g}&\leq\int_{\Omega}2|Rc||\nabla
u||\nabla^{3}u|+|Rc|^{2}|\nabla u|^{2}+|Rc||\nabla^{2}u|^{2}dV_{g}\\\
&+\int_{\Omega}|Rm||\nabla^{3}u||\nabla u|+|Rm||\nabla^{2}u|^{2}dV_{g}\\\
&+\int_{\partial\Omega}\nabla\nabla\nabla u(\nabla\nabla u,\nu)-\nabla\nabla
u(\Delta\nabla u,\nu)dA_{g}.\end{split}$

Applying Hölder’s inequality with $p>1$, we find

(170) $\displaystyle\int_{\Omega}|Rc|^{2}|\nabla
u|^{2}dV_{g}\leq\left(\int_{\Omega}|Rc|^{2p}dV_{g}\right)^{1/p}\left(\int_{\Omega}|\nabla
u|^{\frac{2p}{p-1}}dV_{g}\right)^{\frac{p-1}{p}}.$

We can deal with the remaining terms which are not quadratic in the Hessian of
$u$ by applying Young’s inequality with $\varepsilon$ and Hölder’s inequality
with $p>1$ to find

(171) $\displaystyle\int_{\Omega}$ $\displaystyle 2|Rc||\nabla^{3}u||\nabla
u|+|Rm||\nabla^{3}u||\nabla u|dV_{g}$ (172)
$\displaystyle\leq\int_{\Omega}2\left(C(\varepsilon)|Rc|^{2}|\nabla
u|^{2}+\varepsilon|\nabla^{3}u|^{2}\right)+(C(\varepsilon)|Rm|^{2}|\nabla
u|^{2}+\varepsilon|\nabla^{3}u|^{2}dV_{g}$ (173)
$\displaystyle=\int_{\Omega}C^{\prime}(\varepsilon)(|Rc|^{2}|\nabla
u|^{2}+|Rm|^{2}|\nabla u|^{2})+5\varepsilon|\nabla^{3}u|^{2}dV_{g}$ (174)
$\displaystyle\leq
C^{\prime}(\varepsilon)\left(\int_{\Omega}|Rc|^{2p}+|Rm|^{2p}dV_{g}\right)^{1/p}\left(\int_{\Omega}|\nabla
u|^{\frac{2p}{p-1}}dV_{g}\right)^{\frac{p-1}{p}}+\int_{\Omega}5\varepsilon|\nabla^{3}u|^{2}dV_{g}.$

For the terms which are quadratic in the Hessian we first apply Hólder’s
inequality with $p=\frac{3}{2}$ to find

(175) $\displaystyle\int_{\Omega}(|Rm|+|Rc|)|\nabla^{2}u|^{2}dV_{g}$
$\displaystyle\leq\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}}\left(\int_{\Omega}|\nabla^{2}u|^{3}dV_{g}\right)^{\frac{2}{3}}$
(176)
$\displaystyle\leq\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}}\max\left(1,\|\nabla^{2}u\|_{L^{3}}^{2}\right),$

and now the problem is that one cannot expect to have control on
$\|\nabla^{2}u\|_{L^{3}}^{2}$ and hence we will control this term using
$\|\nabla u\|_{L^{6}}$ and $\|\nabla^{3}u\|_{L^{2}}$ as follows.

From Proposition 5.4, we have

(177)
$\begin{split}\max\left(1,\|\nabla^{2}u\|_{L^{3}(\Omega)}^{2}\right)&\leq\max\left(1,5\|\nabla
u\|_{L^{6}(\Omega)}\|\nabla^{3}u\|_{L^{2}(\Omega)}+\int_{\partial\Omega}\left|\nabla^{2}u\right|^{2}|\nabla
u|\right)dA_{g},\end{split}$

and hence

(178) $\displaystyle\int_{\Omega}(|Rm|+|Rc|)|\nabla\nabla u|^{2}dV_{g}$ (179)
$\displaystyle\leq\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}}\max\left(1,5\|\nabla
u\|_{L^{6}}\|\nabla^{3}u\|_{L^{2}}+\int_{\partial\Omega}\left|\nabla^{2}u\right|^{2}|\nabla
u|dA_{g}\right)$ (180)
$\displaystyle\leq\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}}\left(\max\left(1,5\|\nabla
u\|_{L^{6}}\|\nabla^{3}u\|_{L^{2}}\right)+\max\left(1,\int_{\partial\Omega}\left|\nabla^{2}u\right|^{2}|\nabla
u|dA_{g}\right)\right).$

Now, we may apply Young’s inequality to the first term in the last expression
to find

(181)
$\displaystyle\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}}\max\left(1,5\|\nabla
u\|_{L^{6}}\|\nabla^{3}u\|_{L^{2}}\right)$ (182)
$\displaystyle\leq\max\left(\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}},C\|\nabla
u\|_{L^{6}}^{2}\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{2}{3}}+\frac{1}{2}\|\nabla^{3}u\|_{L^{2}}^{2}\right),$
(183)
$\displaystyle\leq\max\left(\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}},C\|\nabla
u\|_{L^{6}}^{2}\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{2}{3}}\right)$
(184)
$\displaystyle+\max\left(\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}},\frac{1}{2}\|\nabla^{3}u\|_{L^{2}}^{2}\right).$

Notice that if $\|\nabla^{3}u\|_{L^{2}}^{2}\leq
2\left(\int_{\Omega}|Rm|^{3}+|Rc|^{3}dV_{g}\right)^{\frac{1}{3}}$ then we are
done and if not then we also have our desired bound by subtracting off
$\tfrac{1}{2}\|\nabla^{3}u\|_{L^{2}(\Omega)}$ from both sides of Equation
(169) and then keeping track of the sequence of inequalities which follow from
the above equation. ∎

##  6\. Mass Estimates For Harmonic Functions

In this section our goal is to use the estimates of the previous section
combined with the mass formula to obtain Hölder control on Harmonic functions
and their inner products. One should notice that each of the estimates of the
previous section involve boundary terms which involve control in the
asymptotically flat region and so we require control on these terms. Since the
boundaries of the $M_{r}$ are in the asymptotically flat region and well
understood, in this section we will apply the results of Section 5 to
asymptotically linear harmonic functions over the regions $M_{r}$.

