# Convergence of the Weighted Yamabe Flow

Zetian Yan

###### Abstract

We introduce the weighted Yamabe flow

$\left\\{\begin{array}[]{ll}\frac{\partial g}{\partial
t}&=(r^{m}_{\phi}-R^{m}_{\phi})g\\\ \frac{\partial\phi}{\partial
t}&=\frac{m}{2}(R^{m}_{\phi}-r^{m}_{\phi})\end{array}\right.$

on a smooth metric measure space $(M^{n},g,e^{-\phi}{\rm dvol}_{g},m)$, where
$R^{m}_{\phi}$ denotes the associated weighted scalar curvature, and
$r^{m}_{\phi}$ denotes the mean value of the weighted scalar curvature. We
prove long-time existence and convergence of the weighted Yamabe flow if the
dimension $n$ satisfies $n\geqslant 3$.

Key points: Yamabe flow, Convergence, Smooth measure metric spaces.
AMS classification (2010). Primary: 35G25. Secondary: 35K90, 47J35, 53A30.

##  1 Introduction

The Yamabe flow was first introduced by Richard Hamilton in [Ham88]. Hamilton
conjectured that, for every initial metric, the flow converges to a conformal
metric of constant scalar curvature. In case $Y(M^{n},g_{0})\leqslant 0$, it
is not difficult to show that the conformal factor is uniformly bounded above
and below. Moreover, the flow convergences to a metric of constant scalar
curvature as $t\to\infty$.

The case $Y(M^{n},g_{0})>0$ is more interesting. Chow [Cho92] proved the
convergence of the flow for locally conformally flat metrics with positive
Ricci curvature. Ye [Ye94] later extended the result to all locally conformal
flat metrics. Later, Brendle [Bre05] proved convergence of the flow for all
conformal classes and arbitrary initial metrics, and extended the results to
higher dimensions [Bre07].

In this paper, we generalize the Yamabe flow to smooth metric measure spaces.

To explain the results of this article requires some terminology. A smooth
metric measure space is a four-tuple $(M^{n},g,e^{-\phi}{\rm dvol}_{g},m)$ of
a Riemannian manifold $(M^{n},g)$, a smooth measure $e^{-\phi}{\rm dvol}_{g}$
determined by a function $\phi\in C^{\infty}(M)$ and the Riemannian volume
element of $g$, and a dimensional parameter $m\in[0,\infty]$. In the case
$m=0$, we require $\phi=0$. We freuqently denote a smooth metric measure space
by the triple $(M^{n},g,v^{m}{\rm dvol}_{g})$, where the measure is $v^{m}{\rm
dvol}_{g}$ and the dimensional parameter is encoded as the exponent of $v$. In
accordance with this convention, $v$ and $\phi$ will denote throughout this
article functions which are related by $v^{m}=e^{-\phi}$; when $m=\infty$,
this is to be interpreted as the formal definition of the symbol $v^{\infty}$.

The weighted scalar curvature $R^{m}_{\phi}$ of a smooth metric measure space
is

$R^{m}_{\phi}:=R+2\Delta\phi-\frac{m+1}{m}|\nabla\phi|^{2},$ (1.1)

where $R$ and $\Delta$ are the scalar curvature and the Laplacian associated
to the metric $g$, respectively. More generally, the weighted scalar curvature
is

$R^{m}_{\phi}:=R+2\Delta\phi-\frac{m+1}{m}|\nabla\phi|^{2}+m(m-1)\mu
e^{\frac{2\phi}{m}}.$ (1.2)

We can regard $(M^{n},g,e^{-\phi}{\rm dvol}_{g},m)$ as the base of the warped
product

$\left(M^{n}\times F^{m}(\mu),g\oplus e^{-\frac{2\phi}{m}}h\right)$ (1.3)

where $(F^{m}(\mu),h)$ is the $m$-dimensional simply connected spaceform with
constant sectional curvature $\mu$ [Cas19]. Among the reasons that
$R^{m}_{\phi}$ defined in (1.2) is the natural analogue of the scalar
curvature are that it is the scalar curvature of the warped product (1.3), it
plays the role of the scalar curvature in O’Neill’s submersion theorem [Lot07]
and in the Weitzenböck formula for the Dirac operator on spinors [Per02], and
the variational properties of the total weighted scalar curvature functional
are closely related to sharp Sobolev inequalities [Cas15] and special
Einstein-type structures [Cas13b, Cas15].

From above discussion, the constancy of $R^{m}_{\phi}$ defined in (1.1) is
equivalent to the fact that the warped product $\left(M^{n}\times
F^{m}(0),g\oplus e^{-\frac{2\phi}{m}}h\right)$ is constant scalar curvature.
Throughout this article, we will adopt (1.1) as the definition of the weighted
scalar curvature.

Conformal equivalence between smooth metric measure spaces are defined as the
following, see [Cas15] for more details.

######  Definition 1.1.

Smooth metric measure spaces $(M^{n},g,e^{-\phi}{\rm dvol}_{g},m)$ and
$(M^{n},\hat{g},e^{-\hat{\phi}}{\rm dvol}_{\hat{g}},m)$ are pointwise
conformally equivalent if there is a function $\sigma\in C^{\infty}(M)$ such
that

$(M^{n},\hat{g},e^{-\hat{\phi}}{\rm
dvol}_{\hat{g}},m)=(M^{n},e^{\frac{2}{m+n-2}\sigma}g,e^{\frac{m+n}{m+n-2}\sigma}e^{-\phi}{\rm
dvol}_{g},m).$

In the case $m=0$, conformal equivalence is defined in the classical sense.

In this article, we study a Yamabe-type flow on the smooth metric measure
space $(M^{n},g,e^{-\phi}{\rm dvol}_{g},m)$, $m\in(0,\infty)$, called the
weighted Yamabe flow. The definiton of the weighted Yamabe flow arises from
the following observation.

We consider a family of pairs $(g(t),\phi(t))\in\mathcal{M}_{+}\times
C^{\infty}(M)$, where $\mathcal{M}_{+}$ is the space of Riemannian metrics on
$M^{n}$. The fact $(g(t),\phi(t))$ varies within the conformal class
$[(g_{0},\phi_{0})]$ implies that the metric $(e^{\phi(t)})^{\frac{2}{m}}g(t)$
will be fixed. Denoting $\frac{\partial\phi(t)}{\partial t}=\psi(t)$ and
$\frac{\partial g(t)}{\partial t}=h(t)g(t)$, we conclude that

$\frac{2}{m}\psi(t)+h(t)=0.$

Based on this observation, we define the (normalized) weighted Yamabe flow as:

$\left\\{\begin{array}[]{ll}\frac{\partial g}{\partial
t}&=(r^{m}_{\phi}-R^{m}_{\phi})g,\\\ \frac{\partial\phi}{\partial
t}&=\frac{m}{2}(R^{m}_{\phi}-r^{m}_{\phi}),\end{array}\right.$ (1.4)

where $r^{m}_{\phi}$ is the mean value of $R^{m}_{\phi}$; i.e.

$r^{m}_{\phi}=\frac{\int_{M}R^{m}_{\phi}e^{-\phi}{\rm
dvol}_{g}}{\int_{M}e^{-\phi}{\rm dvol}_{g}}.$

By formulas of first variations in [Bes08, Theorem 1.174], direct calculation
shows that evolution equations for $R_{g}$, $\Delta\phi$ and
$|\nabla\phi|^{2}$ are

$\displaystyle\frac{\partial R_{g}}{\partial t}$ $\displaystyle=(n-1)\Delta
R^{m}_{\phi}+(R^{m}_{\phi}-r^{m}_{\phi})R_{g}$
$\displaystyle\frac{\partial\Delta\phi}{\partial t}$
$\displaystyle=(R^{m}_{\phi}-r^{m}_{\phi})\Delta\phi-\frac{n-2}{2}g(dR^{m}_{\phi},d\phi)+\frac{m}{2}\Delta
R^{m}_{\phi}$ $\displaystyle\frac{\partial g(\nabla\phi,\nabla\phi)}{\partial
t}$
$\displaystyle=(R^{m}_{\phi}-r^{m}_{\phi})g(\nabla\phi,\nabla\phi)+mg(dR^{m}_{\phi},d\phi).$

Hence, by the definition of $R^{m}_{\phi}$,

$\frac{\partial R^{m}_{\phi}}{\partial
t}=(n+m-1)\Delta_{g(t),\phi(t)}R^{m}_{\phi}+R^{m}_{\phi}(R^{m}_{\phi}-r^{m}_{\phi}),$
(1.5)

where $\Delta_{g(t),\phi(t)}$ is the weighted Laplacian on
$(M^{n},g(t),e^{-\phi(t)}{\rm dvol}_{g(t)},m)$.

######  Remark 1.2.

The weighted Laplacian: Let $(M^{n},g,v^{m}{\rm dvol}_{g})$ be a smooth metric
measure space. The weighted Laplacian $\Delta_{\phi}:C^{\infty}(M)\to
C^{\infty}(M)$ is the operator

$\Delta_{\phi}:=\Delta-\nabla\phi.$

It is formally self-adjoint with respect to the measure $e^{-\phi}{\rm
dvol}_{g}$, see [Cas15] for more details.

Equation (1.5) is analogous to the evolution of the scalar curvature along the
normalized Yamabe flow. We adapt an argument of Brendle [Bre05] to establish
long-time existence and convergence of the weighted Yamabe flow.

######  Theorem 1.3.

On a smooth metric measure spaces $(M^{n},g,e^{-\phi}{\rm dvol}_{g},m)$, where
$(M^{n},g)$ is a closed Riemannian manifold of dimension $n\geqslant 3$, for
every choice of the initial metric and the measure, the weighted Yamabe flow
(1.4) exists for all time and converges to a metric with constant weighted
scalar curvature.

This article is organized as follows.

In Section 2 we prove that the conformal factor $w(t)$ can not blow up in
finite time by controlling $w(t)$ from above and below on the interval
$[0,T]$. The long-time existence follows from this.

