# Existence and Uniqueness for the Non-Compact Yamabe Problem of Negative
Curvature Type

Joseph Hogg and Luc Nguyen 111Mathematical Institute and St Edmund Hall,
University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter,
Woodstock Road, Oxford OX2 6GG, UK. Email<EMAIL_ADDRESS>
###### Abstract

We study existence and uniqueness results for the Yamabe problem on non-
compact manifolds of negative curvature type. Our first existence and
uniqueness result concerns those such manifolds which are asymptotically
locally hyperbolic. In this context, our result requires only a partial
$C^{2}$ decay of the metric, namely the full decay of the metric in $C^{1}$
and the decay of the scalar curvature. In particular, no decay of the Ricci
curvature is assumed. In our second result we establish that a local volume
ratio condition, when combined with negativity of the scalar curvature at
infinity, is sufficient for existence of a solution. Our volume ratio
condition appears tight. This paper is based on the DPhil thesis of the first
author.

###### Contents

  1. 1 Introduction
  2. 2 The Yamabe problem for ALH manifolds
    1. 2.1 Definitions and notation
    2. 2.2 Existence of a solution for the Yamabe problem
    3. 2.3 Asymptotic properties of the conformal factor
      1. 2.3.1 An upper bound at infinity
      2. 2.3.2 A lower bound at infinity
      3. 2.3.3 First derivative estimates at infinity
    4. 2.4 Uniqueness and completion of the proof of Theorem 1.1
    5. 2.5 Asymptotically warped product manifolds
  3. 3 Volume ratio conditions for solvability of the Yamabe problem
    1. 3.1 An upper bound on the eigenvalue for the conformal Laplacian
    2. 3.2 Sharpness of the eigenvalue estimate

##  1 Introduction

We are interested in the Yamabe problem on non-compact manifolds: given some
complete non-compact Riemannian manifold $(M,g)$ of dimension $n\geq 3$, does
there exist a corresponding complete conformal metric whose scalar curvature
is constant? Equivalently, we would like to find a complete metric
$\tilde{g}=u^{\frac{4}{n-2}}g$ where $u$ is some strictly positive smooth
function on $M$ solving the Yamabe equation

$S_{\tilde{g}}=u^{-\frac{n+2}{n-2}}\left(-c_{n}\Delta_{g}u+S_{g}u\right)\equiv\text{constant,}\qquad\text{
$c_{n}:=\frac{4(n-1)}{n-2}$}.$

Here, $S_{g}$ and $S_{\tilde{g}}$ refer to the scalar curvatures of the
corresponding metrics. The operator $-c_{n}\Delta_{g}+S_{g}$ is known as the
conformal Laplacian.

In the case that $M$ is compact, the Yamabe problem has been extensively
studied. The existence of a solution was established through the combined
works of Yamabe [Yam60], Trudinger [Tru68], Aubin [Aub76] and Schoen [Sch84].
For other aspects of the Yamabe problem, see e.g. [Bre08, BM09, CGS89, CL99,
GNN79, GNN81, KMS09, KMPS99, LZ04, LZ05, LZ07, Mar08, XZ22] and references
therein.

Our work is focused on the Yamabe problem of “negative curvature type” on non-
compact manifolds, namely we are interested in obtaining conformal changes to
constant negative scalar curvature. Consequently, asserting that
$S_{\tilde{g}}\equiv-n(n-1)$, the equation for the conformal change in scalar
curvature yields the Yamabe equation

$-c_{n}\Delta_{g}u+S_{g}u=-n(n-1)u^{\frac{n+2}{n-2}}\ .$ (Ya)

Finding a solution to the Yamabe problem thus amounts to finding a positive
solution of the equation (Ya) above for which the corresponding conformal
metric is complete.

The Yamabe problem of negative curvature type on non-compact manifolds has
been studied extensively in the literature. Important progress has been made
by Andersson, Chruściel and Friedrich [ACF92], Gover and Waldron [GW17],
Graham [Gra17] and Loewner and Nirenberg [LN74]. For further literature see
e.g. [Avi82, AM85, AM88a, AM88b, CP08, Fin98, Fin99, HS20, HL21, Jia21, Jin88,
Lab05, LN74, Maz91, MP96, RRV97, SY88, Vér81] and references therein.

In the present work we consider conditions for existence of a solution to the
Yamabe problem on a given non-compact Riemannian manifold $(M,g)$ where $g$
has asymptotically negative scalar curvature. We assume throughout the paper
that $g$ is complete and satisfies a condition of the type

$\limsup S_{g}\leq-\varepsilon<0$ (1.1)

for some $\varepsilon>0$ and where the limit is taken along any divergent
sequence in the manifold.

It is known that (1.1) is insufficient to conclude that the Yamabe problem can
be solved. For example, in [AM88b], alongside a number of existence results,
Aviles and McOwen give an example of a complete metric $g$ on
$\mathbb{R}\times\mathbb{T}^{n-1}$ satisfying (1.1) for which the Yamabe
problem has no solution. It is therefore of interest to understand what
conditions, in addition to (1.1), are necessary and/or sufficient for the
Yamabe problem to have a solution.

Our first result concerns asymptotically locally hyperbolic (ALH) manifolds, a
well-studied class of manifolds satisfying (1.1), in a weaker sense than
considered in the existing literature (e.g. [ACF92] and [AILA18]). In
particular, our notion of ALH requires only $C^{1}$ decay in the metric
components to those of the model space and a bound on the scalar curvature of
the form

$\displaystyle S_{g}\leq-n(n-1)+C$ $\displaystyle e^{-\alpha r}\qquad\text{ or
}$ (1.2) $\displaystyle|S_{g}+n(n-1)|\leq C$ $\displaystyle e^{-\alpha r}$
(1.3)

for $\alpha\in(0,n)$ and some constant $C>0$, without requiring decay of the
full curvature tensor. For the precise definition, see Section 2.1. We
establish:

######  Theorem 1.1.

Suppose $(M,g)$ is an ALH manifold of regularity order $1$ and with decay
exponent $\alpha\in(0,n)$. If the scalar curvature satisfies (1.2) on $M$,
then there exists a positive smooth solution $u$ to (Ya) on $M$ satisfying
$\liminf_{r\rightarrow\infty}u\geq 1$ corresponding to a complete conformal
metric $\tilde{g}$ with $S_{\tilde{g}}\equiv-n(n-1)$.

If the scalar curvature satisfies the stronger condition (1.3) on $M$, then
the solution $u$ above is the unique solution satisfying
$\lim_{r\rightarrow\infty}u=1$, the corresponding conformal manifold
$(M,\tilde{g})$ is ALH of regularity order $0$ and with decay exponent
$\alpha$, and $u$ is maximal among all solutions to (Ya), that is any other
solution $\tilde{u}$ satisfies $\tilde{u}\leq u$.

Some comments on the conclusions of the theorem are in order. Concerning the
uniqueness, we are able to slightly relax the condition
$\lim_{r\rightarrow\infty}u=1$; see Proposition 2.15. However, this necessary
condition for uniqueness cannot be dropped in its entirety:

######  Example 1.2.

Take $M=B_{1}$ to be the unit ball in $\mathbb{R}^{n}$, $\delta$ to be the
Euclidean metric and

$g=\frac{4}{(1-|x|^{2})^{2}}\delta\quad\text{ and }\quad
g^{\prime}=\frac{4R^{2}}{(R^{2}-|x|^{2})^{2}}\delta$

to be the Poincaré metrics on $B_{1}$ and $B_{R}$ for some $R>1$. Clearly, on
$B_{1}$, if we write $g^{\prime}=u^{\frac{4}{n-2}}g$ then $u$ must solve the
Yamabe equation for $(M=B_{1},g)$ and $\lim_{r(x)\rightarrow\infty}u(x)=0$. In
fact, $u(x)e^{\frac{n-2}{2}r(x)}$ tends to
$\left(\frac{4R}{R^{2}-1}\right)^{\frac{n-2}{2}}$ as $r(x)\rightarrow\infty$.

Regarding the loss in regularity order of the resulting conformal ALH metric,
this is expected due to the assumption that the scalar curvature decays only
in $L^{\infty}$ in (1.3). Under stronger decay assumptions, one may appeal to
elliptic regularity theory to obtain correspondingly stronger regularity of
the resulting conformal metric at infinity.

We place our result in the context of the existing literature on the Yamabe
problem for asymptotically hyperbolic manifolds. A fundamental and pioneering
work of Loewner and Nirenberg [LN74] showed that, on every open subset $M$ of
$\mathbb{R}^{n}$ with regular boundary, there exists a complete and
conformally flat metric $g$ which is ALH and has constant negative scalar
curvature. Of special relevance to Theorem 1.1 is the work of Andersson,
Chruściel and Friedrich [ACF92] which proved, roughly speaking and among other
things, the existence and uniqueness of a solution to the Yamabe problem when
$(M,g)$ is ALH and admits a $C^{2}$ conformal compactification222The related
question of when intrinsic definitions of non-compact ALH manifolds imply the
existence of a corresponding conformal compactification of a certain
regularity is addressed, for example, in [BG11] and [Gic13].. We note that the
existence of such a conformal compactification mandates the full decay of all
sectional curvatures to a negative constant near infinity. Under weaker
regularity assumptions on the conformal compactification, while still
maintaining the full curvature decay, a similar result has been obtained in
[AILA18]. We remark that in Theorem 1.1, we require instead an assumption on
the scalar curvature alone. It is not hard to give examples of metrics which
satisfy the conditions of Theorem 1.1 but whose Ricci curvature is not
asymptotic to a constant multiple of the metric; see Example 2.3.

In our second main result, we give a local condition which, together with
(1.1), ensures the existence of a solution to the Yamabe problem. Our result
is motivated by a result of Aviles and McOwen in [AM88b] which asserts that
the negativity of the first eigenvalue of the conformal Laplacian on some
compact domain, in combination with (1.1), implies the existence of a solution
to the Yamabe problem. We prove:

######  Theorem 1.3.

Let $(M,g)$ be a Riemannian manifold and suppose that there exist two open
sets $\Omega_{1}\subset\Omega_{2}$ with $C^{1}$ boundary which satisfy (taking
$d_{g}$ to be the geodesic distance with respect to $g$) that

$d_{g}(x,\partial\Omega_{2})=R\text{ for each }x\in\partial\Omega_{1}\text{
for some }R>0,$ (1.4)
$\frac{Vol_{g}(\Omega_{2}\setminus\Omega_{1})}{Vol_{g}(\Omega_{1})}\leq\sinh^{2}\left(\frac{\sqrt{n(n-2)}}{2}R\right),$
(1.5)

and the scalar curvature satisfies $S_{g}\leq-n(n-1)$ on $\Omega_{2}$. Then,
the conformal Laplacian $-c_{n}\Delta_{g}+S_{g}$ for $(M,g)$ has a negative
first eigenvalue on $\Omega_{2}$. Consequently, if $g$ satisfies (1.1), there
exists a complete metric $\tilde{g}$ conformal to $g$ on $M$ of constant
scalar curvature $-n(n-1)$.

Our Theorem 1.3 gives a new sufficient condition for existence, relating a
local volume growth to the solvability of the Yamabe problem on the whole
manifold. The local condition (1.5) is tight in the following sense:

######  Remark 1.4.

The constant $\frac{\sqrt{n(n-2)}}{2}$ in (1.5) is sharp with respect to the
existence of a negative first eigenvalue for the conformal Laplacian. Namely,
for any $\beta>\sqrt{n(n-2)}$ there exists a manifold $(M,g)$ and concentric
annuli $\Omega_{1}\subset\Omega_{2}$ satisfying (1.4) with volume ratio

$\frac{Vol_{g}(\Omega_{2}\setminus\Omega_{1})}{Vol_{g}(\Omega_{1})}\leq\sinh^{2}\Big{(}\frac{\beta}{2}R\Big{)}$
(1.6)

and with scalar curvature satisfying $S_{g}\leq-n(n-1)$ on $\Omega_{2}$ but
for which the first eigenvalue for the conformal Laplacian on $\Omega_{2}$ is
positive. See Proposition 3.6 and the comment following it.

While the relationship between volume comparison and the first eigenvalue for
the Laplacian $-\Delta_{g}$ has been well observed in the literature, the
relation between volume comparison and the solvability of the Yamabe problem
is less understood. Although it is sharp for the existence of a negative first
eigenvalue for the conformal Laplacian, we do not know whether our bound in
(1.5) is optimal for the solvability of the Yamabe problem. It would be
interesting to understand what the sharp volume ratio bound of the form (1.5)
would be for existence of solutions to the Yamabe problem.

Theorem 1.3 is of a different flavour to Theorem 1.1. We first note that, for
the ALH manifolds treated in Theorem 1.1, the volume ratio in the asymptotic
region behaves differently than (1.5); in particular if we fix $\Omega_{1}$ in
the asymptotic region and consider large $R$, the volume ratio satisfies

$\frac{Vol_{g}(\Omega_{2}\setminus\Omega_{1})}{Vol_{g}(\Omega_{1})}\approx
e^{(n-1)R}\gg\sinh^{2}\left(\frac{\sqrt{n(n-2)}}{2}R\right).$ (1.7)

Furthermore, we have the following remark:

######  Remark 1.5.

Consider a manifold $(M,g)$ of the form $M=\mathbb{R}\times N$ where $(N,h)$
is a manifold of constant negative scalar curvature and $g=dr^{2}+e^{-\beta
r}h\text{ for some }0\leq\beta<\frac{\sqrt{n(n-2)}}{n-1}.$ By (3.15), a
computation gives $S_{g}=-(n-1)(n-2)e^{2\beta
r}+\mathcal{O}(1)\rightarrow-\infty$ as $r\rightarrow\infty$. Additionally,
taking $\Omega_{1}=(r_{0},r_{0}+1)\times N$,
$\Omega_{2}=(r_{0}-R,r_{0}+1+R)\times N$ for some $r_{0}\gg R\gg 1$ we have
that (1.5) is satisfied. And lastly, by Proposition 2.17, $g$ is not conformal
to a locally hyperbolic metric. Consequently, $(M,g)$ provides an example
which is not covered by Theorem 1.1 and for which we may solve the Yamabe
problem on $(M,g)$.

We briefly discuss the proofs of our results. The proof of the existence part
of Theorem 1.1 makes use of the procedures in [ACF92] and [AM88b] where
barrier functions play an important role. Due to the weaker asymptotic
conditions on the scalar curvature in (1.2), or (1.3), additional work is
required to construct such barrier functions. For the uniqueness part, we
treat separately the lower and upper bounds at infinity for solutions of (Ya).
For the upper bound, we adapt certain ideas from [LN74] using certain
reference solutions. The main difference is that we lack an explicit closed
form of the reference solutions and so the final conclusion is drawn from an
asymptotic analysis of these solutions, exploiting the conformal invariance of
the equation. For the lower bound, we highlight that our treatment is very
different from approaches seen in the literature, where one typically uses the
existence of a sufficiently regular conformal compactification in order to
compare the solution to some reference solution exterior to the compactified
manifold. Owing to our lack of such a compactification, we instead perform a
blow-up analysis intrinsically to the given manifold, identifying a limiting
equation from which we deduce the desired lower bound at infinity.

