On Huber’s type theorems in general dimensions

Shiguang Ma$\,{}^{\dagger}$111The author is the corresponding author and
partially supported by NSFC 11571185 and Jie Qing$\,{}^{\ddagger}$222The
author is partially supported by NSF DMS-1608782

$\mbox{}^{\dagger}$ School of Mathematical Science and LPMC, Nankai
University, Tianjin, China;
e-mail<EMAIL_ADDRESS>
$\mbox{}^{\ddagger}$ Department of Mathematics, University of California,
Santa Cruz, CA 95064;
e-mail<EMAIL_ADDRESS>
###### Abstract.

In this paper we present some extensions of the celebrated finite point
conformal compactification theorem of Huber [Hu57] for complete open surfaces
to general dimensions based on the n-Laplace equations in conformal geometry.
We are able to conclude a domain in the round sphere has to be the sphere
deleted finitely many points if it can be endowed with a complete conformal
metric with the negative part of the smallest Ricci curvature satisfying some
integrable conditions. Our proof is based on the strengthened version of the
Arsove-Huber’s type theorem on n-superharmonic functions in our earlier work
[MQ18]. Moreover, using p-parabolicity, we push the injectivity theorem of
Schoen-Yau to allow some negative curvature and therefore establish the finite
point conformal compactification theorem for manifolds that have a conformal
immersion into the round sphere. As a side product we establish the
injectivity of conformal immersions from n-parabolicity alone, which is
interesting by itself in conformal geometry.

##  1\. Introduction

In this paper we want to present extensions of the celebrated Huber Theorem on
complete open surfaces. Huber [Hu57] in 1957 showed that a complete open
surface whose negative part of the Gaussian curvature is integrable is a
closed surface with finitely many points removed. This Huber theorem made the
highlight of applications of subharmonic functions in differential geometry
after the seminal work of Cohn-Vossen [CV35]. In 2 dimensions the Gaussian
curvature equation and the Gauss-Bonnet formula are the essentials.

It has been highly desirable to extend this celebrated Huber Theorem to higher
dimensions. Notably there is a Huber’s type theorem of Chang, Qing and Yang
[CQY00] in 4 dimensions, based on the so-called $Q$-curvature equations and
the involvement of $Q$ curvature in Chern-Gauss-Bonnet formula. Also notably,
Carron and Herzlich [CH02] (also see [Car20]) obtained a Huber’s type theorem
in general dimensions for domains in a given compact Riemannian manifold that
admit complete conformal metrics with growth conditions on the volume and
integral conditions on Ricci or scalar curvature.

It was observed in [BMQ17, MQ18] the following n-Laplace equation in conformal
geometry

(1.1) $-\Delta_{n}u+Ric(\nabla u)|\nabla u|^{n-2}=(Ric(\nabla u)|\nabla
u|^{n-2})_{g}e^{nu}.$

plays a similar role in general dimensions to the Gaussian curvature equation
in 2 dimensions, where $Ric(v)$ is the Ricci curvature in the $v$ direction,
the term $(|\nabla u|^{n-2}Ric(\nabla u))_{g}$ is calculated for the conformal
metric $g=e^{2u}\bar{g}$, and other terms are calculated under the background
metric $\bar{g}$. Particularly, in [MQ18], Arsove-Huber’s type theorem [ArH73]
for n-superharmonic functions have been established, where Wolff potential
theory provides the important analytic tool in general dimensions as the
Newton potential did in 2 dimensions.

In this paper, first, by capacity estimates, we obtain

######  Theorem 1.1.

For $n\geq 3$, let $\Omega$ be a domain in a compact manifold
$(M^{n},\bar{g})$ and endowed with a complete conformal metric
$g=e^{2u}\bar{g}$. Assume that,

$R^{-}_{g}\in L^{p}(\Omega,g)\text{ for some $p\in(\frac{n}{2},+\infty]$ and
}\left\\{\begin{aligned} &\text{either }Ric^{-}_{g}\in
L^{\frac{n}{2}}(\Omega,g),\\\ &\text{or }Ric^{-}_{g}|\nabla u|^{n-2}e^{2u}\in
L^{1}(\Omega,\bar{g}).\end{aligned}\right.$

where $R^{-}_{g}$ is the negative part of the scalar curvature $R$. Then
$cap_{n}(\partial\Omega,D)=0$, where $D$ is a neighborhood of $\partial\Omega$
in $M^{n}$. Hence $(\Omega,g)$ is $n$-parabolic and $\partial\Omega$ is of
Hausdorff dimension 0 in $(M^{n},\bar{g})$.

This theorem may be compared with [CH02, Corollary 2.2]. Notice that [Gal88,
Theorem 1] (cf. Lemma 2.3 in Section 2) is not applicable under the
assumptions of Theorem 1.1. An example, similar to [Car20, Theorem 2.7], is
discussed at the end of Section 3.1 to illustrate it is not possible to
conclude $\partial\Omega$ to be finite under the assumption $Ric^{-}\in
L^{\frac{n}{2}}(\Omega,g)$.

To achieve finite point conformal compactification, we strengthen [MQ18,
Theorem 1.1] in two aspects. The first is to drop the assumption of the origin
being isolated in [MQ18, Theorem 1.1] for $m\geq 1$ from the completeness (cf.
Theorem 3.3). The second is to confirm that $m$ is related to the point mass
at the origin (cf. Lemma 3.6). In order to use Theorem 3.3 and Lemma 3.6, we
introduce the n-superharmonic function $\mathcal{A}(v)$ (cf. (3.21)) to
overshadow the conformal factor $v$ based on the idea in the proof of [KM92,
Theorem 2.4] on solving n-Laplace equations with nonnegative Radon measure on
the right. For this purpose, we first need to show that the conformal factor
$v$ satisfies the n-Laplace equation with a signed Radon measure on the right.
We therefore obtain the following finite point conformal compactification
theorem of Huber’s type.

######  Theorem 1.2.

For $n\geq 3$, let $\Omega$ be a domain in the standard unit round sphere
$(\mathbb{S}^{n},g_{\mathbb{S}})$. Suppose that, on $\Omega$, there is a
conformal metric $g=e^{2u}g_{\mathbb{S}}$ satisfying

$\lim_{x\to\partial\Omega}u(x)=+\infty\text{ and }Ric^{-}_{g}|\nabla
u|^{n-2}e^{2u}\in L^{1}(\Omega,g_{\mathbb{S}}).$

Then $\partial\Omega=\mathbb{S}^{n}\setminus\Omega$ is a finite point set.

It is easily seen that Theorem 1.2 holds when the Ricci is nonnegative outside
a compact subset or Cheng-Yau’s gradient estimates (cf. [SY94, Theorem 2.12,
Chapter VI] and [CQY00, Lemma 2.3]) applies. Here we use [CHY04, Proposition
8.1] to derive $\lim_{x\to\partial\Omega}u(x)=+\infty$. We therefore have the
following finite point conformal compactification theorem of Huber’s type in a
more intrinsically geometric fashion.

######  Corollary 1.1.

For $n\geq 3$, let $\Omega$ be a domain in the standard unit round sphere
$(\mathbb{S}^{n},g_{\mathbb{S}})$. Suppose that, on $\Omega$, there is a
complete conformal metric $g=e^{2u}g_{\mathbb{S}}$ satisfying either $Ric_{g}$
is nonnegative outside a compact subset or

  1. (1)

$Ric^{-}_{g}\in L^{1}(\Omega,g)\cap L^{\infty}(\Omega,g)$

  2. (2)

$R_{g}\in L^{\infty}(\Omega,g)\text{ and }|\nabla^{g}R_{g}|\in
L^{\infty}(\Omega,g)$.

Then $\partial\Omega=\mathbb{S}^{n}\setminus\Omega$ is a finite point set.

Corollary 1.1 should compared with [CQY00, Theorem 2]. [CQY00, Theorem 2] uses
the $Q$-curvature equation in 4 dimensions while Corollary 1.1 uses the
n-Laplace equation (1.1) in general dimenions. At the end of Section 3.2, we
will present a detailed comparison of Theorem 1.2 and Corollary 1.1 with
[CH02, Theorem 2.1] and [Car20, Corollary 2.4].

To push for locally conformally flat manifolds that are not a priorly known to
be domains in $\mathbb{S}^{n}$, we develop injectivity theorems for conformal
immersions following the PDE approach in the seminal paper [SY88]. To
substitute the nonnegativity of the scalar curvature, we consider volume
growth conditions, p-parabolicity, and integral lower bounds of Ricci
curvature. First we establish, as a side product, a very clean injectivity
theorem based on n-parabolicity alone, which should be interesting by itself
in conformal geometry and geometric group theory.

######  Theorem 1.3.

Suppose that $(M^{n},g)$ ($n\geq 3$) is a complete noncompact manifold that
has a conformal immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}}).$

Assume that $(M^{n},g)$ is n-parabolic. Then $\Phi$ is injective.

Following closely from [SY88, Theorem 3.1], we prove an injectivity theorem
using integral lower bound on Ricci curvature, where the volume growth
estimates [Gal88] are employed. Namely,

######  Theorem 1.4.

Suppose that $(M^{n},g)$ is a complete, noncompact manifold. And suppose that
there is a conformal immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}}).$

Assume that $Ric_{g}^{-}\in L^{\frac{p}{2}}(M,g)$ for some $p\in(n,\infty]$.
Then $\Phi$ is injective provided that

$d(M)<\frac{(n-2)^{2}p}{(p-2)n}.$

For the definition of $d(M)$, please see (4.1) and [SY88]. Again, with a
refined argument of the proof of [SY88, Proposition 2.4], we have

######  Proposition 1.1.

Suppose that $(M^{n},g)$ ($n\geq 3$) is a manifold that has a conformal
immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}}).$

Assume that $R_{g}^{-}\in L^{\frac{n}{2}}(M,g)$. Then $d(M)\leq\frac{n}{2}$.

Combining Theorem 1.4 and Proposition 1.1 we establish

######  Corollary 1.2.

Suppose that $(M^{n},g)$ is a complete, noncompact manifold. And suppose that
there is a conformal immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}}).$

Assume that $Ric^{-}_{g}\in L^{\frac{p}{2}}(M,g)\cap L^{\frac{n}{2}}(M,g)$ for
some $p\in(n,\infty]$. Then $\Phi$ is injective provided that $n\geq 5$.

Finally, in the light of Corollary 1.1 and Corollary 1.2, we are able to show

######  Main Theorem.

Suppose that $(M^{n},g)$ ($n\geq 5$) is a complete noncompact manifold. And
suppose that there is a conformal immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}}).$

Assume that either $Ric_{g}$ is nonnegative outside a compact subset or

  1. (1)

$Ric^{-}_{g}\in L^{1}(\Omega,g)\cap L^{\infty}(\Omega,g)$

  2. (2)

$R_{g}\in L^{\infty}(\Omega,g)\text{ and }|\nabla^{g}R_{g}|\in
L^{\infty}(\Omega,g)$.

Then $\Phi$ is an embedding and
$\partial\Phi(M)=\mathbb{S}^{n}\setminus\Phi(M)$ is a finite point set.

On the organization of this paper, we discuss some preliminaries and
background in Section 2. We present proofs for Theorem 1.1 and Theorem 1.2 in
Section 3. The proof of Theorem 1.2 takes most of Section 3 and is divided
into 4 steps. In Section 4 we follow quite closely from [SY88] and prove
Theorem 1.3, Theorem 1.4 and Proposition 1.1. Finally we conclude Main Theorem
at the end of Section 4.

##  2\. Preliminaries and background

In this section we collect some basic definitions and facts that are relevant
to us in this paper. We first recall the celebrated Huber Theorem [Hu57]. Then
we introduce the n-Laplace equations in conformal geometry and recall our
previous work that extended Arsove-Huber Theorem [ArH73] to n-superharmonic
functions in general dimensions. We also recall definitions and basic
properties of p-capacity and p-parabolicity for our use later [AM73, SY94].

###  2.1. Gaussian curvature equations and Huber’s Theorem

On a surface $(M^{2},\bar{g})$ with Gaussian curvature $K$, let the Gaussian
curvature for a conformal metric

$g=e^{2u}\bar{g}$

be $K_{g}$. Then

$-\Delta u+K=K_{g}\,e^{2u}$

is the celebrated Gaussian curvature equation. Recall

######  Theorem 2.1.

(Huber [Hu57]) Suppose that $(M^{2},g)$ is a complete open surface and that

$\int_{M}(K^{-}dvol)_{g}<\infty,$

where $K^{-}_{g}$ is the negative part of Gaussian curvature of the metric
$g$. Then $M$ is a compact surface with finitely many points removed and
$(M^{2},g)$ is parabolic.

