# Geodesic Equations on asymptotically locally Euclidean Kähler manifolds

Qi Yao

###### Abstract.

In this paper, We solve the geodesic equation in the space of Kähler metrics
under the setting of asymptotically locally Euclidean (ALE) Kähler manifolds
and establish global $\mathcal{C}^{1,1}$ regularity of the solution. The
solution of the geodesic equation is then related to the uniqueness of scalar-
flat ALE metrics. To this end, we study the asymptotic behavior of
$\varepsilon$-geodesics at spatial infinity. We will prove the convexity of
Mabuchi $K$ energy along $\varepsilon$-geodesics under the assumption that the
Ricci curvature of a reference ALE Kähler metric is non-positive. However, by
testing the Ricci curvature of ALE Kähler metrics, we find that on the line
bundle $\mathcal{O}(-k)$ over $\mathbb{C}\mathbb{P}^{n-1}$ with $n\geq 2$ and
$k\neq n$, all ALE Kähler metrics cannot have non-positive (or non-negative)
Ricci curvature.

##  1\. Introduction

In the paper, we study the geodesic equation in the setting of ALE Kähler
cases, assuming relatively weak fall-off conditions. Let $(X,J,g)$ be a
complete non-compact Kähler manifold of complex dimension $n$ ($n\geq 2$), we
say $(X,J,g)$ is ALE if there is a compact subset $K\subseteq X$ such that
$\psi:X\setminus K\rightarrow(\mathbb{C}^{n}\setminus B_{R})/\Gamma$ is a
diffeomorphism, where $B_{R}$ is a closed ball in $\mathbb{C}^{n}$ with radius
$R$ and $\Gamma$ is a finite subgroup of $U(n)$ (any ALE Kähler manifold has
only one end according to [19, Proposition 1.5, 3.2]) and the metric $g$
satisfies the following condition on the end $X\setminus K$:

  1. $\bullet$

The metric $g$ is asymptotic to the Euclidean metric $\delta_{ij}$ at the end
with decay rate $-\tau$ for some $\tau>n-1$, i.e., for $i=0,1,\ldots,k$,

$\displaystyle
g_{ij}=\delta_{ij}+O(r^{-\tau}),\qquad|\nabla^{i}((\psi^{-1})^{*}g)|_{g_{0}}=O(r^{-\tau-i}).$
(1.1)

The fall-off condition $\tau>n-1$ is the weakest decay rate to make the ADM
mass coordinate-invariant in general, referring to Bartnik [4] and Chrus̀ciel
[11].

One of the difficulties in building up a general theory of scalar-flat Kähler
metrics in the ALE setting is that the decay rate of such metrics to their
asymptotic models is not good enough compared to the Ricci-flat case. For
instance, consider the family of scalar-flat Kähler metric constructed on
$\mathcal{O}_{\mathbb{C}\mathbb{P}^{1}}(-k)$ by LeBrun [23],

$\displaystyle
g=\frac{ds^{2}}{1+A/s^{2}+B/s^{4}}+s^{2}\Big{[}\sigma_{1}^{2}+\sigma_{2}^{2}+\Big{(}1+\frac{A}{s^{2}}+\frac{B}{s^{4}}\Big{)}\sigma_{3}^{2}\Big{]},$

where $A$, $B$ are constants, $\sigma_{1}$, $\sigma_{2}$, $\sigma_{3}$ are
three invariant vector fields on $3$-sphere and $s$ is a radial function on
$\mathcal{O}(-k)$. It can be checked that $g-g_{euc}=O(r^{-2})$, where $r$
denotes the geodesic distance from a fixed basepoint, indicating that the
Kähler potential function should be of $\log$ growth. In Arezzo-Pacard [2,
Lemma 7.2], an expansion theorem is proved for scalar-flat Kähler metrics in
the complement of $B_{\Gamma}=\\{z\in\mathbb{C}^{n}/\Gamma:|z|\leq 1\\}$ in
$\mathbb{C}^{n}/\Gamma$, where $\Gamma$ is a finite subgroup of $U(n)$,
assuming that the $dd^{c}$-lemma holds in this situation. In [30], the author
proved a $dd^{c}$ lemma and an expansion theorem under the setting of
asymptotically conical (AC) Kähler manifolds. Here, we only need a theorem of
weaker version under the setting of ALE Kähler manifolds.

######  Theorem 1.1.

(Yao 2022) Let $(X,J)$ be an ALE Kähler manifold asymptotic to
$\mathbb{C}^{n}/\Gamma$. Let $\omega_{1}$, $\omega_{2}$ be Kähler forms in the
same Kähler class of $(X,J)$ with the corresponding metrics satisfying (1.1)
and such that the scalar curvatures of $\omega_{1}$ and $\omega_{2}$ are
equal, $R_{1}\equiv R_{2}$. Then

$\displaystyle\omega_{2}=\omega_{1}+dd^{c}\varphi,\quad\text{ with the
potential }\varphi\in\mathcal{C}^{\infty}_{2-2\tilde{\tau}}$ (1.2)

for some $\tilde{\tau}>n-1$ depending on $(n,\tau)$.

Let $\omega$ be the corresponding Kähler form of $g$. According to Theorem
1.1, given two Kähler forms $\omega_{1},\omega_{2}\in[\omega]$, if the
corresponding ALE Kähler metrics $g_{1},g_{2}$ satisfy the decay condition
(1.1) and that the scalar curvatures of $g_{1}$ and $g_{2}$ are identically
equal, $R(g_{1})\equiv R(g_{2})$, then $\omega_{1}-\omega_{2}=dd^{c}f$ and $f$
decays at infinity with higher rate $-\gamma$, with $\gamma=2\tilde{\tau}-2$,
for some $\tilde{\tau}>n-1$. Hence, for the prescribed scalar curvature
problem, we consider the following restricted weighted Kähler potential space,

$\displaystyle\mathcal{H}_{-\gamma}(\omega)=\\{\varphi\in\hat{\mathcal{C}}^{\infty}_{-\gamma}:\omega_{\varphi}=\omega+dd^{c}\varphi>0\\}\quad(\gamma>2n-4\geq
0),$

where the class of functions, $\hat{\mathcal{C}}^{\infty}_{s}$, is defined as
follows