Convention: In the following statement and throughout this section, $r_{0}$
denotes the radius provided by the conclusions of Propositions 4.2 and 4.3.
Also, throughout this section, we will fix parameters
$b,\Lambda,\kappa,\bar{m}>0$ and $\tau>\tfrac{1}{2}$.

######  Theorem 6.1.

Let $(M,g)$ be a member of the family
$\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$ and let $u$ be an asymptotically
linear harmonic function. Then, for any $r>r_{0}$, we have

(185) $\displaystyle\left(\int_{M_{r}}|\nabla u|^{6}dV_{g}\right)^{1/6}$
$\displaystyle\leq C(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m}),$ (186)
$\displaystyle\left(\int_{M_{r}}|\nabla^{2}u|^{2}dV_{g}\right)^{1/2}$
$\displaystyle\leq C(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m}).$

###### Proof.

First, assume that $u$ is an asymptotically linear harmonic function from
Proposition 4.1. By combining Proposition 4.1 with Proposition 5.1 we find
that

(187) $\displaystyle\|\nabla u\|_{L^{2}(M_{r})}\leq
C_{1}(r,r_{0},b,\tau,\kappa,\Lambda).$

Now, by the mass formula, see Equation (1), we know

(188) $\displaystyle 4\int_{M_{r}}|\nabla\sqrt{|\nabla
u|}|^{2}dV_{g}\leq\int_{M_{r}}\frac{|\nabla^{2}u|^{2}}{|\nabla u|}dV_{g}\leq
16\pi m\leq 16\pi\bar{m}.$

It follows from the fact that $M_{r}$ can be contained in $B_{2d}(x)$ for some
$x\in M_{1}$ and $d=d(r,r_{0},b,\tau,\bar{m})$, the Sobolev inequality on
metric balls, Theorem 3.7, and Equation (187) that we have

(189) $\displaystyle\left(\int_{M_{r}}|\nabla u|^{3}dV_{g}\right)^{1/6}\leq
C_{2}(r,b,\tau,\kappa,\Lambda,\bar{m}).$

Similarly, by combining Theorem 4.3 and Theorem 5.2 with $p=3$, we find that

(190) $\displaystyle\|\nabla^{2}u\|_{L^{2}(M_{r})}\leq
C_{3}(r,b,\tau,\kappa,\Lambda,\bar{m}).$

Hence by applying the fact that $|\nabla|\nabla u||^{2}\leq|\nabla^{2}u|^{2}$
along with Theorem 3.7 we find that

(191) $\displaystyle\|\nabla u\|_{L^{6}(M_{r})}\leq
C_{4}(r,b,\tau,\kappa,\Lambda,\bar{m}).$

Now, since any asymptotically linear harmonic function differs from the
harmonic functions of Proposition 4.1 by subtracting a constant, we obtain the
desired result. ∎

Now we move to a bound on the Hessian of asymptotically linear harmonic
functions in terms of the mass.

######  Theorem 6.2.

Let $(M,g)$ be in $\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$ and let $u$ be
an asymptotically linear harmonic function. Then, for $r>r_{0}$, $2\leq q\leq
6$, and $\theta=\frac{6-q}{2q}$, the following hold

(192) $\displaystyle\left(\int_{M_{r}}|\nabla^{3}u|^{2}dV_{g}\right)^{1/2}$
$\displaystyle\leq C(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m}),$ (193)
$\displaystyle\left(\int_{M_{r}}|\nabla^{2}u|^{q}dV_{g}\right)^{1/q}$
$\displaystyle\leq
C(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m})m^{\frac{\theta}{2}}.$

###### Proof.

If we combine Theorem 4.3, Theorem 6.1, Theorem 3.8, Theorem 5.5 with
$p=\frac{3}{2}$, Remark 1.4, and the fact that $M_{r}$ can be contained in
$B_{2d}(x)$ for some $x\in M_{1}$ and $d=d(r,r_{0},b,\tau,\bar{m})$, then we
find that

(194) $\displaystyle\|\nabla^{3}u\|_{L^{2}(M_{r})}\leq
C_{1}(r,r_{0},b,\tau,\kappa,\Lambda).$

Now the fact that $|\nabla|\nabla^{2}u||^{2}\leq|\nabla^{3}u|^{2}$ together
with Theorem 3.8 implies that

(195) $\displaystyle\left(\int_{M_{r}}|\nabla^{2}u|^{6}dV_{g}\right)^{1/6}$
$\displaystyle\leq
C_{2}(r,r_{0},b,\tau,\kappa,\Lambda)\left(\int_{M_{r}}|\nabla^{3}u|^{2}dV_{g}\right)^{1/2}$
(196) $\displaystyle\leq C_{3}(r,r_{0},b,\tau,\kappa,\Lambda).$

At this point we may use the mass formula with an $L^{q}$ interpolation
inequality where $2<q<6$ and $\frac{1}{q}=\frac{\theta}{2}+\frac{1-\theta}{6}$
to see that

(197) $\displaystyle\left(\int_{M_{r}}|\nabla^{2}u|^{q}dV_{g}\right)^{1/q}$
$\displaystyle\leq\left(\int_{M_{r}}|\nabla^{2}u|^{2}dV_{g}\right)^{\frac{\theta}{2}}\left(\int_{M_{r}}|\nabla^{2}u|^{6}dV_{g}\right)^{\frac{1-\theta}{6}}$
(198) $\displaystyle\leq
C_{4}(r,r_{0},b,\tau,\kappa,\Lambda)m^{\frac{\theta}{2}}.$

∎

The above gives us an $L^{\infty}$ bound on $|\nabla u|$, which will be of
great help later.