Huber in [Hu57] extended the works of Cohn-Vossen [CV35] and marked the
highlight of applications of linear potential theory in differential geometry
of surfaces then.

###  2.2. n-Laplace equations in conformal geometry and n-superharmonic
functions

Suppose that $(M^{n},\bar{g})$ is a Riemannian manifold and that
$g=e^{2u}\bar{g}$ is a conformal metric. In an attempt to search for Huber’s
type theorem in higher dimensions in conformal geometry, we recall the formula
for the transformation of Ricci curvature tensor under conformal change:

$Ric_{ij}-(n-2)u_{i,j}+(n-2)u_{i}u_{j}-(\Delta u)\bar{g}_{ij}-(n-2)|\nabla
u|^{2}\bar{g}_{ij}=(Ric_{g})_{ij},$

where geometric quantities with a subscript $g$ are taken with respect to the
metric $g$, while those without a subscription are taken with respect to the
background metric $\bar{g}$. Multiplying $|\nabla u|^{n-4}$ and contracting
with $\nabla u$ on both sides, one gets

$\displaystyle-((n-2)|\nabla u|^{n-4}u_{i}u_{j}u_{i,j}$ $\displaystyle+|\nabla
u|^{n-2}\Delta u)+|\nabla u|^{n-4}Ric(\nabla u,\nabla u)$
$\displaystyle=|\nabla u|^{n-4}Ric_{g}(\nabla u,\nabla u)$

which is

(2.1) $-\Delta_{n}u+|\nabla u|^{n-2}Ric(\nabla u)=(|\nabla u|^{n-2}Ric(\nabla
u))_{g}\,e^{nu}$

where $Ric(v)$ is the Ricci curvature in the direction of $v$. The right hand
side $(|\nabla u|^{n-2}Ric(\nabla u))_{g}$ is calculated for the conformal
metric $g=e^{2u}\bar{g}$. This n-Laplace equation reduces to Gaussian
curvature equation in dimension 2 and therefore is a natural extension of
Gaussian curvature equation in higher dimensions. Applications of n-Laplace
equations (2.1) have been initiated and studied in [BMQ17, MQ18]. By the
theory of Wolff potential in relation to n-Laplace equations developed in, for
instance, [KM94] and references therein, in [MQ18], we obtained

######  Theorem 2.2.

(Ma-Qing [MQ18, Theorem 1.1]) Let $w$ be a nonnegative lower semi-continuous
function that is $n$-superharmonic in $B(0,2)\subset\hbox{\bb R}^{n}$ and

$-\Delta_{n}u=\mu\geq 0$

for a Radon measure $\mu\geq 0$ and the origin is in the support $\mathcal{S}$
of the Radon measure $\mu$. Then there is a set $E\subset\mathbb{R}^{n}$,
which is $n$-thin at the origin, such that

$\lim_{x\notin E\text{ and }|x|\rightarrow
0}\frac{u(x)}{\log\frac{1}{|x|}}=\liminf_{|x|\rightarrow
0}\frac{u(x)}{\log\frac{1}{|x|}}=m\geq 0$

and

$u(x)\geq m\log\frac{1}{|x|}-C\ \text{ for $x\in B(0,1)\setminus\\{0\\}$ and
some $C$}.$

Moreover, if $u\in C^{2}(B(0,2)\setminus\\{0\\})$ and
$(B(0,2)\setminus\\{0\\},e^{2u}|dx|^{2})$ is complete at the origin, then
$m\geq 1$.

This theorem generalizes Arsove-Huber theorem [ArH73] from 2 dimensions to
general dimensions for n-superharmonic functions. For the definitions of
n-thinness readers are referred to [AM72, KM94, MQ18], for example. In [MQ18],
there are detailed discussions on the differences of the definition in [MQ18]
to the previous, even though both reduce to the same in 2 dimensions. In this
paper, inspired by Huber’s theorem (Theorem 2.1), we want to study
differential geometric properties in large for certain manifolds without
assuming Ricci nonnegative. Theorem 2.2 is one of the important ingredients.

We also want to recall some result on solving n-Laplace equations when the
inhomogeneous term is a nonnegative Radon measure.

######  Theorem 2.3.

(Kilpelainen-Maly [KM92, Theorem 2.4]) Let $U$ be a bounded domain in
$\mathbb{R}^{n}$ and $\mu$ be a nonnegative Radon measure on $U$. Then there
is a nonnegative n-superharmonic function $u$ such that

$-\Delta_{n}u=\mu$

and $\min\\{u,k\\}\in H^{1,n}_{0}(U)$ for all positive number $k$.

###  2.3. p-capacity and p-parabolicity

In this subsection we review materials about p-capacity and p-parabolicity.
First, let us recall the definition of p-capacity.

######  Definition 2.1.

For a compact subset $K$ in a domain $\Omega$ in a Riemannian manifold
$(M^{n},g)$, we define, for $1<p<\infty$,

$cap_{p}(K,\Omega)=\inf\\{\int_{\Omega}|\nabla u|^{p}dvol:u\in
C^{\infty}_{0}(\Omega)\text{ and $u\geq 1$ on $K$}\\}.$

The following relation between the Hausdorff measure and the p-capacity is
well known. Readers are referred to [AM73] and references therein.

######  Lemma 2.1.

([SY94, Theorem 2.10, Chapter VI]) Let $K$ be a compact subset of the
Euclidean space $\mathbb{R}^{n}$, and $1<q\leq n$. We know

  1. (1)

if $H_{n-q}(K)<\infty$ then $cap_{q}(K)=0$;

  2. (2)

$ifH_{n-q+\epsilon}(K)>0$ for some $\epsilon>0$ then $cap_{q}(K)>0$.

For our purpose, we adopt the following definition of p-parabolicity for
noncompact manifolds.

######  Definition 2.2.

Suppose that $(M^{n},g)$ is a complete noncompact Riemannian manifold.
$(M^{n},g)$ is said to be p-parabolic if $cap_{p}(\bar{U},M^{n})=0$ for any
bounded domain $U\subset M$.

This definition of p-parabolicity is known to be equivalent to the more
conventional definition, that is, a complete noncompact manifold $(M^{n},g)$
is p-parabolic if there is no positive Green function for p-Laplacian
$\Delta_{p}$ on $M$. Readers are referred to [Hol90, Tro99] and [CHS00, page
13]. From Definition 2.2, it is easily seen that the p-parabolicity is quasi-
isometrically invariant, that is, if $C^{-1}g_{2}\leq g_{1}\leq Cg_{2}$, then
$(M,g_{1})$ is p-parabolic if and only if $(M,g_{2})$ is. It turns out that
$n$-parabilicity is actually conformally invariant, since n-capacity is
conformally invariant. It is well known that volume growth is closely related
to p-parabolicity. We recall, for instance, the following reference [CHS00,
Corollary 3.2].

######  Lemma 2.2.

(Coulhon-Holopainen-Saloff-Coste [CHS00, Corollary 3.2]) Suppose that
$(M^{n},g)$ is a complete manifold. for $1<p<\infty$, assume that, for a point
$x_{0}\in M$,

$\liminf_{r\to\infty}r^{-p}Vol(B(x_{0},r))=0\text{ or }Vol(B(x_{0},r)\leq
Cr^{p}(\log r)^{p-1}.$

Then $(M^{n},g)$ is p-parabolic.

We also want to mention a volume growth estimate from integral lower bound of
Ricci curvature.

######  Lemma 2.3.

(Gallot [Gal88, Theorem 1]) Suppose that $(M^{n},g)$ is a complete, noncompact
manifold with infinite volume. For $p\in(n,\infty]$, assume $Ric^{-}_{g}\in
L^{\frac{p}{2}}(M^{n},g)$. Then, for any point $x_{0}\in M^{n}$,

(2.2) $\lim_{r\to\infty}\frac{Vol(B(x_{0},r))^{\frac{1}{p}}}{r}=0.$

Recently the critical case when $p=n$ has been resolved with the additional
and appropriate assumption in [Car20, Theorem 2.1]. Consequently from Lemma
2.3, we observe

######  Lemma 2.4.

Suppose that $(M^{n},g)$ is a complete, noncompact manifold. Assume
$Ric^{-}_{g}\in L^{\frac{p}{2}}(M^{n},g)$ for some $p\in(n,\infty]$. Then
$(M^{n},g)$ is p-parabolic

It turns out that n-parabolicity is out of reach in this approach, even though
n-parabolicity is more favorable in conformal geometry.

##  3\. Finite point compactifications for domains in $\mathbb{S}^{n}$

In this section we discuss extensions of Huber’s type theorem in general
dimensions using n-Laplace equations (2.1). First we derive a Huber’s type
theorem in the line of Carron-Herzlich [CH02]. We will study examples to
compare our Theorem 3.1 to [CH02, Theorem 2.1]. To derive a finite point
compactification we focus our attention to domains in the round sphere
$\mathbb{S}^{n}$ based on our earlier work [MQ18].

###  3.1. Domains in a compact manifold

In this subsection we state and prove a Huber’s type theorem for domains in a
given compact Riemannian manifold. We will study examples where one could not
improve Theorem 3.1 as it is stated. Theorem 3.1 and Corollary 3.1 should be
compared with [CH02, Corollary 2.2]. Instead of the volume growth assumptions,
we will use the integral lower bound on Ricci curvature. For this purpose, we
let $Ric^{-}$ be the negative part of the smallest Ricci curvature.

######  Theorem 3.1.

For $n\geq 3$, let $\Omega$ be a domain in a compact manifold
$(M^{n},\bar{g})$ and endowed with a conformal metric $g=e^{2u}\bar{g}$.
Assume

$\lim_{x\to\partial\Omega}u(x)=+\infty\text{ and }\left\\{\begin{aligned}
&\text{either }Ric^{-}_{g}\in L^{\frac{n}{2}}(\Omega,g),\\\ &\text{or
}Ric^{-}_{g}|\nabla u|^{n-2}e^{2u}\in
L^{1}(\Omega,\bar{g}).\end{aligned}\right.$

Then $cap_{n}(\partial\Omega,D)=0$, where $D$ is a neighborhood of
$\partial\Omega$ in $M$. Hence $(\Omega,g)$ is $n$-parabolic and
$\partial\Omega$ is of Hausdorff dimension 0 in $(M^{n},\bar{g})$.