$\displaystyle\mathcal{C}^{\infty}_{s}$
$\displaystyle=\\{f\in\mathcal{C}^{\infty}(X):|\nabla_{g_{0}}^{j}f|_{g_{0}}=O(r^{s-j})\;\,\text{for
all}\;\,j\geq 0\\},$ $\displaystyle\hat{\mathcal{C}}^{\infty}_{s}$
$\displaystyle=\\{\hat{f}\in\mathcal{C}^{\infty}(X):\hat{f}=f+c,\text{ for
}f\in\mathcal{C}^{\infty}_{s}\text{ and }c\text{ is a constant}\\}.$

Define $\omega_{0}=\omega+dd^{c}\psi_{0}$, $\omega_{1}=\omega+dd^{c}\psi_{1}$,
for any two boundary data $\psi_{0,1}\in\mathcal{H}_{-\gamma}(\omega)$. Also
introduce the linear reference path $\psi(t)=(1-t)\psi_{0}+t\psi_{1}$ in
$\mathcal{H}_{-\gamma}(\omega)$. Another path $\varphi(t)$ in
$\mathcal{H}_{-\gamma}(\omega)$ with the same endpoints $\psi_{0},\psi_{1}$ is
called a geodesic in $\mathcal{H}_{-\gamma}(\omega)$ if

$\displaystyle\ddot{\varphi}(t)-\frac{1}{2}|\nabla_{\omega_{\varphi(t)}}\dot{\varphi}(t)|_{\omega_{\varphi(t)}}^{2}=0.$
(1.3)

As observed by Donaldson [15] and Semmes [26], the geodesic equation is
equivalent to a homogeneous complex Monge-Ampère equation in the product space
$X\times\Sigma$, where $\Sigma\cong[0,1]\times S^{1}$ can be embedded as an
annulus in $\mathbb{C}$. Notice that any path $\varphi(t)$ of functions on $X$
can be viewed as a function $\Phi$ on $X\times\Sigma$ via
$\Phi(\cdot,t,e^{is})=\varphi(t)$. Let $\Omega_{\Phi}=p^{*}\omega+dd^{c}\Phi$,
where $p$ is the projection from $X\times\Sigma$ to $X$ and $dd^{c}\Phi$ is
computed on $X\times\Sigma$. Then the equation (1.3) can be rewritten as
follows:

$\displaystyle\Omega_{\Phi}^{n+1}=0,$ (1.4) $\displaystyle\Omega_{\Phi}\geq
0,$ (1.5) $\displaystyle\Phi|_{t=0,1}=\psi_{0,1}.$ (1.6)

In [15], Donaldson proposed a program to attack the existence and uniqueness
problems regarding canonical metrics by studying the geometric structure of
the potential space $\mathcal{H}$, where the geodesic equation plays a central
role. In the cases of compact Kähler manifolds, Chen [9] showed that for any
$\psi_{0}$, $\psi_{1}\in\mathcal{H}$, the geodesic equation has a unique
solution up to $dd^{c}$-regularity. Blocki [6] and He [18] built up direct
calculations to prove the gradient estimate and Laplacian estimate. The full
$\mathcal{C}^{1,1}$ estimate was proved by Chu-Tosatti-Weinkove in [12]. In
the other direction, Lempert-Vivas [24] and Darvas-Lempert [14] constructed
counter-examples to assert that $dd^{c}\Psi$ is not continuous in general,
hence the $\mathcal{C}^{1,1}$ regularity is optimal in general. In [3], Auvray
generalized the $dd^{c}$-regularity to singular cases (precisely, there exist
cusp singularities along simple normal crossings). The main theorem of
sections 2-5 is to generalize the full $\mathcal{C}^{1,1}$ estimates to ALE
Kähler manifolds.

######  Theorem A.

Let $X$ be an ALE Kähler manifold and
$\psi_{0},\psi_{1}\in\mathcal{H}_{-\gamma}(\omega)$ $(\gamma>0)$. Then
$\psi_{0}$ and $\psi_{1}$ can be connected by a $\mathcal{C}^{1,1}$ geodesic
$\Phi$ solving (1.4), (1.5), (1.6). Moreover, there is a uniform constant $C$
depending only on $\|\psi_{0}\|_{\mathcal{C}^{1,1}(X,\omega)}$,
$\|\psi_{1}\|_{\mathcal{C}^{1,1}(X,\omega)}$ and on the geometry of
$(X,\omega)$ such that

$\displaystyle\sup_{X\times\Sigma}\big{(}|\Phi|+|\nabla_{\Theta_{\Psi}}\Phi|_{\Theta_{\Psi}}+|{\nabla}^{2}_{\Theta_{\Psi}}\Phi|_{\Theta_{\Psi}}\big{)}\leq
C.$ (1.7)

Here, $\Theta_{\Psi}$ is a Kähler form on $X\times\Sigma$ given by
$\Theta_{\Psi}=\Theta+dd^{c}\Psi$ with
$\Psi(\cdot,t,e^{is})=\psi(t)=(1-t)\psi_{0}+t\psi_{1}$ the linear path
introduced above, and with $\Theta=p^{*}\omega+Add^{c}t(t-1)$, where $A>0$ is
fixed depending only on $\|\psi_{0}\|_{\mathcal{C}^{1,1}(X,\omega)}$,
$\|\psi_{1}\|_{\mathcal{C}^{1,1}(X,\omega)}$ such that $\Theta_{\Psi}>0$.

Then, we relate the solution of the geodesic equation to the uniqueness of
scalar-flat ALE Kähler metrics in each Kähler class. The main idea is to
follow the framework of Chen [9] in the compact case, under the assumption
that the Ricci curvature of the reference metric is non-positive. This was
extended to the non-compact case with Poincaré cusp ends by Auvray [3]. In the
ALE case, it is first necessary to prove sufficient decay at infinity of
solutions to the $\varepsilon$-geodesic equation.