Convergence of the weighted Yamabe flow (1.4) will be based on the following
crucial proposition.
Proposition 3.3. Let $\\{t_{i}:i\in\mathbb{N}\\}$ be a sequence of times such
that $t_{i}\to\infty$ as $i\to\infty$. Then, we can find a real number
$0<\gamma<1$ and a constant $C$ such that, after passing to a sequence, we
have

$r^{m}_{\phi}(t_{i})-r^{m}_{\infty}\leqslant
C\left(\int_{M}w(t_{i})^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}(1+\gamma)}$ (1.6)

for all integers $i$ in that subsequence.

In Section 3 under Proposition 3.3, we can obtain decay rates of
$r^{m}_{\phi}(t)$ and the uniform upper bound of
$|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|$ in $L^{2}$ norm. Together with the
interior regularity theorem, $w(t)$ is uniformaly bounded above and below on
$[0,\infty)$, such that the weighted Yamabe flow can converge smoothly.

In Section 4 we complete the proof of Proposition 3.3 using the spectral
theorem of self-adjoint operators and asymptotic analysis.

##  2 Longtime existence

In this section we collect some basic facts for smooth metric measure spaces
and prove various properties of the weighted Yamabe flow that will be used
throughout this article.

######  Definition 2.1.

On a smooth metric measure space $(M^{n},g,e^{-\phi}{\rm dvol}_{g},m)$, which
is conformal to $(M^{n},g_{0},e^{-\phi_{0}}{\rm dvol}_{g_{0}},m)$ in the sense
of Definition 1.1,

$(M^{n},g,e^{-\phi}{\rm
dvol}_{g},m)=(M^{n},w^{\frac{4}{n+m-2}}g_{0},w^{\frac{2(m+n)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}},m),$

analogous to the classical Yamabe problem, we define the normalized energy
functional $E(w)$ as

$E_{(g_{0},\phi_{0})}(w)=\frac{\int_{M}\left(\frac{4(n+m-1)}{n+m-2}L^{m}_{\phi_{0}}w,w\right)e^{-\phi_{0}}{\rm
dvol}_{g_{0}}}{\left(\int_{M}w^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m-2}{n+m}}},$ (2.1)

where $L^{m}_{\phi_{0}}$ is the weighted conformal Laplacian on
$(M^{n},g_{0},e^{-\phi_{0}}{\rm dvol}_{g_{0}},m)$

$L^{m}_{\phi_{0}}=-\Delta_{\phi_{0}}+\frac{n+m-2}{4(n+m-1)}R^{m}_{\phi_{0}}.$

######  Remark 2.2.

The normalized total weighted scalar curvature: Under the setting of
Definition 2.1, by the transformation law of the weighted scalar curvature in
[Cas13a],

$R^{m}_{\phi}=\frac{4(n+m-1)}{n+m-2}w^{-\frac{m+n+2}{m+n-2}}L^{m}_{\phi_{0}}w,$
(2.2)

the normalized energy $E_{(g_{0},\phi_{0})}(w)$ is exactly the normalized
total weighted scalar curvature of $(M^{n},g,e^{-\phi}{\rm dvol}_{g},m)$; i.e.

$E_{(g_{0},\phi_{0})}(w)=E_{(g,\phi)}(1)=\frac{\int_{M}R^{m}_{\phi}e^{-\phi}{\rm
dvol}_{g}}{\left({\rm Vol}(M^{n},e^{-\phi}{\rm
dvol}_{g})\right)^{\frac{n+m-2}{n+m}}}.$

We set

$Y_{n,m}[(g,\phi)]=\inf\\{E_{(g_{0},\phi_{0})}(w)|w\in
C^{\infty}(M;\mathbb{R}_{+})\\}.$ (2.3)

In light of the discussion in [Ye94], in case $Y_{n,m}[(g,\phi)]\leqslant 0$,
it is not difficult to show convergence of the weighted Yamabe flow (1.4) as
$t\to\infty$. We are more interested in the positive case.

Throughout this article, we choose $(M^{n},g_{0},e^{-\phi_{0}}{\rm
dvol}_{g_{0}},m)$ to be the initial metric measure space with
$Y_{n,m}[(g_{0},\phi_{0})]>0$. Since the weighted Yamabe flow preserves the
conformal structure, we may write $g(t)=w(t)^{\frac{4}{n+m-2}}g_{0}$ as the
solution of (1.4) with $(g(0),\phi(0))=(g_{0},\phi_{0})$. Hence, the weighted
Yamabe flow reduces to the following evolution equation for the conformal
factor:

$\frac{\partial}{\partial
t}w(t)^{\frac{n+m+2}{n+m-2}}=\frac{n+m+2}{4}\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}w-R^{m}_{\phi_{0}}w+r^{m}_{\phi}w^{\frac{n+m+2}{n+m-2}}\right).$
(2.4)

Since

$\frac{d}{dt}\int_{M}e^{-\phi(t)}{\rm
dvol}_{g(t)}=\frac{n+m}{2}\int_{M}(r^{m}_{\phi}-R^{m}_{\phi})e^{-\phi}{\rm
dvol}_{g}=0,$ (2.5)

we may assume that

$\int_{M}e^{-\phi(t)}{\rm dvol}_{g(t)}=1$

for all $t\geqslant 0$. With this normalization, the mean value of the
weighted scalar curvature can be written as

$r^{m}_{\phi}(t)=\int_{M}R^{m}_{\phi}(t)e^{-\phi(t)}{\rm dvol}_{g(t)}.$

Using the evolution equation (1.5), we obtain

$\frac{d}{dt}r^{m}_{\phi}(t)=-\frac{n+m-2}{2}\int_{M}(r^{m}_{\phi}-R^{m}_{\phi})^{2}e^{-\phi}{\rm
dvol}_{g}<0.$ (2.6)

Observe that $r^{m}_{\phi}(t)>0$ since $Y_{n,m}[(g,\phi)]>0$. Hence,
$r^{m}_{\phi}(t)$ can be bounded above and below i.e.

$0<r^{m}_{\phi}(t)\leqslant r^{m}_{\phi}(0).$ (2.7)

In particular, the function $t\mapsto r^{m}_{\phi}(t)$ is decreasing.

######  Proposition 2.3.

The weighted scalar curvature of the pair $(g(t),\phi(t))$ satisfies

$\inf_{M}R^{m}_{\phi}(t)\geqslant\min\left\\{\inf_{M}R^{m}_{\phi}(0),0\right\\}$
(2.8)

for all $t\geqslant 0$.

###### Proof.

According to the evolution equation (1.5),

$\begin{split}\frac{\partial R^{m}_{\phi}}{\partial
t}&=(n+m-1)\Delta_{g(t),\phi(t)}R^{m}_{\phi}+R^{m}_{\phi}(R^{m}_{\phi}-r^{m}_{\phi})\\\
&\geqslant(n+m-1)\Delta_{g(t),\phi(t)}R^{m}_{\phi}-r^{m}_{\phi}R^{m}_{\phi}.\end{split}$

Since $r^{m}_{\phi}(t)>0$, the assertion follows from the maximum principle. ∎

######  Remark 2.4.

Positivity along the flow: In particular, if $R^{m}_{\phi}(0)>0$, we have
$R^{m}_{\phi}(t)>0$ for all $t\geqslant 0$.

######  Remark 2.5.

Lower bound of $\phi$: From above discussion, we obtain that

$\frac{\partial\phi}{\partial
t}\geqslant\frac{m}{2}(\inf_{M}R^{m}_{\phi}(0)-r^{m}_{\phi}(0)),$ (2.9)

which provides a lower bound of $\phi$.

For abbreviation, let

$\sigma=\max\left\\{\sup_{M}(1-R^{m}_{\phi}(0)),1\right\\},$

so that $R^{m}_{\phi}(t)+\sigma\geqslant 1$ for all $t\geqslant 0$.

The following result is similar to that in [Bre05]. Our arguments mostly
follow those of Brendle, but in our setting, all integrals are computed using
the weighted measure.

######  Lemma 2.6.

For every $p>2$, we have

$\begin{split}\frac{d}{dt}&\int_{M}(R^{m}_{\phi}(t)+\sigma)^{p-1}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
=&-\frac{4(n+m-1)(p-2)}{p-1}\int_{M}|d(R^{m}_{\phi}(t)+\sigma)^{\frac{p-1}{2}}|^{2}_{g(t)}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&-(\frac{n+m+2}{2}-p)\int_{M}((R^{m}_{\phi}(t)+\sigma)^{p-1}-(r^{m}_{\phi}(t)+\sigma)^{p-1})(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&-(p-1)\int_{M}\sigma((R^{m}_{\phi}(t)+\sigma)^{p-2}-(r^{m}_{\phi}(t)+\sigma)^{p-2})(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))e^{-\phi(t)}{\rm
dvol}_{g(t)}.\end{split}$

###### Proof.

This follows immediately from the evolution equation (1.5) and integration by
parts. Notice that $r^{m}_{\phi}(t)$ and $\sigma$ are independent of points in
$M$.

$\begin{split}\frac{d}{dt}&\int_{M}(R^{m}_{\phi}(t)+\sigma)^{p-1}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\ =&\int_{M}(p-1)(R^{m}_{\phi}(t)+\sigma)^{p-2}\frac{\partial
R^{m}_{\phi}}{\partial t}e^{-\phi(t)}{\rm dvol}_{g(t)}\\\
&+\int_{M}(R^{m}_{\phi}(t)+\sigma)^{p-1}\frac{n+m}{2}(r^{m}_{\phi}-R^{m}_{\phi})e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
=&-\frac{4(n+m-1)(p-2)}{p-1}\int_{M}|d(R^{m}_{\phi}(t)+\sigma)^{\frac{p-1}{2}}|^{2}_{g(t)}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&-(\frac{n+m+2}{2}-p)\int_{M}((R^{m}_{\phi}(t)+\sigma)^{p-1}-(r^{m}_{\phi}(t)+\sigma)^{p-1})(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&-(p-1)\int_{M}\sigma((R^{m}_{\phi}(t)+\sigma)^{p-2}-(r^{m}_{\phi}(t)+\sigma)^{p-2})(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))e^{-\phi(t)}{\rm
dvol}_{g(t)}.\end{split}$

∎

######  Lemma 2.7.

For every $p>\max\\{\frac{n+m}{2},2\\}$, we have

$\begin{split}\frac{d}{dt}\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant&C\left(\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{2p-(n+m)+2}{2p-(n+m)}}\\\
&+C\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\end{split}$ (2.10)

for some uniform constant C independent of $t$.