###### Proof.

The proof is based on a rather straightforward estimate of n-capacity when
test functions are chosen right. To construct appropriate test functions, for
$\alpha$ large and fixed and $\beta$ large, we consider the cut-off

$u_{\alpha,\beta}=\left\\{\begin{aligned} \beta&\text{ when
$u-\alpha\geq\beta$}\\\ u-\alpha&\text{ when
$u-\alpha<\beta$}\end{aligned}\right.\quad\text{ and
}\quad\phi=u_{\alpha,\beta}-\beta\eta$

where $\eta\in C^{\infty}_{0}(\Sigma_{\alpha})$ is a fixed usual cut-off that
is equal to one in a neighborhood of $\partial\Omega$ and
$\Sigma_{\lambda}=\\{x\in\Omega:u(x)>\lambda\\}$ is the level set of the
conformal factor function $u$. It is easily seen that

$u_{\alpha,\beta}\in(0,\beta)\text{ in $\Sigma_{\alpha}$ and }\phi=0\text{ on
}\left\\{\begin{aligned} &\\{x\in\Omega:u=\alpha\\}\text{ or}\\\
&\\{x\in\Omega:u\geq\alpha+\beta\\}\end{aligned}\right.$

for $\beta$ large. We multiple $\phi$ to both sides of (2.1) and integrate

$\int_{\Sigma_{\alpha}}|\nabla u|^{n-2}\nabla
u\cdot\nabla\phi+\int_{\Sigma_{\alpha}}|\nabla u|^{n-2}Ric(\nabla
u)\phi=\int_{\Sigma_{\alpha}}|\nabla u|^{n-2}Ric_{g}(\nabla u)e^{2u}\phi.$

Therefore

$\displaystyle\int_{\Sigma_{\alpha}}|\nabla u_{\alpha,\beta}|^{n}$
$\displaystyle\leq\beta\int_{\Sigma_{\alpha}}|\nabla u|^{n-2}\nabla
u\cdot\nabla\eta-\int_{\Sigma_{\alpha}}|\nabla u|^{n-2}Ric(\nabla u)\phi$
$\displaystyle\quad+\beta\int_{\Sigma_{\alpha}}|\nabla u|^{n-2}Ric_{g}(\nabla
u)e^{2u}(1-\eta)$ $\displaystyle\quad+\int_{\Sigma_{\alpha}}|\nabla
u|^{n-2}Ric_{g}(\nabla u)e^{2u}(u_{\alpha,\beta}-\beta).$

Because

$\phi=u_{\alpha,\beta}-\beta\eta=u_{\alpha,\beta}-\beta+\beta(1-\eta).$

Notice that $\eta$ is independent of $\beta$, which implies the terms except
the last term in the right hand side is bounded by $C\beta$. In the mean time,
the last term in the right

(3.1) $\int_{\Sigma_{\alpha}}|\nabla u|^{n-2}Ric_{g}(\nabla
u)e^{2u}(u_{\alpha,\beta}-\beta)\leq\beta\int_{\Sigma_{\alpha}}Ric^{-}_{g}|\nabla
u|^{n-2}e^{2u}dvol\leq C\beta$

if we use the assumption $Ric^{-}_{g}|\nabla u|^{n-2}e^{2u}\in
L^{1}(\Omega,\bar{g})$. Otherwise, we use the alternative assumption and apply
HöLder inequality in (3.1)

$\displaystyle\int_{\Sigma_{\alpha}}|\nabla u|^{n-2}Ric_{g}(\nabla
u)e^{2u}(u_{\alpha,\beta}-\beta)$ $\displaystyle\leq
C\beta(\int_{\Sigma_{\alpha}}|Ric^{-}_{g}|^{\frac{n}{2}}e^{nu})^{\frac{2}{n}}(\int_{\Sigma_{\alpha}}|\nabla
u_{\alpha,\beta}|^{n})^{1-\frac{2}{n}}$
$\displaystyle\leq\frac{1}{2}\int_{\Sigma_{\alpha}}|\nabla
u_{\alpha,\beta}|^{n}+C\beta^{\frac{n}{2}}\int_{\Sigma_{\alpha}}(|Ric^{-}|^{n}\,dvol)_{g}$

Thus, in both cases,

$\int_{\Sigma_{\alpha}}|\nabla(\frac{u_{\alpha,\beta}}{\beta})|^{n}\leq
C\beta^{1-n}+C\beta^{-\frac{n}{2}}\text{ as $\beta\to\infty$}$

which complete the proof of this theorem. ∎

Next we state and prove an improvement of [CHY04, Proposition 8.1].

######  Lemma 3.1.

Suppose that $(M^{n},\bar{g})$ is a compact manifold and that $\Omega$ is a
domain in $M$. Let $g=e^{2w}\bar{g}$ be a complete conformal metric on
$\Omega$. Assume that $R^{-}_{g}\in L^{p}(\Omega,g)$ for some $p>\frac{n}{2}$.
Then

(3.2) $w(x)\to\infty\text{ as $x\to\partial\Omega$}.$

###### Proof.

The proof of (3.2) is inspired by the Moser’s iteration argument of [CHY04,
Proposition 8.1] applied to the scalar curvature equation

(3.3)
$-\Delta_{g}u+\frac{n-2}{4(n-1)}R^{+}_{g}u=\frac{n-2}{4(n-1)}R^{-}_{g}u+\frac{n-2}{4(n-1)}R_{\bar{g}}u^{\frac{n+2}{n-2}}$

for $u=e^{-\frac{n-2}{2}w}$. To get ride of the nonlinear term in the right
hand side, without the loss of generality, we may assume

  1. (1)

$R_{\bar{g}}\leq 0$ when $(M^{n},\bar{g})$ is of nonpositive Yamabe type;

  2. (2)

$R_{\bar{g}}=0$ in an open domain $D$ that includes $\partial\Omega$ in $M$
when $(M^{n},\bar{g})$ is of positive Yamabe type.

(2) in the above can be achieved by using the Green function of the conformal
Laplacian $-\Delta+\frac{n-2}{4(n-1)}R$ with a pole outside of the domain $D$.
Now (3.3) becomes

(3.4)
$-\Delta_{g}u+\frac{n-2}{4(n-1)}R^{+}_{g}u\leq\frac{n-2}{4(n-1)}R^{-}_{g}u.$

To apply the Moser’s iteration argument, we need the Sobolev inequality. In
fact, from [CHY04, Proposition 8.1], what we really need is the Yamabe-Sobolev
(3.5) in below. After we have the Yamabe-Sobolev (3.5), the difference from
[CHY04, Proposition 8.1] is that we have $R^{-}_{g}\in L^{p}(\Omega,g)$
instead of $R_{g}$ bounded from below. For this, one simply needs to observe
that $R^{-}_{g}\in L^{p}(\Omega,g)$ for $\frac{n}{2}<p\leq\infty$ is enough
for Moser’s iteration in (3.4) (cf. (8.5) of [CHY04, Proposition 8.1]).

Therefore this lemma follows from the following Yamabe-Soboev

(3.5) $(\int v^{\frac{2n}{n-2}}dvol_{g})^{\frac{n-2}{n}}\leq C(D)\int(|\nabla
v|^{2}+\frac{n-2}{4(n-1)}R_{g}v^{2})dvol_{g}\text{ for all $v\in
C^{\infty}_{c}(D\setminus\partial\Omega)$}$

for some open neighborhood $D$ of $\partial\Omega$ in $M$. To prove (3.5) when
$(M^{n},,\bar{g})$ is of nonpositive Yamabe type, i. e. $R_{\bar{g}}\leq 0$ on
$M$ (cf. (1) in the above), without the loss of generality, for the compact
manifold $(M^{n},\bar{g})$, we may assume

  * •

$R_{\bar{g}}\geq-c_{1}$ on $M$ for some nonnegative constant $c_{1}$;

  * •

the Sobolev inequality

(3.6) $(\int v^{\frac{2n}{n-2}}dvol_{\bar{g}})^{\frac{n-2}{n}}\leq
S(M^{n},\bar{g})\int(|\nabla v|^{2}dvol)_{\bar{g}}\text{ for all $v\in
C^{\infty}_{c}(M)$}.$

By the definition of Yamabe constant, it is equivalent to prove, for the
metric $\bar{g}$, the following Yamabe-Sobolev holds

(3.7) $(\int v^{\frac{2n}{n-2}}dvol_{\bar{g}})^{\frac{n-2}{n}}\leq
C_{D}\int((|\nabla v|^{2}+\frac{n-2}{4(n-1)}Rv^{2})dvol)_{\bar{g}}\text{ for
all $v\in C^{\infty}_{c}(D\setminus\partial\Omega)$}$

for some open neighborhood $D$ of $\partial\Omega$ in $M$. To do so, one only
needs to control

$\displaystyle-\frac{n-2}{4(n-1)}\int R_{\bar{g}}v^{2}dvol_{\bar{g}}$
$\displaystyle\leq c_{1}\frac{n-2}{4(n-1)}\int v^{2}dvol_{\bar{g}}$
$\displaystyle\leq c_{1}\frac{n-2}{4(n-1)}(\int
v^{\frac{2n}{n-2}}dvol_{\bar{g}})^{\frac{n-2}{n}}Vol(D\setminus\partial\Omega,\bar{g})^{\frac{2}{n}}$
$\displaystyle\leq
c_{1}\frac{n-2}{4(n-1)}Vol(D\setminus\partial\Omega,\bar{g})^{\frac{2}{n}}S(M,\bar{g})\int(|\nabla
v|^{2}dvol)_{\bar{g}},$

which implies (3.7) from (3.6), provided that

(3.8)
$Vol(D\setminus\partial\Omega,\bar{g})^{\frac{2}{n}}<\frac{2(n-1)}{c_{1}(n-2)S(M,\bar{g})}.$

Because

$\displaystyle\int(|\nabla v|^{2}dvol)_{\bar{g}}$ $\displaystyle=\int((|\nabla
v|^{2}+\frac{n-2}{4(n-1)}Rv^{2})dvol)_{\bar{g}}-\frac{n-2}{4(n-1)}\int
R_{\bar{g}}v^{2}dvol_{\bar{g}}$ $\displaystyle\leq\int((|\nabla
v|^{2}+\frac{n-2}{4(n-1)}Rv^{2})dvol)_{\bar{g}}+\frac{1}{2}\int(|\nabla
v|^{2}dvol)_{\bar{g}}.$

Notice that (3.8) holds for some domain $D$ regardless of the Lebesgue measure
of $\partial\Omega$ is positive or not.

On the other hand, when $(M^{n},,\bar{g})$ is of positive Yamabe type, i. e.
$R_{\bar{g}}=0$ in $D\subset M$ (cf. (2) in the above), (3.7) is the same as
(3.6) and implies (3.5). ∎

######  Corollary 3.1.

For $n\geq 3$, let $\Omega$ be a domain in a compact manifold
$(M^{n},\bar{g})$ and endowed with a complete conformal metric
$g=e^{2u}\bar{g}$. Assume Assume that,

$R^{-}_{g}\in L^{p}(\Omega,g)\text{ for some $p\in(\frac{n}{2},+\infty]$ and
}\left\\{\begin{aligned} &\text{either }Ric^{-}_{g}\in
L^{\frac{n}{2}}(\Omega,g),\\\ &\text{or }Ric^{-}_{g}|\nabla u|^{n-2}e^{2u}\in
L^{1}(\Omega,\bar{g}).\end{aligned}\right.$

where $R^{-}_{g}$ is the negative part of the scalar curvature $R$. Then
$cap_{n}(\partial\Omega,D)=0$, where $D$ is a neighborhood of $\partial\Omega$
in $M^{n}$. Hence $(\Omega,g)$ is $n$-parabolic and $\partial\Omega$ is of
Hausdorff dimension 0 in $(M^{n},\bar{g})$.

In the following we give an example to illustrate that it is impossible to
push for $\partial\Omega$ to be a finite point set in Theorem 3.1, more
precisely, not under the assumption $Ric^{-}_{g}\in L^{\frac{n}{2}}(\Omega,g)$
(also see a similar example [Car20, Theorem 2.7]). The idea of such a
construction comes from the paper of Gallot [Gal88]. We refer the readers to
Appendix A1 and A2 of [Gal88] for details. We are constructing a complete
conformal metric $g=e^{2u}g_{\mathbb{S}}$ on a domain
$\Omega\subset\mathbb{S}^{n}$ such that

$\int_{\Omega}|Ric^{-}_{g}|^{n/2}dvol_{g}<+\infty$

but $\partial\Omega$ consists of infinite many points. For an easy visual
presentation, we consider $\Omega=\Phi(M)$ for a conformal embedding

$\Phi:M\to\mathbb{S}^{n}.$

We first construct the manifold $(M^{n},g)$ and then the conformal embedding
will be easily seen. Let

$C=[-1,1]\times\mathbb{S}^{n-1},g_{\alpha,\varepsilon,\eta}=dt^{2}+b^{2}(t)g_{\mathbb{S}^{n-1}},$

where $b(t)=\eta(t^{2}+\varepsilon^{2})^{\frac{\alpha}{2}}.$ For $k\geq 1$, we
choose

$\alpha=\alpha_{k}=1+k\cdot
e^{-k},\varepsilon=\varepsilon_{k}=e^{-e^{k}},\eta=\eta_{k}=\frac{1}{\alpha_{k}}(1+\varepsilon_{k}^{2})^{\frac{2-\alpha_{k}}{2}}.$

We denote $g_{\alpha_{k},\varepsilon_{k},\eta_{k}}$ by $g_{k}$ and $(C,g_{k})$
by $C_{k}$. The manifold $M$ is obtained by gluing countable many $B_{k}$,
which is the Euclidean space with a ball removed, to $A$, which is the
Euclidean space with countably many balls removed, using $C_{k}$ to form a
complete noncompact manifold. The following figure indicates what is $M$ and
what is the image $\Phi(M)$ in $\mathbb{S}^{n}$. From an argument similar to
that in [Gal88, A2] we have

$\int_{M}|Ric^{-}_{g}|^{\frac{n}{2}}dvol_{g}\leq C\sum_{k}[(k\cdot
e^{-k})^{\frac{n-2}{2}}+e^{-(n-1)k}]<+\infty.$

We therefore conclude that $\Omega=\mathbb{S}^{n}\backslash\\{{\rm
countable\,\,many\,\,points}\\}.$

###  3.2. Finite point conformal compactification for domains in
$\mathbb{S}^{n}$

In this subsection we will study domains in the standard round sphere
$\mathbb{S}^{n}$. Using n-Laplace equations and what have been developed in
[MQ18] we state and prove a Huber’s type finite point compactification theorem
for domains in $\mathbb{S}^{n}$.