In Section 6, we discuss the asymptotic behavior of $\varepsilon$-geodesics.
Given any two functions

$\displaystyle\psi_{0},\psi_{1}\in\mathcal{H}_{-\gamma}(\omega)=\\{\varphi\in\hat{\mathcal{C}}^{\infty}_{-\gamma}:\omega_{\varphi}=\omega+dd^{c}\varphi>0\\}\quad(\gamma>0),$

we set $\psi(t)=(1-t)\psi_{0}+t\psi_{1}$ and let $\Psi$ denote the
corresponding function on $X\times\Sigma$. We fix $A$ large depending on
$\|\psi_{0}\|_{\mathcal{C}^{1,1}(X,\omega)}$,
$\|\psi_{1}\|_{\mathcal{C}^{1,1}(X,\omega)}$ such that
$\Theta_{\Psi}:=\Theta+dd^{c}\Psi$ is positive on $X\times\Sigma$, where
$\Theta:=p^{*}\omega+Add^{c}t(t-1)$ with $p:X\times\Sigma\to X$ the
projection. Then, we introduce the following $\varepsilon$-geodesic equations

$\displaystyle(E_{\varepsilon})\qquad\begin{cases}(\Theta+dd^{c}\Phi_{\varepsilon})^{n+1}=\upsilon(\varepsilon)\Theta_{\Psi}^{n+1},\quad&\text{
in }X\times\Sigma,\\\ \Theta+dd^{c}\Phi_{\varepsilon}>0,&\text{ in
}X\times\Sigma,\\\ \Phi_{\varepsilon}|_{t=0,1}=\phi_{0,1},&\text{ on
}X\times\partial\Sigma,\end{cases}$

where $\upsilon(\varepsilon)$ is a smooth nonnegative function defined in
$X\times\Sigma\times[0,1]$ satisfies the following conditions

$\displaystyle\begin{split}&\upsilon(0)\equiv 0,\quad\upsilon(1)\equiv 1;\\\
&\upsilon(\varepsilon)>0,\qquad\qquad\qquad\qquad\ \text{ for
}\varepsilon\in(0,1];\\\
&C^{-1}\varepsilon\leq\upsilon(\varepsilon)\leq\min(C\varepsilon,1),\quad\text{for
}\varepsilon\in[0,1];\\\ &|\nabla^{k}\upsilon(y,\varepsilon)|\leq C\varepsilon
r(y)^{-\varsigma-k},\quad\text{ for }(y,\varepsilon)\in
X\times\Sigma\times[0,1],\quad k\geq 1\end{split}$ (1.8)

where $\varsigma$ is an real number with $\varsigma\geq\gamma$. In particular,
we are interested in the following case. By taking

$\displaystyle\upsilon(\varepsilon)=\varepsilon((1-\chi(\varepsilon))f+\chi(\varepsilon)),$
(1.9)

where $\chi$ is a smooth increasing function in $[0,1]$ equal to $0$ (resp.
$1$) in a neighborhood of $0$ (resp. $1$) and $f$ is defined as follows

$\displaystyle
f=A^{-1}\frac{\Theta^{n+1}}{\Theta_{\psi}^{n+1}}\in\mathcal{C}^{\infty}(X\times\Sigma),$
(1.10)

and in this case, $|\nabla^{k}f|\leq Cr^{-\gamma-2-k}$, $\varsigma=\gamma+2$.
By taking $\varepsilon$ to be small enough, $(E_{\varepsilon})$ can be written
as

$\displaystyle\Big{(}\ddot{\varphi}-\frac{1}{2}|\nabla_{\omega_{\varphi}}\dot{\varphi}|_{\omega_{\varphi}}^{2}\Big{)}\omega_{\varphi}^{n}=\varepsilon\omega^{n}.$

Due to the positivity of the right hand side of $(E_{\varepsilon})$, it is
well known that for every $\varepsilon\in(0,1]$ there exists a solution
$\Phi_{\varepsilon}\in\bigcap_{k,\alpha}\mathcal{C}^{k,\alpha}$. We now prove:

######  Theorem B.

Let $\Phi_{\varepsilon}$ be the $\varepsilon$-geodesic constructed above.
Then, there exists a constant $C(k,\varepsilon^{-1})$ depending on $k\geq 1$
and on an upper bound for $\varepsilon^{-1}$ such that

$\displaystyle\left(|\nabla^{k}_{X,\omega}\Phi_{\varepsilon}|_{\omega}+|\nabla^{k}_{X,\omega}\dot{\Phi}_{\varepsilon}|_{\omega}+|\nabla^{k}_{X,\omega}\ddot{\Phi}_{\varepsilon}|_{\omega}\right)\leq
C(k,\varepsilon^{-1})r^{-\gamma-k}\quad\text{for all}\ k\geq 1,$

where $\nabla_{X,\omega}$ denotes the Levi-Civita connection of the ALE Kähler
metric $\omega$ on $X$, acting as a differential operator in the $X$
directions on $X\times\Sigma$. And

$\displaystyle|\Phi_{\varepsilon}-c(t)|\leq C(\varepsilon^{-1})r^{-\gamma},$

where $c(t)$ is a function only depending on $t$. Hence, for any two
potentials $\psi_{0}$, $\psi_{1}$ in $\mathcal{H}_{-\gamma}(\omega)$, there
exist $\varepsilon$-geodesics in $\mathcal{H}_{-\gamma}(\omega)$ connecting
$\psi_{0}$ and $\psi_{1}$.

In section 6, we prove a stronger statement. Let
$\varphi_{\varepsilon}=\Phi_{\varepsilon}-\Psi$, then
$\varphi_{\varepsilon}\in\mathcal{H}_{\max\\{-2\gamma-2,-\varsigma\\}}$ due to
the fact that $\Psi$ was chosen to be linear in $t$ (see section 6 for
details).

Hence, while we still cannot define the Mabuchi $K$-energy along geodesics,
the Mabuchi $K$-energy is now actually well-defined along
$\varepsilon$-geodesics assuming $\gamma=2\tilde{\tau}-2>2n-4$.

In Section 7, the second derivative of the Mabuchi $K$-energy will be
calculated. Throughout section 7, we assume $\gamma=2\tilde{\tau}-2>2n-4$
(Here it turns out that if $\psi_{0}$, $\psi_{1}$ are only in
$\mathcal{H}_{4-2n}(\omega)$, there would be boundary terms at infinity
breaking the positivity of the second derivative. This is a new phenomenon
compared to Chen [9] and Auvray [3]). However, under the assumption that the
Ricci curvature of some reference ALE Kähler metric, $\omega$, is non-
positive, we can then prove the convexity of Mabuchi $K$-energy:

######  Theorem C.

Assume that $\omega$ is an ALE Kähler metric on $X$ such that the Ricci
curvature of $\omega$ is non-positive, $\textup{Ric}(\omega)\leq 0$. Then,
along each $\varepsilon$-geodesic in $\mathcal{H}_{-\gamma}(\omega)$ with
$\gamma>2n-4$, $\varphi(t)$, the Mabuchi $K$-energy is convex.