###### Proof.

Using (1.5) and (2.6), we obtain

$\begin{split}\frac{d}{dt}&\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
=&p\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p-2}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))\left((n+m-1)\Delta_{g(t),\phi(t)}R^{m}_{\phi}+R^{m}_{\phi}(R^{m}_{\phi}-r^{m}_{\phi})\right)e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&-\frac{n+m}{2}\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&+\frac{(n+m-2)p}{2}\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p-2}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&\times\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{2}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
=&-\frac{4(p-1)(n+m-1)}{p}\int_{M}\left(L^{m}_{\phi}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{\frac{p}{2}},|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{\frac{p}{2}}\right)e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&+\left(\frac{(n+m-2)(p-1)}{p}+p-\frac{n+m}{2}\right)\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&+\left(\frac{(n+m-2)(p-1)}{p}+p\right)\int_{M}r^{m}_{\phi}(t)|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&+\frac{(n+m-2)p}{2}\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p-2}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&\times\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{2}e^{-\phi(t)}{\rm
dvol}_{g(t)}.\end{split}$

Since $Y_{n,m}[(g_{0},\phi_{0})]>0$ and the function $t\mapsto
r^{m}_{\phi}(t)$ is decreasing, we obtain

$\begin{split}\frac{d}{dt}&\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
\leqslant&-\frac{(n+m-2)(p-1)}{p}Y_{n,m}[(g_{0},\phi_{0})]\left(\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{\frac{p(n+m)}{n+m-2}}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{n+m-2}{n+m}}\\\
&+\left(\frac{(n+m-2)(p-1)}{p}+p-\frac{n+m}{2}\right)\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p+1}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&+\left(\frac{(n+m-2)(p-1)}{p}+p\right)\int_{M}r^{m}_{\phi}(0)|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&+\frac{(n+m-2)p}{2}\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p-2}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&\times\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{2}e^{-\phi(t)}{\rm
dvol}_{g(t)}.\end{split}$

By Hölder’s inequality and (2.5), we have

$\begin{split}\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))&|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p-2}e^{-\phi(t)}{\rm
dvol}_{g(t)}\times\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{2}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
\leqslant&\left(\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{p+1}{p}},\end{split}$

and

$\begin{split}&\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p+1}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant\\\
&\left(\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{2p-(n+m)+2}{2p}}\cdot\left(\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{\frac{p(n+m)}{n+m-2}}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{n+m-2}{2p}}.\end{split}$

Moreover, by Young’s inequality, we obtain

$\begin{split}\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p+1}e^{-\phi(t)}{\rm
dvol}_{g(t)}&\leqslant
C_{1}\left(\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{2p-(n+m)+2}{2p-(n+m)}}\\\
&+C_{2}\left(\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{\frac{p(n+m)}{n+m-2}}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{n+m-2}{n+m}}.\end{split}$

From this, the assertion follows. ∎

In order to bound the solution $w(t)$ above and below in the interval $[0,T]$,
we need the following two lemmas.

######  Lemma 2.8.

Let $P$ be a smooth function on $(M^{n},g,e^{-\phi}{\rm dvol}_{g},m)$.
Moreover, assume that $w$ is a positive function on $M$ such that

$-\frac{4(n+m-1)}{n+m-2}\Delta_{g,\phi}w+Pw\geqslant 0.$

Then, there exists a constant $C$, depending only on $g$, $\phi$ and $P$, such
that

$\int_{M}we^{-\phi}{\rm dvol}_{g}\leqslant C\inf_{M}w.$ (2.11)

Moreover, we have

$\int_{M}w^{\frac{2(n+m)}{n+m-2}}e^{-\phi}{\rm dvol}_{g}\leqslant
C\inf_{M}w(\sup_{M}w)^{\frac{n+m+2}{n+m-2}}.$ (2.12)

###### Proof.

Fix $r>0$ sufficiently small. Notice that the weighted Laplacian
$\Delta_{g,\phi}$ has the same leading term as the classical Laplacian. The
difference only occurs on lower order terms. Therefore, the weak Harnack
inequality for linear elliptic equations [GT01, Theorem 8.18] can still hold
in the weighted case, i.e. we obtain

$\int_{B_{2r}(x)}we^{-\phi}{\rm dvol}_{g}\leqslant
e^{-\inf\phi}\int_{B_{2r}(x)}w{\rm dvol}_{g}\leqslant
e^{-\inf\phi}L_{0}\inf_{B_{r}(x)}w$

for some constant $L_{0}$.
The assertion follows from the same argument as that in [Bre05, Proposition
A.2]. ∎

######  Proposition 2.9.

Given any $T>0$, we can find positive constants $C(T)$ and $c(T)$ such that

$\sup_{M}w(t)\leqslant C(T)$

and

$\inf_{M}w(t)\geqslant c(T)$

for all $0\leqslant t\leqslant T$.

###### Proof.

The function $w(t)$ satisfies

$\frac{\partial}{\partial
t}w(t)=-\frac{n+m-2}{4}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))w(t)\leqslant\frac{n+m-2}{4}(r^{m}_{\phi}(0)+\sigma)w(t).$

Hence,

$\frac{\partial}{\partial
t}\ln(w(t))\leqslant\frac{n+m-2}{4}(r^{1}_{\phi}(0)+\sigma).$

We conclude that $\sup_{M}w(t)\leqslant C(T)$ for all $0\leqslant t\leqslant
T$. Hence, if we define

$P=R^{m}_{\phi_{0}}+\sigma(\sup_{0\leqslant t\leqslant
T}\sup_{M}w(t))^{\frac{4}{n+m-2}},$

then we obtain

$\begin{split}&-\frac{4(n+m-1)}{n+m-2}\Delta_{g_{0},\phi_{0}}w(t)+Pw(t)\\\
&\geqslant-\frac{4(n+m-1)}{n+m-2}\Delta_{g_{0},\phi_{0}}w(t)+R^{m}_{\phi_{0}}w(t)+\sigma
w(t)^{\frac{n+m+2}{n+m-2}}\\\
&=(R^{m}_{\phi}(t)+\sigma)w(t)^{\frac{n+m+2}{n+m-2}}\geqslant 0\end{split}$
(2.13)

for all $0\leqslant t\leqslant T$. By Lemma 2.8 and (2.5), we can find a
positive constant $c(T)$ such that

$\inf_{M}w(t)(\sup_{M}w(t))^{\frac{n+m+2}{n+m-2}}\geqslant c(T)$

for all $0\leqslant t\leqslant T$. Since $\sup_{M}w(t)\leqslant C(T)$, the
assertion follows. ∎

######  Proposition 2.10.

Let $0<\alpha<\frac{2m}{n+m}$. Given any $T>0$, there exists a constant $C(T)$
such that

$|w(x_{1},t_{1})-w(x_{2},t_{2})|\leqslant
C(T)((t_{1}-t_{2})^{\frac{\alpha}{2}}+d(x_{1},x_{2})^{\alpha})$ (2.14)

for all $x_{1},x_{2}\in M$ and all $t_{1},t_{2}\in[0,T]$ satisfying
$0<t_{1}-t_{2}<1$.

###### Proof.

Using Lemma 2.6 with $p=\frac{n+m+2}{2}$, we obtain for all $0\leqslant
t\leqslant T$

$\frac{d}{dt}\int_{M}(R^{m}_{\phi}(t)+\sigma)^{\frac{n+m}{2}}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant 0,$

which implies for all $0\leqslant t\leqslant T$

$\int_{M}(R^{m}_{\phi}(t)+\sigma)^{\frac{n+m}{2}}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant C.$

Hence,

$\begin{split}&\left(\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{\frac{n+m}{2}}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{2}{n+m}}\\\
&\leqslant\left(\int_{M}(R^{m}_{\phi}(t)+\sigma)^{\frac{n+m}{2}}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{2}{n+m}}+(r^{m}_{\phi}(t)+\sigma)\\\ &\leqslant
C.\end{split}$ (2.15)

Let $\alpha=2-\frac{n}{p}$, where $\frac{n}{2}<p<\frac{n+m}{2}$, $m>0$. Using
(2.4), (2.15) and Proposition 2.9, we obtain

$\int_{M}\left|-\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}w(t)+R^{m}_{\phi_{0}}w(t)\right|^{p}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\leqslant C(T)$ (2.16)

and

$\int_{M}\left|\frac{\partial}{\partial t}w(t)\right|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant C(T)$ (2.17)

for all $t\in[0,T]$. By the embedding $W^{2,p}(M)\hookrightarrow
C^{0,\alpha}(M)$, the first inequality implies that

$|w(x_{1},t)-w(x_{2},t)|\leqslant C(T)d(x_{1},x_{2})^{\alpha}$

for all $x_{1},x_{2}\in M$ and all $t\in[0,T]$. Using the second inequality,
we obtain

$\begin{split}&|w(x,t_{1})-w(x,t_{2})|\\\ &\leqslant
C(t_{1}-t_{2})^{-\frac{n}{2}}\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}|w(x,t_{1})-w(x,t_{2})|e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\\\ &\leqslant
C(t_{1}-t_{2})^{-\frac{n}{2}}\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}|w(t_{1})-w(t_{2})|e^{-\phi_{0}}{\rm
dvol}_{g_{0}}+C(T)(t_{1}-t_{2})^{\frac{\alpha}{2}}\\\ &\leqslant
C(t_{1}-t_{2})^{-\frac{n-2}{2}}\sup_{t_{2}\leqslant t\leqslant
t_{1}}\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}\left|\frac{\partial}{\partial
t}w(t)\right|e^{-\phi_{0}}{\rm
dvol}_{g_{0}}+C(T)(t_{1}-t_{2})^{\frac{\alpha}{2}}\\\ &\leqslant
C(t_{1}-t_{2})^{\frac{\alpha}{2}}\sup_{t_{2}\leqslant t\leqslant
t_{1}}\left(\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}\left|\frac{\partial}{\partial
t}w(t)\right|^{p}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{1}{p}}+C(T)(t_{1}-t_{2})^{\frac{\alpha}{2}}\\\
&\leqslant C(T)(t_{1}-t_{2})^{\frac{\alpha}{2}},\end{split}$

for all $x\in M$ and all $t_{1},t_{2}\in[0,T]$ satisfying $0<t_{1}-t_{2}<1$.
This proves the assertion. ∎

We can now use the standard regularity theory for parabolic equations [Fri64,
Section 3, Theorem 5] to show that all higher order derivatives of $w$ are
uniformly bounded on every fixed time interval $[0,T]$. Therefore, the flow
exists for all time.