######  Theorem 3.2.

For $n\geq 3$, let $\Omega$ be a domain in the standard unit round sphere
$(\mathbb{S}^{n},g_{\mathbb{S}})$. Suppose that, on $\Omega$, there is a
conformal metric $g=e^{2u}g_{\mathbb{S}}$ satisfying

$\lim_{x\to\partial\Omega}u(x)=+\infty\text{ and }Ric^{-}_{g}|\nabla
u|^{n-2}e^{2u}\in L^{1}(\Omega,g_{\mathbb{S}}).$

Then $\partial\Omega=\mathbb{S}^{n}\setminus\Omega$ is a finite point set.

###### Proof.

Our proof of this theorem is divided into the following four steps. But,
first, in order to employ estimates of Wolff potential, we consider n-Laplace
equations on Euclidean space $\mathbb{R}^{n}$ via a stereographic projection
$\mathcal{P}$ with respect to an interior point of $\Omega$. We have then a
complete conformal metric $g=e^{2v}|dx|^{2}$ and n-Laplace equations

(3.9) $-\Delta_{n}v=Ric_{g}(\nabla v)|\nabla v|^{n-2}e^{2v}=f(x)\qquad\text{
in $\mathcal{P}(\Omega)\subset\mathbb{R}^{n}$}$

thanks to the conformal invariance of the n-Laplace equation (2.1).
$\mathcal{S}=\partial\mathcal{P}(\Omega)\subset\mathbb{R}^{n}$ becomes a
compact subset of zero Hausdorff dimensions in the light of Theorem 3.1. To
focus on the asymptotic behavior near $\mathcal{S}$, we will consider a
bounded open neighborhood $D\subset\mathbb{R}^{n}$ of $\mathcal{S}$. The right
hand side of (3.9) satisfies that $f^{-}\in L^{1}(D,|dx|^{2})$ as well as

$\lim_{x\to\mathcal{S}}v(x)=\infty$

according to the assumptions.

Step 1. In this step we show that in fact

$f\in L^{1}(D,|dx|^{2})$

from $f^{-}\in L^{1}(D,|dx|^{2})$. This situation is similar to what was dealt
with in [BV89, Lemma 1](see also [KV86, V17]), particularly the idea in [BV89,
Lemma 1] seems to work. But we adopt a more straightforward argument here. The
following cutoff function is very useful to our investigations and has been
used in [DHM97, Lemma 14] and [MQ18]. Let

$\alpha_{s}(t)=\begin{cases}t&t\leq s,\\\ {\rm increasing}&s\leq t\leq 10s\\\
2s&t\geq 10s.\end{cases}$

We may ask $0\leq\alpha_{s}^{\prime}(t)\leq 1$ and
$\alpha^{\prime\prime}_{s}(t)\leq 0$. It is easily seen that

  * •

$\alpha_{s}(t)\leq t$

  * •

$\lim_{s\to+\infty}\alpha_{s}(t)=t\text{ for any $t\in(-\infty,\infty)$}$

  * •

$\alpha_{s}(v)=2s$ in the neighborhood $\\{x\in\mathbb{R}^{n}:v(x)>10s\\}$ of
$\partial D$

  * •

$-\Delta_{n}\alpha_{s}(v)=0$ in the neighborhood
$\\{x\in\mathbb{R}^{n}:v(x)>10s\\}$ of $\partial D$

  * •

$\alpha_{s}(v)\in C^{\infty}(\mathbb{R}^{n})$.

More importantly,

(3.10) $\int_{D}-\Delta_{n}\alpha_{s}(v)=\int_{\partial D}|\nabla
v|^{n-2}\frac{\partial v}{\partial\nu}$

provided $s>\max_{\partial D}v$. Let

(3.11) $f_{s}=-\Delta_{n}a_{s}(v).$

We shall prove that

(3.12) $\|f_{s}\|_{L^{1}(D,|dx|^{2})}\leq C\text{ for a constant $C$ that is
independent of $s$}.$

We first calculate

(3.13)
$-\Delta_{n}\alpha_{s}(v)=(\alpha^{\prime}_{s}(v))^{n-1}(-\Delta_{n}v)-(n-1)(\alpha^{\prime}_{s}(v))^{n-2}\alpha^{\prime\prime}_{s}(v)|\nabla
v|^{n},$

where the second term in the right side
$-(n-1)(\alpha^{\prime}_{s}(v))^{n-2}\alpha^{\prime\prime}_{s}(v)|\nabla
v|^{n}$ is nonnegative because $\alpha_{s}^{\prime\prime}\leq 0$. Hence

$f_{s}^{-}=((\alpha^{\prime}_{s}(v))^{n-1}(-\Delta_{n}v)-(n-1)(\alpha^{\prime}_{s}(v))^{n-2}\alpha^{\prime\prime}_{s}(v)|\nabla
v|^{n})^{-}\leq(\alpha^{\prime}_{s}(v))^{n-1}f^{-}\leq f^{-},$

where we use the fact that $\alpha_{s}^{\prime}\in[0,1]$. This implies

$\|f_{s}^{-}\|_{L^{1}(D,|dx|^{2})}=\int_{D}f_{s}^{-}dx\leq\|f^{-}\|_{L^{1}(D,|dx|^{2})}.$

Therefore

$\displaystyle\int_{D}f_{s}^{+}dx$
$\displaystyle=\int_{D}f_{s}dx+\int_{D}f_{s}^{-}dx$
$\displaystyle=\int_{\partial D}|\nabla v|^{n-2}\frac{\partial
v}{\partial\nu}+\int_{D}f_{s}^{-}dx$ $\displaystyle\leq\int_{\partial
D}|\nabla v|^{n-2}\frac{\partial
v}{\partial\nu}+\|f^{-}\|_{L^{1}(D,|dx|^{2})}.$

The claim (3.12) is proven. Next we apply Fatou lemma

$\int_{D}f^{+}dx\leq\liminf_{s\to\infty}\int_{D}f_{s}^{+}\leq\int_{\partial
D}|\nabla v|^{n-2}\frac{\partial
v}{\partial\nu}+\|f^{-}\|_{L^{1}(D,|dx|^{2})}$

and complete the first step. We now collect what we have got in this step as a
lemma.

######  Lemma 3.2.

Under the assumptions in Theorem 3.2,

$\displaystyle\|f_{s}\|_{L^{1}(D,|dx|^{2})}$ $\displaystyle\leq\int_{\partial
D}|\nabla v|^{n-2}\frac{\partial
v}{\partial\nu}+2\|f^{-}\|_{L^{1}(D,|dx|^{2})}\text{ and }$
$\displaystyle\|f\|_{L^{1}(D,|dx|^{2})}$ $\displaystyle\leq\int_{\partial
D}|\nabla v|^{n-2}\frac{\partial
v}{\partial\nu}+2\|f^{-}\|_{L^{1}(D,|dx|^{2})}.$

Moreover $-\Delta_{n}\alpha_{s}(v)$ is a Radon measure on $D$ and

$\displaystyle|(-\Delta_{n}\alpha_{s}(v))(\phi)|$
$\displaystyle=|\int_{\mathbb{R}^{n}}(-\Delta_{n}\alpha_{s}(v))\phi dx|$
$\displaystyle\leq(\int_{\partial D}|\nabla v|^{n-2}\frac{\partial
v}{\partial\nu}+2\|f^{-}\|_{L^{1}(D,|dx|^{2})})\|\phi\|_{C^{0}(D)}$

for any continuous function with compact support $\phi\in C_{c}^{0}(D)$,
provided $s$ is appropriately large.

Step 2. In this step we want to show that in fact $-\Delta_{n}v$ is a Radon
measure in $D$. The analysis in this step is very subtle and one needs to be
very careful to understand the Radon measure $-\Delta_{n}v$ from
$-\Delta_{n}\alpha_{s}(v)$. We start with $\phi\in C^{\infty}_{c}(D)$

$(-\Delta_{n}v)(\phi)=(|\nabla v|^{n-2}\nabla v)(\nabla\phi)$

by the definition of distributional derivatives. We then claim that $|\nabla
v|^{n-1}\in L^{1}(D,|dx|^{2})$ (cf. Lemma 3.4). Notice that
$\nabla\alpha_{s}(v)=\alpha_{s}^{\prime}(v)\nabla v$ and
$\alpha_{s}^{\prime}(v)\in[0,1]$. Therefore, by the dominated convergence
theorem, we have

(3.14) $\displaystyle(-\Delta_{n}v)(\phi)$ $\displaystyle=\int|\nabla
v|^{n-2}\nabla v\cdot\nabla\phi$
$\displaystyle=\lim_{s\to\infty}\int|\nabla\alpha_{s}(v)|^{n-2}\nabla\alpha_{s}(v)\cdot\nabla\phi$
$\displaystyle=\lim_{s\to\infty}(-\Delta_{n}\alpha_{s}(v))(\phi).$

Therefore, by Lemma 3.2,

$|(-\Delta_{n}v)(\phi)|=|\lim_{s\to\infty}\int_{D}-\Delta_{n}\alpha_{s}(v)\phi
dx|\leq 2\|f^{-}\|_{L^{1}(D,|dx|^{2})}\|\phi\|_{C^{0}(D)},$

which implies that $-\Delta_{n}v$ is a Radon measure on $D$. In fact we want
to be a bit more careful and calculate the following according to (3.13) and
Step 1

$\displaystyle\int_{D}(-\Delta_{n}v)\phi dx+$
$\displaystyle(-\Delta_{n}v)|_{\mathcal{S}}(\phi)=(-\Delta_{n}v)(\phi)=\lim_{s\to\infty}\int_{D}-\Delta_{n}\alpha_{s}(v)\phi
dx$
$\displaystyle=\lim_{s\to\infty}\int_{D}[(\alpha^{\prime}_{s}(v))^{n-1}(-\Delta_{n}v)-(n-1)(\alpha^{\prime}_{s}(v))^{n-2}\alpha^{\prime\prime}_{s}(v)|\nabla
v|^{n}]dx$
$\displaystyle\geq\lim_{s\to\infty}\int_{D}(\alpha^{\prime}_{s}(v))^{n-1}(-\Delta_{n}v)dx$
$\displaystyle\geq\int_{D}(-\Delta_{n}v)\phi dx,$

which implies that $-\Delta_{n}v$ is nonnegative on its support $\mathcal{S}$.
Let us first collect what we have derived from the claim in this step as the
following lemma.

######  Lemma 3.3.

Under the assumptions in Theorem 3.2, $-\Delta_{n}v$ is a Radon measure on
$\mathbb{R}^{n}$ and is nonnegative on its support $\partial D$.

To complete the proof of Lemma 3.3, we state a lemma to ensure the claim we
made in the above. This lemma is well known for n-superharmonic functions (cf.
[KM92] and references therein). We want to extend it to our cases.

######  Lemma 3.4.

Under the assumptions in Theorem 3.2, for any $1\leq q<n$,

$\|\nabla v\|_{L^{q}(D,|dx|^{2})}\leq C\text{ and
}\|\nabla\alpha_{s}(v)\|_{L^{q}(D,|dx|^{2})}\leq C$

for a constant $C$ that is independent of $s$.

###### Proof.

First we introduce the notations

$D_{k}=\\{x\in\mathbb{R}^{n}:v(x)>k\\}$

where we consider $v(x)=\infty$ on $\mathcal{S}$. We may assume that
$D_{k}\subset D$ for $k=1,2,\cdots$. We start with the estimates using Hölder
inequality

(3.15) $\displaystyle\int_{D_{l}}|\nabla\alpha_{s}(v)|^{q}dx$
$\displaystyle=\int_{D_{l}}\alpha_{s}(v)^{q}|\alpha_{s}(v)^{-1}\nabla\alpha_{s}(v)|^{q}dx$
$\displaystyle\leq(\int_{D_{l}}\alpha_{s}(v)^{\frac{nq}{n-q}}dx)^{\frac{n-q}{n}}(\int_{D_{l}}|\nabla\alpha_{s}(v)|^{n}\alpha_{s}(v)^{-n}dx)^{\frac{q}{n}}.$

For the first factor in the right hand side of (3.15), we rely on the Sobolev
inequality.