A quick corollary of Theorem C is that assuming $\textup{Ric}(\omega)\leq 0$,
the scalar-flat Kähler metric, if it exists, is unique in
$\mathcal{H}_{-\gamma}(\omega)$. However, if there exists a scalar-flat Kähler
metric $\omega_{0}$ in $\mathcal{H}_{-\gamma}(\omega)$, the condition,
$\textup{Ric}(\omega)\leq 0$ implies $\textup{Ric}(\omega)=0$. Hence, the
uniqueness of scalar-flat ALE metric can be reduced to the uniqueness result
of Ricci-flat ALE Kähle metric, which can be found in reference [20, 28, 13].
The point is that $\omega_{0}=\omega+O(r^{-\gamma-2})$ implies by definition
that the ADM masses of $\omega$ and $\omega_{0}$ are equal,
$\mathfrak{m}(\omega)=\mathfrak{m}(\omega_{0})$. According to mass formula by
Hein-LeBrun [19], it follows that $\int R(\omega)=\int R(\omega_{0})=0$. The
assumption that $\textup{Ric}(\omega)\leq 0$ implies that
$\textup{Ric}(\omega)=0$ (see Remark 7.4 for details). In fact, in Section 8,
we will prove that many ALE Kähler manifolds do not admit any ALE Kähler
metrics with $\textup{Ric}\leq 0$ (or $\textup{Ric}\geq 0$) at all:

######  Theorem D.

Let $\mathcal{O}(-k)$ be the standard negative line bundle over
$\mathbb{C}\mathbb{P}^{n-1}$ with $n\geq 2$, $k\neq n$, and let $\omega$ be an
ALE Kähler metric on $\mathcal{O}(-k)$ with decay rate $-\tau$, $\tau>0$.
Then, the Ricci form of $\omega$, is of mixed type, i.e., neither
$\textup{Ric}(\omega)\geq 0$ nor $\textup{Ric}(\omega)\leq 0$ is true.

In Riemannian geometry, AE metrics of negative Ricci curvature are well-known
to exist in $\mathbb{R}^{n}$ by explicit construction in Lohkamp [25]. Theorem
D gives a negative answer to this question in the setting of ALE Kähler
metrics.

An interesting question in this context is to ask whether some version of the
Nonexistence Theorem D holds in general ALE Kähler manifolds or even AC Kähler
manifolds.

######  Question.

Is it true in any ALE Kähler manifold that the Ricci curvature form of an ALE
Kähler metric can only be identically zero or of mixed type?

This paper is a part of the Ph.D. thesis of the author. The author would like
to express his gratitude to Professor Hans-Joachim Hein and Professor Bianca
Santoro for suggesting the problem, and for constant support, many helpful
comments, as well as much enlightening conversation. The author is also
thankful to professor Gustav Holzegel for providing financial support via his
Alexander von Humboldt Professorship during the last semester at University of
Münster. The whole project is Funded by the DFG under Germany’s Excellence
Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure,
and by the CRC 1442, Geometry: Deformations and Rigidity, of the DFG.

##  2\. $\varepsilon$-geodesic equations and openness

Recall that $\varepsilon$ geodesic equations can be written as follows,

$\displaystyle(E_{\varepsilon})\qquad\begin{cases}(\Theta+dd^{c}\widetilde{\Phi})^{n+1}=\upsilon(\varepsilon)(\Theta+dd^{c}\Psi)^{n+1},\quad&\text{
in }X\times\Sigma,\\\
\lambda\Theta<\Theta+dd^{c}\widetilde{\Phi}<\Lambda\Theta,&\text{ in
}X\times\Sigma,\\\ \widetilde{\Phi}|_{t=0,1}=\psi_{0,1},&\text{ on
}X\times\partial\Sigma,\end{cases}$

where $\varepsilon\in(0,1]$ and $0<\lambda<\Lambda$ are constants depending on
$\varepsilon$. The family of equations $(E_{\varepsilon})$ is called the
$\varepsilon$-geodesic equations. The idea to solve the equation
$(E_{\varepsilon})$ is the following. Firstly, we apply the continuity method
to show that there exists a solution of $(E_{\varepsilon})$ in
$\mathcal{C}^{k,\alpha}$. In particular, consider the family of equations
$(E_{s})$, $s\in[\varepsilon,1]$. There is a trivial solution at $(E_{1})$.
Then, we shall prove the openness and closedness of $(E_{s})$ in certain
regularity. In the current section, we deal with the openness of $(E_{s})$.

Assuming that there exists a solution of $(E_{s_{0}})$ in
$\mathcal{C}^{k,\alpha}$ for some $s_{0}\in[\varepsilon,1]$, we will show in
this subsection that $(E_{s})$ can be solved for all $s$ in a small open
neighborhood of $s_{0}$. For simplicity, we write
$\Theta_{\Psi}=\Theta+dd^{c}\Psi$ as in Theorem A and
$\widetilde{\varphi}=\widetilde{\Phi}-\Psi$. Then, the equation
$(E_{\varepsilon})$ can be written as,
$(\Theta_{\Psi}+dd^{c}\widetilde{\varphi})^{n+1}=\varepsilon\Theta_{\Psi}^{n+1}$
in $X\times\Sigma$, with the boundary condition $\widetilde{\varphi}=0$ on
$X\times\partial\Sigma$. Then, the Monge-Ampère operator is defined to be

$\displaystyle\mathcal{M}(\chi)=\frac{(\Theta_{\Psi}+dd^{c}\chi)^{n+1}}{\Theta_{\Psi}^{n+1}}.$

Let $\widetilde{\varphi}$ be a solution of $(E_{s_{0}})$ for some
$s_{0}\in[\varepsilon,1]$. By assumption, $\widetilde{\varphi}$ is
$\Theta_{\Psi}$-plurisubharmonic satisfying
$c\Theta\leq\Theta_{\Psi}+dd^{c}\widetilde{\varphi}\leq C\Theta$. Then, the
linearization of Monge-Ampère operator at $\widetilde{\varphi}$ is uniformly
elliptic, and is given by