To prove Theorem 1.3, we make use of a particular test function for the
variational functional corresponding to the first eigenvalue of the conformal
Laplacian. This test function is constructed by minimising the sup-norm of the
integrand in the variational functional, following the procedure in [Aro65]
and using the volume ratio condition (1.5). It was surprising to the authors
that condition (1.5) for the volume ratio turned out to be sharp in the sense
of Remark 1.4 above. While falling outside the scope of the present paper, it
would be interesting to understand what other geometric quantities similar in
nature to the volume ratio used here could have relevance to the behaviour of
the first eigenvalue of the conformal Laplacian and the solvability of the
Yamabe problem.

The paper is structured as follows. In Section 2, we address Theorem 1.1 and
our work in the context of ALH manifolds. In Section 3, we address Theorem 1.3
and our related study of the volume ratios of multiply warped product
manifolds.

#### Acknowledgments.

J.H. was partially supported by the EPSRC Centre for Doctoral Training in
Partial Differential Equations [grant number EP/L015811/1]. The authors would
like to thank the anonymous referee for their constructive comments.

##  2 The Yamabe problem for asymptotically locally hyperbolic manifolds

In this section, we prove Theorem 1.1. We first make precise our setting in
Section 2.1. Our results are then split into two main sections; Section 2.2
addresses the existence of a solution to the Yamabe problem and Section 2.3
focuses on an understanding of the asymptotic behaviour of the obtained
conformal factor. We then summarise how the results of the previous two
sections combine to prove Theorem 1.1 in Section 2.4. Finally, in Section 2.5,
we provide a discussion of how Theorem 1.1 can be applied to the broader class
of asymptotically warped product manifolds, in particular via a result
regarding the conformal classes of reference warped product metrics.

###  2.1 Definitions and notation

We will consider manifolds which may be decomposed as a union $M=M_{0}\cup
M^{+}$, where $M_{0}$ is some compact interior region, $M^{+}$ is a non-
compact exterior region and both parts are disjoint apart form their common
boundary. We assume further that we may express $M^{+}=\mathbb{R}_{\geq
0}\times N$ where $N$ is some $(n-1)$ \- dimensional compact manifold. On the
end $\mathbb{R}_{\geq 0}\times N$, we denote by $r$ a coordinate on the
$\mathbb{R}_{\geq 0}$ fibre. Additionally, we denote the coordinates on any
local (angular) chart on $N$ with a $\theta^{a}$, where $a=1,\ldots,n-1$, and
we use $a,b,c,\ldots$ to index angular coordinates. When referring to the full
set of coordinates on $M^{+}$ we use the notation
$x^{1}=\theta^{1},\ldots,x^{n-1}=\theta^{n-1}$, $x^{n}=r$ and we use
$i,j,k,\ldots$ to index over all coordinates.

We define a reference locally hyperbolic metric $\mathring{g}$ on the exterior
region $M^{+}$ to be

$\mathring{g}=dr^{2}+f_{k}^{2}(r+r_{0})\mathring{h}$ (2.1)

for some $r_{0}>0$ where $\mathring{h}$ is a metric on $N$ of constant scalar
curvature $(n-1)(n-2)k$ for $k\in\\{-1,0,1\\}$ and

$f_{k}(r)=\begin{cases}\sinh(r)&\ k=1,\\\ e^{r}&\ k=0,\\\ \cosh(r)&\
k=-1.\end{cases}$

In particular, when $k=1$ and $N=\mathbb{S}^{n-1}$, $\mathring{g}$ is the
standard hyperbolic metric.

We additionally note that the scalar curvature of a warped product metric like
$\mathring{g}$ may be computed via the formula

$S_{\mathring{g}}=-2(n-1)\frac{f_{k}^{\prime\prime}}{f_{k}}-(n-1)(n-2)\left(\frac{f_{k}^{\prime}}{f_{k}}\right)^{2}+\frac{S_{\mathring{h}}}{f_{k}^{2}}.$

From this, one may compute that $S_{\mathring{g}}\equiv-n(n-1)$ for each $k$
in the above definition.

For the next definition below, we choose a finite set of preferred charts
$U_{i}$ covering $N$, each with a preferred choice of local coordinates
$\\{\theta^{1},\ldots,\theta^{n-1}\\}$. We extend these charts to $M^{+}$ by
defining $V_{i}=\mathbb{R}_{\geq 0}\times U_{i}$ with coordinates
$\\{r,\theta^{1},\ldots,\theta^{n-1}\\}$ and fix them from hereon. For a
function $\varphi$ on the end $\mathbb{R}_{\geq 0}\times N$ we use the
notation $\varphi=\mathcal{O}_{m}(e^{-\alpha r})$ to indicate that $\varphi$
and all of its first $m$ derivatives in the coordinates defined above decay as
$e^{-\alpha r}$, that is there exists a constant $C>0$ such that $\varphi$
satisfies $|\partial_{\beta}\varphi|\leq Ce^{-\alpha r}$ where $\beta$
indicates any multi-index with $0\leq|\beta|\leq m$.

We adopt the following definition of ALH manifolds which, while differing
slightly, is consistent with those found in the literature (e.g. in [AILA18,
CH03, CGNP18, Wan01]).

######  Definition 2.1 (Asymptotically Locally Hyperbolic).

We say a Riemannian manifold $(M,g)$ is _asymptotically locally hyperbolic of
regularity order $m\in\mathbb{N}$ and with decay exponent $\alpha>0$_ if we
can write $M=M_{0}\cup(\mathbb{R}_{\geq 0}\times N)$ and we can write the
metric components of $g$ on any preferred coordinate chart as

$\displaystyle g_{rr}=\mathring{g}_{rr}=1,\quad
g_{ab}=\mathring{g}_{ab}+\mathcal{O}_{m}\left(e^{-(\alpha-2)r}\right),\quad
g_{ra}=\mathcal{O}_{m}\left(e^{-(\alpha-1)r}\right)$

where $\mathring{g}$ is a reference locally hyperbolic metric of the form
(2.1) for some $k\in\\{-1,0,1\\}$. If $(N,\mathring{h})$ is the round sphere
then we simply say that $g$ is asymptotically hyperbolic.

It is clear that if $(M,g)$ is ALH with decay exponent $\alpha$ then it is
also ALH with decay exponent $\alpha^{\prime}$ for any
$0<\alpha^{\prime}<\alpha$. We note that if $\alpha<1$, then $(M,g)$ is
$C^{0,\alpha}$ conformally compactifiable and if $\alpha\geq 1$ then $(M,g)$
is $C^{1}$ conformally compactifiable. For terminology, see for example
[CGNP18].

In the remainder of this paper, we will make regular use of the coordinate
function $r$ corresponding to the $\mathbb{R}_{\geq 0}$ fibre of the exterior
region $M^{+}$. We note that the particular choice of $r$ is not unique, in
that the reference metric $\mathring{g}$ defined above may be expressed in the
form (2.1) for arbitrarily many choices of coordinate function $r$ via
diffeomorphism of $M^{+}$ or by an altogether different choice of splitting of
$M$ into the interior and exterior regions $M_{0}$ and $M^{+}$. To avoid this
complication, whenever we speak of an ALH manifold as defined above, we
implicitly assume that there is a pre-chosen $r$.

We provide the following computational lemma establishing the corresponding
decay of the metric inverse and Christoffel symbols which will be useful to us
later.

######  Lemma 2.2.

Suppose $(M,g)$ is an ALH manifold of regularity order $1$ and decay exponent
$\alpha$. Then, in any of the preferred charts

$\displaystyle\begin{split}g^{rr}&=1+\mathcal{O}_{1}(e^{-2\alpha r}),\\\
\Gamma^{r}_{rr}&=\mathcal{O}\left(e^{-\alpha r}\right),\\\
\Gamma^{a}_{br}&=\mathring{\Gamma}^{a}_{br}+\mathcal{O}\left(e^{-\alpha
r}\right),\end{split}\qquad\begin{split}g^{ra}&=\mathcal{O}_{1}(e^{-(\alpha+1)r}),\\\
\Gamma^{a}_{rr}&=\mathcal{O}\left(e^{-(\alpha+1)r}\right),\\\
\Gamma^{r}_{ab}&=\mathring{\Gamma}^{r}_{ab}+\mathcal{O}\left(e^{-(\alpha-2)r}\right),\end{split}\qquad\begin{split}g^{ab}&=\mathring{g}^{ab}+\mathcal{O}_{1}(e^{-(\alpha+2)r}),\\\
\Gamma^{r}_{ar}&=\mathcal{O}\left(e^{-(\alpha-1)r}\right),\\\
\Gamma^{a}_{bc}&=\mathring{\Gamma}^{a}_{bc}+\mathcal{O}\left(e^{-\alpha
r}\right),\end{split}$

where we use the notation $\mathring{\Gamma}$ to denote the Christoffel
symbols of the reference locally hyperbolic metric $\mathring{g}$ which
satisfy

$\mathring{\Gamma}^{r}_{ab}=\mathcal{O}_{2}\left(e^{2r}\right),\qquad\mathring{\Gamma}^{a}_{br}=\mathcal{O}_{2}\left(1\right),\qquad\mathring{\Gamma}^{a}_{bc}=\mathcal{O}_{2}\left(1\right).$

We provide an example of an ALH manifold in the sense of Definition 2.1 which
is not covered by the existing literature for the Yamabe problem on
asymptotically hyperbolic manifolds. In particular, the class described below
includes metrics where not all of the sectional curvatures are decaying to a
negative constant at infinity.

######  Example 2.3.

Consider the warped product manifold $M^{3}=\mathbb{R}_{\geq
0}\times\mathbb{T}^{2}$ where $\mathbb{T}^{2}$ is the $2$-dimensional flat
torus with metric $(dx^{1})^{2}+(dx^{2})^{2}$ and standard coordinates
$\\{x^{a}\\}$. We endow $M^{3}$ with the diagonal metric

$g=dr^{2}+e^{2r}\left(p(r)(dx^{1})^{2}+p^{-1}(r)(dx^{2})^{2}\right).$

This metric is ALH in the sense of Definition 2.1 provided, for example,
$p(r)=1+\mathcal{O}(e^{-\alpha r})$ and $p^{\prime}(r)=\mathcal{O}(e^{-\alpha
r})$, which we assume in this example. The metric $g$ has Ricci curvature

$\displaystyle
R_{rr}=-2-\frac{1}{2}\left(\frac{p^{\prime}}{p}\right)^{2},\qquad
R_{11}=e^{2r}p\left(-2+\frac{1}{2}\left(\frac{p^{\prime}}{p}\right)^{2}-\frac{p^{\prime}}{p}-\frac{1}{2}\left(\frac{p^{\prime\prime}}{p}\right)\right),$
$\displaystyle
R_{22}=e^{2r}p^{-1}\left(-2-\frac{1}{2}\left(\frac{p^{\prime}}{p}\right)^{2}+\frac{p^{\prime}}{p}+\frac{1}{2}\left(\frac{p^{\prime\prime}}{p}\right)\right).$

We see that the Ricci curvature does not necessarily decay to a constant
multiple of the metric (note the presence of the $p^{\prime\prime}$ term in
$R_{11}$ and $R_{22}$). In contrast, the metric $g$ has scalar curvature

$S_{g}=-6-\frac{1}{2}\left(\frac{p^{\prime}}{p}\right)^{2}=-6-\mathcal{O}(e^{-\alpha
r}).$ (2.2)

The example above demonstrates a class which does not satisfy the requirements
(discussed in more detail in the following section) in [ACF92] or [AILA18] but
which falls under Definition 2.1. There are certainly many such $p$ which
behave wildly in $C^{2}$ and so have poor behaviour of the Ricci curvature,
for example take $p(r)=1+e^{-2\alpha r}\sin(e^{\alpha r})$.

###  2.2 Existence of a solution for the Yamabe problem

In this section we will exhibit a sub-solution to (Ya), from which existence
of a solution follows via the arguments of Aviles and McOwen in [AM88b].

######  Lemma 2.4.

Let $(M,g)$ be ALH of regularity order $1$ and with decay exponent $\alpha<n$
and satisfy (1.2). For any $0<\beta<\min(n-1,\alpha)$, there exist constants
$0<\delta<1$, close to $1$, and $\theta>0$, large, such that the function
$u_{-}\in H^{1}_{\operatorname{loc}}(M)$ defined by

$u_{-}:=\begin{cases}1-C(\theta,\delta)e^{-\alpha r}\qquad&\text{ on
}\\{r_{\delta}\leq r\\}\times N\\\ 1-\theta e^{-\beta r}\qquad&\text{ on
}\\{r_{\theta}\leq r\leq r_{\delta}\\}\times N\\\
0\qquad\qquad\qquad\qquad&\text{ on }M_{0}\cup(\\{r\leq r_{\theta}\\}\times
N)\ ,\end{cases}$

where $r_{\delta}:=\frac{1}{\beta}\log\left(\frac{\theta}{1-\delta}\right)$,
$r_{\theta}:=\frac{1}{\beta}\log\left(\theta\right)$ and
$C(\theta,\delta)=\left(\frac{\theta}{1-\delta}\right)^{\frac{\alpha}{\beta}}$,
is a sub-solution of (Ya) on $M$ which is positive outside of some compact
set.

We note here that the choice of $C(\theta,\delta)$ ensures that $u_{-}$ is
continuous and therefore belongs to $H^{1}_{\operatorname{loc}}(M)$. The
change at $r_{\delta}$ is used only in the case that $n-1\leq\alpha<n$.

We have a sub-solution of the form $u_{-}=u_{-}(r)$ so we are led to consider
the ordinary differential inequality arising immediately from (Ya)

$\displaystyle-c_{n}\left(1+\mathcal{O}(e^{-2\alpha
r})\right)u_{-}^{\prime\prime}$
$\displaystyle+c_{n}\left((n-1)\frac{f_{k}^{\prime}}{f_{k}}+\mathcal{O}(e^{-\alpha
r})\right)u_{-}^{\prime}$ $\displaystyle-$
$\displaystyle\left(n(n-1)+\mathcal{O}(e^{-\alpha
r})\right)u_{-}\leq-n(n-1)u_{-}^{\frac{n+2}{n-2}}.$ (2.3)

###### Proof.

Fix some $0<\beta<\min(n-1,\alpha)$. To establish that $u_{-}$ is a sub-
solution, we must show that (2.2) holds on $\\{r_{\theta}\leq r\leq
r_{\delta}\\}$ and $\\{r_{\delta}\leq r\\}$ and check the following
transmission conditions at $r_{\theta}$ and at $r_{\delta}$,

$\lim_{r\nearrow r_{\theta}}u_{-}^{\prime}(r)\leq\lim_{r\searrow
r_{\theta}}u_{-}^{\prime}(r)\qquad\text{and}\qquad\lim_{r\nearrow
r_{\delta}}u_{-}^{\prime}(r)\leq\lim_{r\searrow r_{\delta}}u_{-}^{\prime}(r).$

The first condition is immediate as $\beta>0$ and the second condition holds
true as $\beta<\alpha$.