######  Corollary 6.3.

Let $(M^{3},g)$ be in $\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$ and let $u$
be an asymptotically linear harmonic function. Then, for any $r>r_{0}$, we
have

(199) $\|\nabla u\|_{C^{0}(M_{r})}\leq
C(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m})$

###### Proof.

To begin, we may use Lemma 4.5 to find a $d=d(b,\tau,\Lambda,r)$ and $x_{0}$
in $M_{1}$ such that $B_{d}=B_{d}(x_{0})$ contains $M_{r}$, and we may pick
$s=s(b,\tau,d+R)$ such that $B_{d+R}=B_{d+R}(x_{0})$ is contained in $M_{s}$.
Then, from Theorem 3.14 we have

(200) $\|\nabla u\|_{C_{R}^{0,\frac{p-n}{p}}(B_{d})}\leq
C_{1}(b,\tau,\kappa,R,p)\left(\|\nabla
u\|_{L^{1}(B_{d+R})}+\|\nabla^{2}u\|_{L^{p}(B_{d+R})}\right).$

Now, using Hölder’s inequality and the results of Theorem 6.2 with $p=6$, we
see that

(201) $\|\nabla u\|_{C_{R}^{0,\frac{1}{2}}(B_{d})}\leq
C_{2}(b,\tau,\kappa,R,p)\left(|B_{d+R}|^{\frac{5}{6}}\|\nabla
u\|_{L^{6}(B_{d+R})}+\|\nabla^{2}u\|_{L^{6}(B_{d+R})}\right)$

From Theorem 6.2, we see that the above is bounded by a constant
$C_{3}(r,r_{0},b,\tau,\bar{m},\kappa,\Lambda)$. Of course, this bounds the
supremum of $|\nabla u|$ independently of $R$, and the result follows. ∎

Having established control on the $L^{p}$ norm of $\nabla^{2}u^{i}$ and on the
$C^{0}$ norm of $|\nabla u^{i}|$, we may begin to analyze the inner products
of the gradients of $u^{i}$.

######  Theorem 6.4.

Let $(M^{3},g)$ be in $\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$ and let
$u^{i},u^{j}$ be a pair of its asymptotically linear harmonic functions. Then,
for $r>r_{0}$, $1\leq q\leq 6$, and $\theta=\tfrac{6-q}{2q}$, we have

(202) $\displaystyle\left(\int_{M_{r}}|\nabla g(\nabla u^{i},\nabla
u^{j})|^{q}dV_{g}\right)^{\frac{1}{q}}\leq
C(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m},q)m^{\frac{\theta}{2}}.$

###### Proof.

We calculate that

(203) $\int_{M_{r}}|\nabla g(\nabla u^{i},\nabla u^{j})|^{q}dV_{g}\leq
C_{1}(q)\int_{M_{r}}|\nabla^{2}u^{i}|^{q}|\nabla
u^{j}|^{q}+|\nabla^{2}u^{j}|^{q}|\nabla u^{i}|^{q}dV_{g}.$

Since both $|\nabla u^{i}|$ and $|\nabla u^{j}|$ are bounded above by some
constant $C_{2}(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m})$ by Corollary 6.3, it
follows that we have

(204) $\int_{M_{r}}|\nabla g(\nabla u^{i},\nabla u^{j})|^{q}dV_{g}\leq
C_{1}(q)C_{2}(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m})^{q}\int_{M_{r}}|\nabla^{2}u^{i}|^{q}+|\nabla^{2}u^{j}|^{q}dV_{g}.$

By Theorem 6.2 and the above, we see that

(205) $\|\nabla g(\nabla u^{i},\nabla u^{j})\|_{L^{q}}\leq
2^{\frac{1}{q}}C_{1}(q)C_{2}(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m})m^{\frac{6-q}{4q}}.$

∎

Next we apply Theorem 6.4 with our Morrey’s inequality to show the inner
products of the gradients converge to their average.

######  Corollary 6.5.

Let $(M^{3},g)$ be in $\mathcal{M}(b,\tau,\bar{m},\Lambda,\kappa)$ and let
$u^{i},u^{j}$ be any two asymptotically linear harmonic functions. If $x\in
M$, $r>r_{0}$, and $R,d>0$ are such that $B_{d+R}(x)\subset M_{r}$, then

(206) $\displaystyle\|g(\nabla u^{i},\nabla u^{j})-g(\nabla u^{i},\nabla
u^{j})_{B_{d+R}(x)}\|_{C_{R}^{0,\gamma}(B_{d}(x))}\leq
C(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m})m^{\frac{1}{8}}.$

###### Proof.

If we choose $q=4$ and apply Theorem 3.14 to Theorem 6.4, where
$\theta=\frac{1}{4}$, we find

(207) $\displaystyle\|g(\nabla u^{i},\nabla u^{j})-g(\nabla u^{i},\nabla
u^{j})_{B_{d+R}(x)}\|_{C_{R}^{0,\gamma}(B_{d}(x))}$ (208) $\displaystyle\leq
C_{1}(r,b,\tau,\kappa,\Lambda)\int_{B_{d+R}(x)}|g(\nabla u^{i},\nabla
u^{j})-g(\nabla u^{i},\nabla u^{j})_{B_{d+R}(x)}|dV_{g}$ (209)
$\displaystyle+C_{1}(r,b,\tau,\kappa,\Lambda)\left(\int_{B_{d+R}(x)}|\nabla
g(\nabla u^{i},\nabla u^{j})|^{4}dV_{g}\right)^{1/4}$ (210) $\displaystyle\leq
C_{2}(r,b,\tau,\kappa,\Lambda)\left(|B_{d+R}(x)|\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{d+R}(x)}|\nabla g(\nabla u^{i},\nabla
u^{j})|^{4}dV_{g}\right)^{1/4}+m^{\frac{1}{8}}\right)$ (211)
$\displaystyle\leq
C_{3}(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m})m^{\frac{1}{8}},$

where we use that $B_{d+R}(x)\subset M_{r}$ and Lemma 4.4 in the last line. ∎

Now the goal is to put the estimates of the previous sections together in
order to prove Theorem 1.5. The first main step will be to show that since the
average integral of $g(\nabla u^{i},\nabla u^{j})$ over a large coordinate
annulus is close to $\delta^{ij}$ we can show that the average over $M_{r}$ is
close as well. Then we notice that the previous estimate implies that
$(u^{1},u^{2},u^{3})$ are actually a global coordinate system on $M$, which
will finish the proof.