$\displaystyle\mathcal{L}_{\widetilde{\varphi}}(\chi)=\big{(}\Delta_{{\widetilde{\varphi}}}\chi\big{)}\cdot\frac{(\Theta_{\Psi}+dd^{c}\widetilde{\varphi})^{n+1}}{\Theta_{\Psi}^{n+1}}=s_{0}\Delta_{\widetilde{\varphi}}\chi,$

where $\Delta_{\widetilde{\varphi}}$ represents the Laplacian with respect to
$\Theta_{\Psi}+dd^{c}\widetilde{\varphi}$. Let $(\mathcal{C}^{k,\alpha})_{0}$
be the functions in $\mathcal{C}^{k,\alpha}$ vanishing on the boundary
$X\times\partial\Sigma$. Then, we have the following property of
$\mathcal{L}_{\widetilde{\varphi}}$, from which the desired openness is clear
by the implicit function theorem.

######  Proposition 2.1.

Let $\widetilde{\varphi}$ be the solution of $(E_{s_{0}})$, then the
linearized operator
$\mathcal{L}_{\widetilde{\varphi}}:({\mathcal{C}}^{k,\alpha})_{0}\rightarrow{\mathcal{C}}^{k-2,\alpha}$
is an isomorphism for all integers $k\geq 2$ and $\alpha\in(0,1)$.

###### Proof.

Let us first prove the surjectivity. Fixing $f\in{\mathcal{C}}^{k-2,\alpha}$,
the exhaustion argument will be applied to solve the equation
$\mathcal{L}_{\widetilde{\varphi}}u=f$. Take an exhaustive sequence of pre-
compact sets, $\Omega_{k}\subseteq X\times\Sigma$, with smooth boundary. In
particular, by taking a sequence of subsets, $B_{r_{k}}\times\Sigma$ where
$B_{r_{k}}=\\{x\in X:r(x)\leq r_{k}\\}$, and smoothing the corners, we can
obtain the exhaustive sequence $\\{\Omega_{k}\\}$. Then, we can solve the
following Dirichlet problems,

$\displaystyle(L_{k})\quad\begin{cases}\mathcal{L}_{\widetilde{\varphi}}u_{k}=f\quad&\text{
in }\Omega_{k},\\\ u_{k}=0&\text{ on }\partial\Omega_{k},\end{cases}$

where $f\in{\mathcal{C}}^{k-2,\alpha}$. The existence of the solution of
$(L_{k})$ is a classic result of the Dirichlet problem on compact Riemannian
manifolds with boundary. The key to complete the proof is to give the uniform
estimates of $u_{k}$. The main idea to show the $\mathcal{C}^{0}$ uniform
estimates is to construct barrier functions. Consider the function $At(1-t)$.
The fact that
$\lambda\Theta\leq\Theta_{\Psi}+dd^{c}\widetilde{\varphi}\leq\Lambda\Theta$
implies $\Delta_{\widetilde{\varphi}}At(1-t)\leq-\lambda A$. If we suppose
that $\|f\|_{L^{\infty}}\leq C_{0}$ and take $A=C_{0}/\lambda$, then we have
$\Delta_{\widetilde{\varphi}}At(1-t)\leq f=\Delta_{\widetilde{\varphi}}u_{k}$.
Combining with the fact that $At(1-t)\geq 0$ on the boundary
$\partial\Omega_{k}$, the maximum principle implies that,

$\displaystyle\|u_{k}\|_{L^{\infty}}\leq\frac{C_{0}}{\lambda}t(1-t)\leq\frac{C_{0}}{4\lambda}.$
(2.1)

The uniform $\mathcal{C}^{k,\alpha}$ estimates follows directly from the
standard Schauder estimates. Precisely, for interior points $p\in\Omega_{k}$
away from the boundary, we pick a pair of balls centered at $p$,
$B_{\frac{1}{4}}(p)\subseteq B_{\frac{1}{2}}(p)\subset\Omega_{k}$. Then, the
interior Schauder estimates implies that
$\|u_{k}\|_{k,\alpha;B_{\frac{1}{16}}(p)}\leq
C(\|u_{k}\|_{L^{\infty}(B_{\frac{1}{8}}(p))}+\|f\|_{k-2,\alpha;B_{\frac{1}{8}}(p)})$.
If $p\in\Omega_{k}$ is close to the boundary, we can apply the boundary
Schauder estimate. After straightening the boundary in case the boundary
portion on $\partial\Omega_{k}$ is not flat, we can pick half balls, $p\in
B^{+}_{\frac{1}{4}}(q)\subseteq B^{+}_{\frac{1}{2}}(q)$ for some
$q\in\partial\Omega_{k}$. Together with the interior estimates, we have

$\displaystyle\|u_{k}\|_{k,\alpha;\Omega_{k}}\leq
C(\|u_{k}\|_{L^{\infty}(X\times\Sigma)}+\|f\|_{k-2,\alpha;X\times\Sigma}),$
(2.2)

where $C$ depends only on $n,k,\alpha,\lambda,\Lambda$. After passing to a
subsequence, we conclude that the limit function, $u$, satisfies
$\mathcal{L}_{\widetilde{\varphi}}u=f$ in $X\times\Sigma$ and $u\equiv 0$ on
$X\times\partial\Sigma$. The uniqueness directly follows from the following
maximum principle, Lemma 2.3. ∎

The following lemma comes from Yau’s generalized maximum principle, referring
to [10, 29]. To describe the model metric on $X\times\Sigma$, we introduce the
asymptotic coordinates of $X\times\Sigma$. Let $\\{z_{1},\ldots,z_{n}\\}$ be
asymptotic coordinates of the end of $X$ and let $w=t+is$ be the complex
coordinate of $\Sigma$. Real asymptotic coordinates are given by
$\\{x_{1},\ldots,x_{2n},x_{2n+1}=t,x_{2n+2}=s\\}$, where the complex
coordinates are written as $z_{i}=x_{2i-1}+ix_{2i}$. The asymptotic coordinate
system will be applied to describe the asymptotic behavior of prescribed
Kähler metrics on $X\times\Sigma$.