To establish (2.2) it suffices to show, for some constant $C_{1}>0$ depending
only on $g$, that there exist $\theta$ and $\delta$ (possibly depending on
$\beta$) such that

$\displaystyle\begin{split}L_{-}u_{-}:=&-c_{n}u_{-}^{\prime\prime}-c_{n}\left((n-1)\frac{f_{k}^{\prime}}{f_{k}}-C_{1}e^{-\alpha
r}\right)u_{-}^{\prime}\\\ &-\left(n(n-1)-C_{1}e^{-\alpha
r}\right)u_{-}+C_{1}e^{-2\alpha
r}|u_{-}^{\prime\prime}|\leq-n(n-1)u_{-}^{\frac{n+2}{n-2}}\end{split}$ (2.4)

holds on $\\{r_{\theta}<r<r_{\delta}\\}$ and $\\{r_{\delta}<r\\}$. We note
here that $u_{-}^{\prime}\geq 0$ for all $r$.

In the following, we write $C$ to indicate a constant changing from line to
line but depending only on $g$. For $\\{r_{\theta}<r<r_{\delta}\\}$, we have
that $n(n-1)u_{-}^{\frac{n+2}{n-2}}\leq n(n-1)\delta^{\frac{4}{n-2}}u_{-}$ as
$0\leq u_{-}\leq\delta$. We compute

$\displaystyle L_{-}u_{-}+n(n-1)u_{-}^{\frac{n+2}{n-2}}<$ $\displaystyle\
c_{n}\theta\left(\beta^{2}-(n-1)\beta+\frac{1}{4}n(n-2)(1-\delta^{\frac{4}{n-2}})\right)e^{-\beta
r}$ $\displaystyle+C(\underbrace{e^{(\beta-\alpha)r}+\theta e^{-\alpha
r}+\theta e^{-2r}}_{A})e^{-\beta r}$

where we used that $\frac{f_{k}^{\prime}(r+r_{0})}{f_{k}(r+r_{0})}>1-Ce^{-2r}$
and $u_{-}^{\prime}\geq 0$. Note first that, for $\delta$ close to $1$, the
first term of the RHS of the above is a negative multiple of $e^{-\beta r}$.
In addition, as $\beta<\alpha$ we have, for $r\geq r_{\theta}$, that
$|A(r)|\leq A(r_{\theta})\leq
C(\theta^{1-\frac{\alpha}{\beta}}+\theta^{1-\frac{2}{\beta}}).$ Consequently,
for $\theta$ sufficiently large, the first term on the RHS dominates and, for
all $\delta$ sufficiently close to 1, we have
$L_{-}u_{-}+n(n-1)u_{-}^{\frac{n+2}{n-2}}<0\text{ in
}\\{r_{\theta}<r<r_{\delta}\\}.$ We choose such a $\theta$ and fix it from
here on.

On $\\{r>r_{\delta}\\}$, note that we have $\delta\leq u_{-}\leq 1$. As

$\displaystyle\left|x^{\frac{n+2}{n-2}}-1-\frac{n+2}{n-2}(x-1)\right|=$
$\displaystyle\left|\int_{x}^{1}4\frac{n+2}{(n-2)^{2}}t^{-\frac{n-6}{n-2}}(x-t)\
dt\right|<4\frac{n+2}{(n-2)^{2}}\delta^{-\frac{n-6}{n-2}}(x-1)^{2}$

for $\delta<x\leq 1$. We have,

$\displaystyle
n(n-1)u_{-}^{\frac{n+2}{n-2}}<n(n-1)-c_{n}\frac{n}{4}(n+2)C(\theta,\delta)e^{-\alpha
r}+c_{n}\frac{n(n+2)}{(n-2)}\delta^{-\frac{n-6}{n-2}}C(\theta,\delta)^{2}e^{-2\alpha
r}\ .$

Using again that $\frac{f_{k}^{\prime}(r+r_{0})}{f_{k}(r+r_{0})}>1-Ce^{-2r}$
and $u_{-}^{\prime}\geq 0$, we obtain

$\displaystyle L_{-}u_{-}+n(n-1)u_{-}^{\frac{n+2}{n-2}}$ $\displaystyle\leq
c_{n}e^{-\alpha r}C(\theta,\delta)\Big{[}\alpha^{2}-(n-1)\alpha-n$
$\displaystyle+C\Big{(}\underbrace{\delta^{-\frac{n-6}{n-2}}C(\theta,\delta)e^{-\alpha
r}+e^{-\alpha r}+e^{-2r}+\frac{1}{C(\theta,\delta)}}_{B}\Big{)}\Big{]}.$ (2.5)

As $\alpha<n$, $\alpha^{2}-(n-1)\alpha-n<0$. We note that $B(r)$ is non-
increasing. We have for $r>r_{\delta}$.

$\displaystyle
0<B(r)<B(r_{\delta})=\Bigg{[}\delta^{-\frac{n-6}{n-2}}(1-\delta)+\left(\frac{\theta}{1-\delta}\right)^{-\frac{\alpha}{\beta}}+\left(\frac{\theta}{1-\delta}\right)^{-\frac{2}{\beta}}+\frac{1}{C(\theta,\delta)}\Bigg{]}\rightarrow
0\text{ as }\delta\nearrow 1.$

It follows that for $\delta$ close to 1,
$L_{-}u_{-}+n(n-1)u_{-}^{\frac{n+2}{n-2}}<0$ as required. ∎

We note that if $\alpha=n$, the leading term in (2.2) vanishes and so the
above computation does not work . However, it is possible to adapt the proof
to include the case that $\alpha=n$ by using a decay rate of $re^{-nr}$ for
$r>r_{\delta}$ and making the corresponding adjustments. A similar obstacle
occurs in the later construction of a corresponding super-solution and can be
treated with the same adjustment.

We now use the above sub-solution to prove the existence part of Theorem 1.1.

######  Proposition 2.5.

Let $(M,g)$ be ALH of regularity order $1$ and with decay exponent
$\alpha\in(0,n)$ satisfying (1.2). There exists a positive smooth solution $u$
of (Ya) on $M$ satisfying $u\geq 1-Ce^{-\alpha r}$ for some $C>0$.
Consequently, there exists a complete conformal metric $\tilde{g}$ such that
$S_{\tilde{g}}\equiv-n(n-1)$ on $M$.

###### Proof.

By Lemma 2.4 there exists a sub-solution $u_{-}$ of (Ya) which satisfies
$u_{-}\geq 1-Ce^{-\alpha r}$. The sub- and super-solution argument of Aviles
and McOwen in [AM88b, Proposition 2.1] yields a smooth solution $u$ of (Ya)
satisfying $u\geq u_{-}$ on $M$. It remains to show that $u$ is positive on
all of $M$; once this is established, we will have obtained a conformal metric
$\tilde{g}=u^{\frac{4}{n-2}}g$ which is complete, from the sub-solution lower
bound, and has constant scalar curvature.

As $u\geq u_{-}$, $u$ is non-negative and positive outside of some compact
set. Let $B$ be a large ball on which $u\not\equiv 0$ and outside of which
$u>0$. On $B$, the scalar curvature $S_{g}\leq A$ for some constant $A>0$.
Consequently, taking a sufficiently large constant $C>0$ such that $u$
satisfies

$c_{n}\Delta_{g}u-Cu\leq[n(n-1)u^{\frac{n+2}{n-2}}+Au]-Cu\leq 0,$

we can apply the strong maximum principle ([GT01, Theorem 3.5]) to deduce
that, as $u\not\equiv 0$ on $B$, $u$ is strictly positive in $B$ and so on all
of $M$. ∎

###  2.3 Asymptotic properties of the conformal factor

In order to complete the proof of the remaining parts of Theorem 1.1, we study
the asymptotic properties of solutions of the Yamabe equation as well as the
asymptotic properties of the particular conformal factor obtained as the
solution to (Ya) in the previous sub-section.

####  2.3.1 An upper bound at infinity

We will first establish an a priori upper bound on solutions to the Yamabe
equation for ALH manifolds in the sense of Definition 2.1.

######  Proposition 2.6.

Let $(M,g)$ be ALH in the sense of Definition 2.1 and suppose that

$\liminf_{r(x)\rightarrow\infty}S_{g}(x)\geq-n(n-1).$ (2.6)

Then all solutions of the (Ya) on $(M,g)$ satisfy
$\limsup_{r(x)\rightarrow\infty}u(x)\leq 1$.

A key point for our discussion will be to compare the Laplacian of the
perturbed metric $g$ to that of the reference metric $\mathring{g}$. In
particular, where we use $\nabla$ and $\mathring{\nabla}$ to denote the
covariant derivative with respect to $g$ and $\mathring{g}$ respectively, we
compute directly that

$\displaystyle\Delta_{g}\varphi$
$\displaystyle=\mathring{g}^{ij}\mathring{\nabla}_{i}\mathring{\nabla}_{j}+(g^{ij}-\mathring{g}^{ij})\mathring{\nabla}_{i}\mathring{\nabla}_{j}\varphi+g^{ij}(\Gamma^{k}_{ij}-\mathring{\Gamma}^{k}_{ij})\partial_{k}\varphi$
$\displaystyle=\Delta_{\mathring{g}}\varphi+a^{ij}\mathring{\nabla}_{ij}\varphi+b^{i}\mathring{\nabla}_{i}\varphi$
(2.7)

where $a^{ij}:=g^{ij}-\mathring{g}^{ij}$ and
$b^{i}:=g^{jk}(\Gamma^{i}_{jk}-\mathring{\Gamma}^{i}_{jk})$ satisfy
$a^{ij}=\mathcal{O}_{1}(e^{-(\alpha+2)r})$ and $b^{i}=\mathcal{O}(e^{-\alpha
r})$ using the estimates for the Christoffel symbols found in the proof of
Lemma 2.2.

We will make use of the following consequence of the maximum principle:

######  Lemma 2.7.

Let $u>0$ be a bounded smooth solution to (Ya) on a bounded open set
$\Omega\subset M$ and $\bar{u}>0$ be a smooth super–solution to (Ya) on
$\Omega$ satisfying $\bar{u}(x)\rightarrow\infty$ as
$x\rightarrow\partial\Omega$. Then necessarily $u<\bar{u}$ on $\Omega$.

###### Proof.

Due to the facts that $\bar{u}\rightarrow\infty$ as
$x\rightarrow\partial\Omega$, $u$ is bounded and both $u>0$ and $\bar{u}>0$,
there must exist some $C>0$ such that the difference $w_{C}:=C\bar{u}-u$
satisfies that $w_{C}\geq 0$ and achieves $0$ at some point in $\Omega$. We
claim that $C<1$. If $C\geq 1$, then $w_{C}$ would satisfy,

$\displaystyle-c_{n}\Delta_{g}w_{C}-n(n-1)w_{C}$
$\displaystyle\geq-n(n-1)\left(C\bar{u}^{\frac{n+2}{n-2}}-u^{\frac{n+2}{n-2}}\right)$
$\displaystyle\geq-n(n-1)\left((C\bar{u})^{\frac{n+2}{n-2}}-u^{\frac{n+2}{n-2}}\right)=c(x)w_{C}$

where

$c(x):=\begin{cases}\frac{(C\bar{u})^{\frac{n+2}{n-2}}-u^{\frac{n+2}{n-2}}}{C\bar{u}-u}\text{
for }w_{C}\neq 0,\\\ \frac{n+2}{n-2}(C\bar{u}(x))^{\frac{4}{n-2}}\text{ if
}w_{C}=0.\end{cases}$

We could then apply the strong maximum principle to the linear differential
inequality $-c_{n}\Delta_{g}w_{C}-\left(n(n-1)+c(x)\right)w_{C}\geq 0,$ noting
that the sign of $\left(n(n-1)+c(x)\right)$ does not matter as the value of
the minimum in question is $0$, see [GT01, Section 3.2]. We would then have
that $w_{C}\equiv 0$, a contradiction as $w_{C}\rightarrow\infty$ at the
boundary. We conclude that $C<1$ as claimed and so $u<\bar{u}$ in $\Omega$. ∎

In order to prove the upper bound in Proposition 2.6, we will study a family
of ODE solutions on annuli in the following series of lemmas.

######  Lemma 2.8.

There exists a positive solution $u_{1}$ of the equation

$-c_{n}\left(u_{1}^{\prime\prime}+(n-1)u_{1}^{\prime}\right)-n(n-1)u_{1}+n(n-1)u_{1}^{\frac{n+2}{n-2}}=0\text{
on }(-1,1)$ (2.8)

satisfying $u_{1}(r)\rightarrow\infty$ as $r\rightarrow\pm 1$ and there exists
a constant $C>0$ depending on $n$ such that $u_{1}$ satisfies

$\displaystyle\begin{split}\frac{1}{C}(1-|r|)^{-\frac{n-2}{2}}\leq\
&u_{1}(r)\leq C(1-|r|)^{-\frac{n-2}{2}},\\\ &u_{1}^{\prime}(r)\leq
C(1-|r|)^{-\frac{n}{2}},\\\ &u_{1}^{\prime\prime}(r)\leq
C(1-|r|)^{-\frac{n+2}{2}}.\end{split}$ (2.9)

###### Proof.

This result is essentially due to [ACF92]. Consider the model locally
hyperbolic manifold $(\mathbb{R}\times\mathbb{T}^{n-1},g_{\delta})$ where
$(\mathbb{T}^{n-1},\mathring{h})$ is the flat Torus. Define the annulus
$A_{1}:=\\{|r|\leq 1\\}\subset M$. We note here that any solution of (2.8)
would automatically be a solution of the Yamabe equation

$-c_{n}\Delta_{g_{\delta}}u_{1}-n(n-1)u_{1}+n(n-1)u_{1}^{\frac{n+2}{n-2}}=0$

on the annulus in this particular model space. From [ACF92, Theorem 1.2], we
know that, on the annulus $A_{1}$, there exists a unique solution $u_{1}$ of
the Yamabe equation satisfying $u_{1}\rightarrow\infty$ at the boundary
$\partial A_{1}$. Additionally, from Theorem 1.3 of [ACF92], we know the
behaviour of $u_{1}$ at the boundary satisfies (2.9).