$\displaystyle\|\alpha_{s}(v)\|_{L^{\frac{nq}{n-q}}(D_{l},|dx|^{2})}$
$\displaystyle\leq\|\alpha_{s}(v)-l\|_{L^{\frac{nq}{n-q}}(D_{l},|dx|^{2})}+\|l\|_{L^{\frac{nq}{n-q}}(D_{l},|dx|^{2})}$
$\displaystyle\leq
C(n)\|\nabla\alpha_{s}(v)\|_{L^{q}(D_{l},|dx|^{2})}+lVol(D_{l})^{\frac{1}{q}-\frac{1}{n}},$

which implies

(3.16) $\|\alpha_{s}(v)\|_{L^{\frac{nq}{n-q}}(D_{l})}^{q}\leq
2^{q-1}(C(n)^{q}\int_{D_{l}}|\nabla\alpha_{s}(v)|^{q}dx+l^{q}Vol(D_{l})^{1-\frac{q}{n}}).$

Next, we want to estimate the second factor in the right hand side of (3.15).
Thanks to Theorem 3.1, we know

$\lim_{l\to\infty}cap_{n}(D_{l},D_{k})=0$

for any fixed and large $k$. $k$ is chosen such that, for $\epsilon>0$,

(3.17)
$\frac{1}{k^{n-1}}\int_{D_{k}}|f_{s}|dx\leq\frac{1}{2}\epsilon^{\frac{n}{q}}.$

Then, for this fixed and large $k$, there exists $l$ and a test function
$\zeta(x)$ such that

$\zeta(x)\in C^{\infty}_{c}(D_{k}),\zeta\in[0,1],\zeta\equiv 1\text{ on
}D_{l}$

and moreover

(3.18)
$\int|\nabla\zeta|^{n}dx\leq\frac{1}{2}(\frac{n-2}{n-1})^{n-1}\epsilon^{\frac{n}{q}}.$

Multiplying $\zeta^{n}\alpha_{s}(v)^{-n+1}$ to both sides of (3.11) and
integrating, we have

$\displaystyle\int f_{s}\zeta^{n}\alpha_{s}(v)^{-n+1}dx$
$\displaystyle=\int(-\Delta_{n}\alpha_{s}(v))\zeta^{n}\alpha_{s}(v)^{-n+1}dx$
$\displaystyle=\int|\nabla\alpha_{s}(v)|^{n-2}\nabla\alpha_{s}(v)\cdot\nabla(\zeta^{n}\alpha_{s}(v)^{-n+1})dx$
$\displaystyle=n\int|\nabla\alpha_{s}(v)|^{n-2}\alpha_{s}(v)^{-n+1}$
$\displaystyle\zeta^{n-1}\nabla\alpha_{s}(v)\cdot\nabla\zeta
dx-(n-1)\int|\nabla\alpha_{s}(v)|^{n}\zeta^{n}\alpha_{s}(v)^{-n}dx,$

wich implies

$\displaystyle(n-1)\int|\nabla\alpha_{s}(v)|^{n}$
$\displaystyle\zeta^{n}\alpha_{s}(v)^{-n}dx\leq\frac{1}{k^{n-1}}\int_{D_{k}}|f_{s}|dx$
$\displaystyle+(\frac{n}{n-1}(n-2)\int$
$\displaystyle|\nabla\alpha_{s}(v)|^{n}\zeta^{n}\alpha_{s}(v)^{-n}dx)^{\frac{n-1}{n}}(n(\frac{n-1}{n-2})^{n-1}\int|\nabla\zeta|^{n}dx)^{\frac{1}{n}}.$

Therefore, applying Young’s inequality, we get

(3.19)
$\displaystyle\int|\nabla\alpha_{s}(v)|^{n}\zeta^{n}\alpha_{s}(v)^{-n}dx$
$\displaystyle\leq\frac{1}{k^{n-1}}\int_{D_{k}}|f_{s}|dx+(\frac{n-1}{n-2})^{n-1}\int|\nabla\zeta|^{n}dx$
$\displaystyle\leq\epsilon^{\frac{n}{q}}.$

in the light of (3.17) and (3.18). Combining (3.15), (3.16) and (3.19), we
arrive at

$\int_{D_{l}}|\nabla\alpha_{s}(v)|^{q}dx\leq
2^{q-1}C(n)^{q}\epsilon\int_{D_{l}}|\nabla\alpha_{s}(v)|^{q}dx+2^{q-1}l^{q}\epsilon
Vol(D_{l})^{1-\frac{q}{n}},$

For appropriate choice of $\epsilon,k,l$ so that
$2^{q-1}C(n)^{q}\epsilon\leq\frac{1}{2}$ (independent of $s$), the above
implies

(3.20) $\int_{D_{l}}|\nabla\alpha_{s}(v)|^{q}dx\leq 2^{q}l^{q}\epsilon
Vol(D_{l})^{1-\frac{q}{n}}<\infty$

and finishes the proof of this lemma by Fatou lemma. ∎

Step 3. In this step we want to build an n-superharmonic function as a sharp
upper bound to $v$ so that the strengthened [MQ18, Theorem 1.1] is applicable.
For this purpose we introduce the desired n-superharmonic function
$\mathcal{A}(v)$ to be the solution to

(3.21) $\left\\{\begin{aligned}
-\Delta_{n}\mathcal{A}(v)&=(-\Delta_{n}v)^{+}\text{ in $D$}\\\
\mathcal{A}(v)&=v\text{ on $\partial D$}.\end{aligned}\right.$

For the existence and properties of $\mathcal{A}(v)$, we follow [KM92].
Specifically we follow the idea in the proof of [KM92, Theorem 2.4] to find
the n-superharmonic function $\mathcal{A}(v)$. From Step 2, we know
$(-\Delta_{n}\alpha_{s}(v))$ as Radon measures converges to the Radon measure
$(-\Delta_{n}v)$, which implies $(-\Delta\alpha_{s}(v))^{+}$ converges to
$(-\Delta_{n}v)^{+}$. Hence we consider

(3.22) $\left\\{\begin{aligned}
-\Delta_{n}\mathcal{A}_{s}(v)&=(-\Delta_{n}\alpha_{s}(v))^{+}\text{ in $D$}\\\
\mathcal{A}_{s}(v)&=\alpha_{s}(v)\text{ on $\partial D$}.\end{aligned}\right.$

The existences and uniqueness of $\mathcal{A}_{s}(v)$ is due to [BG89, Theorem
1 and Remark 1], for instance. Notice that we may let

$D=\\{x\in\mathbb{S}^{n}:v(x)>1\\}.$

Hence $\alpha_{s}(v)=v=1$ on $\partial D$ when $s$ is appropriately large.
Regarding regularity, for example, thanks to [Te14, Theorem 3.3],
$\mathcal{A}_{s}(v)\in C^{1,\gamma}$. Readers are referred to [BG89, Te14] for
more references and history. As in the proof [KM92, Theorem 2.4], from [KM92,
Theorem 1.17], at least for a sequence $s\to\infty$, $\mathcal{A}_{s}(v)$
converges to the n-superharmonic function $\mathcal{A}(v)$ that satisfies
(3.21).

######  Lemma 3.5.

For $s$ appropriately large, in $D$,

$\mathcal{A}_{s}(v)+1\geq\alpha_{s}(v)$

where $\mathcal{A}_{s}(v)$ is the solution to (3.22).

###### Proof.

For simplicity, we substitute $\alpha_{s}(v)$ by $a_{s}$ and
$\mathcal{A}_{s}(v)+1$ by $b_{s}$. Assume otherwise that there is a point
$x_{0}\in D$ such that $b_{s}(x_{0})<a_{s}(x_{0})$. Then we can find a domain
$U\subset D$ such that $b_{s}<a_{s}$ in $U$ and $b_{s}=a_{s}$ on $\partial U$.
Obviously $\partial U$ is in the interior of $D$ because
$\mathcal{A}_{s}(v)=\alpha_{s}(v)$ on $\partial D$. We now choose a
nonnegative cutoff function $\phi_{\varepsilon}\in C_{0}^{\infty}(U)$ such
that

$\phi_{\varepsilon}(x)=\begin{cases}1,&x\in U_{2\varepsilon}=\\{x\in
U;d(x,\partial U)\geq 2\varepsilon\\},\\\ 0,&x\in U\backslash
U_{\varepsilon}=\\{x\in U:d(x,\partial U)<\varepsilon\\}\\\ \in[0,1]&\text{
otherwise}.\end{cases}$

We then calculate, from (3.22) and (3.11),

$\displaystyle 0$ $\displaystyle\leq\int_{U}f_{s}^{-}(a_{s}-b_{s})dx$
$\displaystyle=\int_{U}(|\nabla b_{s}|^{n-2}\nabla b_{s}-|\nabla
a_{s}|^{n-2}\nabla a_{s})\cdot(\nabla a_{s}-\nabla b_{s})\phi_{\varepsilon}$
$\displaystyle\quad\quad\quad+\int_{U}(|\nabla b_{s}|^{n-2}\nabla
b_{s}-|\nabla a_{s}|^{n-2}\nabla
a_{s})\cdot(a_{s}-b_{s})\nabla\phi_{\varepsilon}.$

That is

(3.23) $0\leq-2^{1-n}\int_{U}|\nabla b_{s}-\nabla
a_{s}|^{n}\phi_{\varepsilon}+C\int_{U}(|\nabla b_{s}|^{n-1}+|\nabla
a_{s}|^{n-1})(a_{s}-b_{s})|\nabla\phi_{\varepsilon}|,$

where we use the following inequality

$(|\vec{a}|^{n-2}\vec{a}-|\vec{b}|^{n-2}\vec{b})\cdot(\vec{a}-\vec{b})\geq
2^{1-n}|\vec{a}-\vec{b}|^{n}$

for any vectors $\vec{a},\vec{b}\in\mathbb{R}^{n}$. Since $a_{s}=b_{s}$ on
$\partial U$, we may assume $|a_{s}-b_{s}|\leq C\varepsilon$ in
$supp(\nabla\phi_{\varepsilon})$ by their regularities (as we mentioned
$\bar{U}$ is in the interior of $D$). We also naturally have
$|\nabla\phi_{\varepsilon}|\leq\frac{C}{\varepsilon}$ for the cutoff function
$\phi_{\varepsilon}$. Hence

$(a_{s}-b_{s})|\nabla\phi_{\varepsilon}|\leq C.$

Let $\varepsilon\to 0$ and derive a contradiction in (3.23) (we may need to
adjust $b_{s}=\mathcal{A}_{s}(v)+\lambda$ for $\lambda\in(0,1]$ so that
$\partial U$ is at least $C^{1}$ if necessary). Unless

$\nabla b_{s}\equiv\nabla a_{s}\,\,{\rm in}\,\,U$

which is again a contradiction to the assumption of the proof
$b_{s}(x_{0})<a_{s}(x_{0})$. ∎

Consequently, we know that

(3.24) $v\leq\mathcal{A}(v)+1.$

Step 4. In this step we want to show that the point mass
$(-\Delta_{n}v)(\\{p\\})$ has a uniform lower bound for any $p$ in the support
$\mathcal{S}$ of the Radon measure $-\Delta_{n}v$, which completes the proof
of Theorem 3.2. By Step 2, we know that
$(-\Delta_{n}v)(\\{p\\})=(-\Delta_{n}v)^{+}(\\{p\\})$ for $p\in\mathcal{S}$.
Our argument in this step therefore relies on the following strengthened two
aspects of [MQ18, Theorem 1.1]. The first is

######  Theorem 3.3.

Let $w$ be a nonnegative lower semi-continuous function that is
$n$-superharmonic in $B(0,2)\subset\hbox{\bb R}^{n}$ and

$-\Delta_{n}u=\mu\geq 0$

for a Radon measure $\mu\geq 0$ and the origin is in the support $\mathcal{S}$
of the Radon measure $\mu$. Then there is a set $E\subset\mathbb{R}^{n}$,
which is $n$-thin at the origin, such that

$\lim_{x\notin E\text{ and }|x|\rightarrow
0}\frac{u(x)}{\log\frac{1}{|x|}}=\liminf_{|x|\rightarrow
0}\frac{u(x)}{\log\frac{1}{|x|}}=m\geq 0$

and

$u(x)\geq m\log\frac{1}{|x|}-C\ \text{ for $x\in B(0,1)\setminus\mathcal{S}$
and some $C$}.$

Moreover, if $(B(0,2)\setminus\mathcal{S},e^{2u}|dx|^{2})$ is complete at the
origin, then $m\geq 1$.