######  Lemma 2.2.

Let $(X\times\Sigma,\Theta_{\widetilde{\Phi}})$ be the noncompact Kähler
manifold as above with the Kähler metric $\widetilde{g}$ associated with
$\widetilde{\Phi}$ satisfying, for some uniform constant $0<\lambda<\Lambda$,

$\displaystyle\lambda\delta_{ij}\leq\widetilde{g}_{ij}\leq\Lambda\delta_{ij}$

in the asymptotic coordinates of $X\times\Sigma$. Let $u$ be a
$\mathcal{C}_{loc}^{2}$ function bounded from above on $X\times\Sigma$.
Suppose that $\sup_{X\times\Sigma}u>\sup_{X\times\partial\Sigma}u$, then there
exists a sequence $\\{x_{k}\\}$ in $X\times\Sigma^{\circ}$ such that

$\displaystyle\lim_{k\rightarrow\infty}u(x_{k})=\sup_{X\times\Sigma}u,\quad\lim_{k\rightarrow\infty}|du(x_{k})|_{\widetilde{g}}=0,\quad\limsup_{k\rightarrow\infty}\Delta_{\widetilde{g}}u(x_{k})\leq
0.$ (2.3)

###### Proof.

Let $r$ be the radial function inherited from the asymptotic chart of $X$, for
instance, $r=(\sum_{i=1}^{n}|z_{i}|^{2})^{1/2}$. The radial function can be
extended to a non-negative smooth function in the whole space $X\times\Sigma$
satisfying the estimate

$\displaystyle|\nabla_{\widetilde{g}}r|_{\widetilde{g}}\leq
C,\qquad|\Delta_{\widetilde{g}}r|\leq C,$ (2.4)

for some uniform constant $C$. Consider the function
$u_{\mathbf{e}}=u-\mathbf{e}r$. Since $u_{\mathbf{e}}$ tends to negative
infinity as $r$ goes to infinity, $u_{\mathbf{e}}$ achieves its maximum at
some point $x_{\mathbf{e}}$. And $x_{\mathbf{e}}$ must be an interior point in
$X\times\Sigma$ based on the assumption that
$\sup_{X\times\Sigma}u>\sup_{X\times\partial\Sigma}u$. At $x_{\mathbf{e}}$,
the function $u_{\mathbf{e}}$ satisfies

$\displaystyle
0=du_{\mathbf{e}}(x_{\mathbf{e}})=du(x_{\mathbf{e}})-\mathbf{e}dr(x_{\mathbf{e}}),$
$\displaystyle
0\geq\Delta_{\widetilde{g}}u_{\mathbf{e}}(x_{\mathbf{e}})=\Delta_{\widetilde{g}}u(x_{\mathbf{e}})-\mathbf{e}\Delta_{\widetilde{g}}r(x_{\mathbf{e}})$

and

$\displaystyle u_{\mathbf{e}}(x_{\mathbf{e}})\geq
u(x)-\mathbf{e}r(x),\quad\text{ for all }x\in X\times\Sigma.$

Choosing $\\{x_{k}\\}$ to be points achieving the maximum of $u_{1/k}$, then
combining with (2.4) and letting $k$ go to infinity, we complete the proof of
(2.3). ∎

The following lemma is a strengthened version of the above maximum principle,
based on solving the Dirichlet problem in $X\times\Sigma$.

######  Lemma 2.3.

Let $(X\times\Sigma,\widetilde{g})$ be the same as in Lemma 2.2. Suppose that
$u$ is a function in $\mathcal{C}^{2}_{loc}(X\times\Sigma)$ and bounded from
above. Suppose that $u$ satisfies $\Delta_{\widetilde{g}}u\geq 0$ in
$X\times\Sigma$ and $u\leq 0$ on $X\times\partial\Sigma$. Then $u\leq 0$ in
$X\times\Sigma$.

###### Proof.

Assuming $u$ satisfies $\sup_{X\times\Sigma}u\geq\delta>0$. According to the
surjectivity part of the proof of Proposition 2.1, there exists a function $v$
satisfying

$\displaystyle\begin{cases}\Delta_{\widetilde{g}}v=-1,\quad&\text{ in
}X\times\Sigma,\\\ v=0,&\text{ on }X\times\partial\Sigma,\end{cases}$

and $\|v\|_{L^{\infty}}\leq C(n,\lambda,\Lambda)$. Consider the function
$u_{\mathbf{e}}=u-\mathbf{e}v$ for
$\displaystyle\mathbf{e}=\frac{\delta}{2C}$. Then
$\displaystyle\sup_{X\times\Sigma}u_{\mathbf{e}}\geq\frac{\delta}{2}>0$ and
$\displaystyle\Delta_{\widetilde{g}}u_{\mathbf{e}}\geq\mathbf{e}$. According
to Lemma 2.2, there exists a sequence $\\{x_{k}\\}$ in $X\times\Sigma^{\circ}$
such that
$\lim_{k\rightarrow\infty}u_{\mathbf{e}}(x_{k})=\sup_{X\times\Sigma}u_{\mathbf{e}}$,
$\lim_{k\rightarrow\infty}|du_{\mathbf{e}}(x_{k})|_{\widetilde{g}}=0$,
$\limsup_{k\rightarrow\infty}\Delta_{\widetilde{g}}u_{\mathbf{e}}(x_{k})\leq
0$. However, $\Delta_{\widetilde{g}}u_{\mathbf{e}}>0$, which leads to the
contradiction. ∎

##  3\. A priori estimate up to $\mathcal{C}^{0}$

From section 3 to 5, we complete the proof of Theorem A. The key ingredient is
to prove uniform a priori estimates up to order $\mathcal{C}^{1,1}$ for the
solution $\widetilde{\varphi}=\widetilde{\Phi}-\Psi$ of the
$\varepsilon$-geodesic equation ($E_{\varepsilon}$). These estimates will be
uniform with respect to $\varepsilon\in(0,1]$ and with respect to the distance
from a fixed point in $X$. (In section 6, we will also see that for a fixed
$\varepsilon>0$ it can be proved that $\widetilde{\varphi}$ is decaying at
spatial infinity. However, we are currently unable to make these decay
estimates uniform with respect to $\varepsilon$.)