We note that the Yamabe equation is invariant under symmetries of the Torus
and so, by uniqueness, $u_{1}$ must be radial and so the Yamabe equation
reduces to the ODE (2.8) as desired. ∎

Equation (2.8) has the following important scaling property that we will make
use of: if we define the family $\\{u_{R}\\}$ by

$u_{R}(r)=\left(\frac{(e^{2}-1)e^{R}}{e^{2+R}+e^{2R+r}-e^{2+r}-e^{R}}\right)^{\frac{n-2}{2}}u_{1}\left(\log\left(\frac{(e^{2R}-1)e^{r+1}}{e^{2+R}+e^{2R+r}-e^{2+r}-e^{R}}\right)\right),$
(2.10)

then each $u_{R}$ provides a solution of the ODE

$-c_{n}\left(u_{R}^{\prime\prime}+(n-1)u_{R}^{\prime}\right)-n(n-1)u_{R}+n(n-1)u_{R}^{\frac{n+2}{n-2}}=0\text{
on }(-R,R).$ (2.11)

Furthermore, from the growth rate of $u_{1}$ in (2.9) we have that each
$u_{R}$ satisfies that $u_{R}(r)\rightarrow\infty$ as $r\rightarrow\pm R$ and,
for some constant $C_{R}>0$ depending on $R$ and $n$, that

$\displaystyle\begin{split}\frac{1}{C_{R}}(R-|r|)^{-\frac{n-2}{2}}\leq\
&u_{R}(r)\leq C_{R}(R-|r|)^{-\frac{n-2}{2}},\\\ &u_{R}^{\prime}(r)\leq
C_{R}(R-|r|)^{-\frac{n}{2}},\\\ &u_{R}^{\prime\prime}(r)\leq
C_{R}(R-|r|)^{-\frac{n+2}{2}}.\end{split}$ (2.12)

We now establish locally uniform convergence of this family to $1$ as
$R\rightarrow\infty$.

######  Lemma 2.9.

The family $\\{u_{R}\\}_{R>0}$ of positive solutions of the equation (2.11)
defined above is decreasing with respect to $R$ and satisfies $u_{R}\searrow
1$ uniformly on compact sets as $R\rightarrow\infty$.

###### Proof.

If $R_{1}<R_{2}$, we may apply Lemma 2.7 on the annulus $A_{R_{1}}(0)$ to see
that $u_{R_{2}}<u_{R_{1}}$. Consequently, $\\{u_{R}\\}$ is monotone
decreasing.

Define the pointwise limit $u_{\infty}(r):=\lim_{R\rightarrow\infty}u_{R}(r)$.
We now show that $\\{u_{R}\\}$ converges locally in $C^{2}$ to $u_{\infty}$.
To this end, a locally uniform bound on the first derivative will suffice, as
from this we immediately obtain local boundedness of $u_{R}^{\prime\prime}$
and $u_{R}^{\prime\prime\prime}$ (the latter after differentiating the ODE
once). The Arzela-Ascoli theorem then provides the desired convergence. To
obtain such a bound on the first derivative, note that on any compact domain
$[-R_{0},R_{0}]$ and for $R>R_{0}+\varepsilon$, the fact that $\\{u_{R}\\}$ is
monotone decreasing implies

$u_{R}^{\prime\prime}+(n-1)u_{R}^{\prime}=\frac{n(n-2)}{4}\left(u_{R}^{\frac{n+2}{n-2}}-u_{R}\right)$

is bounded in $[-R_{0},R_{0}]$ uniformly as $R\rightarrow\infty$. Integrating
the above from some point $s\in[-R_{0},R_{0}]$ to $r\in[-R_{0},R_{0}]$ we
obtain that $u_{R}^{\prime}(r)-u_{R}^{\prime}(s)$ is bounded in
$[-R_{0},R_{0}]$ uniformly as $R\rightarrow\infty$. Integrating the above, now
in $s$, from $-R_{0}$ to $R_{0}$ we see that $u_{R}^{\prime}$ is bounded in
$[-R_{0},R_{0}]$ uniformly as $R\rightarrow\infty$.

We proceed to show that $u_{\infty}$ is constant by using the scaling property
(2.10) of the family $u_{R}$. In particular, for any $R,S>0$ we can verify
that

$u_{R}(r)=\left(\frac{(e^{2S}-1)e^{R}}{e^{2S+R}+e^{2R+r}-e^{2S+r}-e^{R}}\right)^{\frac{n-2}{2}}u_{S}\left(\log\left(\frac{(e^{2R}-1)e^{r+S}}{e^{2S+R}+e^{2R+r}-e^{2S+r}-e^{R}}\right)\right).$
(2.13)

Consider the value of $u_{R}$ at $r=0$ and write $R=S+\Lambda$. We then have

$u_{S+\Lambda}(0)=\left(\frac{e^{3S+\Lambda}-e^{S+\Lambda}}{e^{3S+\Lambda}+e^{2(S+\Lambda)}-e^{2S}-e^{S+\Lambda}}\right)^{\frac{n-2}{2}}u_{S}\left(\log\left(\frac{e^{3S+2\Lambda}-e^{S}}{e^{3S+\Lambda}+e^{2(S+\Lambda)}-e^{2S}-e^{S+\Lambda}}\right)\right).$

Taking the limit as $S\rightarrow\infty$ while keeping $\Lambda$ fixed we then
obtain that for all $\Lambda$,
$u_{\infty}(0)=u_{\infty}(\log\left(e^{\Lambda}\right))=u_{\infty}(\Lambda)$.
Consequently, $u_{\infty}\equiv C_{\infty}$ for some constant $C_{\infty}\geq
0$.

Recall that $u_{R}$ converges locally in $C^{2}$ to $u_{\infty}$ and so
$u_{\infty}$ solves the Yamabe equation. Consequently, as $u_{\infty}$ is
constant, either $u_{\infty}\equiv 0$ or $u_{\infty}\equiv 1$.

It then suffices to show that $u_{\infty}(0)>0$ to conclude the proof. We take
the limit in (2.10) as $R\rightarrow\infty$ at $r=0$ to obtain

$\lim_{R\rightarrow\infty}u_{R}(0)=\lim_{R\rightarrow\infty}e^{-\frac{n-2}{2}R}u_{1}\left(1-(e^{2}-1)e^{-R}+\mathcal{O}(e^{-2R})\right)$

and we note also that $1-(e^{2}-1)e^{-R}+\mathcal{O}(e^{-2R})\rightarrow 1.$
Then, from the asymptotic rates (2.9) for $u_{1}$, we conclude that

$u_{\infty}(0)=\lim_{R\rightarrow\infty}u_{R}(0)\geq\lim_{R\rightarrow\infty}Ce^{-\frac{n-2}{2}R}\left((e^{2}-1)e^{-R}+\mathcal{O}(e^{-2R})\right)^{-\frac{n-2}{2}}=C(e^{2}-1)^{-\frac{n-2}{2}}>0.$

Consequently, we conclude that $u_{\infty}\equiv 1$. ∎

Having established the properties of the family $u_{R}$ above, we now prove
Proposition 2.6 by showing that these functions can be used to generate super-
solutions with the desired properties on annuli in the asymptotic region.

###### Proof of Proposition 2.6.

First, we consider the Yamabe equation for $g$ as a perturbation of an
elliptic equation with respect to the reference metric $\mathring{g}$ as in
(2.7). Fix $R>0$ and consider some point $x_{\ast}\in M^{+}$ such that
$r_{\ast}:=r(x_{\ast})>R$. To bound the value of $u$ at $x_{\ast}$, we define
the annulus $\Omega_{R}:=\Omega(R,r_{\ast})=\\{x\in M^{+}\ :\
|r(x)-r_{\ast}|<R\\}$ and define a candidate super-solution to (Ya) on
$\Omega_{R}$,

$\bar{u}(x):=Au_{R}(r(x)-r_{\ast})$

where $u_{R}$ is defined in (2.10) and $A>0$ is some constant to be
determined. In the argument below, all implicit constants in various
$\mathcal{O}$ terms are independent of both $A$ and $R$.

We first note that

$-c_{n}\Delta_{\mathring{g}}\bar{u}=-c_{n}\left(\partial_{rr}\bar{u}+(n-1)q_{k}(r)\partial_{r}\bar{u}\right)=A\left(-c_{n}\left(u_{R}^{\prime\prime}+(n-1)u_{R}^{\prime}\right)+\mathcal{O}(e^{-2r}|u_{R}^{\prime}|)\right)$

where

$q_{k}(r)=\begin{cases}\coth(r)&\ k=1,\\\ 1&\ k=0,\\\ \tanh(r)&\
k=-1,\end{cases}$

and $k$ is as in the definition of reference metrics in (2.1) and we use only
that $q_{k}=1+\mathcal{O}(e^{-2r})$.

Consequently, from (2.11), we have that $\bar{u}$ satisfies

$\displaystyle-
c_{n}\Delta_{\mathring{g}}\bar{u}-n(n-1)\bar{u}+n(n-1)\bar{u}^{\frac{n+2}{n-2}}=(A^{\frac{n+2}{n-2}}-A)u_{R}^{\frac{n+2}{n-2}}+A\mathcal{O}(e^{-2r}|u_{R}^{\prime}|)$

for the reference metric $\mathring{g}$ on $\Omega_{R}\subset M^{+}$ and
$\bar{u}$ blows up as $x\rightarrow\partial\Omega_{R}$. If follows that

$\displaystyle-$ $\displaystyle
c_{n}\Delta_{\mathring{g}}\bar{u}+S_{g}\bar{u}+n(n-1)\bar{u}^{\frac{n+2}{n-2}}+a^{ij}\mathring{\nabla}_{ij}\bar{u}+b^{i}\mathring{\nabla}_{i}\bar{u}$
$\displaystyle=n(n-1)(A^{\frac{n+2}{n-2}}-A)u_{R}^{\frac{n+2}{n-2}}+A\left(a^{ij}\mathring{\nabla}_{ij}u_{R}+b^{i}\mathring{\nabla}_{i}u_{R}+(S_{g}+n(n-1))u_{R}\right)$
$\displaystyle\geq
n(n-1)(A^{\frac{n+2}{n-2}}-A)u_{R}^{\frac{n+2}{n-2}}-A\varepsilon(r_{\ast})\left(|u_{R}^{\prime\prime}|+|u_{R}^{\prime}|+|u_{R}|\right)$

where $\varepsilon(r)\geq 0$ and $\lim_{r\rightarrow\infty}\varepsilon(r)=0$;
here we have used (2.6) as well as the asymptotic local hyperbolicity.

From the asymptotic rates in (2.12), we see that there exists a constant
$C_{R}>0$ depending on $R$ such that
$|u_{R}^{\prime\prime}|+|u_{R}^{\prime}|+|u_{R}|\leq
C_{R}u_{R}^{\frac{n+2}{n-2}}$ near the boundary. Therefore,

$\displaystyle-
c_{n}\Delta_{\mathring{g}}\bar{u}+S_{g}\bar{u}+n(n-1)\bar{u}^{\frac{n+2}{n-2}}$
$\displaystyle+a^{ij}\mathring{\nabla}_{ij}\bar{u}+b^{i}\mathring{\nabla}_{i}\bar{u}$
$\displaystyle\geq
n(n-1)\left(A^{\frac{n+2}{n-2}}-A\left(1+C_{R}\varepsilon(r_{\ast})\right)\right)u_{R}^{\frac{n+2}{n-2}}.$

We can now choose $A=1+\tilde{C}\varepsilon(r_{\ast})$ with a sufficiently
large constant $\tilde{C}$ independent of $x_{\ast}$ so that the RHS above is
positive and so $\bar{u}$ is a strict super-solution to (Ya).

We may now apply Lemma 2.7 to conclude that $u<\bar{u}$ on $\Omega_{R}$ and so
we have shown that any solution to the Yamabe equation $u$ satisfies
$u(x_{\ast})<(1+\tilde{C}\varepsilon(r_{\ast}))u_{R}(0).$ Consequently, as
$x_{\ast}$ was arbitrary and $\varepsilon(r_{\ast})\rightarrow 0$, we have
$\limsup_{r(x)\rightarrow\infty}u(x)\leq u_{R}(0).$ As $R>0$ was arbitrary, we
may now take the limit as $R\rightarrow\infty$ to conclude, from Lemma 2.9,
that $\limsup_{r(x)\rightarrow\infty}u(x)\leq 1$ as desired. ∎

####  2.3.2 A lower bound at infinity

In this sub-section, we provide an a priori lower bound for solutions $u$ of
the Yamabe equation for ALH manifolds in the sense of Definition 2.1.

We highlight that Example 1.2 demonstrates, in a similar way as for uniqueness
of the conformal factor, that we must impose some additional condition before
attempting to bound the conformal factor $u$ from below. We first provide a
lower bound under the assumption that the solution is bounded away from zero
at infinity in the sense of (2.15) below. We then refine this first result by
weakening the strictly positive requirement of (2.15) to allow some degree of
decay of $u$ to zero.

######  Lemma 2.10.

Let $(M,g)$ be ALH in the sense of Definition 2.1 and suppose that

$\lim_{r(x)\rightarrow\infty}|S_{g}(x)-n(n-1)|=0.$ (2.14)

Then any solution $u$ of (Ya) on $(M,g)$ satisfying

$\liminf_{r(x)\rightarrow\infty}u(x)>0$ (2.15)

must satisfy $\lim_{r(x)\rightarrow\infty}u(x)=1.$

###### Proof.

Suppose, for the sake of contradiction that the conclusion fails. Since, by
Proposition 2.6, we have that $\limsup_{r(x)\rightarrow\infty}u(x)\leq 1$, we
must have that $\delta=\liminf_{r(x)\rightarrow\infty}u(x)\in(0,1)$.

Let $\\{(r_{i},\theta_{i})\\}\subset M^{+}$ be a sequence such that
$r_{i}\rightarrow\infty$ and $u(r_{i},\theta_{i})\rightarrow\delta$. Passing
to a subsequence, we may assume that $\theta_{i}\rightarrow\theta_{\infty}$
for some $\theta_{\infty}\in N$. Take a normal coordinate chart $U$ on $N$
around $\theta_{\infty}$ with coordinate functions
$\theta^{a}:U\rightarrow\mathbb{R}^{n}$ so that
$\mathring{h}_{ab}(\theta_{\infty})=\delta_{ab}$. Consider the region
$\Omega:=\mathbb{R}_{\geq 0}\times U\subset M^{+}.$ For each $i$, define a map
$\Psi_{i}:\Omega\rightarrow\mathbb{R}\times\mathbb{R}^{n-1}=\\{(\tilde{r},\tilde{\theta}):\tilde{r}\in\mathbb{R},\tilde{\theta}\in\mathbb{R}^{n-1}\\}$
by

$\Psi_{i}(r,\theta)=(r-r_{i},e^{r_{i}}(\theta-\theta_{i}))$

and define $\tilde{V}_{i}:=\Psi_{i}(\Omega).$ We also write
$\psi_{i}(\theta):=e^{r_{i}}(\theta-\theta_{i})$. Define
$v_{i}:=u\circ\Psi_{i}^{-1}:\tilde{V}_{i}\rightarrow\mathbb{R}.$ We note here
that $\tilde{V}_{1}\subset\tilde{V}_{2}\subset\dots$ and
$\cup_{i}\tilde{V}_{i}=\mathbb{R}\times\mathbb{R}^{n-1}$.