The difference of Theorem 3.3 from [MQ18, Theorem 1.1] is that, when using
completeness at the origin to derive $m\geq 1$, the assumption of the origin
being isolated singularity is dropped. Theorem 3.3 indeed applies to
$\mathcal{A}(v)$, since the completeness at the origin for the metric
$e^{2v}|dx|^{2}$ implies the completeness at the origin for the metric
$e^{2\mathcal{A}(v)}|dx|^{2}$ by (3.24).

###### Proof of Theorem 3.3.

We will only need to focus on the last statement to conclude $m\geq 1$. All
the rest goes exactly as in [MQ18]. It is easily seen that, if we can find a
ray connecting the origin to some point in $B(0,2)\setminus\mathcal{S}$, then
the length of such ray under the metric $e^{2u}|dx|^{2}$ will be finite if
$m<1$. In the following we present a proof of the existence of such ray in the
way similar to the idea used in the proof of [ArH73, Theorem 5.2] in 2
dimensions.

Let $P$ be the radial project

$P=\frac{x}{|x|}:B(0,r)\setminus\\{0\\}\to\mathbb{S}^{n-1}\subset\mathbb{R}^{n}.$

Clearly, $P$ is a Lipschitz map. Recall

$\omega_{i}=\\{x\in\mathbb{R}^{n}:2^{-i-1}\leq|x|\leq 2^{-i}\\}\text{ and
}\Omega_{i}=\\{x\in\mathbb{R}^{n}:2^{-i-2}\leq|x|\leq 2^{-i+1}\\}.$

Applying [AdH99, Theorem 5.2.1], we have

$cap_{n}(P(\omega_{i}\cap E),\Omega_{-1})\leq Ccap_{n}(\omega_{i}\cap
E,\Omega_{i})$

and

$\displaystyle cap_{n}(\bigcup_{i=0}^{\infty}P(\omega_{i}\cap E),\Omega_{-1})$
$\displaystyle\leq\sum_{i=1}^{\infty}cap_{n}(P(\omega_{i}\cap E),\Omega_{-1})$
$\displaystyle\leq C\sum_{i=1}^{\infty}cap_{n}(\omega_{i}\cap E,\Omega_{i})$
$\displaystyle\leq C\sum_{i=1}^{\infty}2^{i-1}cap_{n}(\omega_{i}\cap
E,\Omega_{i}),$

which is finite since $E$ is n-thin at the origin under [MQ18, Definition
3.1]. Therefore

$cap_{n}(\bigcup_{i=i_{0}}^{\infty}P(\omega_{i}\cap E),\Omega_{-1})$

is arbitrarily small as $i_{0}$ chosen to be large. From
$cap_{n}(\mathbb{S}^{n-1}(1),\Omega_{-1})>0$ (cf. Lemma 2.1), it then implies
that there is a ray from $\omega_{i_{0}}\setminus E$ to the origin in
$B(0,2^{-i_{0}})\setminus E$, which completes the proof of Theorem 3.3. ∎

Notice that here $\mathcal{S}\subset E$ in Theorem 3.3. The second
strengthened aspect of the [MQ18, Theorem 1.1] is the following

######  Lemma 3.6.

Under the assumptions of Theorem 3.2, when applying Theorem 3.3,

$n\omega_{n}m^{n-1}\leq\mu(\\{0\\}).$

###### Proof.

We continue using the notations in the proof of Theorem 3.3. For $\epsilon>0$,
let

$G_{m-\epsilon,L}=(m-\epsilon)\log\frac{1}{|x|}+L.$

We want to pick up sequence of domains $\\{U_{k}\\}$ for a sequence of numbers
$L_{k}\to\infty$ such that

  * •

$\mathcal{A}(v)\geq G_{\epsilon,L_{k}}$ in domain $U_{k}$;

  * •

$\mathcal{A}(v)=G_{\epsilon,L_{k}}$ on $\partial U_{k}$;

  * •

$\bigcap_{k=1}^{\infty}U_{k}=\\{0\\}$.

Notice that this requires that $\partial U_{k}\cap E=\emptyset$. Assume such
$\\{U_{k}\\}$ exists. Then one observes that, on the boundary $\partial
U_{k}$,

(3.25) $\displaystyle\nabla\mathcal{A}(v)-\nabla G_{\epsilon,k}$
$\displaystyle=(\frac{\partial(\mathcal{A}(v)-G_{\epsilon,k})}{\partial
n},0,\cdots,0)$ $\displaystyle\frac{\partial\mathcal{A}(v)}{\partial n}$
$\displaystyle\geq\frac{\partial G_{\epsilon,k}}{\partial n}.$

Therefore, by the divergence theorem,

(3.26) $\displaystyle(-\Delta_{n}\mathcal{A}(v))(U_{k})$
$\displaystyle=\int_{\partial
U_{k}}|\nabla\mathcal{A}(v)|^{n-2}\frac{\partial\mathcal{A}(v)}{\partial n}$
$\displaystyle\geq\int_{\partial U_{k}}|\nabla
G_{\epsilon,k}|^{n-2}\frac{\partial G_{\epsilon,k}}{\partial n}$
$\displaystyle=n\omega_{n}(m-\epsilon)^{n-1}$

which implies this lemma. So the rest is to construct the sequence
$\\{U_{k}\\}$.

In the light of [MQ18, Lemma 5.1] (also see [Re94, Lemma 1.4 page 212] and
[Ge61, Theorem 4]), for a straight line segment $l_{i}$ connecting the inner
sphere to the outer sphere of $\omega_{i}$, we have

$cap_{n}(l_{i},\Omega_{i})=cap_{n}(l_{0},\Omega_{0})\geq\frac{c_{n}}{(\log
3)^{n-1}}>0.$

Hence there are plenty shperes
$\mathbb{S}^{n-1}(\lambda)\subset\omega_{i}\setminus E$ at least for $i$
sufficiently large. We may consider a sequence $r_{k}\to 0$ such that the
spheres $\mathbb{S}^{n-1}(r_{k})\bigcap E=\emptyset$. Let

$L_{k}=\min_{\mathbb{S}^{n-1}(r_{k})}(\mathcal{A}(v)-(m-\epsilon)\log\frac{1}{|x|})-1.$

Then

$\mathcal{A}(v)\geq G_{\epsilon,k}+1>G_{\epsilon,k}\text{ on
$\mathbb{S}^{n-1}(r_{k})$}.$

Because

  * •

$G_{\epsilon,k}(x)\to-\infty$ as $x\to\infty$;

  * •

$\mathcal{A}(v)(x)\to+\infty$ as $x$ approaches $\mathcal{S}$ away from the
origin;

  * •

$\mathcal{A}(v)(x)\geq m\log\frac{1}{|x|}-C$ as $x\to 0$ away from
$\mathcal{S}$.

There exists $U_{k}$ as desired. Finally, notice that $\mathcal{A}(v)$ is at
least $C^{1,\gamma}$ away from the singular set $\mathcal{S}$ due to [Te14,
Theorem 3.3]. The regularity of $U_{k}$ therefore is guaranteed. This
completes the proof of this lemma. ∎

Finally Lemma 3.6 and Theorem 3.3 together produce a contradiction if the
singular set $\mathcal{S}$ is infinite. Now we complete the proof of Theorem
3.2. ∎

To end this subsection we state a corollary to Theorem 3.2 in a more geometric
fashion.

######  Corollary 3.2.

For $n\geq 3$, let $\Omega$ be a domain in the standard unit round sphere
$(\mathbb{S}^{n},g_{\mathbb{S}})$. Suppose that, on $\Omega$, there is a
complete conformal metric $g=e^{2u}g_{\mathbb{S}}$ satisfying either $Ric_{g}$
is nonnegative outside a compact subset or

  1. (1)

$Ric^{-}_{g}\in L^{1}(\Omega,g)\cap L^{\infty}(\Omega,g)$

  2. (2)

$R_{g}\in L^{\infty}(\Omega,g)\text{ and }|\nabla^{g}R_{g}|\in
L^{\infty}(\Omega,g)$.

Then $\partial\Omega=\mathbb{S}^{n}\setminus\Omega$ is a finite point set.

###### Proof.

We only need to verify the two assumptions of Theorem 3.2. First, since $g$ is
complete and the scalar curvature is bounded, directly applying [CHY04,
Proposition 8.1], we know

$\lim_{x\to\partial\Omega}w(x)=+\infty.$

Now, via a stereographic projection, let
$e^{2w}g_{\mathbb{S}}=e^{2u}|dx|^{2}.$ We want to prove

$Ric_{g}^{-}|\nabla u|^{n-2}e^{2u}\in L^{1}(\Omega,|dx|^{2})$

instead, which is equivalent to one of the second assumptions in Theorem 3.2.
From the scalar curvature equation

$0=-\frac{4(n-1)}{(n-2)}\Delta_{g}(e^{-\frac{n-2}{2}u})+R_{g}e^{-\frac{n-2}{2}u},$

and Cheng-Yau’s gradient estimate (cf. [SY94, Theorem 2.12, Chapter VI] and
[CQY00, Lemma 2.3]), under the assumptions

  1. (1)

$Ric_{g}$ is bounded from below;

  2. (2)

both $R_{g}\in L^{\infty}(\Omega,g)\text{ and }|\nabla^{g}R_{g}|\in
L^{\infty}(\Omega,g)$,

we know that

$|\nabla_{g}u|\leq C\text{ and equivalently }|\nabla u|\leq e^{u},$

which implies

$\int_{M}Ric_{g}^{-}|\nabla
u|^{n-2}e^{2u}\leq\int_{M}Ric_{g}^{-}e^{nu}\leq\|Ric^{-}\|_{L^{1}(\Omega,g)}.$

So Theorem 3.2 applies. ∎

Before we end this subsection we want to share examples to compare Theorem 3.2
and Corollary 3.2 to [CH02, Theorem 2.1] and its improvement [Car20, Corollary
2.4] in cases when ambient manifolds are round spheres. Particularly, no
condition on the upper bound of Ricci is assumed in Theorem 3.2 and Corollary
3.2. Let us consider a conformal metric
$g=e^{2u}|dx|^{2}=v^{\frac{4}{n-2}}|dx|^{2}$ on the entire Euclidean space
$\mathbb{R}^{n}=\mathbb{S}^{n}\setminus\\{p_{0}\\}$, where

$v(x)=v(|x|)=(\frac{2}{1+|x|^{2}})^{\frac{m(n-2)}{4}}\text{ and
}u=\frac{m}{2}\log\frac{2}{1+|x|^{2}}\text{ when $|x|$ is large}.$

We may deform $g$ to have negative Ricci in the region when $|x|$ is not
large. For completeness and noncompactness, $m\in[0,1]$. Hence its scalar
curvature is

$\displaystyle R_{g}$ $\displaystyle=\frac{4(n-1)}{n-2}(-\Delta
v)v^{-\frac{n+2}{n-2}}$
$\displaystyle=n(n-1)(\frac{m}{2}+\frac{m(2-m)(n-2)}{4n}|x|^{2})(\frac{2}{1+|x|^{2}})^{(2-m)}$

and Ricci curvature tensor is

$\displaystyle(Ric_{g})_{ij}$
$\displaystyle=-(n-2)u_{ij}+(n-2)u_{i}u_{j}-\Delta u\delta_{ij}-(n-2)|\nabla
u|^{2}\delta_{ij}$
$\displaystyle=\frac{m}{2}(n-1)(\frac{2}{1+|x|^{2}})^{2}\delta_{ij}+\frac{m(2-m)(n-2)|x|^{2}}{4}(\frac{2}{1+|x|^{2}})^{2}\delta_{ij}$
$\displaystyle\quad\quad-\frac{m(2-m)(n-2)x_{i}x_{j}}{4}(\frac{2}{1+|x|^{2}})^{2}.$

For any $m\in(0,1)$, we have

$\displaystyle Vol^{g}(B_{r}(0))$
$\displaystyle\lesssim\int_{0}^{r}|x|^{-mn}|x|^{n-1}d|x|\lesssim r^{n-mn}$
$\displaystyle\text{ and }Vol^{g}(B_{r}(x_{0}))$ $\displaystyle\lesssim
Vol^{|dx|^{2}}(B_{r}(x_{0}))\lesssim r^{n}\text{ for any
$x_{0}\in\mathbb{R}^{n}$}$
$\int_{\mathbb{R}^{n}}|R^{+}_{g}|^{\frac{n}{2}}dvol_{g}\simeq\int_{0}^{\infty}|x|^{-n-mn}|x|^{mn}|x|^{n-1}\simeq\infty$

and similarly

$\int_{\mathbb{R}^{n}}|Ric_{g}|^{\frac{n}{2}}dvol_{g}\simeq\int_{0}^{\infty}|x|^{-n-mn}|x|^{mn}|x|^{n-1}\simeq\infty.$

On the other hand, $Ric_{g}$ clearly turns nonnegative when $|x|$ is
sufficiently large and is not asymptotically vanishing. Therefore the
assumption $Ric\in L^{\frac{n}{2}}(M,g)$ in [CH02, Theorem 2.1] and [Car20,
Corollary 2.4] is not satisfied. But Theorem 3.2 and Corollary 3.2 cover the
cases in these examples.