###### Proof of Theorem 1.5.

Let $r_{0}$ be given by Proposition 4.2 and suppose $r>r_{0}$ is a radius,
which will be chosen successively larger in the arguments below. Denote
$\mathcal{A}_{r_{0},r}=M_{r}\setminus M_{r_{0}}$, and then consider

(212) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{M_{r}}|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}|dV_{g}$ (213)
$\displaystyle=\frac{1}{|M_{r}|}\left(\int_{\mathcal{A}_{r_{0},r}}|g(\nabla
u^{i},\nabla u^{j})-\delta^{ij}|dV_{g}+\int_{M_{r_{0}}}|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}|dV_{g}\right)$ (214)
$\displaystyle\leq\frac{1}{|\mathcal{A}_{r_{0},r}|}\left(\int_{\mathcal{A}_{r_{0},r}}|g(\nabla
u^{i},\nabla u^{j})-\delta^{ij}|dV_{g}+\max_{1\leq i\leq
3}\left(\int_{M_{r_{0}}}|\nabla u^{i}|^{2}dV_{g}\right)+|M_{r_{0}}|\right)$
(215) $\displaystyle\leq\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{\mathcal{A}_{r_{0},r}}|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}|dV_{g}+\frac{|M_{r_{0}}|}{|\mathcal{A}_{r_{0},r}|}C(r_{0},b,\tau,\kappa,\Lambda),$

where we have used Theorem 6.1 and Proposition 4.2 in the last line. We know
that since $\mathcal{A}_{r_{0},r}$ is contained in the asymptotically flat
region of $M$ that $|\mathcal{A}_{r_{0},r}|\geq
C_{1}(b,\tau)(r^{3}-r_{0}^{3})$ and by Lemma 4.4 we know $|M_{r_{0}}|\leq
C_{2}(r_{0},b,\tau,\Lambda)r_{0}^{3}$ hence

(216) $\displaystyle\begin{split}\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{M_{r}}|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}|dV_{g}&\leq\frac{1}{C_{1}(b,\tau)(r^{3}-r_{0}^{3})}\int_{\mathcal{A}_{r_{0},r}}|g(\nabla
u^{i},\nabla u^{j})-\delta^{ij}|dV_{g}\\\
{}&\qquad+C_{2}(r_{0},b,\tau,\Lambda)\frac{r_{0}^{3}}{(r^{3}-r_{0}^{3})}.\end{split}$

Notice that, by Proposition 4.2, choose $m$ small enough so that the first
term on the right side of (216) is smaller than $\frac{\varepsilon}{2}$. It
follows that we may choose $r$ large enough so that the second term on the
right side of (216) is less than $\frac{\varepsilon}{2}$. So, given an
$\varepsilon>0$, there is an $r_{1}=r_{1}(r_{0},b,\tau,\Lambda,\varepsilon)$
and $m_{1}=m_{1}(r_{0},b,\tau,\varepsilon)$ such that when $r>r_{1}$ and
$m\leq m_{1}$, we have

(217) $\displaystyle\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{M_{r}}|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}|dV_{g}$ $\displaystyle\leq\varepsilon.$

Fix a point $x\in M_{1}$. By the first statement in Lemma 4.5 there is a
radius $d=d(b,\tau,\Lambda,r)$ so that $M_{r}\subset B_{d}(x)$. We estimate

(218) $\displaystyle\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{d+1}(x)}g(\nabla u^{i},\nabla
u^{j})dV_{g}-\delta^{ij}\right|$ (219)
$\displaystyle=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{M_{r}}|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}|dV_{g}$ (220)
$\displaystyle\quad+\left(\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{d+1}(x)}|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}|dV_{g}-\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{M_{r}}|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}|dV_{g}\right)$ (221)
$\displaystyle\leq\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{M_{r}}|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}|dV_{g}$ (222)
$\displaystyle\quad+\frac{1}{|B_{d+1}(x)|}\int_{B_{d+1}(x)\setminus
M_{r}}|g(\nabla u^{i},\nabla u^{j})-\delta^{ij}|dV_{g}$ (223)
$\displaystyle\leq\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{M_{r}}|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}|dV_{g}+\frac{C_{3}(r_{0},b,\tau,\bar{m})|B_{d+1}(x)\setminus
M_{r}|}{r^{\tau}|B_{d+1}(x)|}$ (224)
$\displaystyle\leq\varepsilon+C_{3}(r_{0},b,\tau,\bar{m})r^{-\tau}$

where in the penultimate inequality we have made use of the pointwise estimate
(104). Notice that the estimate (224) is independent of $d$. It follows there
is an $r_{2}=r_{2}(r_{0},b,\tau,\Lambda,\varepsilon)$ so that if $r>r_{2}$, we
have

(225) $\left|\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{B_{d+1}(x)}g(\nabla u^{i},\nabla
u^{j})dV_{g}-\delta^{ij}\right|\leq 2\varepsilon.$

Now, by the second statement of Lemma 4.5, there is an
$\hat{r}=\hat{r}(r,r_{0},b,\tau,\Lambda)$ so that $B_{d+1}(x)\subset
M_{\hat{r}}$. Combining Corollary 6.5 and equation (225), we find that there
is an $m_{2}=m_{2}(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m})$ such that if
$m<m_{2}$, then

(226) $\displaystyle\|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}\|_{C_{1}^{0,\gamma}(B_{d}(x))}$
$\displaystyle\leq\|g(\nabla u^{i},\nabla u^{j})-g(\nabla u^{i},\nabla
u^{j})_{B_{d+1}(x)}\|_{C_{1}^{0,\gamma}(B_{d}(x))}$ (227)
$\displaystyle\quad+|g(\nabla u^{i},\nabla u^{j})_{B_{d+1}(x)}-\delta^{ij}|$
(228) $\displaystyle\leq
C(r,r_{0},b,\tau,\kappa,\Lambda,\bar{m})m^{\tfrac{1}{8}}+2\varepsilon$ (229)
$\displaystyle\leq 3\varepsilon.$