##  3 Proof of the main result assuming Proposition 3.3

In this section, we will prove Theorem 1.3 based on Proposition 3.3. In the
following, $c$ and $C$ are positive constants whose value are independent of
$t$ and may change from line to line.

######  Proposition 3.1.

Fix $\max\\{\frac{n+m}{2},2\\}<p<\frac{n+m+2}{2}$. Then, we have

$\lim_{t\to\infty}\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}=0.$

###### Proof.

It follows from Lemma 2.6 that

$\begin{split}&\frac{d}{dt}\int_{M}(R^{m}_{\phi}(t)+\sigma)^{p-1}e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&\leqslant-\left(\frac{n+m+2}{2}-p\right)\int_{M}((R^{m}_{\phi}(t)+\sigma)^{p-1}-(r^{m}_{\phi}(t)+\sigma)^{p-1})(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))e^{-\phi(t)}{\rm
dvol}_{g(t)}.\end{split}$

Since $p>2$, we have

$((R^{m}_{\phi}(t)+\sigma)^{p-1}-(r^{m}_{\phi}(t)+\sigma)^{p-1})(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))\geqslant
c|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}$

for a suitable constant $c>0$. Since $p<\frac{n+m+2}{2}$, it follows that

$\frac{d}{dt}\int_{M}(R^{m}_{\phi}(t)+\sigma)^{p-1}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant-c\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}.$

Integrating with respect to $t$ yields

$\int_{0}^{\infty}\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant C,$

hence,

$\liminf_{t\to\infty}\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}=0.$

On the other hand, since $p>\max\\{\frac{n+m}{2},2\\}$, by the evolution
equation (2.7), we have

$\begin{split}\frac{d}{dt}\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant&C\left(\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{2p-(n+m)+2}{2p-(n+m)}}\\\
&+C\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\end{split}$ (3.1)

From this, the assertion follows. ∎

Hence, if we define

$r^{m}_{\infty}=\lim_{t\to\infty}r^{m}_{\phi}(t),$ (3.2)

then we obtain the following result:

######  Corollary 3.2.

For every $1<p<\frac{n+m+2}{2}$, we have

$\lim_{t\to\infty}\int_{M}|R^{m}_{\phi}(t)-r^{m}_{\infty}|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}=0.$ (3.3)

The proof of the main result will be based on the following proposition. The
proof of this critical proposition will occupy Section 4.

######  Proposition 3.3.

Let $\\{t_{i}:i\in\mathbb{N}\\}$ be a sequence of times such that
$t_{i}\to\infty$ as $i\to\infty$. Then, we can find a real number $0<\gamma<1$
and a constant $C$ such that, after passing to a sequence, we have

$r^{m}_{\phi}(t_{i})-r^{m}_{\infty}\leqslant
C\left(\int_{M}w(t_{i})^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}(1+\gamma)}$ (3.4)

for all integers $i$ in that subsequence.

Note that $\gamma$ and $C$ may depend on the sequence
$\\{t_{i}:i\in\mathbb{N}\\}$. The following result is an immediate consequence
of Proposition 3.3.

######  Proposition 3.4.

There exists real numbers $0<\gamma<1$ and $t_{0}>0$ such that

$r^{m}_{\phi}(t)-r^{m}_{\infty}\leqslant\left(\int_{M}w(t)^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t)-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}(1+\gamma)}$ (3.5)

for all $t\geqslant t_{0}$.

###### Proof.

Suppose this is not true. Then, there exists a sequence of times
$\\{t_{i}:i\in\mathbb{N}\\}$ such that $t_{i}\geqslant i$ and

$r^{m}_{\phi}(t_{i})-r^{m}_{\infty}\geqslant\left(\int_{M}w(t_{i})^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}(1+\frac{1}{i})}.$

We now apply Proposition 3.3 to this sequence $\\{t_{i}:i\in\mathbb{N}\\}$.
Hence, there exists an infinite subset $I\subset\mathbb{N}$, real numbers
$0<\gamma<1$ and $C$ such that

$r^{m}_{\phi}(t_{i})-r^{m}_{\infty}\leqslant
C\left(\int_{M}w(t_{i})^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}(1+\gamma)}$

for all $i\in I$. Thus, we conclude that

$1\leqslant
C\left(\int_{M}w(t_{i})^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}(\gamma-\frac{1}{i})}$

for all $i\in I$.

On the other hand, using Corollary 3.2 with
$p=\frac{2(n+m)}{n+m+2}<\frac{n+m+2}{2}$ and
$w(t_{i})^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}=e^{-\phi(t_{i})}{\rm dvol}_{g(t_{i})}$, we have

$\lim_{i\to\infty}\int_{M}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi(t_{i})}{\rm
dvol}_{g(t_{i})}=0.$

Therefore, if $i$ is sufficiently large,

$\begin{split}&\lim_{i\to\infty}\left(\int_{M}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi(t_{i})}{\rm
dvol}_{g(t_{i})}\right)^{\frac{n+m+2}{2(n+m)}(\gamma-\frac{1}{i})}\\\
&\leqslant\lim_{i\to\infty}\left(\int_{M}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi(t_{i})}{\rm
dvol}_{g(t_{i})}\right)^{\frac{n+m+2}{2(n+m)}\frac{\gamma}{2}}=0.\end{split}$

This is a contradiction. ∎

######  Proposition 3.5.

We have

$\int_{0}^{\infty}\left(\int_{M}w(t)^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{2}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{1}{2}}dt\leqslant C.$ (3.6)

###### Proof.

It follows from Proposition 3.4 that

$\begin{split}r^{m}_{\phi}(t)-r^{m}_{\infty}&\leqslant\left(\int_{M}w(t)^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}(1+\gamma)}\\\
&+C(r^{m}_{\phi}(t)-r^{m}_{\infty})^{1+\gamma},\end{split}$

hence,

$r^{m}_{\phi}(t)-r^{m}_{\infty}\leqslant\left(\int_{M}w(t)^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}(1+\gamma)}$ (3.7)

if $t$ is sufficiently large. Therefore, by Hölder’s inequality and (2.5), we
have

$\begin{split}\frac{d}{dt}(r^{m}_{\phi}(t)-r^{m}_{\infty})&=-\frac{n+m-2}{2}\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))^{2}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\\\
&\leqslant-\frac{n+m-2}{2}\left(\int_{M}w(t)^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t)-r^{m}_{\phi}(t)|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{n+m}}\\\
&\leqslant-\frac{n+m-2}{2}(r^{m}_{\phi}(t)-r^{m}_{\infty})^{\frac{2}{1+\gamma}}.\end{split}$
(3.8)

This implies

$\frac{d}{dt}(r^{m}_{\phi}(t)-r^{m}_{\infty})^{-\frac{1-\gamma}{1+\gamma}}\geqslant
c.$ (3.9)

From this, it follows that if $t$ is sufficiently large,

$r^{m}_{\phi}(t)-r^{m}_{\infty}\leqslant Ct^{-\frac{1-\gamma}{1+\gamma}}.$
(3.10)

Moreover, integrating the first equality in (3.8) from $T$ to $2T$ yields

$r^{m}_{\phi}(T)-r^{m}_{\phi}(2T)=\frac{n+m-2}{2}\int_{T}^{2T}\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))^{2}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}dt.$

Using Hölder’s inequality, we obtain

$\begin{split}\int_{T}^{2T}&\left(\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))^{2}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{1}{2}}dt\\\
&\leqslant\left(T\int_{T}^{2T}\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))^{2}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}dt\right)^{\frac{1}{2}}\\\
&\leqslant\left(\frac{2}{n+m-2}T(r^{m}_{\phi}(T)-r^{m}_{\phi}(2T))\right)^{\frac{1}{2}}\\\
&\leqslant CT^{-\frac{\gamma}{1-\gamma}}\end{split}$ (3.11)

if $T$ is sufficiently large. Since $0<\gamma<1$, we conclude that

$\begin{split}\int_{0}^{\infty}&(\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))^{2}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}})^{\frac{1}{2}}dt\\\
&=\int_{0}^{1}(\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))^{2}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}})^{\frac{1}{2}}dt\\\
&+\sum^{\infty}_{k=0}\int_{2^{k}}^{2^{k+1}}(\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))^{2}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}})^{\frac{1}{2}}dt\\\ &\leqslant
C\left(1+\sum_{k=0}^{\infty}2^{-\frac{\gamma}{1-\gamma}k}\right)\leqslant
C.\end{split}$ (3.12)

This proves the assertion. ∎

######  Proposition 3.6.

Given any $\eta_{0}>0$, we can find a real number $r>0$ such that

$\int_{B_{r}(x)}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\leqslant\eta_{0}$ (3.13)

for all $x\in M$ and $t\geqslant 0$.