We next compute the equation corresponding to (Ya) that $v_{i}$ satisfies. We
denote partial derivatives in the coordinates $(\tilde{r},\tilde{\theta})$ by
$\tilde{\partial}$, that is we write
$\frac{\partial}{\partial\tilde{r}}=\tilde{\partial_{r}}$ and
$\frac{\partial}{\partial\tilde{\theta}^{a}}=\tilde{\partial_{a}}$. On
$\mathbb{R}\times\mathbb{R}^{n-1}$ let
$g_{i}=\left(\Psi_{i}^{-1}\right)^{\ast}g$. Noting that, by asymptotic local
hyperbolicity, we may express $g$ in the form

$g=dr^{2}+\varepsilon_{ra}drd\theta^{a}+\left(f_{k}(r+r_{0})\mathring{h}_{ab}+\varepsilon_{ab}\right)d\theta^{a}d\theta^{b},$

where $\varepsilon_{ra}(r,\theta)=\mathcal{O}_{m}(e^{-(\alpha-1)r})$ and
$\varepsilon_{ab}(r,\theta)=\mathcal{O}_{m}(e^{-(\alpha-2)r})$. We thus have

$\displaystyle g_{i}$
$\displaystyle=d\tilde{r}^{2}+e^{-r_{i}}\varepsilon_{ra}\circ\Psi_{i}^{-1}d\tilde{r}d\tilde{\theta}^{a}+e^{-2r_{i}}\left(f_{k}^{2}(\tilde{r}+r_{i}+r_{0})\mathring{h}_{ab}\circ\psi_{i}^{-1}+\varepsilon_{ab}\circ\Psi_{i}^{-1}\right)d\tilde{\theta}^{a}d\tilde{\theta}^{b}$
$\displaystyle=d\tilde{r}^{2}+e^{2(\tilde{r}+r_{0})}\mathring{h}_{ab}\circ\psi_{i}^{-1}d\tilde{\theta}^{a}d\tilde{\theta}^{b}+e^{-2r_{i}}\hat{\varepsilon}_{ab}d\tilde{\theta}^{a}d\tilde{\theta}^{b}+e^{-\alpha
r_{i}}\left(\tilde{\varepsilon}_{ra}d\tilde{r}d\tilde{\theta}^{a}+\tilde{\varepsilon}_{ab}d\tilde{\theta}^{a}d\tilde{\theta}^{b}\right)$

where
$\hat{\varepsilon}_{ab}=\hat{\varepsilon}_{ab}^{i}=\left(f_{k}^{2}(\tilde{r}+r_{i}+r_{0})-e^{2(r_{i}+\tilde{r}+r_{0})}\right)\mathring{h}_{ab}\circ\psi_{i}^{-1}$,
$\tilde{\varepsilon}_{ra}=\tilde{\varepsilon}_{ra}^{i}=e^{(\alpha-1)r_{i}}\varepsilon_{ra}\circ\Psi_{i}^{-1}$
and
$\tilde{\varepsilon}_{ab}=\tilde{\varepsilon}_{ab}^{i}=e^{(\alpha-2)r_{i}}\varepsilon_{ab}\circ\Psi_{i}^{-1}$.
In particular, as $i\rightarrow\infty$, the sequence of matrices of metric
components of $g_{i}$ in $(\tilde{r},\tilde{\theta})$ is uniformly bounded in
$C^{1}$ and uniformly positive definite on any compact subset of
$\mathbb{R}\times\mathbb{R}^{n-1}$. In addition, $\\{g_{i}\\}$ converges in
$C^{1}_{\operatorname{loc}}(\mathbb{R}\times\mathbb{R}^{n-1})$ to
$g_{\infty}=d\tilde{r}^{2}+e^{2(\tilde{r}+r_{0})}\delta_{ab}d\tilde{\theta}^{a}d\tilde{\theta}^{b}$.

Each $v_{i}$ solves the equation

$-c_{n}\Delta_{g_{i}}v_{i}=s_{i}v_{i}-n(n-1)\left(v_{i}^{\frac{n+2}{n-2}}-v_{i}\right)=c_{i}v_{i}$
(2.16)

where $s_{i}=S_{g_{i}}\circ\Psi_{i}^{-1}+n(n-1)$ which, by (2.14), satisfies
$s_{i}\rightarrow 0$ in $C^{0}_{\operatorname{loc}}$ as $i\rightarrow\infty$
and
$c_{i}=s_{i}-n(n-1)\left(\left(u\circ\Psi_{i}^{-1}\right)^{\frac{4}{n-2}}-1\right)$.
By Proposition 2.6, $u$ is bounded and so $\\{c_{i}\\}$ is uniformly bounded
on any compact subset of $\mathbb{R}\times\mathbb{R}^{n-1}$. Consequently,
fixing some $R>0$, we may apply the Harnack inequality of [GT01, Theorem 8.20]
to the equation $L_{i}v_{i}=0$ where $L_{i}:=c_{n}\Delta_{g_{i}}+c_{i}$ to
conclude that, on the ball
$B_{R}(0,0)\subset\mathbb{R}\times\mathbb{R}^{n-1}$,

$\sup_{B_{R}(0,0)}v_{i}\leq C_{R}\inf_{B_{R}(0,0)}v_{i}\leq C_{R}\delta$
(2.17)

where the final inequality follows from the fact that
$v_{i}(0,0)\rightarrow\delta$ by construction and $C$ is some constant
independent of $i$ (depending only on $n$, $R$ and bounds on the ellipticity
of $L_{i}$ and sup-norm of the coefficients of $L_{i}$ which we have shown may
be chosen independently of $i$). We conclude that $\\{v_{i}\\}$ is uniformly
bounded on $B_{R}(0,0)$.

After recasting (2.16) to non-divergence form and noting that $g$ is ALH of
regularity order $1$, we may now apply standard $W^{2,p}$ estimates ([GT01,
Theorem 9.11]) to conclude that $v_{i}$ are locally uniformly bounded in
$W^{2,p}$ for all $p\in(1,\infty)$ and hence in $C^{1,\beta}$ for all
$\beta\in(0,1)$. Therefore, passing to a subsequence if necessary, we may
assume that $v_{i}\rightarrow v_{\infty}\text{ in
}C^{1}_{\operatorname{loc}}(\mathbb{R}\times\mathbb{R}^{n-1}).$ Furthermore,
for any point
$(\tilde{r}_{0},\tilde{\theta}_{0})\in\mathbb{R}\times\mathbb{R}^{n-1}$ and
corresponding sequence
$(\bar{r}_{i},\bar{\theta}_{i}):=\Psi_{i}^{-1}(\tilde{r}_{0},\tilde{\theta}_{0})\in
M^{+}$, we have that

$\displaystyle v_{\infty}(\tilde{r}_{0},\tilde{\theta}_{0})$
$\displaystyle=\lim_{i\rightarrow\infty}(v_{i}\circ\Psi_{i})(\bar{r}_{i},\bar{\theta}_{i})=\lim_{i\rightarrow\infty}u(\bar{r}_{i},\bar{\theta}_{i})\geq\delta.$

We now pass to the limit as $i\rightarrow\infty$ in (2.16) to obtain

$-c_{n}\Delta_{g_{\infty}}v_{\infty}=-n(n-1)\left(v_{\infty}^{\frac{n+2}{n-2}}-v_{\infty}\right)\text{
on }\mathbb{R}\times\mathbb{R}^{n-1},$ (2.18)

and so $v_{\infty}$ is in fact in
$C^{\infty}(\mathbb{R}\times\mathbb{R}^{n-1})$ (as $v_{\infty}>0$). By
construction, we have that
$v_{\infty}(0,0)=\lim_{i\rightarrow\infty}v_{i}(\Psi(r_{i},\theta_{i}))=\lim_{i\rightarrow\infty}u(r_{i},\theta_{i})=\delta$.
Since $\delta$ is a strict sub-solution of (2.18) and $v_{\infty}\geq\delta$
on $\mathbb{R}\times\mathbb{R}^{n-1}$, we obtain a contradiction to the strong
comparison principle. ∎

We are now able to prove our main result of the section, expanding the extent
of the lower bound result by allowing for the conformal factor to exhibit
decay to zero at infinity.

######  Proposition 2.11.

Let $(M,g)$ be an ALH manifold in the sense of Definition 2.1 which satisfies
(2.14). Suppose that $u$ is a solution of the Yamabe problem on $(M,g)$ and
that there exist an $S>0$ and $r_{1}>r_{0}$ such that

$\frac{\underline{u}(r+S)}{\underline{u}(r)}>\frac{n}{2}e^{-\frac{n-2}{2}S}-\frac{n-2}{2}e^{-\frac{n}{2}S}\text{
for all }r>r_{1},$ (2.19)

where $\underline{u}(r):=\min_{N}u(r,\cdot)$. Then
$\lim_{r(x)\rightarrow\infty}u(x)=1.$

We note that the asymptotic exponential rate of $-\frac{n-2}{2}$ is also seen
in Example 1.2 and represents a borderline rate where a solution to (Ya)
exists with such a rate of decay but notably the corresponding conformal
metric fails to be complete. Indeed, for any slower decay, we obtain the
following immediate corollary of Proposition 2.11:

######  Corollary 2.12.

Let $(M,g)$ be an ALH manifold in the sense of Definition 2.1 which satisfies
(2.14). There does not exist a solution $u$ of (Ya) on $M$ satisfying
$e^{\beta r(x)}u(x)\rightarrow 0$ as $r(x)\rightarrow\infty$ for any
$0<\beta<\frac{n-2}{2}$.

###### Proof of Proposition 2.11.

By Lemma 2.10, it suffices to show that
$\liminf_{r(x)\rightarrow\infty}u(x)>0$. Suppose not, then there exists a
sequence $r_{i}\rightarrow\infty$ such that $\underline{u}(r_{i})\rightarrow
0$. It follows that

$\min_{r(x)\in[r_{1},R]}u(x)\rightarrow 0\text{ as }R\rightarrow\infty.$

Consequently, for $R$ sufficiently large the minimum cannot be attained at
$r_{1}$. Furthermore, the fact that $u\leq 1$ (by Proposition 2.6) and
$\liminf u=0$ together imply, via the strong maximum principle, that $u<1$ and
satisfies

$-c_{n}\Delta_{g}u=-n(n-1)\left(u^{\frac{n+2}{n-2}}-u\right)>0.$

The strong maximum principle again implies that the minimum of $u$ on
$\\{r(x)\in[r_{1},R]\\}$ is achieved only on $\\{r(x)=R\\}$ and so
$\underline{u}$ is strictly decreasing for large $r$.

Define $\alpha_{i}:=\underline{u}(r_{i})^{-1}\rightarrow\infty$ and consider
$w_{i}:=\alpha_{i}v_{i}$ where $v_{i}$ is as in the proof of Lemma 2.10. Note
that, by (2.16), $w_{i}$ satisfies $-c_{n}\Delta_{g_{i}}w_{i}=c_{i}w_{i}.$ As
in the proof of Lemma 2.10, we have that $w_{i}\rightarrow w_{\infty}$ in
$C^{1}_{\operatorname{loc}}(\mathbb{R}\times\mathbb{R}^{n-1})$. Since
$\alpha_{i}\rightarrow\infty$, the limit $w_{\infty}$ now solves

$-c_{n}\left(\tilde{\partial}_{rr}w_{\infty}+(n-1)\tilde{\partial}_{r}w_{\infty}+e^{-2(\tilde{r}+r_{0})}\Delta_{n-1}w_{\infty}\right)=n(n-1)w_{\infty}\text{
on }\mathbb{R}\times\mathbb{R}^{n-1}.$ (2.20)

Moreover, by construction, $w_{\infty}(r,\theta)\geq 1$ for $r\leq 0$ and
there exists a $\theta_{0}$ such that $w_{\infty}(0,\theta_{0})=1$. We will
now see that the above provides a contradiction to (2.19). Indeed, consider
the radial function

$\underline{w}(r)=\frac{n}{2}e^{-\frac{n-2}{2}r}-\frac{n-2}{2}e^{-\frac{n}{2}r}$

which solves (2.20) and satisfies $\underline{w}(0)=1$ and
$\underline{w}^{\prime}(0)=0$. By assumption (2.19), there exists an $S>0$
such that $w_{\infty}(S,\theta)>\underline{w}(S)$ for all $\theta\in N$.
Consequently, as $w_{\infty}(0,\theta)\geq 1=\underline{w}(0)$, the strong
maximum principle and the Hopf Lemma imply that we must have
$w_{\infty}>\underline{w}$ in the region $\\{r(x)\in(0,S)\\}$ and
$\partial_{r}w_{\infty}(0,\theta_{0})>0$. The latter contradicts the fact that
$w_{\infty}(r,\theta)\geq 1$ for $r\leq 0$ and $w_{\infty}(0,\theta_{0})=1$. ∎

####  2.3.3 First derivative estimates at infinity

In this sub-section, we prove results for the first derivative decay of the
conformal factor.

We begin by defining a particular super-solution on $M$. The following super-
solution will provide an asymptotic upper bound on our solution obtained in
Proposition 2.5 and will allow us to conclude that the conformal metric
remains locally asymptotically hyperbolic to the same order.

######  Lemma 2.13.

Let $(M,g)$ be ALH of regularity order $1$ and with decay exponent
$\alpha\in(0,n)$ and $S_{g}\geq-n(n-1)-Ce^{-\alpha r}$ for some constant
$C>0$. There exist constants $A>0$ and $R>0$ such that the function $u_{+}\in
H^{1}_{\operatorname{loc}}(M)$ defined by

$u_{+}(r):=\begin{cases}1+Ae^{-\alpha r}\qquad\qquad&\text{ on }\\{r\geq
R\\}\times N\\\ 1+Ae^{-\alpha R}\qquad\qquad&\text{ on
}M_{0}\cup(\\{r<R\\}\times N)\end{cases},$

is a super-solution to (Ya) on $M$.

###### Proof.

We first note that the transmission condition $0=\lim_{r\nearrow
R}u_{+}^{\prime}(r)\geq\lim_{r\searrow R}u_{+}^{\prime}(r)=-\alpha Ae^{-\alpha
R}$ certainly holds. Thus, we only need to find large $A$ and $R$ such that

$-c_{n}\Delta_{g}u_{+}+S_{g}u_{+}\geq-n(n-1)u_{+}^{\frac{n+2}{n-2}}$

holds in the two regions $r\leq R$ and $r>R$.