These uniform $\mathcal{C}^{1,1}$ estimates are then used in two ways:

  * $\bullet$

First, they allow us to solve $(E_{\varepsilon})$ for any fixed
$\varepsilon\in(0,1]$ via the continuity method in
$(\mathcal{C}^{k,\alpha})_{0}$ for any $k\geq 2$. Recall this is done by
considering the family of equations $(E_{s})$ with $s\in[\varepsilon,1]$,
where openness in $(\mathcal{C}^{k,\alpha})_{0}$ follows from Proposition 2.1.
The uniform $\mathcal{C}^{1,1}$ estimates that we will prove, together with
the general regularity theory of the Monge-Ampère equation, then imply
closedness. Here, it is not yet important that the $\mathcal{C}^{1,1}$
estimates are uniform in $\varepsilon$, and the higher
$\mathcal{C}^{k,\alpha}$ estimates will depend on $\varepsilon$ because the
ellipticity of the equation does. Also, note that these higher-order estimates
follow from standard local regularity in the interior and from [8, Section
2.1–2.2] near the boundary because we already have a true $\mathcal{C}^{1,1}$
bound.

  * $\bullet$

Once $(E_{\varepsilon})$ is actually solved, we can then let $\varepsilon$ go
to zero and use the uniformity of the $\mathcal{C}^{1,1}$ estimates of the
$\varepsilon$-geodesic solution $\widetilde{\varphi}$ to extract a
subsequential limit $\varphi\in\mathcal{C}^{1,1}$ such that
$\Phi=\Psi+\varphi$ solves the geodesic equation (1.4), (1.5), (1.6).

We omit these standard arguments and instead focus on the proof of the uniform
$\mathcal{C}^{1,1}$ a priori estimates of the $\varepsilon$-geodesic solution
$\widetilde{\varphi}$. For this we follow the outline of [7] in the compact
case. However, we provide all the necessary details that are required to
generalize this theory to the ALE case. In addition, we also make use of the
recent advance [12] to obtain a $\mathcal{C}^{1,1}$ estimate which is uniform
in $\varepsilon$.

In this section, we only deal with the uniform $\mathcal{C}^{0}$ estimate. We
begin with a standard comparison principle [5, Proposition 3.1].

######  Lemma 3.1.

Let $D$ be a bounded connected domain in $\mathbb{C}^{n}$ with smooth boundary
and $u,v\in\mathcal{C}^{2}(D)$, plurisubharmonic functions in $D$. If $u=v$ on
$\partial D$ and $u\geq v$, then we have

$\displaystyle\int_{\Omega}(dd^{c}u)^{n}\leq\int_{\Omega}(dd^{c}v)^{n}.$

Then we can prove the following maximum principle for Monge-Ampère operators.

######  Theorem 3.2.

Let $\Theta$ be a fixed reference Kähler form and $\Omega$, the pull-back of a
semipositive $(1,1)$-form in $X$. Assume that
$u,v\in\mathcal{C}^{2}(X\times\Sigma)$ are bounded functions with
$\Omega+dd^{c}v$, $\Omega+dd^{c}u\geq 0$. If for some positive constants
$\lambda$, $\Lambda$, we have the following properties:

$\displaystyle(\Omega+dd^{c}v)^{n+1}\leq(\Omega+dd^{c}u)^{n+1}$
$\displaystyle\quad\text{ in }X\times\Sigma,$ (3.1)
$\displaystyle\lambda\Theta\leq\Omega+dd^{c}u\leq\Lambda\Theta$
$\displaystyle\quad\text{ in }X\times\Sigma,$ (3.2) $\displaystyle u\leq v$
$\displaystyle\quad\text{ on }X\times\partial\Sigma,$ (3.3)

then $u\leq v$ in $X\times\Sigma$.

###### Proof.

Assume $u(z_{0})>v(z_{0})$ at some point $z_{0}\in X\times\Sigma$. Let
$2h=u(z_{0})-v(z_{0})$. Then, we can modify $u,v$ to be $\tilde{u},\tilde{v}$
as follows:

$\displaystyle\begin{split}&\tilde{v}=v+h,\\\
&\tilde{u}=u+\frac{h}{2}|\tau|^{2}.\end{split}$ (3.4)

It can be checked that $\tilde{u}$, $\tilde{v}$ are bounded functions
satisfying that $\tilde{u}\leq\tilde{v}$ on $X\times\partial\Sigma$ and
$\tilde{u}(z_{0})\geq\tilde{v}(z_{0})+h$. By Wu-Yau’s generalized maximum
principle, there exists a sequence $\\{p_{k}\\}$ in $X\times\Sigma$ such that

$\lim_{k\rightarrow\infty}(\tilde{u}-\tilde{v})(p_{k})=\sup_{X\times\Sigma}(\tilde{u}-\tilde{v})\geq
h,\quad\limsup_{k\rightarrow\infty}dd^{c}(\tilde{u}-\tilde{v})(p_{k})\leq 0.$

For a sufficiently small constant $\delta>0$, there exist a point $p\in
X\times\Sigma$, $dd^{c}\tilde{u}(p)-dd^{c}\tilde{v}(p)\leq\delta\Theta$ and
$\eta_{0}=\tilde{u}(p)-\tilde{v}(p)\geq\sup_{X\times\Sigma}(\tilde{u}-\tilde{v})-\delta$.
Fix a local holomorphic chart around $p$, $\\{U,z^{i}:i=1,\ldots,n+1\\}$ with
$z^{n+1}=\tau$. Without loss of generality, we assume $U$ contains the unit
disk in $\mathbb{C}^{n+1}$ and for any local vector field $V\in T^{1,0}U$,

$\displaystyle C^{-1}|V|^{2}\leq\Theta(V,\overline{V})\leq C|V|^{2},$

where the constant $C$ only depends on the geometry of $X$ and the reference
metric $\Theta$. Let $\mathbf{e}=2C\delta$ and
$\eta=\eta_{0}-\frac{C\delta}{2}$. To derive the contradiction, we construct
the following local functions in $U$,

$\displaystyle\begin{split}&\accentset{\approx}{u}=\tilde{u}-\mathbf{e}|z|^{2},\\\
&\accentset{\approx}{v}=\tilde{v}+\eta.\end{split}$ (3.5)