##  4\. Injectivity of conformal immersions

In [CQY00, Theorem 3], in order to identify the compactification, it invoked
the injectivity theorem of Schoen-Yau [SY88] (cf. [CH06, LUY20, CL20]). The
injectivity theorem of Schoen-Yau is available for [CQY00, Theorem 3] because
the scalar curvature is assumed to be strictly positive. In this section we
consider manifolds that have a conformal immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}})$

instead simply are domains in $\mathbb{S}^{n}$. Our approach follows closely
to what was in [SY88] but we would like to have the injectivity when the
scalar curvature is allowed to have some negativity. We believe the discussion
in this section has importance in its own, besides it strengthens Theorem 3.2.

To describe the end behavior that is related to the injectivity, Schoen-Yau in
[SY88] introduced the following very important quantity:

(4.1) $d(M)=\frac{n-2}{2}\inf\\{\gamma:\text{ such that
}\int_{M^{\setminus}\mathcal{O}}G_{o}^{\gamma}dvol_{g}<\infty\\}$

where $G_{o}$ is the positive minimal Green function for the conformal
Laplacian at a base point $o\in M^{n}$ and $\mathcal{O}$ is an open
neighborhood of $o$ in $M^{n}$. First, Schoen-Yau made the following estimate
of $d(M)$ when the scalar curvature is nonnegative.

######  Proposition 4.1.

(Schoen-Yau [SY88, Proposition 2.4]) Suppose that $(M^{n},g)$ is complete and
that there is a conformal immersion
$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}})$. Assume the scalar
curvature is nonnegative. Then $d(M)\leq\frac{n}{2}$.

It is remarkable that Schoen-Yau managed to use $d(M)$ to gain the
injectivity.

######  Theorem 4.1.

(Schoen-Yau [SY88, Theorem 3.1]) Suppose that $(M^{n},g)$ is complete and that
there is a conformal immersion
$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}})$. Assume the scalar
curvature of $(M^{,}g)$ is bounded from below and $d(M)<\frac{(n-2)^{2}}{n}$.
For the dimension $n=3,4$, assume in addition, the scalar curvature is
bounded. Then $\Phi$ is an embedding.

As a consequence of Proposition 2.4 and Theorem 3.1 in [SY88], they derived
the injectivity when the scalar curvature is nonnegative and the dimension is
greater than 6. The proof of the rest of the injectivity theorem of Schoen-Yau
[SY88] used a different argument based on a variant of positive mass theorem
(cf. [CH06, LUY20, CL20]).

We will produce a version of Proposition 4.1 and Theorem 4.1 under different
assumptions. We will then prove the main theorem as an improvement of Theorem
3.2.

###  4.1. Estimates of $d(M)$

In this subsection we want to state and prove a variant of Proposition 4.1 to
give an estimate of $d(M)$ under integral lower bound of the scalar curvature.
The proof is a modificationof the proof of Proposition 2.4 in [SY88].

######  Proposition 4.2.

Suppose that $(M^{n},g)$ ($n\geq 3$) is a manifold that has a conformal
immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}}).$

Assume that $R_{g}^{-}\in L^{\frac{n}{2}}(M,g)$. Then $d(M)\leq\frac{n}{2}$.

###### Proof.

Given $o\in M$, we let $G_{o}$ be the minimal Green function with pole at $o$.
Let $\mathcal{O}$ be an open neighborhood of $o$. Let
$\\{U_{i}\\}_{i=1}^{\infty}$ be an exhaustion of $M$ such that
$\bar{U}_{i}\subset U_{i+1}$ and $\bar{U}_{i}$ is compact for each
$i=1,2,\cdots$. Consider the positive Green function $G_{i}$ that satisfies

$\left\\{\begin{aligned} -\Delta
G_{i}+\frac{n-2}{4(n-1)}R_{g}G_{i}&=\delta_{o}.\\\ G_{i}|_{\partial
U_{i}}&=0\end{aligned}\right.$

Then $G_{i}\to G_{o}$ on $M$ in $C^{\infty}$ over any compact subset since
$G_{o}$ is the minimal Green function. On $U_{i}\backslash\mathcal{O}$, we
have

$-\Delta
G_{i}+\frac{n-2}{4(n-1)}R^{+}_{g}G_{i}=\frac{n-2}{4(n-1)}R^{-}_{g}G_{i},$

where $R_{g}^{+}$ and $R_{g}^{-}$ are the positive and negative part of the
scalar curvature $R_{g}$ respectively. For simplicity, we will use
$c(n)=\frac{n-2}{4(n-1)}$ in the following. Multiplying by
$G_{i}^{\varepsilon}$ for any small $\varepsilon>0$ and integrating on
$U_{i}\backslash\mathcal{O}$ on both sides, we have

$\int_{U_{i}\backslash\mathcal{O}}-G_{i}^{\varepsilon}\Delta
G_{i}+c(n)R_{g}^{+}G_{i}^{1+\varepsilon}=c(n)\int_{U_{i}\backslash\mathcal{O}}R_{g}^{-}G_{i}^{1+\varepsilon}.$

Then we have

(4.2)
$\int_{U_{i}\backslash\mathcal{O}}\frac{4\varepsilon}{(1+\varepsilon)^{2}}|\nabla
G_{i}^{\frac{1+\varepsilon}{2}}|^{2}+c(n)R_{g}^{+}G_{i}^{1+\varepsilon}\leq
c(n)\|R_{g}^{-})\|_{L^{\frac{n}{2}}(M\setminus\mathcal{O})}\,(\int_{U_{i}\backslash\mathcal{O}}G_{i}^{\frac{n(1+\varepsilon)}{n-2}})^{\frac{n-2}{n}}+C,$

where we use $\int_{\partial\mathcal{O}}\frac{\partial
G_{i}}{\partial\nu}G_{i}^{\varepsilon}\to\int_{\partial\mathcal{O}}\frac{\partial
G}{\partial\nu}G^{\varepsilon}$. Notice that, different from [SY88], we use
the integral lower bound of the scalar curvature instead.

Suppose $\mathcal{O}\subset U_{1}$, let $\zeta$ be a cut off function which is
$1$ in $\mathcal{O}$ and $0$ outside $U_{1}$. By the Sobolev inequality from
[SY88, Proposition 2.2],

$\displaystyle(\int_{U_{i}\backslash\mathcal{O}}G_{i}^{\frac{n(1+\varepsilon)}{n-2}})^{\frac{n-2}{n}}$
$\displaystyle\leq
C\int_{U_{i}}[(1-\zeta)G_{i}]^{\frac{n(1+\varepsilon)}{n-2}})^{\frac{n-2}{n}}+C$
$\displaystyle\leq
C(\int_{U_{i}}|\nabla((1-\zeta)G_{i})^{\frac{1+\varepsilon}{2}}|^{2}+c(n)R_{g}^{+}((1-\zeta)G_{i})^{1+\varepsilon})+C$
$\displaystyle\leq
C(\int_{U_{i}}|\nabla(G_{i})^{\frac{1+\varepsilon}{2}}|^{2}+c(n)R_{g}^{+}G_{i}^{1+\varepsilon})+C$

Therefore we arrive at

$\displaystyle\int_{U_{i}\backslash\mathcal{O}}\frac{4\varepsilon}{(1+\varepsilon)^{2}}|\nabla
G_{i}^{\frac{1+\varepsilon}{2}}|^{2}+c(n)R_{g}^{+}G_{i}^{1+\varepsilon}$
$\displaystyle\leq$ $\displaystyle
C\|R_{g}^{-}\|_{L^{\frac{n}{2}}(M\setminus\mathcal{O})}[\int_{U_{i}\backslash\mathcal{O}}\frac{4\varepsilon}{(1+\varepsilon)^{2}}|\nabla
G_{i}^{\frac{1+\varepsilon}{2}}|^{2}+c(n)R_{g}^{+}G_{i}^{1+\varepsilon}]+C$

Thus, by the assumption $R_{g}^{-}\in L^{\frac{n}{2}}(M,g)$, we choose
$\mathcal{O}$ such that
$\|R^{-}_{g}\|_{L^{\frac{n}{2}}(M\backslash\mathcal{O})}$ is sufficiently
small and get

$\int_{U_{i}\setminus\mathcal{O}}|\nabla
G_{i}^{\frac{1+\varepsilon}{2}}|^{2}+c(n)R_{g}^{+}G_{i}^{1+\varepsilon}\leq
C(\varepsilon,\mathcal{O},\|R^{-}_{g}\|_{L^{\frac{n}{2}}(M\backslash\mathcal{O})}),$

and

$\int_{U_{i}\setminus\mathcal{O}}G_{i}^{\frac{n(1+\varepsilon)}{n-2}}\leq
C(\varepsilon).$

This finishes the proof of this proposition. ∎

###  4.2. Injectivity from the integral lower bound on Ricci curvature

Let us first state the injectivity theorem in term of integral lower bound on
Ricci curvature. Recall from Lemma 2.4 in Section 2.3, the integral lower
bound on Ricci curvature implies p-parabolicity.

######  Theorem 4.2.

Suppose that $(M^{n},g)$ is a complete, noncompact manifold. And suppose that
there is a conformal immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}}).$

Assume that $Ric^{-}_{g}\in L^{\frac{p}{2}}(M,g)$ for some $p\in(n,\infty]$.
Then $\Phi$ is injective provided that

$d(M)<\frac{(n-2)^{2}p}{(p-2)n}.$

###### Proof.

Our proof is a modification of the proof of [SY88, Theorem 3.1]. Readers are
referred to more details in [SY88]. As in [SY88], for a point $o\in M$, let
$G_{o}$ be the minimal Green function and $\bar{G}_{o}$ be the pul-back of the
Green function on the sphere. And let

$v=\frac{G_{o}}{\bar{G}_{o}}.$

Then, as observed in [SY88],

$\tilde{g}=G_{o}^{\frac{4}{n-2}}g=v^{\frac{4}{n-2}}\bar{g}\text{ and
}\bar{g}=\bar{G}_{o}^{\frac{4}{n-2}}g$

where $\bar{g}$ is the pull-back from the Euclidean metric on
$\mathbb{S}^{n}\setminus\\{\Phi(o)\\}$. Hence $v$ is a harmonic function with
respect to the flat metric $\bar{g}$ because both $\tilde{g}$ and $\bar{g}$
are scalar-flat. By the definition, it is easily seen that $0<v\leq 1$. The
goal is to show that $v\equiv 1$. Starting from the Bochner formula [SY88,
(3.1)] for $v$

$\bar{\Delta}|\bar{\nabla}v|^{2}=2|\bar{\nabla}\bar{\nabla}v|^{2},$

for $\alpha=\frac{2(n-2)}{n}$, it is shown that

$\int_{M}\phi^{2}|\bar{\nabla}v|^{\alpha-2}|\bar{\nabla}|\bar{\nabla}v||^{2}d\bar{vol}\leq
C\int_{M}|\bar{\nabla}\phi|^{2}|\bar{\nabla}v|^{\alpha}d\bar{vol}=C\int_{M}|\nabla\phi|^{2}\bar{G}_{o}^{\alpha}|\nabla
v|^{\alpha}dvol$

(cf. [SY88, (3.4)]), where quantities with an upper bar are those taken with
respect to the metric $\bar{g}$, for any smooth function $\phi$ with compact
support. Hence

(4.3)
$\int_{M}\phi^{2}|\bar{\nabla}v|^{\alpha-2}|\bar{\nabla}|\bar{\nabla}v||^{2}d\bar{vol}\leq
C(\int_{M}|\nabla\phi|^{p}dvol)^{\frac{2}{p}}(\int_{supp\nabla\phi}(\bar{G}_{o}^{\alpha}|\nabla
v|^{\alpha})^{\frac{p}{p-2}}dvol)^{1-\frac{2}{p}}$

for any smooth function with compact support on $M$. (4.3) is where we deviate
from [SY88]. We will handle the first factor of the right hand side of (4.3)
by p-parabolicity, which is available thanks to Lemma 2.4 in Section 2.3.
According to Definition 2.1 and Definition 2.2 in Section 2.3, there is a
smooth function $\phi$ of compact support that is equal to $1$ inside a big
ball $B(o,2r)$ and equal to $0$ outside an even bigger ball $B(o,R)$ such that
$\int_{M}|\nabla\phi|^{p}dvol$ is as small as desired for appropriate choice
of $r$ and $R$.