###### Proof.

We can find a real number $T>0$ such that

$\int_{T}^{\infty}\left(\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))^{2}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{1}{2}}dt\leqslant\frac{\eta_{0}}{n}.$ (3.14)

We now choose a real number $r>0$ such that

$\int_{B_{r}(x)}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\leqslant\frac{\eta_{0}}{2}$ (3.15)

for all $x\in M$ and $0\leqslant t\leqslant T$. Using the evolution equation
(2.5) and Hölder’s inequality, we have

$\begin{split}\frac{d}{dt}\int_{B_{r}(x)}e^{-\phi(t)}{\rm
dvol}_{g(t)}&=\frac{n+m}{2}\int_{B_{r}(x)}(r^{m}_{\phi}(t)-R^{m}_{\phi}(t))e^{-\phi(t)}{\rm
dvol}_{g(t)}\\\
&\leqslant\frac{n+m}{2}\left(\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))^{2}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{1}{2}}.\end{split}$ (3.16)

Integrating (3.16) from $T$ to $t$ yields

$\begin{split}&\int_{B_{r}(x)}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\\\
&\leqslant\int_{B_{r}(x)}w(T)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\\\
&+\frac{n+m}{2}\int_{T}^{\infty}\left(\int_{M}(R^{m}_{\phi}(t)-r^{m}_{\phi}(t))^{2}w(t)^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{1}{2}}dt\leqslant\eta_{0}\end{split}$ (3.17)

for all $x\in M$ and $t\geqslant T$. This proves the assertion. ∎

######  Lemma 3.7.

Let $p=\frac{2(n+m)}{n+m-2}$ and $q>\frac{n}{2}$. There are positive constants
$\eta_{1}$ and $C$ such that if

$\begin{cases}&g=w^{\frac{4}{n+m-2}}g_{0},\\\
&e^{-\phi}=w^{\frac{2m}{n+m-2}}e^{-\phi_{0}},\end{cases}$ (3.18)

is pointwise conformal to $(g_{0},\phi_{0})$ and

$\displaystyle\int_{B_{r}(x)}e^{-\phi}{\rm dvol}_{g}\leqslant 1,$
$\displaystyle\int_{B_{r}(x)}|R^{m}_{\phi}|^{q}e^{-\phi}{\rm
dvol}_{g}\leqslant\eta_{1},$

where $B_{r}(x)$ is the ball with respect to $g_{0}$ and $r<1$, then

$w(x)\leqslant Cr^{-\frac{n}{p}}\left(\int_{B_{r}(x)}e^{-\phi}{\rm
dvol}_{g}\right)^{\frac{1}{p}}.$ (3.19)

###### Proof.

By the smoothness of the conformal factor $w(t)$, there exists $r_{0}$ a real
number such that $r_{0}<r$ and

$(r-s)^{\frac{n}{p}}\sup_{B_{s}(x)}w\leqslant(r-r_{0})^{\frac{n}{p}}\sup_{B_{r_{0}}(x)}w$

for all $s<r$. Moreover, we choose a point $x_{0}\in\overline{B_{r_{0}}(x)}$
such that

$\sup_{B_{r_{0}}(x)}w=w(x_{0}).$

Notice that in (2.2), the conformal weighted Laplacian $L^{m}_{\phi_{0}}$ has
the same leading term as the classical Laplacian $\Delta_{g_{0}}$. The
difference only occurs on lower order terms. Using a standard interior
estimate for linear elliptic equations in [GT01, Theorem 8.17], we obtain

$\begin{split}s^{\frac{n}{p}}w(x_{0})&\leqslant
C\left(\int_{B_{s}(x_{0})}w^{p}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{1}{p}}\\\
&+Cs^{\frac{n}{p}+2-\frac{n}{q}}\left(\int_{B_{s}(x_{0})}\left|\frac{4(n+m-1)}{(n+m-2)}L^{m}_{\phi_{0}}w\right|^{q}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{1}{q}}\end{split}$ (3.20)

for $s\leqslant\frac{r-r_{0}}{2}$. From this, it follows that

$\begin{split}s^{\frac{n}{p}}w(x_{0})&\leqslant
C\left(\int_{B_{s}(x_{0})}e^{-\phi}{\rm dvol}_{g}\right)^{\frac{1}{p}}\\\
&+Cs^{\frac{n}{p}+2-\frac{n}{q}}\left(\int_{B_{s}(x_{0})}w^{(p-1)q-p}|R^{m}_{\phi}|^{q}e^{-\phi}{\rm
dvol}_{g}\right)^{\frac{1}{q}}\end{split}$ (3.21)

for $s\leqslant\frac{r-r_{0}}{2}$. By definition of $r_{0}$ and $x_{0}$, we
have

$\sup_{B_{\frac{r-r_{0}}{2}}(x_{0})}w\leqslant\sup_{B_{\frac{r+r_{0}}{2}}(x_{0})}w\leqslant
2^{\frac{n}{p}}\sup_{B_{r_{0}}(x_{0})}w=2^{\frac{n}{p}}w(x_{0}).$ (3.22)

Notice that $2-n+\frac{2n}{p}>0$ for $p=\frac{2(n+m)}{n+m-2}<\frac{2n}{n-2}$,
and $s<r<1$. Hence, we can find a fixed constant $K$ such that

$\begin{split}s^{\frac{n}{p}}w(x_{0})&\leqslant
K\left(\int_{B_{s}(x_{0})}e^{-\phi}{\rm dvol}_{g}\right)^{\frac{1}{p}}\\\
&+K(s^{\frac{n}{p}}w(x_{0}))^{(p-1)-\frac{p}{q}}\left(\int_{B_{s}(x_{0})}|R^{m}_{\phi}|^{q}e^{-\phi}{\rm
dvol}_{g}\right)^{\frac{1}{q}}\end{split}$ (3.23)

for $s\leqslant\frac{r-r_{0}}{2}$. We now choose $\eta_{1}>0$ such that

$(2K)^{(p-1)-\frac{p}{q}}\eta_{1}^{\frac{1}{q}}\leqslant\frac{1}{2}.$

We claim that $(\frac{r-r_{0}}{2})^{\frac{n}{p}}w(x_{0})\leqslant 2K$. Indeed,
if $(\frac{r-r_{0}}{2})^{\frac{n}{p}}w(x_{0})\geqslant 2K$, then we may apply
inequality (3.23) with
$s=(\frac{2K}{w(x_{0})})^{\frac{p}{n}}\leqslant\frac{r-r_{0}}{2}$. This yields

$\begin{split}2K\leqslant&K\left(\int_{B_{r}(x_{0})}e^{-\phi}{\rm
dvol}_{g}\right)^{\frac{1}{p}}\\\
&+K(2K)^{(p-1)-\frac{p}{q}}\left(\int_{B_{r}(x_{0})}|R^{m}_{\phi}|^{q}e^{-\phi}{\rm
dvol}_{g}\right)^{\frac{1}{q}},\end{split}$

hence

$2K\leqslant K+(2K)^{(p-1)-\frac{p}{q}}\eta_{1}^{\frac{1}{q}}.$

Using (3.23) with $s=\frac{r-r_{0}}{2}$, we obtain

$\begin{split}(\frac{r-r_{0}}{2})^{\frac{n}{p}}w(x_{0})\leqslant&K\left(\int_{B_{r}(x_{0})}e^{-\phi}{\rm
dvol}_{g}\right)^{\frac{1}{p}}\\\
&+K(2K)^{(p-2)-\frac{p}{q}}\left(\int_{B_{r}(x_{0})}|R^{m}_{\phi}|^{q}e^{-\phi}{\rm
dvol}_{g}\right)^{\frac{1}{q}}\cdot(\frac{r-r_{0}}{2})^{\frac{n}{p}}w(x_{0}).\end{split}$
(3.24)

This implies

$\begin{split}(\frac{r-r_{0}}{2})^{\frac{n}{p}}w(x_{0})\leqslant&K\left(\int_{B_{r}(x_{0})}e^{-\phi}{\rm
dvol}_{g}\right)^{\frac{1}{p}}\\\
&+\frac{1}{2}(2K)^{(p-1)-\frac{p}{q}}\eta_{1}^{\frac{1}{q}}\cdot(\frac{r-r_{0}}{2})^{\frac{n}{p}}w(x_{0}),\end{split}$
(3.25)

hence

$(\frac{r-r_{0}}{2})^{\frac{n}{p}}w(x_{0})\leqslant
2K\left(\int_{B_{r}(x_{0})}e^{-\phi}{\rm dvol}_{g}\right)^{\frac{1}{p}}.$

Thus, we conclude that

$r^{\frac{n}{p}}w(x)\leqslant(r-r_{0})^{\frac{n}{p}}w(x_{0})\leqslant
2^{\frac{n}{p}+1}K\left(\int_{B_{r}(x_{0})}e^{-\phi}{\rm
dvol}_{g}\right)^{\frac{1}{p}}.$

This proves the assertion. ∎

######  Proposition 3.8.

The function $w(t)$ satisfies

$\sup_{M}w(t)\leqslant C$ (3.26)

and

$\inf_{M}w(t)\geqslant c$ (3.27)

for all $t\geqslant 0$. Here, $C$ and $c$ are positive constants independent
of $t$.