Notice that, using Proposition 3.11 (taking $R=1$) the estimate in (229), we
have the global Hölder estimate

(230) $\|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}\|_{C^{0,\gamma}(B_{d}(x))}<6\varepsilon.$

On the other hand, Lemma 4.1 implies there is an
$r_{3}=r_{3}(r_{0},b,\tau,\bar{m})$ so that when $r\geq r_{3}$, we have

(231) $\displaystyle\|g(\nabla u^{i},\nabla
u^{j})-\delta^{ij}\|_{C^{0,\gamma}(M\setminus M_{r})}\leq\varepsilon.$

Finally, we fix a value of $r\geq\max(r_{2},r_{3})$. This fixes the size of
$m_{2}$ required for (230) to hold. It follows that when $m$ is sufficiently
small relative to $r_{0},b,\tau,\kappa,$ and $\Lambda$, equations (231) and
(231) combine to show the global Hölder closeness

(232) $\|g(\nabla u^{i},\nabla u^{j})-\delta^{ij}\|_{C^{0,\gamma}(M)}\leq
7\varepsilon.$

As a consequence of (232), for sufficiently small $\varepsilon$, the map
$U:M\rightarrow\mathbb{R}^{3}$ given by $U(x)=(u^{1}(x),u^{2}(x),u^{3}(x))$
becomes a diffeomorphism, giving a global coordinate system on
$M\cong\mathbb{R}^{3}$. Then, in these coordinates $g^{ij}=g(\nabla
u^{i},\nabla u^{j})$ and (232) becomes
$\|g^{ij}-\delta^{ij}\|_{C^{0,\gamma}(M)}\leq 7\varepsilon.$ By perhaps taking
$\varepsilon$ smaller, we may assume $\varepsilon<\frac{1}{100}$. Now since
$g^{-1}$ is within $7\varepsilon$ of the identity matrix in this global
coordinate system, its inverse, that is $\\{g_{ij}\\}$, is within
$14\varepsilon$ of the identity in these coordinates, yielding the desired
estimate. ∎

##  7\. Estimates of Harmonic Functions on the Torus

In a similar way that we used the mass formula to control the geometry of
asymptotically flat manifolds in terms of their mass, we can use the formula
Theorem 1.7 to control the geometry of $3$-manifolds in
$\mathcal{N}(\Lambda,V,\kappa)$ in terms of the $L^{1}$ norm of the negative
part of their scalar curvature. Before approaching Theorem 1.11, some
preliminary observations are required. The following is an important result
which comes from combining information about isoperimetric profiles with the
result [Petersen-97, Theorem 5.4].

######  Proposition 7.1.

Let $\Lambda,V,\kappa>0$. Then $\mathcal{N}\left(\Lambda,V,\kappa\right)$ is
pre-compact in the $C^{0,\gamma}$-topology for any $\gamma\in(0,1)$. In
particular, the underlying manifolds in
$\mathcal{N}\left(\Lambda,V,\kappa\right)$ range over a finite number of
diffeomorphism types.

###### Proof.

According to Lemma 3.2, we see that for any region $\Omega\subset M$ such that
$|\Omega|\leq\frac{1}{2V}$, we have

(233) $ID_{\frac{3}{2}}\left(\Omega\right)\geq
IN_{\frac{3}{2}}\left(M,g\right)\geq\Lambda.$

In particular, by Corollary 2.4 we see there is an $R(\kappa,V)$ such that if
$r\leq R(\kappa,V)$, then $|B_{r}(x)|\leq\frac{1}{2V}$. Combining this with
(233), Lemma 2.2, and another application of Corollary 2.4, we obtain the
following control above and below for $r\leq R(\kappa,V)$

(234) $\frac{r^{3}}{C(\Lambda,\kappa)}\leq|B_{r}(x)|\leq
C(\Lambda,\kappa)r^{3}.$

This volume growth property shows that
$\mathcal{N}\left(\Lambda,V,\kappa\right)$ satisfies the assumptions of
[Petersen-97, Theorem 5.4], allowing us to conclude that this class has
bounded harmonic $\|*\|_{3,2,r}$-norm, see [Petersen-2006, Petersen-97] for
the definition of this norm. As a consequence, this class is precompact in the
$C^{0,\gamma}$-topology for any $\gamma\in(0,1)$. Since a sequence of
manifolds converging in the $C^{0,\gamma}$ sense must have a tail whose
underlying manifolds are diffeomorphic, the class
$\mathcal{N}\left(\Lambda,V,\kappa\right)$ supports only finitely many
diffeomorphism types. ∎

For each diffeomorphism type $M$ supporting a metric in
$\mathcal{N}\left(\Lambda,V,\kappa\right)$, fix once and for all three closed
differential forms $\omega_{1},\omega_{2},\omega_{3}\in\Omega^{1}(M)$ so that
$\int_{M}\omega_{1}\wedge\omega_{2}\wedge\omega_{3}=1$. The existence of such
forms follows from condition $(5)$ in Definition 1.9. Proposition 7.1 allows
us to uniformly control the $L^{2}$-norm of the forms
$\\{\omega_{i}\\}_{i=1}^{3}$.