We proceed similarly as in the proof of Lemma 2.4 with the difference that, as
$u_{+}>1$, we can estimate directly on the RHS that

$-n(n-1)u_{+}^{\frac{n+2}{n-2}}\leq-n(n-1)\left(1+\frac{n+2}{n-2}(u_{+}-1)\right).$

In the following the value of $C$ may change from line to line but always
depends only on $g$ and we choose $R$ larger if necessary so that $R>1$. We
compute for $r>R$, using that
$\frac{f_{k}^{\prime}}{f_{k}}(r+r_{0})>1-Ce^{-2r}$ and $u_{+}^{\prime}<0$,

$\displaystyle-c_{n}\Delta_{g}u_{+}+S_{g}u_{+}+$ $\displaystyle
n(n-1)u_{+}^{\frac{n+2}{n-2}}\geq$ $\displaystyle-c_{n}Ae^{-\alpha
r}\Bigg{[}\underbrace{\alpha^{2}-(n-1)\alpha-n+Ce^{-\alpha
r}+Ce^{-2r}+\frac{C}{A}}_{B}\Bigg{]}.$

Thus, we must choose $A$ and $R$ such that $B\leq 0$. As
$\alpha^{2}-(n-1)\alpha-n<0$, we can select an $A_{0}$ and $R$ such that the
RHS of the above is positive for all $A>A_{0}$. Fix $R$ from hereon. It
remains to find an $A>A_{0}$ such that (2.2) holds on $r\leq R$ which can be
verified by direct computation and noting that the scalar curvature must be
bounded on this compact interior region. ∎

Having established the super- solution above, we use it to gain control on the
solution $u$ obtained in the Section 2.2.

######  Lemma 2.14.

Let $(M,g)$ be an ALH manifold of regularity order $1$ and with decay exponent
$\alpha\in(0,n)$ satisfying (1.3). Then the smooth solution $u$ of (Ya) on $M$
obtained in Proposition 2.5 satisfies
$|u-1|+|\mathring{\nabla}u|_{\mathring{g}}=\mathcal{O}(e^{-\alpha r})$.
Furthermore, $u$ is maximal in the sense that any solution $\tilde{u}$ of (Ya)
satisfies $\tilde{u}\leq u$.

###### Proof.

We first prove the sup-norm decay. From Proposition 2.5, we have that $u\geq
u_{-}\geq 1-Ce^{-\alpha r}$ where $u_{-}$ is the sub-solution constructed in
Lemma 2.4. Additionally, Proposition 2.6 gives that
$\limsup_{r(x)\rightarrow\infty}u\leq 1$. Consequently, we have that
$\lim_{r(x)\rightarrow\infty}u(x)=1$

To obtain the rate of decay from above, we show that $u\leq u_{+}$ where
$u_{+}$ is the super-solution $u_{+}\leq 1+Ce^{-\alpha r}$ constructed in
Lemma 2.13. For the sake of contradiction, we suppose that
$\inf\frac{u_{+}}{u}=\frac{1}{C}<1$ for some constant $C>1$. As
$\lim_{r(x)\rightarrow\infty}\frac{u_{+}(x)}{u(x)}=1$, the infimum would be
achieved and so we could define $v:=Cu_{+}-u$ which would satisfy $v\geq 0$
and would achieve $0$ at some minimum. We could then apply the same maximum
principle argument as in Lemma 2.7 to conclude that $\inf\frac{u_{+}}{u}\geq
1$ and so $u\leq u_{+}$ as desired.

We next prove the derivative estimates. Let $(r_{\ast},\theta_{\ast})\in
M^{+}$ be some arbitrary point with $r_{\ast}$ sufficiently large. Take a
normal coordinate chart $U$ on $N$ around $\theta_{\ast}$ with coordinate
functions $\theta^{a}:U\rightarrow\mathbb{R}^{n}$ so that
$\mathring{h}_{ab}(\theta_{\infty})=\delta_{ab}$. Consider the region

$\Omega:=(r_{\ast}-1,r_{\ast}+1)\times U\subset M^{+}.$

Define a map
$\Psi:\Omega\rightarrow\mathbb{R}\times\mathbb{R}^{n-1}=\\{(\tilde{r},\tilde{\theta}):\tilde{r}\in\mathbb{R},\tilde{\theta}\in\mathbb{R}^{n-1}\\}$
by

$\Psi(r,\theta)=(r-r_{\ast},e^{r_{\ast}}(\theta-\theta_{\ast}))$

and define $\tilde{V}:=\Psi(\Omega).$ We also write
$\psi(\theta):=e^{r_{\ast}}(\theta-\theta_{\ast})$. Define
$v:=u\circ\Psi^{-1}:\tilde{V}\rightarrow\mathbb{R}.$

Let $g_{\ast}=(\Psi^{-1})^{*}g$. As in the proof of Lemma 2.10, as
$r_{\ast}\rightarrow\infty$, $g_{\ast}$ converges in $C^{1}$ to
$g_{\infty}=d\tilde{r}^{2}+e^{2(\tilde{r}+r_{0})}\delta_{ab}\
d\tilde{\theta}^{a}d\tilde{\theta}^{b}$. Moreover (see equation (2.16)), $v$
satisfies $\Delta_{g_{\ast}}v=\mathcal{O}\left(e^{-\alpha r_{\ast}}\right)$
where the implicit constant in the $\mathcal{O}$ term is independent of
$r_{\ast}$. Applying $W^{2,p}$ estimates to the above equation, after
recasting to non-divergence form, and using Sobolev embeddings, we obtain
$|\nabla v(0,0)|_{g_{\infty}}=\mathcal{O}(e^{-\alpha r_{\ast}})$ and so obtain
decay in the first derivative $|\nabla
u|_{\mathring{g}}=\mathcal{O}(e^{-\alpha r})$.

We finally address the maximality of $u$. Given another solution $\tilde{u}>0$
of (Ya), we know that $\limsup_{r(x)\rightarrow\infty}\tilde{u}(x)\leq 1$,
again from Proposition 2.6, and so we have that
$\liminf_{r(x)\rightarrow\infty}\frac{u}{\tilde{u}}\geq 1$ as
$\lim_{r(x)\rightarrow\infty}u(x)=1$. Consequently, if
$\inf\frac{u}{\tilde{u}}<1$ then the infimum must be attained and so we may
apply the maximum principle (as in the proof of $u\leq u_{+}$) to conclude
that $\tilde{u}\leq u$. ∎

###  2.4 Uniqueness and completion of the proof of Theorem 1.1

Under assumption (1.2), the first part of Theorem 1.1 is proved in Proposition
2.5. For the remainder of the proof, we will assume that (1.3) holds. By Lemma
2.14, we know that $\lim_{r\rightarrow\infty}u=1$,
$|u-1|+|\mathring{\nabla}u|_{\mathring{g}}=\mathcal{O}(e^{-\alpha r})$ and $u$
is the maximal solution of (Ya). It remains to prove the uniqueness of $u$ and
that the conformal metric $\tilde{g}=u^{\frac{4}{n-2}}g$ is ALH of regularity
order $0$ and with decay exponent $\alpha$, which we now treat in turn.

The uniqueness of $u$ is a direct corollary of the stronger result below:

######  Proposition 2.15.

Let $(M,g)$ be ALH of regularity order $1$ and decay exponent $\alpha\in(0,n)$
satisfying (1.3). Then the solution $u$ of the Yamabe problem on $(M,g)$
obtained in Proposition 2.5 is the unique solution such that there exist an
$S>0$ and $r_{1}>r_{0}$ such that

$\frac{\underline{u}(r+S)}{\underline{u}(r)}>\frac{n}{2}e^{-\frac{n-2}{2}S}-\frac{n-2}{2}e^{-\frac{n}{2}S}\text{
for all }r>r_{1},$ (2.21)

where $\underline{u}(r):=\min_{N}\tilde{u}(r,\cdot)$.

###### Proof.

Let $\tilde{u}$ be any solution of (Ya) on $(M,g)$ satisfying (2.21). By
Proposition 2.11, we know that $\lim_{r\rightarrow\infty}\tilde{u}=1$. Observe
that in the proof of the maximality of $u$ in Lemma 2.14 we actually proved
that any solution of (Ya) which tends to $1$ at infinity is maximal.
Consequently, both $u$ and $\tilde{u}$ are maximal and hence equal. ∎

Lastly, we prove that the asymptotic local hyperbolicity of the obtained
complete conformal metric of constant scalar curvature, thus completing the
proof of Theorem 1.1.

######  Lemma 2.16.

Let $(M,g)$ be an ALH manifold of regularity order $1$ and with decay exponent
$\alpha\in(0,n)$ satisfying (1.3). Let $u$ be the solution of (Ya) obtained in
Proposition 2.5, then $u=1+\mathcal{O}(e^{-\alpha r})$ and the corresponding
conformal metric is ALH of regularity order $0$ and with decay exponent
$\alpha$.

###### Proof.

By Lemma 2.14 we have
$|u-1|+|\mathring{\nabla}u|_{\mathring{g}}=\mathcal{O}(e^{-\alpha r})$.
Consider the new coordinate function

$z:=r+\int_{r}^{\infty}\left(1-u^{\frac{2}{n-2}}\right)\ ds\ .$

First note that

$z-r=\int_{r}^{\infty}\left(1-u^{\frac{2}{n-2}}\right)\
ds=\mathcal{O}_{1}(e^{-\alpha r})$

and the map $(r,\theta)\mapsto(z,\theta)$ is a diffeomorphism for large $r$.
We will show that this implies $\tilde{g}=u^{\frac{4}{n-2}}g$ is ALH of
regularity order $0$ and decay exponent $\alpha$ with respect to the
coordinate system $(z,\theta)$.

We compute that

$dz=u^{\frac{2}{n-2}}dr-\left(\int_{r}^{\infty}\frac{4-n}{n-2}u^{\frac{4-n}{n-2}}\partial_{a}u\
ds\right)d\theta^{a}=u^{\frac{2}{n-2}}dr+\mathcal{O}(e^{-(\alpha-1)r})d\theta^{a}.$

Recalling that $g$ is ALH and so may be written
$g=dr^{2}+f_{k}(r+r_{0})^{2}\mathring{h}+\varepsilon_{ra}d\theta^{a}dr+\varepsilon_{ab}d\theta^{a}d\theta^{b}$
where $\varepsilon_{ab}=\mathcal{O}(e^{-(\alpha-2)r})$ and
$\varepsilon_{ra}=\mathcal{O}(e^{-(\alpha-1)r})$, we deduce that

$\tilde{g}=dz^{2}+f_{k}(z+r_{0})^{2}\mathring{h}+\tilde{\varepsilon}_{za}d\theta^{a}dz+\tilde{\varepsilon}_{ab}d\theta^{a}d\theta^{b}$

where $\tilde{\varepsilon}_{za}=\mathcal{O}(e^{-(\alpha-1)z})$ and

$\tilde{\varepsilon}_{ab}=u^{\frac{4}{n-2}}\varepsilon_{ab}+\left(u^{\frac{4}{n-2}}f_{k}(r+r_{0})^{2}-f_{k}(z+r_{0})^{2}\right)\mathring{h}_{ab}+\mathcal{O}(e^{-2(\alpha-1)z}).$

Since
$f_{k}(r+r_{0})^{2}=f_{k}(z+r_{0})^{2}+\mathcal{O}_{1}(e^{-(\alpha-2)z})$, the
conclusion follows. ∎

###  2.5 Asymptotically warped product manifolds

To conclude this section, we briefly outline some remarks on the Yamabe
Problem for asymptotically warped product manifolds.

Recall that, in the definition of ALH manifolds, we supposed that we could
decompose $(M,g)$ in two parts, $M=M_{0}\cup M^{+}$, where $M_{0}$ is some
compact manifold with boundary, $M^{+}=\mathbb{R}^{+}\times N$ with
$(N,\mathring{h})$ some compact manifold of constant scalar curvature
$S_{\mathring{h}}=(n-1)(n-2)k$ for some $k\in\\{-1,0,1\\}$ and $M_{0}$ and
$M^{+}$ coincide on their common boundary. We then considered those metrics
$g$ which asymptote to the locally hyperbolic reference metrics in (2.1).

One may, instead, consider a different choice of reference metric to which $g$
is asymptotic. An immediate generalisation of interest would be the following
warped product metrics

$\mathring{g}_{f}=dz^{2}+f^{2}(z)\mathring{h}$ (2.22)

where $f$ is some positive function, usually referred to as the warping
function. Metrics $g$ which are asymptotic to $\mathring{g}_{f}$ will be
informally referred to as asymptotically warped product (AWP) metrics.

In this section, we discuss the applications of Proposition 1.1 in this
broader category of AWP metrics. This amounts to understanding whether the
warped product metrics of (2.22) are conformal to the reference locally
hyperbolic metrics of (2.1). We establish a necessary and sufficient condition
on the warping function for such conformality in Proposition 2.17. The
condition we identify is an integral condition of Keller-Osserman type (see
[Kel57, Oss57]). Furthermore, we also show that, if the warped product metric
is not conformally locally hyperbolic, then it is conformal to a complete
metric of finite volume.

######  Proposition 2.17.

Let $f:[0,\infty)\rightarrow\mathbb{R}$ be a smooth, positive function. The
metric $\mathring{g}_{f}$ in (2.22) is conformal to a locally hyperbolic
metric (2.1) if and only if

$I(f):=\int_{0}^{\infty}\frac{1}{f(s)}\ ds<\infty.$ (2.23)

Moreover, if (2.23) does not hold, then $\mathring{g}_{f}$ is conformal to
$d\check{z}^{2}+e^{-2\check{z}}\mathring{h}$ which is a complete metric of
finite volume on $M^{+}$.

###### Proof.

Step 1: We first suppose that $I(f)<\infty$ and show that $\mathring{g}_{f}$
is conformal to a locally hyperbolic metric.

Recall the function $f_{k}$ in the definition of the locally hyperbolic
reference metrics $\mathring{g}$ in (2.1) and consider the following separable
ODE for an unknown function $K$:

$f(z)K^{\prime}(z)=f_{k}(K(z))=\begin{cases}\sinh(K(z)+r_{0})&\ k=1,\\\
e^{K(z)+r_{0}}&\ k=0,\\\ \cosh(K(z)+r_{0})&\ k=-1,\end{cases}$ (2.24)

with the initial condition $K(z_{0})=0$ where $z_{0}=0$ and
$r_{0}=2\operatorname{arctanh}\left(\exp(-I(f))\right)$ if $k=1$, $z_{0}=0$
and $r_{0}=\log\left(I(f)\right)$ if $k=0$, and $z_{0}$ satisfies
$\int_{z_{0}}^{\infty}f(s)\ ds<\frac{\pi}{2}$ and
$r_{0}=2\operatorname{arctanh}\left(\tan\left(\frac{\pi}{4}-\frac{1}{2}\int_{z_{0}}^{\infty}f(s)\
ds\right)\right)$ if $k=-1$. It is straightforward to check that the explicit
solution is defined for all $z\geq z_{0}$ and, furthermore, that $K(z)$ is
strictly increasing and $K(\infty)=\infty$. Consequently,
$K:[z_{0},\infty)\rightarrow[0,\infty)$ is a diffeomorphism.