###### Proof.

Fix $\frac{n}{2}<q<p<\frac{n+m+2}{2}$. By Corollary 3.2, we have

$\int_{M}|R^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm dvol}_{g(t)}\leqslant C,$

for some constant $C$ independent of $t$. By Proposition 3.6, we can find a
constant $r>0$ independent of $t$ such that

$\int_{B_{r}(x)}e^{-\phi(t)}{\rm dvol}_{g(t)}\leqslant\eta_{0}$

for all $x\in M$ and $t\geqslant 0$. Using Hölder’s inequality, we obtain

$\int_{B_{r}(x)}|R^{m}_{\phi}(t)|^{q}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant\left(\int_{B_{r}(x)}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{p-q}{p}}\left(\int_{B_{r}(x)}|R^{m}_{\phi}(t)|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{q}{p}}.$

Hence, if we choose $\eta_{0}$ sufficiently small, then we have

$\int_{B_{r}(x)}|R^{m}_{\phi}(t)|^{q}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant\eta_{1}$

for all $x\in M$ and $t\geqslant 0$. Here, $\eta_{1}$ is the constant
appearing in Lemma 3.7. Applying Lemma 3.7 at the maximum point of $w(t)$ over
$M$,

$\sup_{M}w(t)\leqslant Cr^{-\frac{n}{p}}\left(\int_{B_{r}(x)}e^{-\phi(t)}{\rm
dvol}_{g(t)}\right)^{\frac{1}{p}}.$ (3.28)

By (2.5), we conclude that $w(t)$ is uniformly bounded above. Hence, if we
define

$P=R^{m}_{\phi_{0}}+\sigma(\sup_{t\geqslant
0}\sup_{M}w(t))^{\frac{4}{n+m-2}},$

then we obtain

$\begin{split}&-\frac{4(n+m-1)}{n+m-2}\Delta_{g_{0},\phi_{0}}w(t)+Pw(t)\\\
&\geqslant-\frac{4(n+m-1)}{n+m-2}\Delta_{g_{0},\phi_{0}}w(t)+R^{m}_{\phi_{0}}w(t)+\sigma
w(t)^{\frac{n+m+2}{n+m-2}}\\\
&=(R^{m}_{\phi}(t)+\sigma)w(t)^{\frac{n+m+2}{n+m-2}}\geqslant 0.\end{split}$
(3.29)

By Lemma 2.8 and (2.5), we can find a positive constant $c$ independent of $t$
such that

$\inf_{M}w(t)(\sup_{M}w(t))^{\frac{n+m+2}{n+m-2}}\geqslant c$

for all $t\geqslant 0$. Since $\sup_{M}w(t)\leqslant C$, the assertion
follows. ∎

######  Proposition 3.9.

Let $0<\alpha<\frac{2m}{n+m}$. Then, the function $w(t)$ satisfies

$|w(x_{1},t_{1})-w(x_{2},t_{2})|\leqslant
C((t_{1}-t_{2})^{\frac{\alpha}{2}}+d(x_{1},x_{2})^{\alpha})$ (3.30)

for all $x_{1},x_{2}\in M$ and $0<t_{1}-t_{2}<1$. Here, $C$ is a positive
constant independent of $t_{1}$ and $t_{2}$.

###### Proof.

Let $\alpha=2-\frac{n}{p}$, where $\frac{n}{2}<p<\frac{n+m}{2}$, $m>0$. Using
(3.2) and Proposition 3.8, we obtain

$\int_{M}\left|-\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}w(t)+R^{m}_{\phi_{0}}w(t)\right|^{p}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\leqslant C$ (3.31)

and

$\int_{M}\left|\frac{\partial}{\partial t}w(t)\right|^{p}e^{-\phi(t)}{\rm
dvol}_{g(t)}\leqslant C$ (3.32)

where $C$ is a positive constant independent of $t$. By the embedding
$W^{2,p}(M)\hookrightarrow C^{0.\alpha}(M)$, the first inequality implies that

$|w(x_{1},t)-w(x_{2},t)|\leqslant Cd(x_{1},x_{2})^{\alpha}$

for all $x_{1},x_{2}\in M$ and all $t\geqslant 0$. Using the second
inequality, we obtain

$\begin{split}&|w(x,t_{1})-w(x,t_{2})|\\\ &\leqslant
C(t_{1}-t_{2})^{-\frac{n}{2}}\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}|w(x,t_{1})-w(x,t_{2})|e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\\\ &\leqslant
C(t_{1}-t_{2})^{-\frac{n}{2}}\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}|w(t_{1})-w(t_{2})|e^{-\phi_{0}}{\rm
dvol}_{g_{0}}+C(t_{1}-t_{2})^{\frac{\alpha}{2}}\\\ &\leqslant
C(t_{1}-t_{2})^{-\frac{n-2}{2}}\sup_{t_{2}\leqslant t\leqslant
t_{1}}\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}|\frac{\partial}{\partial
t}w(t)|e^{-\phi_{0}}{\rm dvol}_{g_{0}}+C(t_{1}-t_{2})^{\frac{\alpha}{2}}\\\
&\leqslant C(t_{1}-t_{2})^{\frac{\alpha}{2}}\sup_{t_{2}\leqslant t\leqslant
t_{1}}\left(\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}|\frac{\partial}{\partial
t}w(t)|^{p}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{1}{p}}+C(t_{1}-t_{2})^{\frac{\alpha}{2}}\\\
&\leqslant C(t_{1}-t_{2})^{\frac{\alpha}{2}}.\end{split}$

for all $x\in M$ and all $t_{1},t_{2}$ satisfying $0<t_{1}-t_{2}<1$. This
proves the assertion. ∎

In light of the foregoing argument at the end of Section 2, we derive uniform
estimates for all higher order derivatives of $w(t)$, $t\geqslant 0$. The
uniqueness of the asymptotic limit follows from Proposition 3.5. This
completes the proof of the main result.

##  4 Proof of the Critical Proposition

Let $\\{t_{i}:i\in\mathbb{N}\\}$ be a sequence of times such that
$t_{i}\to\infty$ as $i\to\infty$. For abbreviation, let $w_{i}=w(t_{i})$. The
normalization condition implies that

$\int_{M}e^{-\phi_{i}}{\rm dvol}_{g_{i}}=1,$

hence

$\int_{M}w_{i}^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm dvol}_{g_{0}}=1$ (4.1)

for all $i\in\mathbb{N}$. Moreover, it follows from Corollary 3.2 that

$\int_{M}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi(t_{i})}{\rm
dvol}_{g(t_{i})}\to 0,$

hence

$\int_{M}\left|\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}w_{i}-R^{m}_{\phi_{0}}w_{i}+r^{m}_{\infty}w_{i}^{\frac{n+m+2}{n+m-2}}\right|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\to 0$ (4.2)

as $i\to\infty$.

Using the standard elliptic theory, we have the following compactness result.

######  Proposition 4.1.

Let $\\{w_{i}:i\in\mathbb{N}\\}$ be a sequence of positive functions
satisfying (4.1) and (4.2). After passing to a subsequence if necessary,
$\\{w_{i}:i\in\mathbb{N}\\}$ converges to a positive smooth function
$w_{\infty}$ satisfying the equation:

$\frac{4(n+m-1)}{n+m-2}\Delta_{g_{0},\phi_{0}}w_{\infty}-R^{m}_{\phi}(0)w_{\infty}+r^{m}_{\infty}w_{\infty}^{\frac{n+m+2}{n+m-2}}=0.$

###### Proof.

Noticing that $\frac{n+m+2}{n+m-2}<\frac{n+2}{n-2}$, the assertion follows
from the standard elliptic theory [Eva10, Section 8, Theorem 3]. ∎

In order to prove the critical proposition, we need the following results.

######  Proposition 4.2.

The exists a sequence of smooth functions $\\{\psi_{a}:a\in\mathbb{N}\\}$ and
a sequence of positive real numbers $\\{\lambda_{a}:a\in\mathbb{N}\\}$ with
the following properties:
(i) For every $a\in\mathbb{N}$, the function $\psi_{a}$ satisfies the equation

$\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\psi_{a}-R^{m}_{\phi_{0}}\psi_{a}+\lambda_{a}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}=0.$
(4.3)

(ii) For all $a,b\in\mathbb{N}$, we have

$\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}\psi_{b}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}=\left\\{\begin{array}[]{lr}0,&a\neq b\\\
1,&a=b\end{array}\right..$ (4.4)

(iii) The span of $\\{\psi_{a}:a\in\mathbb{N}\\}$ is dense in
$L^{2}(M,e^{-\phi_{0}}{\rm dvol}_{g_{0}})$.
(iv) $\lambda_{a}\to\infty$ as $a\to\infty$.

###### Proof.

Consider the linear operator

$\psi\mapsto
w_{\infty}^{-\frac{4}{n+m-2}}\left(\frac{4(n+m-1)}{n+m-2}\Delta_{g_{0},\phi_{0}}\psi-R^{m}_{\phi_{0}}\psi\right).$

This operator is symmetric with respect to the inner product

$(\psi_{1},\psi_{2})\mapsto\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{1}\psi_{2}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}$

on $L^{2}(M,e^{-\phi_{0}}{\rm dvol}_{g_{0}})$. Hence, the assertion follows
from the spectral theorem. ∎

Let $A$ be a finite subset of $\mathbb{N}$ such that
$\lambda_{a}\leqslant\frac{n+m+2}{n+m-2}r^{m}_{\infty}$ for all $a\in A$. We
denote by $\Pi$ the projection operator

$\begin{split}\Pi f&=\sum_{a\notin A}\left(\int_{M}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}\\\ &=f-\sum_{a\in
A}\left(\int_{M}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}\end{split}$ (4.5)

######  Lemma 4.3.

For every $1\leqslant p<\infty$, we can find a constant $C$ such that

$\begin{split}\|f\|_{L^{p}(M)}&\leqslant
C\left\|\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}f-R^{m}_{\phi_{0}}f+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}f\right\|_{L^{p}(M)}\\\
&+C\sup_{a\in
A}\left|\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right|.\end{split}$ (4.6)

###### Proof.

Assume that is not true. By compactness, we can find a function $f\in
L^{p}(M)$ satisfying $\|f\|_{L^{p}(M)}=1$,

$\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}=0$ (4.7)

for all $a\in A$ and

$\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}f-R^{m}_{\phi_{0}}f+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}f=0$
(4.8)

in the sense of distributions. Hence, if we use the function $\psi_{a}$ as a
test function, then we obtain

$(\lambda_{a}-\frac{n+m+2}{n+m-2}r^{m}_{\infty})\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}=0$

for all $a\in\mathbb{N}$. In particular, we have

$\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}=0$

for all $a\notin A$. Thus, we conclude that $f=0$. This is a contradiction. ∎

######  Lemma 4.4.

(i) There exists a constant $C$ such that

$\begin{split}&\|f\|_{L^{\frac{n+m+2}{n+m-2}}(M)}\leqslant C\sup_{a\in
A}\left|\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right|\\\
&+C\left\|\Pi\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}f-R^{m}_{\phi_{0}}f+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}f\right)\right\|_{L^{s}(M)},\end{split}$
(4.9)

where $s=\frac{n(n+m+2)}{n(n+m-2)+2(n+m+2)}$.