######  Corollary 7.2.

Let $\Lambda,V,\kappa>0$. Then there is a constant $C=C(\Lambda,V,\kappa)$ so
that the following holds: if
$(M,g)\in\mathcal{N}\left(\Lambda,V,\kappa\right)$, then

(235) $\|\omega_{i}\|_{L^{2}}\leq C,\qquad i=1,2,3.$

###### Proof.

We argue by contradiction, using the $C^{0,\gamma}$, for $\gamma\in(0,1)$,
pre-compactness of $\mathcal{N}\left(\Lambda,V,\kappa\right)$ mentioned in the
proof of Proposition 7.1. Assuming there is no constant $C$ uniformly bounding
$\|\omega_{i}\|_{L^{2}}$ for all $(M,g)$ in
$\mathcal{N}\left(\Lambda,V,\kappa\right)$, we may find a sequence
$\\{(M_{l},g_{l})\\}^{\infty}_{l=1}$ so that

(236) $\lim_{l\rightarrow\infty}\|\omega_{i}\|_{L^{2}(M_{l},g_{l})}=\infty$

for $i=1,2,3$. As a consequence of $C^{0,\gamma}$ pre-compactness, we may pass
to a subsequence and assume $M_{l}$ are diffeomorphic to a single manifold $M$
and that $\\{(M,g_{l})\\}$ converges in the $C^{0,\gamma}$ topology to a
limiting Riemannian manifold $(M,g_{\infty})$. By the nature of this
convergence, $g_{\infty}$ is metric of $C^{0,\gamma}$ regularity. Since the
$1$-forms $\omega_{i}$ are independent of the metric and smooth, that there
exists a constant $K$ such that $\max_{M}|\omega_{i}|_{g_{\infty}}\leq K$. It
follows from the Dominated Convergence Theorem that

(237)
$\lim_{l\rightarrow\infty}\|\omega_{i}\|_{L^{2}(g_{l})}=\|\omega_{i}\|_{L^{2}(g_{\infty})}\leq
KV,$

where the last inequality follows from the fact that
$\operatorname{Vol}_{g_{\infty}}(M)\leq V$. This is a contradiction, and we
conclude the desired result. ∎

We are now prepared to prove the stability result Theorem 1.11.

###### Proof of Theorem 1.11.

Observe that the integral bound on $\mathrm{Rc}_{g}$ implies that
$\|R^{-}_{g}\|_{L^{3}}$ is bounded uniformly by $\kappa$. Using Hölder’s
inequality, it follows that $\|R^{-}_{g}\|_{L^{2}}$ is bounded in terms of $V$
and $\kappa$, a fact we will make use of below. Let $\\{u^{i}\\}_{i=1}^{3}$ be
harmonic maps to $S^{1}$ so that the pull-backs $du^{i}:=(u^{i})^{*}d\theta$
are cohomologous to the fixed forms $\\{\omega_{i}\\}_{i=1}^{3}$. Since
harmonic forms minimize the $L^{2}$ norm in their cohomology class, Corollary
7.2 implies that

(238) $\|du^{i}\|_{L^{2}}\leq C_{1}(\Lambda,V,\kappa).$

At this point, we are ready to apply the arguments from Sections 5 and 6. Let
us begin by applying the integral identity (4) to find

(239) $\int_{M}R^{-}(g)|du^{i}|dV_{\bar{g}}\geq\int_{M}\frac{|\nabla
du^{i}|^{2}}{|du^{i}|}dV_{g},\qquad i=1,2,3.$

Similar to (188), we may combine (239), Hölder’s inequality, and the fact that
$R^{-}(g)$ is $L^{2}$ bounded so that we may apply the Sobolev inequality to
$\sqrt{|du^{i}|}$, yielding $\|du^{i}\|_{L^{3}}\leq C_{2}(\Lambda,V,\kappa)$.
Now Theorem 5.2 with $\Omega=M$ shows that $\|\nabla du^{i}\|_{L^{2}}\leq
C_{3}(\Lambda,V,\kappa)$. In turn, the uniform Sobolev constant yields
$\|du^{i}\|_{L^{6}}\leq C_{4}(\Lambda,V,\kappa)$.

Now, we return to the integration by parts

(240) $\begin{split}\int_{M}|\nabla du^{i}|^{3}dV_{g}&=-\int_{M}\nabla
du^{i}\left(du^{i},\nabla|\nabla du^{i}|\right)+|\nabla
du^{i}|\nabla^{*}\nabla du^{i}(du^{i})dV_{g}\\\ &\leq
2\int_{M}\left|du^{i}\right|\left|\nabla
du^{i}\right|\left|\nabla^{2}du^{i}\right|dV_{g}.\end{split}$

We may now use the generalized Hölder’s inequality with exponents $6$, $3$,
and $2$, respectively, to find

(241) $\int_{M}|\nabla du^{i}|^{3}dV_{g}\leq 2\|du^{i}\|_{L^{6}}\|\nabla
du^{i}\|_{L^{3}}\|\nabla^{2}du^{i}\|_{L^{2}}.$

Dividing out both sides of the above by $\|\nabla du^{i}\|_{L^{3}}$ gives us
that

(242) $\|\nabla du^{i}\|^{2}_{L^{3}}\leq
2\|du^{i}\|_{L^{6}}\|\nabla^{2}du^{i}\|_{L^{2}}.$

As in the proof of Theorem 5.5, (242) can be used to show that
$\|\nabla^{2}du^{i}\|_{L^{2}}\leq C_{5}(\Lambda,V,\kappa)$. Since $\|\nabla
du^{i}\|_{L^{2}}\leq C_{3}(\Lambda,V,\kappa)$, the above allows us to apply
the Sobolev inequality to $|\nabla du^{i}|$, yielding $\|\nabla
du^{i}\|_{L^{6}}\leq C_{6}(\Lambda,V,\kappa)$. Finally, this last bound may be
used with Theorem 3.12, showing the pointwise control

(243) $\left|du^{i}\right|_{g}\leq C_{7}(\Lambda,V,\kappa).$

As a second consequence of the $L^{6}$ bound on $|\nabla du^{i}|$, there is
the following interpolation inequality, for any $q\in[2,6]$,

(244) $\|\nabla du^{i}\|_{L^{q}}\leq\|\nabla du^{i}\|_{L^{2}}^{\theta}\|\nabla
du^{i}\|_{L^{6}}^{1-\theta}\leq
C_{8}(\theta,\Lambda,V,\kappa)\|R^{-}(g)\|^{\frac{\theta}{2}}_{L^{1}},$

where $\tfrac{1}{q}=\tfrac{\theta}{2}+\tfrac{1-\theta}{6}$.