Consider a coordinate system $(r,\theta)$ on $M^{+}$ where $\theta$ is some
coordinate system on $N$ and $r=K(z)$ for $z>z_{0}$. For $z>z_{0}$, in view of
(2.1) and (2.24), we have

$\mathring{g}_{f}=(K^{\prime}(z))^{-2}\left(dr^{2}+(K^{\prime}(z)f(z))^{2}h\right)=(K^{\prime}(z))^{-2}\mathring{g}$

as desired.

Step 2: In order to show the sufficiency in the proposition, we first prove
that $I(f)=\infty$ if and only if $\mathring{g}_{f}$ is conformal to a
complete, finite volume metric on $M^{+}$.

To see this, first suppose $I(f)=\infty$ and define
$\check{K}(z)=\log\left(1+\int_{0}^{z}(f(s))^{-1}ds\right)$ so that
$\check{K}^{\prime}(z)=\left(f(z)\right)^{-1}e^{-\check{K}(z)}$. As
$I(f)=\infty$, we may define a new coordinate $\check{z}=\check{K}(z)$ so that

$\mathring{g}_{f}=dz^{2}+f^{2}(z)\mathring{h}=\left(\check{K}^{\prime}(z)\right)^{-2}\left(d\check{z}^{2}+(\check{K}^{\prime}(z)f(z))^{2}\mathring{h}\right)=\left(\check{K}^{\prime}(z)\right)^{-2}\left(d\check{z}^{2}+e^{-2\check{z}}\mathring{h}\right)$

and so $\mathring{g}_{f}$ is conformal to
$d\check{z}^{2}+e^{-2\check{z}}\mathring{h}$ which can readily be seen to be
both complete and of finite volume on $M^{+}$.

Conversely, suppose that there exists some conformal factor $u(z,\theta)$ such
that $\check{g}=u^{\frac{4}{n-2}}(z,\theta)\mathring{g}_{f}$ for some
complete, finite volume metric $\check{g}$. Consider the divergent curves
$\gamma_{\theta}:[0,\infty)\rightarrow M^{+}$ defined by
$\gamma_{\theta}(t)=\left(z(t),\theta\right)$ for $\theta\in N$. By
completeness, we have that the length
$\mathcal{L}_{\check{g}}(\gamma_{\theta})=\infty$ for all $\theta\in N$. We
compute directly the length

$\displaystyle\infty=\mathcal{L}_{\check{g}}(\gamma_{\theta})$
$\displaystyle=\int_{0}^{\infty}\sqrt{\check{g}\left(\dot{\gamma}_{\theta}(t),\dot{\gamma}_{\theta}(t)\right)}\
dt=\int_{0}^{\infty}u^{\frac{2}{n-2}}(z,\theta)\ dz.$

Consequently, we may write

$\displaystyle\infty=\int_{N}\mathcal{L}_{\check{g}}(\gamma_{\theta})\
d\theta$ $\displaystyle=\int_{N}\int_{0}^{\infty}u^{\frac{2}{n-2}}(z,\theta)\
dz\ d\theta$
$\displaystyle\leq\left(\int_{N}\int_{0}^{\infty}u^{\frac{2n}{n-2}}(z,\theta)f^{n-1}(z)\
dz\ d\theta\right)^{\frac{1}{n}}\left(\int_{N}\int_{0}^{\infty}\frac{1}{f(z)}\
dz\ d\theta\right)^{\frac{n-1}{n}}$
$\displaystyle=Vol_{\check{g}}(M^{+})^{\frac{1}{n}}Vol_{h}(N)^{\frac{n-1}{n}}I(f)^{\frac{n-1}{n}}.$

Since $Vol_{\check{g}}(M^{+})<\infty$ we deduce that $I(f)=\infty$.

Step 3: We now suppose that $I(f)=\infty$ and prove that then
$\mathring{g}_{f}$ cannot be conformal to a locally hyperbolic metric.

By Step 2, $\mathring{g}_{f}$ (which has $I(f)=\infty$) cannot be conformal to
another warped product metric $\mathring{g}_{\tilde{f}}$ with
$I(\tilde{f})<\infty$, as the former is conformal to a complete metric of
finite volume, and the latter cannot be. On the other hand, we can compute
that $I(f_{k})<\infty$ for the reference locally hyperbolic metrics in (2.1)
for each $k=-1,0,1$. The conclusion follows. ∎

It should be clear that, via Proposition 2.17, one can apply Theorem 1.1 to
solve the Yamabe problem for AWP metrics with a wide range of warping
functions. For further detail, the reader is referred to [Hog20]. The only
reference warped product metrics for which this line of argument does not
apply are those which are conformal to
$\check{g}=d\check{z}^{2}+e^{-2\check{z}}\mathring{h}$. It can be shown that
$\check{g}$ admits a conformal compactification where infinity corresponds to
a single point, which is very different to the situation for ALH metrics.

##  3 Volume ratio conditions for solvability of the Yamabe problem

In this section of this paper, we prove Theorem 1.3 concerning the negativity
of the first eigenvalue of the conformal Laplacian on sub-domains of non-
compact manifolds of negative curvature type satisfying the volume ratio
condition (1.5). In view of the work of [AM88b], this leads us to an existence
result for the Yamabe problem. We furthermore show that condition (1.5) is
sharp for the negativity of the first eigenvalue (see Proposition 3.6).

###  3.1 An upper bound on the eigenvalue for the conformal Laplacian

As in the setting of Theorem 1.3, let $(M,g)$ be a Riemannian manifold and
suppose that there exist two open sets $\Omega_{1}\subset\Omega_{2}$ with
$C^{1}$ boundary which satisfy (1.4) and that the scalar curvature satisfies
$S_{g}\leq-n(n-1)$ on $\Omega_{2}$.

In proving Theorem 1.3, we relate the first eigenvalue of the conformal
Laplacian on $\Omega_{2}$ to the following sup-norm minimisation problem:

$H_{0}=\inf\left\\{H(\tilde{\varphi}):=\sup_{0\leq r\leq
R}F\left(\tilde{\varphi}(r),\tilde{\varphi}^{\prime}(r)\right):\tilde{\varphi}\in
C^{1}([0,R]),\tilde{\varphi}(0)=0,\tilde{\varphi}(R)=1\right\\}$ (3.1)

where $F(y,z):=c_{n}z^{2}-n(n-1)y^{2}$ and $R$ is given in (1.4). Note that
$H(\tilde{\varphi})\geq F(\tilde{\varphi}(0),\tilde{\varphi}^{\prime}(0))\geq
0$ and so $H_{0}\geq 0$.

We recall the variational formulation of the first eigenvalue $\lambda$ of the
conformal Laplacian with Dirichlet boundary data on the bounded domain
$\Omega_{2}$:

$\lambda=\inf_{\begin{subarray}{c}\varphi\in H^{1}_{0}(\Omega_{2}),\\\
\|\varphi\|_{L^{2}}=1\end{subarray}}\int_{\Omega_{2}}\left(c_{n}|\nabla_{g}\varphi|^{2}+S_{g}\varphi^{2}\right)dV_{g}.$
(3.2)

Define the distance function
$r:\overline{\Omega_{2}\setminus\Omega_{1}}\rightarrow[0,R]$ by
$r(x)=d_{g}(x,\partial\Omega_{2}).$ Central in our discussion will be test
functions $\varphi=\varphi(r)$ of the form

$\varphi(x)=\left\\{\begin{array}[]{ll}\displaystyle 1&x\in\Omega_{1}\\\
\vspace{-0.3cm}\\\
\tilde{\varphi}(r(x))&x\in\overline{\Omega_{2}\setminus\Omega_{1}}\end{array}\right.$
(3.3)

where $\tilde{\varphi}:[0,R]\rightarrow\mathbb{R}$ is a $C^{1}$ function
satisfying $\tilde{\varphi}(0)=0$ and $\tilde{\varphi}(R)=1$. We note that, as
$r$ is Lipschitz, these conditions on $\tilde{\varphi}$ ensure that
$\varphi\in H^{1}_{0}(\Omega_{2})$ and so is a valid test function.

Assuming this form for $\varphi$, we bound the integral in (3.2) by

$\displaystyle\int_{\Omega_{2}}$
$\displaystyle\left(c_{n}|\nabla_{g}\varphi|^{2}+S_{g}\varphi^{2}\right)dV_{g}\leq\int_{\Omega_{2}}\left(c_{n}(\varphi^{\prime})^{2}-n(n-1)\varphi^{2}\right)dV_{g}$
$\displaystyle\qquad\leq-n(n-1)Vol_{g}(\Omega_{1})+\int_{\Omega_{2}\setminus\Omega_{1}}\left(c_{n}(\tilde{\varphi}^{\prime})^{2}-n(n-1)\tilde{\varphi}^{2}\right)dV_{g}$
$\displaystyle\qquad\leq-n(n-1)Vol_{g}(\Omega_{1})+Vol_{g}(\Omega_{2}\setminus\Omega_{1})\sup_{0\leq
r\leq
R}\left(c_{n}(\tilde{\varphi}^{\prime}(r))^{2}-n(n-1)\tilde{\varphi}^{2}(r)\right)$
$\displaystyle\qquad=-n(n-1)Vol_{g}(\Omega_{1})+Vol_{g}(\Omega_{2}\setminus\Omega_{1})H(\tilde{\varphi})$
(3.4)

where we used additionally that $S_{g}\leq-n(n-1)$ on $\Omega_{2}$. In
particular, we have:

######  Lemma 3.1.

Let $(M,g)$ be a Riemannian manifold and suppose that there exist two open
sets $\Omega_{1}\subset\Omega_{2}$ with $C^{1}$ boundary which satisfy (1.4),
that the scalar curvature satisfies $S_{g}\leq-n(n-1)$ on $\Omega_{2}$, and
that

$H_{0}\frac{Vol_{g}(\Omega_{2}\setminus\Omega_{1})}{Vol_{g}(\Omega_{1})}<n(n-1),$
(3.5)

where $H_{0}$ is the infimum defined in (3.1). Then, the conformal Laplacian
$-c_{n}\Delta_{g}+S_{g}$ for $(M,g)$ has a negative first eigenvalue on
$\Omega_{2}$.

###### Proof.

The lemma follows from (3.1) and (3.4). ∎

The proof of Theorem 1.3 will use the explicit form of the solution of the
minimisation problem (3.1) which we will obtain using [Aro65].333In fact, the
work of Aronsson addressed a broader class of sup-norm minimisation problems
and for rougher admissible functions.

######  Lemma 3.2.

The minimisation problem (3.1) is uniquely solved by

$H_{0}=n(n-1)\operatorname{csch}^{2}\left(\frac{\sqrt{n(n-2)}}{2}R\right)\text{
and }\\\
\tilde{\varphi}(r)=\sinh\left(\frac{\sqrt{n(n-2)}}{2}r\right)\operatorname{csch}\left(\frac{\sqrt{n(n-2)}}{2}R\right).$
(3.6)

###### Proof.

Let $\mathcal{A}$ denote the set of admissible functions for (3.1), i.e.
$\psi\in\mathcal{A}$ if $\psi\in C^{1}([0,R])$, $\psi(0)=0$ and $\psi(R)=1$.
Following [Aro65], we show that if $\psi\in\mathcal{A}$, $\psi^{\prime}>0$ in
$(0,R)$ and $\psi$ solves the ODE $F(\psi,\psi^{\prime})\equiv\text{constant}$
in $(0,R)$, then $\psi$ is the unique minimiser of (3.1). To this end, we show
that, for any function $\psi_{2}\in\mathcal{A}$ different from $\psi$, there
exist points $\xi,\xi_{2}\in(0,R)$ such that $\psi_{2}(\xi)=\psi(\xi_{2})$ and
$\psi_{2}^{\prime}(\xi)>\psi^{\prime}(\xi_{2})>0$. This will suffice to show
that $\psi$ is the unique minimiser as

$H(\psi_{2})\geq
F(\psi_{2}(\xi),\psi_{2}^{\prime}(\xi))>F(\psi(\xi_{2}),\psi^{\prime}(\xi_{2}))=\text{
constant }=H(\psi),$

where the second inequality uses the fact that $\partial_{z}F>0$ for $z>0$.

As $\psi^{\prime}>0$ in $(0,R)$, we may define the inverse
$\alpha(y)=\psi^{-1}(y)$. Clearly $\alpha([0,1])=[0,R]$ and $\alpha\in
C([0,1])\cap C^{1}((0,1))$. Define $r_{1}=\sup\\{r:r\in[0,R],\psi_{2}(r)\leq
0\\}$ and $r_{2}=\inf\\{r:r\in(r_{1},R],\psi_{2}(r)\geq 1\\}$. Then
$0\leq\psi_{2}\leq 1$ in $[r_{1},r_{2}]$, $0<\psi_{2}<1$ in $(r_{1},r_{2})$
and we may define the function $g:=\alpha(\psi_{2})$ on $[r_{1},r_{2}]$ where
$g\in C([r_{1},r_{2}])\cap C^{1}((r_{1},r_{2}))$.

We claim that there exists $r_{0}\in(r_{1},r_{2})$ such that
$\psi(r_{0})\neq\psi_{2}(r_{0})$. Suppose not, then $\psi(r)=\psi_{2}(r)$ for
$r\in[r_{1},r_{2}]$ and so $\psi(r_{1})=0$ and $\psi(r_{2})=1$. By the strict
monotonicity of $\psi$, we then have $r_{1}=0$ and $r_{2}=R$, which implies
$\psi\equiv\psi_{2}$, a contradiction.

Note that $g(r_{1})=0\leq r_{1}$, $g(r_{0})\neq r_{0}$ and $g(r_{2})=R\geq
r_{2}$. As $g\in C([r_{1},r_{2}])\cap C^{1}((r_{1},r_{2}))$, we may apply the
mean value theorem to either interval $(r_{1},r_{0})$ or $(r_{0},r_{2})$ to
deduce the existence of $\xi\in(r_{1},r_{2})$ such that $g^{\prime}(\xi)>1$.
Set $\xi_{2}=\alpha(\psi_{2}(\xi))\in(0,R)$ and observe that
$g^{\prime}(\xi)=\frac{\psi_{2}^{\prime}(\xi)}{\psi^{\prime}(\xi_{2})}>1$.
Consequently, $\psi_{2}(\xi)=\psi(\xi_{2})$ and
$\psi_{2}^{\prime}(\xi)>\psi^{\prime}(\xi_{2})>0$, as desired.