(ii) There exists a constant $C$ such that

$\begin{split}\|f\|_{L^{1}(M)}&\leqslant
C\left\|\Pi\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}f-R^{m}_{\phi_{0}}f+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}f\right)\right\|_{L^{1}(M)}\\\
&+C\sup_{a\in
A}\left|\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right|.\end{split}$ (4.10)

###### Proof.

(i) It follows from the embedding $W^{2,s}(M)\hookrightarrow
L^{\frac{n+m+2}{n+m-2}}(M)$ that

$\begin{split}\|f\|_{L^{\frac{n+m+2}{n+m-2}}(M)}&\leqslant
C\|f\|_{L^{s}(M)}\\\
&+C\left\|\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}f-R^{m}_{\phi_{0}}f+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}f\right\|_{L^{s}(M)}.\end{split}$
(4.11)

Using Lemma 4.3, we obtain

$\begin{split}\|f\|_{L^{\frac{n+m+2}{n+m-2}}(M)}&\leqslant C\sup_{a\in
A}\left|\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right|\\\
&+C\left\|\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}f-R^{m}_{\phi_{0}}f+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}f\right\|_{L^{s}(M)}.\end{split}$
(4.12)

By definition of $\Pi$, we have

$\begin{split}\frac{4(n+m-1)}{n+m-2}&\Delta_{\phi_{0}}f-R^{m}_{\phi_{0}}f+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}f\\\
=&\Pi\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}f-R^{m}_{\phi_{0}}f+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}f\right)\\\
&-\sum_{a\in
A}(\lambda_{a}-\frac{n+m+2}{n+m-2}r^{m}_{\infty})\left(\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}.\end{split}$ (4.13)

This implies

$\begin{split}&\left\|\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}f-R^{m}_{\phi_{0}}f+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}f\right\|_{L^{q}(M)}\\\
\leqslant&\left\|\Pi\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}f-R^{m}_{\phi_{0}}f+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}f\right)\right\|_{L^{q}(M)}\\\
&+C\sup_{a\in
A}\left|\int_{M}w_{\infty}^{\frac{4}{n+m-1}}\psi_{a}fe^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right|.\end{split}$ (4.14)

Putting these facts together, the assertion follows.

Similar to (i), (ii) follows from Lemma 4.3 and the definition of $\Pi$. ∎

######  Lemma 4.5.

There exists a positive real number $\xi$ such that for every vector
$z\in\mathbb{R}^{A}$ with $|z|\leqslant\xi$, there exists a smooth function
$\bar{w}_{z}$ such that

$\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}(\bar{w}_{z}-w_{\infty})e^{-\phi_{0}}{\rm
dvol}_{g_{0}}=z_{a}$ (4.15)

for all $a\in A$ and

$\Pi\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z}-R^{m}_{\phi_{0}}\bar{w}_{z}+r^{m}_{\infty}\bar{w}_{z}^{\frac{n+m+2}{n+m-2}}\right)=0.$
(4.16)

Furthermore, the map $z\mapsto\bar{w}_{z}$ is real analytic.

###### Proof.

This is a consequence of the implicit function theorem. ∎

######  Lemma 4.6.

There exists a real number $0<\gamma<1$ such that

$\begin{split}&E(\bar{w}_{z})-E(w_{\infty})\\\ &\leqslant C\sup_{a\in
A}\left|\int_{M}\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z}-R^{m}_{\phi_{0}}\bar{w}_{z}+r^{m}_{\infty}\bar{w}_{z}^{\frac{n+m+2}{n+m-2}}\right)\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right|^{1+\gamma}.\end{split}$ (4.17)

if $z$ is sufficiently small.

###### Proof.

Note that the function $z\mapsto E(\bar{w}_{z})$ is real analytic. According
to results of Lojasiewicz [Sim83, equation (2.4)], there exists a real number
$0<\gamma<1$ such that

$|E(\bar{w}_{z})-E(w_{\infty})|\leqslant\sup_{a\in
A}\left|\frac{\partial}{\partial z_{a}}E(\bar{w}_{z})\right|^{1+\gamma}$
(4.18)

if $z$ is sufficiently small. For convenience, we define the energy functional
$\tilde{E}_{(g_{0},\phi_{0})}(w)$ as

$\begin{split}\tilde{E}_{(g_{0},\phi_{0})}(w)&=\frac{\int_{M}\left(\frac{4(n+m-1)}{n+m-2}L^{m}_{\phi_{0}}w,w\right)e^{-\phi_{0}}{\rm
dvol}_{g_{0}}}{\int_{M}w^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}}\\\ &=\frac{\int_{M}R^{m}_{\phi}e^{-\phi}{\rm dvol}_{g}}{{\rm
Vol}(M^{n},e^{-\phi}{\rm dvol}_{g})}.\end{split}$ (4.19)

The partial derivatives of the function $z\mapsto E(\bar{w}_{z})$ are given by

$\begin{split}\frac{\partial}{\partial
z_{a}}E(\bar{w}_{z})=&-2\frac{\int_{M}\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z}-R^{m}_{\phi_{0}}\bar{w}_{z}+r^{m}_{\infty}\bar{w}_{z}^{\frac{n+m+2}{n+m-2}}\right)\tilde{\psi}_{a,z}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}}{\left(\int_{M}\bar{w}_{z}^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m-2}{n+m}}}\\\
&-2(\tilde{E}(\bar{w}_{z})-r^{m}_{\infty})\frac{\int_{M}\bar{w}_{z}^{\frac{n+m+2}{n+m-2}}\tilde{\psi}_{a,z}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}}{\left(\int_{M}\bar{w}_{z}^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m-2}{n+m}}}\end{split}$ (4.20)

where $\tilde{\psi}_{a,z}=\frac{\partial}{\partial z_{a}}\bar{w}_{z}$ for
$a\in A$. The function $\tilde{\psi}_{a,z}$ satisfies

$\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\tilde{\psi}_{a,z}\psi_{b}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}=\left\\{\begin{array}[]{lr}0,&a\neq b\\\
1,&a=b\end{array}\right.$ (4.21)

for all $a\in A$ and

$\Pi\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\tilde{\psi}_{a,z}-R^{m}_{\phi_{0}}\tilde{\psi}_{a,z}+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}\tilde{\psi}_{a,z}\right)=0.$

Using the identity

$\Pi\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z}-R^{m}_{\phi_{0}}\bar{w}_{z}+r^{m}_{\infty}\bar{w}_{z}^{\frac{n+m+2}{n+m-2}}\right)=0,$

we obtain

$\begin{split}&\frac{\partial}{\partial
z_{a}}E(\bar{w}_{z})=-2\frac{\int_{M}\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z}-R^{m}_{\phi_{0}}\bar{w}_{z}+r^{m}_{\infty}\bar{w}_{z}^{\frac{n+m+2}{n+m-2}}\right){\psi}_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}}{\left(\int_{M}\bar{w}_{z}^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m-2}{n+m}}}\\\ &+2\sum_{b\in
A}\frac{\int_{M}\bar{w}_{z}^{\frac{n+m+2}{n+m-2}}\tilde{\psi}_{a,z}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}}{\left(\int_{M}\bar{w}_{z}^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m-2}{n+m}}}\\\
&\cdot\frac{\left(\int_{M}\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z}-R^{m}_{\phi_{0}}\bar{w}_{z}+r^{m}_{\infty}\bar{w}_{z}^{\frac{n+m+2}{n+m-2}}\right)\psi_{b}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)\left(\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\bar{w}_{z}\psi_{b}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)}{\int_{M}\bar{w}_{z}^{\frac{2(n+m)}{n+m-2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}}\end{split}$ (4.22)

for all $a\in A$. Thus, we obtain that

$\begin{split}\sup_{a\in A}&\left|\frac{\partial}{\partial
z_{a}}E(\bar{w}_{z})\right|\\\ &\leqslant C\sup_{a\in
A}\left|\int_{M}\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z}-R^{m}_{\phi_{0}}\bar{w}_{z}+r^{m}_{\infty}\bar{w}_{z}^{\frac{n+m+2}{n+m-2}}\right){\psi}_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right|.\end{split}$

From this, the assertion follows. ∎

From every $i\in\mathbb{N}$, we can find $\bar{w}_{z_{i}}$ such that
$|z_{i}|\leqslant\xi$ and

$\int_{M}\left(L^{m}_{\phi_{0}}(w_{i}-\bar{w}_{z_{i}}),(w_{i}-\bar{w}_{z_{i}})\right)e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\leqslant\int_{M}\left(L^{m}_{\phi_{0}}(w_{i}-\bar{w}_{z}),(w_{i}-\bar{w}_{z})\right)e^{-\phi_{0}}{\rm
dvol}_{g_{0}}$ (4.23)

for all $|z|\leqslant\xi$. By definition of $\bar{w}_{z_{i}}$, we have

$\int_{M}\left(L^{m}_{\phi_{0}}(w_{i}-\bar{w}_{z_{i}}),(w_{i}-\bar{w}_{z_{i}})\right)e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\leqslant\int_{M}\left(L^{m}_{\phi_{0}}(w_{i}-{w}_{\infty}),(w_{i}-{w}_{\infty})\right)e^{-\phi_{0}}{\rm
dvol}_{g_{0}}.$ (4.24)

By the compactness result in Proposition 4.2, the expression on the right-hand
side tends to $0$ as $i\to\infty$, i.e we have as $i\to\infty$,

$\|w_{i}-\bar{w}_{z_{i}}\|_{H^{1}(M)}\to 0,$ (4.25)

and

$\|\bar{w}_{z_{i}}-w_{\infty}\|_{H^{1}(M)}\to 0.$ (4.26)

######  Lemma 4.7.

The difference $w_{i}-\bar{w}_{z_{i}}$ satisfies

$\|w_{i}-\bar{w}_{z_{i}}\|^{\frac{n+m+2}{n+m-2}}_{L^{\frac{n+m+2}{n+m-2}}(M)}\leqslant
C\left\|w_{i}^{\frac{n+m+2}{n+m-2}}(R^{m}_{\phi_{i}}-r^{m}_{\infty})\right\|^{\frac{n+m+2}{n+m-2}}_{L^{\frac{2(n+m)}{n+m+2}}(M)}+o(1)$
(4.27)

if $i$ is sufficiently large.