To estimate the second factor in the right hand side of (4.3) is to estimate

(4.4) $\int_{supp\nabla\phi}\bar{G}_{o}^{\beta}|\nabla v|^{\beta}\text{ where
}\beta=\frac{2p(n-2)}{n(p-2)}\in(0,2)$

since $p\in(n,\infty]$. Here the same idea of the argument in [SY88] applies.
For the convenience of readers, we include an outline. Notice that

$\bar{G}_{o}^{\beta}|\nabla v|^{\beta}\leq C(|\nabla
G_{o}|^{\beta}+G_{o}^{\beta}|\nabla\log\bar{G}_{o}|^{\beta})$

due to the definition of $v$. Let $\gamma=\frac{1}{2}\beta(2-\beta)$. Then

$|\nabla G_{o}|^{\beta}=G_{o}^{\gamma}G_{o}^{-\gamma}|\nabla
G_{o}|^{\beta}\leq\frac{2-\beta}{2}G_{o}^{\beta}+\frac{\beta}{2}G_{o}^{\beta}|\nabla\log
G_{o}|^{2}$

and

$G_{o}^{\beta}|\nabla\log\bar{G}_{o}|^{\beta}\leq
G_{o}^{\beta}(\frac{2-\beta}{2}+\frac{\beta}{2}|\nabla\log\bar{G}_{o}|^{2})\leq\frac{2-\beta}{2}G_{o}^{\beta}+\frac{\beta}{2}G_{o}^{\beta}|\nabla\log\bar{G}_{o}|^{2}$

by Young’s inequality. To treat $G_{o}^{\beta}|\nabla\log\bar{G}_{o}|^{2}$, we
realize, away from the poles,

$\Delta\log\bar{G}_{o}-\Delta\log G_{o}=|\nabla\log
G_{o}|^{2}-|\nabla\log\bar{G}_{o}|^{2},$

which helps to derive

$\int_{M}\psi^{2}G_{o}^{\beta}|\nabla\log\bar{G}_{o}|^{2}\leq
C\int_{M}\psi^{2}G_{o}^{\beta}|\nabla\log
G_{o}|^{2}+C\int_{M}\psi^{2}G_{o}^{\beta}$

and [SY88, (3.6)]

(4.5) $\int_{M}\psi^{2}\bar{G}_{o}^{\beta}|\nabla v|^{\beta}\leq
C\int_{M}\psi^{2}G_{o}^{\beta-2}|\nabla
G_{o}|^{2}+C\int_{M}\psi^{2}G_{o}^{\beta}$

for any smooth function $\psi$ with a compact support away from the point $o$.
In fact we will choose $\psi$ that is identically one on
$\bar{B}(o,R)\setminus B(0,2r)$ and vanishes outside $\bar{B}(o,2R)\setminus
B(0,r)$ which includes the support of $\phi$. Finally, to estimate the first
term in the right hand side of (4.5), we multiply $G^{\beta-1}\psi^{2}$ to
both sides of the equation

$-\Delta G_{o}=-\frac{n-2}{4(n-1)}R_{g}G_{o}.$

and integrate to get

$\displaystyle\int_{M}\psi^{2}G_{o}^{\beta-2}|\nabla G_{o}|^{2}$
$\displaystyle\leq
C\int_{M}\psi^{2}R_{g}^{-}G_{o}^{\beta}+C\int_{M}G_{o}^{\beta}|\nabla\psi|^{2}$
$\displaystyle\leq
C\|R_{g}^{-}\|_{L^{\frac{p}{2}}(M,g)}(\int_{M}(\psi^{2}G_{o}^{\beta})^{\frac{p}{p-2}})^{\frac{p-2}{p}}+C\int_{M}G_{o}^{\beta}|\nabla\psi|^{2}$

Similar to [SY88], when

$d(M)<\frac{p(n-2)^{2}}{n(p-2)}=\frac{n-2}{2}\beta,$

we have

$\int_{M\setminus\mathcal{O}}G_{o}^{\beta}<\infty\text{ and
}\int_{M\setminus\mathcal{O}}G_{o}^{\beta\frac{p}{p-2}}<\infty.$

Thus we may arrange the sizes of $r$ and $R$ to derive

$\int_{B(o,2r)}|\bar{\nabla}v|^{\alpha-2}|\bar{\nabla}|\bar{\nabla}v||^{2}d\bar{vol}$

is as small as desired and therefore vanishes, which implies $v\equiv 1$ as
argued in [SY88]. ∎

Combining with Proposition 4.2 from the previous subsection we have

######  Corollary 4.1.

Suppose that $(M^{n},g)$ is a complete, noncompact manifold. And suppose that
there is a conformal immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}}).$

Assume that $Ric^{-}_{g}\in L^{\frac{p}{2}}(M,g)\cap L^{\frac{n}{2}}(M,g)$ for
some $p\in(n,\infty]$. Then $\Phi$ is injective provided that $n\geq 5$.

This is because of the interpolation property of $L^{p}$ norms and

$\lim_{p\to n}\frac{p(n-2)^{2}}{n(p-2)}=n-2>\frac{n}{2}\text{ when $n\geq
5$}.$

###  4.3. Injectivity from n-parabolicity

In this subsection we prove an injectivity theorem using n-parabolicity only.
Namely,

######  Theorem 4.3.

Suppose that $(M^{n},g)$ ($n\geq 3$) is a complete noncompact manifold that
has a conformal immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}}).$

Assume that $(M^{n},g)$ is n-parabolic. Then $\Phi$ is injective.

###### Proof.

We continue to use the same set up and notations in the previous subsection.
The goal is again to show that $v\equiv 1$ under the different assumption. The
proof goes exactly the same as the proof of Theorem 3.2 in the previous
subsection (cf. citeSY88) until (4.3), where $p=n$ now, that is,

(4.6)
$\int_{M}\phi^{2}|\bar{\nabla}v|^{\alpha-2}|\bar{\nabla}|\bar{\nabla}v||^{2}d\bar{vol}\leq
C(\int_{M}|\nabla\phi|^{n}dvol)^{\frac{2}{n}}(\int_{supp\nabla\phi}\bar{G}_{o}^{2}|\nabla
v|^{2}dvol)^{1-\frac{2}{n}}$

The first factor in the right hand side of (4.6) can be handled in the same
spirit as before. It is as small as desired by n-parabolicity assumption for
appropriate choices of $\phi$.

The second factor in the right hand side of (4.6) becomes

$\int_{supp\nabla\phi}\bar{G}_{o}^{2}|\nabla
v|^{2}dvol=\int_{supp\nabla\phi}|\bar{\nabla}v|^{2}\bar{dvol}.$

In the following we want to show it is bounded for appropriate choices of
$\phi$. Again using the fact $v$ is harmonic with respect to the flat metric
$\bar{g}$, for a smooth function $\psi$ that has a compact support away from
the other poles of $\bar{G}_{o}$ and is identically equal to one in a
neighborhood $B(o,2\rho)$ of $o$ for some $\rho>0$ (chosen later), we have

$\displaystyle 0$ $\displaystyle=\int_{M\setminus
B(o,\rho)}\psi^{2}v(-\bar{\Delta}v)\bar{dvol}=\int_{M\setminus
B(o,\rho)}\psi^{2}|\bar{\nabla}v|^{2}\bar{dvol}$
$\displaystyle-2\int_{M\setminus B(o,\rho)}\psi v\nabla
v\cdot\nabla\psi\bar{dvol}-\int_{\partial B(o,\rho)}v\frac{\partial
v}{\partial\bar{n}}\bar{d\sigma}$

which implies

$\int_{M\setminus B(o,\rho)}\psi^{2}|\bar{\nabla}v|^{2}\bar{dvol}\leq
4\int_{M}|\bar{\nabla}\psi|^{2}v^{2}\bar{dvol}+2\int_{\partial
B(o,\rho)}v\frac{\partial v}{\partial\bar{n}}\bar{d\sigma}.$

The second term is a boundary integral on $\partial B(o,\rho)$, which is
bounded when $\rho$ is fixed. About $\psi$, applying the same argument as in
[SY88], we may remove the restrictions of vanishing in a neighborhood of the
other poles of $\bar{G}_{o}$. We therefore have

(4.7) $\displaystyle\int_{M}|\bar{\nabla}\psi|^{2}v^{2}\bar{dvol}=$
$\displaystyle\int_{M}G_{o}^{2}|\nabla\psi|^{2}dvol$
$\displaystyle\leq(\int_{supp\nabla\psi}|\nabla\psi|^{n}dvol)^{\frac{2}{n}}(\int_{supp\nabla\psi}G_{o}^{\frac{2n}{n-2}}dvol)^{\frac{n-2}{n}}$

for any smooth functions that has compact support and is identically one in
$B(o,2\rho)$. Now the first factor in the right hand side of (4.7) can be
handled by the n-parabolicity assumption and the second factor of the right
hand side of (4.7) is bound due to [SY88, Corollary 2.3]. Thus the second
factor in the right hand side of (4.6) is bounded for appropriate choices of
$\psi$ after $\phi$ is chosen such as

  * •

$\psi$ is identically one on the support of $\nabla\phi$

  * •

$\psi$ is identically one in $B(o,2\rho)$

  * •

$\rho<r$ fixed appropriately.

One may fix $\rho>0$ first, then fix $r>4\rho$, and finally set $R$ as large
as desired. ∎

n-parabolicity is a very interesting notion in conformal geometry though its
significance needs to be explored more. For one thing, n-parabolicity is
conformally invariant. Therefore it seems to capture that the Euclidean end is
the same as the punctured point most efficiently. The injectivity theorem in
this subsection is worth to mention even though it detached from rest of the
discussions in this paper. It is very desirable to find some geometric
conditions that can induce n-parabolicity.

###  4.4. Improvements of Theorem 3.2

In this subsection, as consequences of results in Section 4.1 and 4.2, we want
to state the improvement of Theorem 3.2, particularly, from Corollary 3.2 and
Corollary 4.1, which is the main theorem of this paper.

######  Theorem 4.4.

Suppose that $(M^{n},g)$ ($n\geq 5$) is a complete noncompact manifold. And
suppose that there is a conformal immersion

$\Phi:(M^{n},g)\to(\mathbb{S}^{n},g_{\mathbb{S}}).$

Assume that either $Ric_{g}$ is nonnegative outside a compact subset or

  1. (1)

$Ric^{-}_{g}\in L^{1}(\Omega,g)\cap L^{\infty}(\Omega,g)$

  2. (2)

$R_{g}\in L^{\infty}(\Omega,g)\text{ and }|\nabla^{g}R_{g}|\in
L^{\infty}(\Omega,g)$.

Then $\Phi$ is an embedding and
$\partial\Phi(M)=\mathbb{S}^{n}\setminus\Phi(M)$ is a finite point set.

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