Since the wedge product of $\\{\omega_{i}\\}_{i=1}^{3}$ generates the top
integral cohomology of $M$,

(245) $\int_{M}du^{1}\wedge du^{2}\wedge du^{3}=1.$

In addition, we have

(246) $\nabla\left(du^{1}\wedge du^{2}\wedge du^{3}\right)=\nabla du^{1}\wedge
du^{2}\wedge du^{3}+du^{1}\wedge\nabla du^{2}\wedge du^{3}+du^{1}\wedge
du^{2}\wedge\nabla du^{3},$

so our $C^{0}$ control on $du^{i}$ allows us to apply Hadamard’s inequality to
find

(247) $\begin{split}&\left|\nabla g\left(\nabla u^{i},\nabla
u^{j}\right)\right|\leq
C_{3}(\Lambda,V,\kappa)\left(\left|\nabla^{2}u^{i}\right|+\left|\nabla^{2}u^{j}\right|\right)\\\
&\left|\nabla\left(du^{1}\wedge du^{2}\wedge du^{3}\right)\right|\leq
C_{4}(\Lambda,V,\kappa)\left(\left|\nabla du^{1}\right|+\left|\nabla
du^{2}\right|+\left|\nabla du^{3}\right|\right),\end{split}$

where we denote by $\nabla u^{i}$ the vector field corresponding to $du^{i}$.
Considering the non-negative function $f$ on $M$ defined by

(248) $du^{1}\wedge du^{2}\wedge du^{3}=fdV_{g},$

then (244) and (247) imply

(249) $\begin{split}&\|f\|_{L^{\infty}}\leq C_{5}(\Lambda,V,\kappa),\\\
&\|\nabla f\|_{L^{q}}\leq C_{6}(\Lambda,V,\kappa)\left(\|\nabla
du^{1}\|_{L^{q}}+\|\nabla du^{2}\|_{L^{q}}+\|\nabla
du^{3}\|_{L^{q}}\right)\end{split}$

for any $q\in[2,6]$.

Combining (247) and (249), and applying Theorem 3.12, we find

(250) $\begin{split}&\|f-\frac{1}{|M|}\|_{C^{0,\gamma}}\leq
C_{7}(\Lambda,V,\kappa)\left(\|\nabla du^{1}\|_{L^{q}}+\|\nabla
du^{2}\|_{L^{q}}+\|\nabla du^{3}\|_{L^{q}}\right)\\\ &\left\|g\left(\nabla
u^{i},\nabla u^{j}\right)-g\left(\nabla u^{i},\nabla
u^{j}\right)_{M}\right\|_{C^{0,\gamma}}\leq C_{8}(\Lambda,V,\kappa)\|\nabla
du^{i}\|_{L^{q}},\end{split}$

where $\gamma=\frac{q-3}{q}$. A technical remark is required concerning the
second line of (250). Note that technically Theorem 3.12 only estimates the
$C^{0,\gamma}_{R}$ norm. However, in the present context, the
$C^{0,\gamma}_{R}$ norm of a function differs from its $C^{0,\gamma}$ norm
only by the uniformly controlled $R=R(\kappa,V)$ and the $C^{0}$ norm of that
function.

As an immediate consequence of the first line of (250), if
$||R^{-}_{g}||_{L^{1}}$ is sufficiently small relative to $\Lambda,\kappa,$
and $V$, $f$ cannot vanish since $\tfrac{1}{|M|}\geq\tfrac{1}{V}$. Therefore,
the map $\Psi:M\to T^{3}$ given by

(251) $\Psi(x):=(u^{1}(x),u^{2}(x),u^{3}(x))$

is a local diffeomorphism. Since the degree of $\Psi$ is $1$, it is in fact a
global diffeomorphism. Now consider the coefficients

(252) $a^{ij}=g\left(\nabla u^{i},\nabla
u^{j}\right)_{M}=\mathchoice{{\vbox{\hbox{$\textstyle-$
}}\kern-7.83337pt}}{{\vbox{\hbox{$\scriptstyle-$
}}\kern-5.90005pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.75003pt}}{{\vbox{\hbox{$\scriptscriptstyle-$
}}\kern-4.25003pt}}\\!\int_{M}g\left(\nabla u^{i},\nabla
u^{j}\right)dV_{{g}}.$

In light of (250), the fields $\left\\{\nabla u^{i}\right\\}_{i=1}^{3}$ forms
a frame for $TM$ if $||R^{-}_{g}||_{L^{1}}$ is sufficiently small relative to
$\Lambda,\kappa,$ and $V$. Consequently, the constant symmetric tensor on the
torus $\mathbb{T}^{3}$ given by

(253)
$\delta_{F}^{-1}=a^{ij}\frac{\partial}{\partial\theta^{i}}\otimes\frac{\partial}{\partial\theta^{j}}$

where $\theta^{i}$ denotes the angular coordinate of the $i^{\mathrm{th}}$
circle in $\mathbb{T}^{3}$, defines a flat metric $\delta_{F}$ for small
$\|R^{-}(g)\|_{L^{1}}$. Now, in the frame $\\{\nabla u^{i}\\}_{i=1}^{3}$ we
may express $g^{ij}=g\left(\nabla u^{i},\nabla u^{j}\right)$. Finally, set
$g_{F}=\Psi^{*}\delta_{F}.$ Applying the second line of (250) with Theorem
3.12, we see that

(254) $\|g^{ij}-g_{F}^{ij}\|_{C^{0,\gamma}(M)}<\frac{\varepsilon}{2}$

when $\|R^{-}(g)\|_{L^{1}}$ is small enough relative to $\Lambda,V,$ and
$\kappa$. Similar to the final lines in the proof of Theorem 1.5, inequality
(254) implies $\|g_{ij}-\left(g_{F}\right)_{ij}\|_{C^{0,\gamma}}<\varepsilon$
for sufficiently small $\varepsilon$, finishing the argument. ∎

Lastly, we show Corollary 1.12.