###### Proof.

Using the identities

$\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}w_{i}-R^{m}_{\phi_{0}}w_{i}+r^{m}_{\infty}w_{i}^{\frac{n+m+2}{n+m-2}}=-w_{i}^{\frac{n+m+2}{n+m-2}}(R^{m}_{\phi_{i}}-r^{m}_{\infty})$

and

$\Pi\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z}-R^{m}_{\phi_{0}}\bar{w}_{z}+r^{m}_{\infty}\bar{w}_{z}^{\frac{n+m+2}{n+m-2}}\right)=0,$

we obtain

$\displaystyle\Pi$
$\displaystyle\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}(w_{i}-\bar{w}_{z_{i}})-R^{m}_{\phi_{0}}(w_{i}-\bar{w}_{z_{i}})+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}(w_{i}-\bar{w}_{z_{i}})\right)$
$\displaystyle=$
$\displaystyle\Pi\left(-w_{i}^{\frac{n+m+2}{n+m-2}}(R^{m}_{\phi_{i}}-r^{m}_{\infty})-\frac{n+m+2}{n+m-2}r^{m}_{\infty}\left(\bar{w}_{z_{i}}^{\frac{4}{n+m-2}}-w_{\infty}^{\frac{4}{n+m-2}}\right)(w_{i}-\bar{w}_{z_{i}})\right.$
$\displaystyle\left.+r^{m}_{\infty}\left(w_{i}^{\frac{n+m+2}{n+m-2}}-\bar{w}_{z_{i}}^{\frac{n+m+2}{n+m-2}}+\frac{n+m+2}{n+m-2}\bar{w}_{z_{i}}^{\frac{4}{n+m-2}}(w_{i}-\bar{w}_{z_{i}})\right)\right)$

Using the inequality,

$\begin{split}&\|w_{i}-\bar{w}_{z_{i}}\|_{L^{\frac{n+m+2}{n+m-2}}(M)}\leqslant
C\sup_{a\in
A}\left|\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}(w_{i}-\bar{w}_{z_{i}})e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right|\\\
&+C\left\|\Pi\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}(w_{i}-\bar{w}_{z_{i}})-R^{m}_{\phi_{0}}(w_{i}-\bar{w}_{z_{i}})+\frac{n+m+2}{n+m-2}r^{m}_{\infty}w_{\infty}^{\frac{4}{n+m-2}}(w_{i}-\bar{w}_{z_{i}})\right)\right\|_{L^{s}(M)},\end{split}$

we conclude that

$\begin{split}\|w_{i}&-\bar{w}_{z_{i}}\|_{L^{\frac{n+m+2}{n+m-2}}(M)}\\\
\leqslant&C\left\|w_{i}^{\frac{n+m+2}{n+m-2}}(R^{m}_{\phi_{i}}-r^{m}_{\infty})\right\|_{L^{s}(M)}\\\
&+C\left\|(\bar{w}_{z_{i}}^{\frac{4}{n+m-2}}-w_{\infty}^{\frac{4}{n+m-2}})(w_{i}-\bar{w}_{z_{i}})\right\|_{L^{s}(M)}\\\
&+C\left\|w_{i}^{\frac{n+m+2}{n+m-2}}-\bar{w}_{z_{i}}^{\frac{n+m+2}{n+m-2}}+\frac{n+m+2}{n+m-2}\bar{w}_{z_{i}}^{\frac{4}{n+m-2}}(w_{i}-\bar{w}_{z_{i}})\right\|_{L^{s}(M)}\\\
&+C\sup_{a\in
A}\left|\int_{M}w_{\infty}^{\frac{4}{n+m-2}}\psi_{a}(w_{i}-\bar{w}_{z_{i}})e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right|.\end{split}$

The last three terms are dominated by the first, and they tend to $0$ as
$i\to\infty$ by the compact result. Since $s<\frac{2(n+m)}{n+m+2}$, the
assertion follows from Hölder’s inequality. ∎

######  Lemma 4.8.

The difference $w_{i}-\bar{w}_{z_{i}}$ satisfies

$\left\|w_{i}-\bar{w}_{z_{i}}\right\|_{L^{1}(M)}\leqslant
C\left\|w_{i}^{\frac{n+m+2}{n+m-2}}(R^{m}_{\phi_{i}}-r^{m}_{\infty})\right\|_{L^{\frac{2(n+m)}{n+m+2}}(M)}+o(1)$
(4.28)

if $i$ is sufficiently large.

###### Proof.

The proof is totally similar to that of Lemma 4.7. We omit it. ∎

######  Lemma 4.9.

We have

$\begin{split}\sup_{a\in
A}&\left|\left(\int_{M}\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z_{i}}-R^{m}_{\phi_{0}}\bar{w}_{z_{i}}+r^{m}_{\infty}\bar{w}_{z_{i}}^{\frac{n+m+2}{n+m-2}}\right)\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)\right|\\\ &\leqslant
C\left(\int_{M}w(t_{i})^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}}+o(1)\end{split}$ (4.29)

if $i$ is sufficiently large.

###### Proof.

Integration by parts yields

$\begin{split}\int_{M}&\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z_{i}}-R^{m}_{\phi_{0}}\bar{w}_{z_{i}}+r^{m}_{\infty}\bar{w}_{z_{i}}^{\frac{n+m+2}{n+m-2}}\right)\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\\\
=&\int_{M}\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}{w}_{i}-R^{m}_{\phi_{0}}{w}_{i}+r^{m}_{\infty}{w}_{i}^{\frac{n+m+2}{n+m-2}}\right)\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\\\
&+\lambda_{a}\int_{M}w_{\infty}^{\frac{4}{n+m-2}}(w_{i}-\bar{w}_{z_{i}})\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}-r^{m}_{\infty}\int_{M}\left(w_{i}^{\frac{n+m+2}{n+m-2}}-\bar{w}_{z_{i}}^{\frac{n+m+2}{n+m-2}}\right)\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}.\end{split}$

Using the identities

$\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}w_{i}-R^{m}_{\phi_{0}}w_{i}+r^{m}_{\infty}w_{i}^{\frac{n+m+2}{n+m-2}}=-w_{i}^{\frac{n+m+2}{n+m-2}}(R^{m}_{\phi_{i}}-r^{m}_{\infty}),$

we obtain

$\begin{split}\int_{M}&\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z_{i}}-R^{m}_{\phi_{0}}\bar{w}_{z_{i}}+r^{m}_{\infty}\bar{w}_{z_{i}}^{\frac{n+m+2}{n+m-2}}\right)\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\\\
=&-\int_{M}w_{i}^{\frac{n+m+2}{n+m-2}}(R^{m}_{\phi_{i}}-r^{m}_{\infty})\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\\\
&+\lambda_{a}\int_{M}w_{\infty}^{\frac{4}{n+m-2}}(w_{i}-\bar{w}_{z_{i}})\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}-r^{m}_{\infty}\int_{M}\left(w_{i}^{\frac{n+m+2}{n+m-2}}-\bar{w}_{z_{i}}^{\frac{n+m+2}{n+m-2}}\right)\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}.\end{split}$

Using the pointwise estimate

$\left|w_{i}^{\frac{n+m+2}{n+m-2}}-\bar{w}_{z_{i}}^{\frac{n+m+2}{n+m-2}}\right|\leqslant
C\bar{w}_{z_{i}}^{\frac{4}{n+m-2}}\left|w_{i}-\bar{w}_{z_{i}}\right|+C\left|w_{i}-\bar{w}_{z_{i}}\right|^{\frac{n+m+2}{n+m-2}},$

we conclude that

$\begin{split}\sup_{a\in
A}&\left|\left(\int_{M}\left(\frac{4(n+m-1)}{n+m-2}\Delta_{\phi_{0}}\bar{w}_{z_{i}}-R^{m}_{\phi_{0}}\bar{w}_{z_{i}}+r^{m}_{\infty}\bar{w}_{z_{i}}^{\frac{n+m+2}{n+m-2}}\right)\psi_{a}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)\right|\\\ &\leqslant
C\left(\int_{M}w(t_{i})^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}}\\\
&+C\left\|w_{i}-\bar{w}_{z_{i}}\right\|_{L^{1}(M)}+C\left\|w_{i}-\bar{w}_{z_{i}}\right\|^{\frac{n+m+2}{n+m-2}}_{L^{\frac{n+3}{n-1}}(M)}.\end{split}$

The assertion follows from Lemma 4.7 and 4.8. ∎

Combining Lemma 4.6 and Lemma 4.9, we immediately obtain that

######  Proposition 4.10.

$E(\bar{w}_{z})$ satisfies the estimate

$\begin{split}E(\bar{w}_{z})&-E(w_{\infty})\\\ &\leqslant
C\left(\int_{M}w(t_{i})^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}(1+\gamma)}+o(1)\end{split}$ (4.30)

if $i$ is sufficiently large.

Now, we can prove our critical proposition.

###### Proof of Proposition 3.3:.

By definition of $r^{m}_{\phi}$ and the reservation of the volume, we have

$r^{m}_{\phi}(t_{i})-r^{m}_{\infty}=E(w_{i})-r^{m}_{\infty}.$

Besides, by the above estimate, we conclude that

$r^{m}_{\phi}(t_{i})-r^{m}_{\infty}\leqslant
C\left(\int_{M}w(t_{i})^{\frac{2(n+m)}{n+m-2}}|R^{m}_{\phi}(t_{i})-r^{m}_{\infty}|^{\frac{2(n+m)}{n+m+2}}e^{-\phi_{0}}{\rm
dvol}_{g_{0}}\right)^{\frac{n+m+2}{2(n+m)}(1+\gamma)}.$

This completes the proof. ∎

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