Direct computation verifies that the function $\tilde{\varphi}$ from the
statement of the lemma solves

$F(\tilde{\varphi},\tilde{\varphi}^{\prime})=n(n-1)\operatorname{csch}^{2}\left(\frac{\sqrt{n(n-2)}}{2}R\right)$

and satisfies $\tilde{\varphi}(0)=0$, $\tilde{\varphi}(R)=1$ and
$\tilde{\varphi}^{\prime}>0$ in $(0,R)$. The conclusion follows. ∎

We may now use the explicit minimiser above with the ideas from Lemma 3.1 to
prove the following bounds on the first eigenvalue:

######  Lemma 3.3.

Let $(M,g)$ be a Riemannian manifold and suppose that there exist two open
sets $\Omega_{1}\subset\Omega_{2}$ with $C^{1}$ boundary which satisfy (1.4)
and that the scalar curvature satisfies $S_{g}\leq-n(n-1)$ on $\Omega_{2}$. If
(1.5) holds, then

$\lambda<n(n-1)\frac{Vol_{g}(\Omega_{2}\setminus\Omega_{1})\operatorname{csch}^{2}\left(\frac{\sqrt{n(n-2)}}{2}R\right)-Vol_{g}(\Omega_{1})}{Vol_{g}(\Omega_{2})}\leq
0.$ (3.7)

On the other hand, if (1.5) does not hold, we have

$\lambda<n(n-1)\frac{Vol_{g}(\Omega_{2}\setminus\Omega_{1})\operatorname{csch}^{2}\left(\frac{\sqrt{n(n-2)}}{2}R\right)-Vol_{g}(\Omega_{1})}{Vol_{g}(\Omega_{1})}.$
(3.8)

###### Proof.

Define $\varphi$ to be the function of the form (3.3) where $\tilde{\varphi}$
is taken to be the minimiser obtained in Lemma 3.2. Substituting $\varphi$ in
the integral in (3.2) and using (3.4) we obtain

$\lambda\leq
n(n-1)\frac{Vol_{g}(\Omega_{2}\setminus\Omega_{1})\operatorname{csch}^{2}\left(\frac{\sqrt{n(n-2)}}{2}R\right)-Vol_{g}(\Omega_{1})}{\|\varphi\|_{L^{2}(\Omega_{2})}}\
.$ (3.9)

If the inequality above were saturated, then $\varphi$ (up to a harmless
normalisation) would be a minimiser of (3.2) and hence would be smooth in
$\Omega_{2}$, contradicting the gradient discontinuity at the boundary
$\partial\Omega_{1}$ observed using the explicit form for $\tilde{\varphi}$
obtained in Lemma 3.2 and recalling (3.3). Consequently, inequality (3.9)
above is, in fact, strict.

In the case that (1.5) holds, the numerator in (3.9) is non-positive and so,
using that $\|\varphi\|_{L^{2}(\Omega_{2})}<Vol_{g}(\Omega_{2})$, we obtain
(3.7). Likewise, if (1.5) does not hold, the numerator is positive and we use
that $\|\varphi\|_{L^{2}(\Omega_{2})}>Vol_{g}(\Omega_{1})$ to obtain (3.8). ∎

We may now prove Theorem 1.3 as an immediate consequence of the lemma above:

###### Proof of Theorem 1.3.

Combined with assumption (1.1), the existence of a solution to the Yamabe
problem now follows from [AM88b, Theorem C] provided that we can show that the
first eigenvalue $\lambda$ of the conformal Laplacian on $\Omega_{2}$ is
negative. This negativity is an immediate consequence of assumption (1.5) and
estimate (3.7) from Lemma 3.3. ∎

We conclude the section with the following corollaries of Theorem 1.3 for
geodesic balls and for annuli which will aid in our discussion regarding
warped product and multiply warped product type manifolds in the next
subsection. In particular, for geodesic balls we have:

######  Corollary 3.4.

Let $(M,g)$ be a Riemannian manifold and suppose there exist constants
$\alpha,R>0$ and some geodesic ball $B_{(1+\alpha)R}$ which satisfies

$\frac{Vol_{g}(B_{(1+\alpha^{-1})R}\setminus
B_{\alpha^{-1}R})}{Vol_{g}(B_{\alpha^{-1}R})}\leq\sinh^{2}\left(\frac{R\sqrt{n(n-2)}}{2}\right)$
(3.10)

and on which the scalar curvature satisfies $S_{g}\leq-n(n-1)$. Then, the
conformal Laplacian $-c_{n}\Delta_{g}+S_{g}$ for $(M,g)$ has a negative first
eigenvalue on $B_{(1+\alpha^{-1})R}$.

In the case of multiply warped product metrics (see (3.14)), which have a
radial fibre whose coordinate we denote by $r$, we fix some value
$r_{0}\in\mathbb{R}$ and define the annular region $A_{R}(r_{0})=\\{x\in
M:\left|r(x)-r_{0}\right|\leq R\\}.$ We may then obtain as a corollary of
Theorem 1.3,

######  Corollary 3.5.

Let $(M,g)$ be a Riemannian manifold and suppose there exist constants
$\alpha,R>0$ and some annulus $A_{(1+\alpha)R}(r_{0})$ which satisfies

$\frac{Vol_{g}(A_{(1+\alpha^{-1})R}(r_{0})\setminus
A_{\alpha^{-1}R}(r_{0}))}{Vol_{g}(A_{\alpha^{-1}R}(r_{0}))}\leq\sinh^{2}\left(\frac{R\sqrt{n(n-2)}}{2}\right)$
(3.11)

and on which the scalar curvature satisfies $S_{g}\leq-n(n-1)$. Then, the
conformal Laplacian $-c_{n}\Delta_{g}+S_{g}$ for $(M,g)$ has a negative first
eigenvalue on $A_{(1+\alpha^{-1})R}(r_{0})$.

The proofs of the two corollaries follow directly from Theorem 1.3 with the
appropriate choices made for $\Omega_{1}$, $\Omega_{2}$ and $R$.

###  3.2 Sharpness of the eigenvalue estimate

In this section we demonstrate that the volume ratio condition (1.5) in
Theorem 1.3 is sharp for the negativity of the first eigenvalue. In
particular, we will show:

######  Proposition 3.6.

Let $\beta>0$. There exists a constant $C>0$, such that for any large $R$
there is a complete, non-compact manifold $(M,g)$ and bounded domains
$\Omega_{1}\subset\Omega_{2}$ satisfying (1.4) with volume ratio

$\frac{Vol_{g}(\Omega_{2}\setminus\Omega_{1})}{Vol_{g}(\Omega_{1})}<Ce^{\beta
R}$ (3.12)

and with scalar curvature satisfying $S_{g}\leq-n(n-1)$ on $M$ for which the
first eigenvalue for the conformal Laplacian on $\Omega_{2}$ satisfies

$\lambda>\begin{cases}\frac{1}{(n-1)(n-2)}\beta^{2}&\text{ if }\beta>n-1,\\\
\frac{n-1}{n-2}\beta^{2}-n(n-1)&\text{ if }\beta\leq n-1.\end{cases}$ (3.13)

In particular, if $\beta>\sqrt{n(n-2)}$, then $\lambda$ is positive.

Note that, for large $R$, (3.12) implies (1.6) with a slightly larger $\beta$.
Remark 1.4 follows.

We consider the product manifolds $M=\mathbb{R}\times N_{1}\times\ldots\times
N_{m}$ where $m\geq 1$ and each $N_{i}$ is a compact manifold of dimension
$n_{i}\geq 1$ with $\sum_{i}n_{i}=n-1$. We endow $M$ with a (multiply) warped
metric $g$ of the form

$g=dr^{2}+\sum_{i}p_{i}^{2}(r)h_{i}$ (3.14)

where $p_{i}:\mathbb{R}\rightarrow(0,\infty)$ are warping functions and
$h_{i}$ are metrics on each of the $N_{i}$. A computation shows that $(M,g)$
has Laplacian
$\Delta_{g}=\partial_{rr}+\sum_{i}n_{i}\frac{p_{i}^{\prime}}{p_{i}}\partial_{r}+\sum_{i}\frac{1}{p^{2}_{i}}\Delta_{h_{i}}$
and

$S_{g}=-2\sum_{i}n_{i}\frac{p_{i}^{\prime\prime}}{p_{i}}-\sum_{i}n_{i}(n_{i}-1)\left(\frac{p_{i}^{\prime}}{p_{i}}\right)^{2}-2\sum_{i<j}n_{i}n_{j}\frac{p_{i}^{\prime}p_{j}^{\prime}}{p_{i}p_{j}}+\sum_{i}\frac{S_{h_{i}}}{p^{2}_{i}}\
.$ (3.15)

A particularly convenient family of warped product metrics for our purposes
will be:

######  Example 3.7.

Let $(M,g)$ be a multiply warped product as in (3.14) with each
$N_{i}=\mathbb{S}^{1}$ (so that $m=n-1$) and $p_{i}=e^{\alpha_{i}r}$ for some
$\alpha_{i}\in\mathbb{R}$. We note that

$S_{g}=-2\left(\sum_{i}\alpha_{i}\right)^{2}+2\sum_{i<j}\alpha_{i}\alpha_{j}.$

As a consequence, for any given $\beta$ and constant
$C\geq-\frac{n}{n-1}\beta^{2}$, there exists a choice of $\alpha_{i}$ such
that $\beta=\sum_{i}\alpha_{i}$ and $S_{g}\equiv-C$. In particular, if
$|\beta|\leq n-1$, then there exists a choice of $\alpha_{i}$ with
$\sum_{i}\alpha_{i}=\beta$ so that $S_{g}\equiv-n(n-1)$.

We now turn to a study of the volume ratios of concentric balls in the example
above.

######  Lemma 3.8.

Let $(M,g)$ be as in Example 3.7. Then there exists a constant $C$ depending
only on $\alpha_{i}$ such that, for any $R$ sufficiently large, one can find
concentric balls $B_{2R}$ and $B_{R}$ such that

$\frac{Vol_{g}(B_{2R}\setminus B_{R})}{Vol_{g}(B_{R})}\leq Ce^{\beta R}.$

###### Proof.

Fixing some large $R$ and taking some ball $B_{R}(p_{0})$, writing
$p_{0}=(r_{0},x_{0})$, we have immediately that

$B_{R}(p_{0})\subset\\{p:|r(p)-r_{0}|\leq R\\}.$

On the other hand, defining $S_{0}:=\\{r_{0}\\}\times\mathbb{T}^{n-1}$ we have
that

$d_{g}(p,p_{0})\leq
d_{g}(p,S_{0})+\text{diam}(S_{0})=r(p)-r_{0}+e^{r_{0}}\text{diam}(\mathbb{T}^{n-1}).$

Consequently, writing $A_{0}=e^{r_{0}}\text{diam}(\mathbb{T}^{n-1})$, we have
that

$\\{|r-r_{0}|\leq R-A_{0}\\}\subset B_{R}(p_{0})\subset\\{|r-r_{0}|\leq R\\}$

from which we obtain

$Vol_{g}(B_{2R}(p_{0}))\leq\int_{\mathbb{T}^{n-1}}\int_{r_{0}-2R}^{r_{0}+2R}e^{\beta
r}dr\ dx=\beta
Vol(\mathbb{T}^{n-1})\left(e^{\beta(r_{0}+2R)}-e^{\beta(r_{0}-2R)}\right)$

and

$\displaystyle Vol_{g}(B_{R}(p_{0}))$
$\displaystyle\geq\int_{\mathbb{T}^{n-1}}\int_{r_{0}-(R-A_{0})}^{r_{0}+(R-A_{0})}e^{\beta
r}dr\ dx=\beta
Vol(\mathbb{T}^{n-1})e^{-A_{0}}\left(e^{\beta(r_{0}+R)}-e^{\beta(r_{0}-R+2A_{0})}\right)$

from which we may estimate

$\frac{Vol_{g}(B_{2R}\setminus B_{R})}{Vol_{g}(B_{R})}\leq
e^{A_{0}}\frac{e^{\beta(r_{0}+2R)}-e^{\beta(r_{0}-2R)}}{e^{\beta(r_{0}+R)}-e^{\beta(r_{0}-R+2A_{0})}}-1.$

Consequently, $(M,g)$ satisfies the following bound on the volume ratio for
large balls

$\frac{Vol_{g}(B_{2R}\setminus B_{R})}{Vol_{g}(B_{R})}\leq Ce^{\beta
R}\left(1+\mathcal{O}(e^{-2(n-1)R})\right)-1$ (3.16)

and so, taking $R$ large, we obtain

$\frac{Vol_{g}(B_{2R}\setminus B_{R})}{Vol_{g}(B_{R})}\leq Ce^{\beta R}.$

for some large constant $C>0$ depending only on $\alpha_{i}$. ∎

We are now ready to prove Proposition 3.6. However, before this, we make the
following remark:

######  Remark 3.9.

As a consequence of Lemma 3.8, we may use Corollary 3.5 of Theorem 1.3 to
prove that any manifold of the form of Example 3.7 admits a solution to the
Yamabe problem, provided that $\beta<\sqrt{n(n-2)}$.

###### Proof of Proposition 3.6.

Let $(M,g)$ be of the form of Example 3.7 where $\alpha_{i}$ are chosen such
that $\sum_{i}\alpha_{i}=\beta$ and

$S_{g}\equiv\begin{cases}-\frac{n}{n-1}\beta^{2}\text{ if }\beta>n-1,\\\
-n(n-1)\text{ if }\beta\leq n-1.\end{cases}$

In particular, $S_{g}\leq-n(n-1)$. We note that, by the same argument as in
the proof of Lemma 3.8, there exists a $C(\alpha_{1},\ldots,\alpha_{n-1})>0$
such that we may choose concentric annuli $\Omega_{2}=A_{2R}$ and
$\Omega_{1}=A_{R}$ satisfying

$\frac{Vol_{g}(A_{2R}\setminus A_{R})}{Vol_{g}(A_{R})}\leq
C(\alpha_{1},\ldots,\alpha_{n-1})e^{\beta R}.$

As the eigenspace corresponding to the first eigenvalue $\lambda$ is
necessarily one dimensional, we must have that the first eigenfunctions
$\varphi$ of the conformal Laplacian on $A_{2R}$ are radially symmetric (i.e.
$\varphi=\varphi(r)$) by virtue of the symmetry of the torus. Consequently,
$\varphi$ satisfies the constant coefficient ODE

$\varphi^{\prime\prime}+\beta\varphi^{\prime}+\frac{\lambda-
S_{g}}{c_{n}}\varphi=0$

on $\Omega_{2}$ subjected to the zero Dirichlet boundary condition.

It follows that the corresponding characteristic equation must have complex,
non-real roots, which implies that

$\beta^{2}-\frac{{(n-2)(\lambda-S_{g})}}{(n-1)}<0.$

Consequently, $\lambda>\frac{n-1}{n-2}\beta^{2}-S_{g}$ and so, recalling our
choice of $S_{g}$, the proposition follows. ∎

To conclude, we remark that one may use Theorem 1.3 to show existence of
solutions to the Yamabe problem for manifolds obtained from the warped product
type metrics discussed above via perturbations which preserve the volume ratio
and scalar curvature conditions.

In a future work, it would be interesting to explore which results similar to
those achieved for ALH manifolds could be obtained in the setting of Theorem
1.3 and to understand to what extent the upper bound on the scalar curvature
could be loosened.

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