###### Proof of Corollary 1.12.

In light of Theorem 1.11, a diagonal argument shows that it suffices to show
that the collection of flat Riemannian $3$-tori in
$\mathcal{N}(\Lambda,V,\kappa)$ is compact in the $C^{0,\gamma}$-topology.
This follows, for instance, by Proposition 7.1. Indeed, one may fix
coordinates on $\mathbb{T}^{3}$ and represent any flat metric $g$ as a
constant matrix $g_{ij}d\theta^{i}\otimes d\theta^{j}$ . ∎

In light of Theorem 1.11, a diagonal argument shows that it suffices to
establish the following.

######  Proposition 7.3.

Given $V,\Lambda>0$ and $\alpha\in[1,\tfrac{3}{2}]$, the collection of flat
Riemannian $3$-tori with Neumann $\alpha$-isoperimetric constant at least
$\Lambda$ and volume within $[V^{-1},V]$ is compact in the
$C^{\infty}$-topology (in particular in $C^{0,\gamma}$).

###### Proof.

Suppose $\mathcal{T}\in\mathcal{N}(\Lambda,V,\kappa)$ is a flat torus, which
we may be represent as a quotient $\mathbb{R}^{3}/\Gamma$ of $3$-space by a
lattice $\Gamma$ generated by three vectors $\\{X_{i}\\}_{i=1}^{3}$. The main
goal is to show $(1)$ the lengths of $X_{i}$ are bounded above and away from
$0$ in terms of $\Lambda,V$ and $(2)$ the angles between $X_{i},X_{j}$ are
bounded away from $0$ and $\pi$ in terms of $\Lambda,V$. Upon accomplishing
this, one obtains the desired sequential compactness of flat tori in
$\mathcal{N}(\Lambda,V,\kappa)$ by the following argument: given a family of
such tori, one applies $(1)$ to find a subsequential limit of the underlying
vectors to three non-zero vectors, then noting that $(2)$ ensures these
vectors form a lattice of full rank. The maps to the limiting torus which
yield the $C^{\infty}$ convergence are given by linear mappings, taking the
lattice elements to their associated limits.

First, we’ll need a few computational observations. The volume of
$\mathcal{T}$ is computed by $|X_{1}\wedge X_{2}\wedge X_{3}|$. Meanwhile, the
planes spanned by $X_{i},X_{j}$ descend to $2$-dimensional tori with areas
$|X_{i}\wedge X_{j}|$. Using one of these tori and an appropriate translate,
one can cut $\mathcal{T}$ into two pieces of equal volumes and conclude

(255) $|X_{i}\wedge X_{j}|\geq\frac{\Lambda}{2}|X_{1}\wedge X_{2}\wedge
X_{3}|^{\frac{1}{\alpha}}$

for $i\neq j$.

For $r>0$, consider the cubical domains $\mathcal{C}_{r}\subset\mathcal{T}$
given by

(256)
$\mathcal{C}_{r}=\left\\{aX_{1}+b\frac{X_{2}}{|X_{2}|}+c\frac{X_{3}}{|X_{3}|}\colon
a\in[0,1],\;b\in[0,r],\;c\in[0,r]\right\\}.$

Notice that the faces of $\mathcal{C}_{r}$ where $a=0$ and $a=1$ are
identified by the lattice $\Gamma$. When $r<\min(|X_{2}|,|X_{3}|)$, one may
compute

(257) $|\mathcal{C}_{r}|=r^{2}\frac{|X_{1}\wedge X_{2}\wedge
X_{3}|}{|X_{2}||X_{3}|},\quad|\partial\;\mathcal{C}_{r}|=2r\frac{|X_{1}\wedge
X_{2}|}{|X_{2}|}+2r\frac{|X_{1}\wedge X_{3}|}{|X_{3}|}.$

On the other hand,
$\partial_{r}|\mathcal{C}_{r}|=\tfrac{1}{2}|\partial\mathcal{C}_{r}|$.
Combining these three equations, we have

(258) $\frac{|X_{1}\wedge X_{2}|}{|X_{2}|}+\frac{|X_{1}\wedge
X_{3}|}{|X_{3}|}=2\frac{|X_{1}\wedge X_{2}\wedge X_{3}|}{|X_{2}||X_{3}|}.$

After performing straight-forward manipulations, the identity (258) used with
two applications of (255) yields

(259) $|X_{1}\wedge X_{2}\wedge
X_{3}|^{\tfrac{\alpha-1}{\alpha}}\geq\frac{\Lambda}{4}\left(|X_{3}|+|X_{2}|\right).$

Since this inequality is symmetric in $X_{1},X_{2},X_{3}$, we may conclude an
upper bound $|X_{i}|\leq 4V^{\tfrac{\alpha-1}{\alpha}}/\Lambda$ for $i=1,2,3$.
Using this, a lower bound for the lengths may be obtained by

(260) $\frac{1}{V}\leq|X_{1}\wedge X_{2}\wedge
X_{3}|\leq|X_{1}||X_{2}||X_{3}|\leq|X_{i}|\frac{16V^{\tfrac{2\alpha-2}{\alpha}}}{\Lambda^{2}}$

for any $i=1,2,3$. To summarize these last two observations, there is a
constant $C(\Lambda,V)>0$ so that $C(\Lambda,V)^{-1}\leq|X_{i}|\leq
C(\Lambda,V)$.

All that is left is to establish that the angles $\theta_{ij}$ between
$X_{i},X_{j}$ are bounded away from $0$ and $\pi$ in terms of $\Lambda,V$.
This is accomplished by combining the isoperimetric inequality with the length
bounds as follows

(261) $\displaystyle\sin^{2}(\theta_{ij})$
$\displaystyle=1-\cos^{2}(\theta_{ij})$ (262)
$\displaystyle=\frac{|X_{i}|^{2}|X_{j}|^{2}-\langle
X_{i},X_{j}\rangle^{2}}{|X_{i}|^{2}|X_{j}|^{2}}$ (263)
$\displaystyle=\frac{|X_{i}\wedge X_{j}|^{2}}{|X_{i}|^{2}|X_{j}|^{2}}$ (264)
$\displaystyle\geq
C(\Lambda,V)^{-4}\left(\frac{\Lambda}{2V^{\frac{1}{\alpha}}}\right)^{2}.$

∎

## References