If we denote the unit ball contained in the coordinate chart of $U$ by
$B_{1}(p)$, we have
$\accentset{\approx}{u}(p)-\accentset{\approx}{v}(p)=\frac{C\delta}{2}>0$ and
$\accentset{\approx}{u}\leq\accentset{\approx}{v}$ on $\partial B_{1}(p)$.
Consider the following subset of $B_{1}(p)$,

$\displaystyle D=\\{z\in
B_{1}(p):\accentset{\approx}{u}(z)>\accentset{\approx}{v}(z)\\}.$

Let $\rho$ be the local potential of $\Omega$ in $U$, $\Omega=dd^{c}\rho$.
According to Lemma 3.1,

$\displaystyle\int_{D}\big{[}dd^{c}(\rho+\accentset{\approx}{v})\big{]}^{n+1}\geq\int_{D}\big{[}dd^{c}(\rho+\accentset{\approx}{u})\big{]}^{n+1}.$
(3.6)

Taking $\mathbf{e}\leq\frac{\lambda}{4C}$,

$\displaystyle dd^{c}(\rho+u-\mathbf{e}|z|^{2})\geq\frac{1}{2}dd^{c}(\rho+u).$

Together with the construction of $\accentset{\approx}{u}$ and
$\accentset{\approx}{v}$ in (3.4), (3.5),

$\displaystyle\int_{D}\big{[}dd^{c}(\rho+v)\big{]}^{n+1}$
$\displaystyle\geq\int_{D}\Big{[}dd^{c}\big{(}\rho+u+\frac{h}{2}|\tau|^{2}-\mathbf{e}|z|^{2}\big{)}\Big{]}^{n+1}$
$\displaystyle\geq\frac{h\lambda^{n}}{2^{n+1}}\int_{D}\Theta^{n+1}+\int_{D}\big{[}dd^{c}(\rho+u)\big{]}^{n+1}-2\mathbf{e}\Lambda^{n}\int_{D}\Theta^{n+1}.$
(3.7)

By picking $\mathbf{e}$ smaller,
$\mathbf{e}\leq\frac{h\lambda^{n}}{2^{n+4}\Lambda^{n}}$, and combining with
(3.1), we have,

$\displaystyle\int_{D}\big{[}dd^{c}(\rho+v)\big{]}^{n+1}$
$\displaystyle\geq\int_{D}\big{[}dd^{c}(\rho+u)\big{]}^{n+1}+\frac{h\lambda^{n}}{2^{n+4}}\int_{D}\Theta^{n+1}$
$\displaystyle\geq\int_{D}\big{[}dd^{c}(\rho+v)\big{]}^{n+1}+\frac{h\lambda^{n}}{2^{n+4}}\int_{D}\Theta^{n+1}.$
(3.8)

Since the second term of (3.8) is strictly positive, which leads to a
contradiction, we complete the proof. ∎

Let $\widetilde{\varphi}=\widetilde{\Phi}-\Psi$ be the solution of
$(E_{\varepsilon})$ after subtracting $\Psi$. According to Theorem 3.2, we
have a uniform lower bound $\widetilde{\varphi}\geq 0$; hence,
$\widetilde{\Phi}\geq\Psi$. The upper bound is easy to construct. Consider the
function defined in $X\times\Sigma$, $H=2t(1-t)$. By restricting to each
section $\Sigma_{x_{0}}=\\{x_{0}\\}\times\Sigma\stackrel{{\scriptstyle
i_{x_{0}}}}{{\hookrightarrow}}X\times\Sigma$, we have

$\displaystyle i_{x_{0}}^{*}(\Theta_{\Psi}+dd^{c}H)\leq
0<i_{x_{0}}^{*}(\Theta_{\Psi}+dd^{c}\varphi).$

Hence, $\Delta_{\Sigma}H\leq dd^{c}\widetilde{\varphi}$ in $\Sigma_{x_{0}}$
and $H=\varphi=0$ on its boundary $\partial\Sigma_{x_{0}}$. The maximum
principle on compact manifolds with boundary implies that
$\widetilde{\varphi}\leq H$ on each section. Hence, we get the desired uniform
$\mathcal{C}^{0}$ estimate,

$\displaystyle\Psi\leq\widetilde{\Phi}\leq\Psi+H.$

##  4\. A priori estimate up to $\mathcal{C}^{1}$

For the $\mathcal{C}^{1}$ bound, Blocki gives an explicit estimate in the
compact setting in [6]. We generalize this estimate to the non-compact case.
The $\mathcal{C}^{1}$ boundary estimate follows directly from the fact that
$\Psi\leq\widetilde{\Phi}\leq\Psi+H$ in $X\times\Sigma$ and $\Psi$,
$\widetilde{\Phi}$, $\Psi+H$ agree along $X\times\partial\Sigma$. Let $\nabla$
be the Levi-Civita connection of $\Theta_{\Psi}$ on $X\times\Sigma$. Then we
have

$\displaystyle|\nabla\widetilde{\Phi}|_{\Theta_{\Psi}}\leq\max\\{|\nabla\Psi|_{\Theta_{\Psi}},|\nabla(\Psi+H)|_{\Theta_{\Psi}}\\},\quad\text{
on }X\times\partial\Sigma.$

Hence,
$\sup_{X\times\partial\Sigma}|\nabla\widetilde{\Phi}|_{\Theta_{\Psi}}\leq C$,
where $C$ is a uniform constant.

######  Proposition 4.1.

Let
$\widetilde{\varphi}=\widetilde{\Phi}-\Psi\in\mathcal{C}^{3}_{loc}(X\times\Sigma)$
be a solution of $(E_{\varepsilon})$ and let $\nabla$ be the Levi-Civita
connection of the Kähler metric $\Theta_{\Psi}$ on $X\times\Sigma$. Assume
that $\widetilde{\varphi}$ lies in the space
$\mathcal{C}^{1}(X\times\Sigma,\Theta_{\Psi})$. Then,

$\displaystyle\sup_{X\times\Sigma}|\nabla\widetilde{\varphi}|_{\Theta_{\Psi}}\leq
C,$

where $C$ is a positive constant depending only on upper bounds for
$|\widetilde{\varphi}|$, on lower bounds for the bisectional curvature of
$\Theta_{\Psi}$, and on $n$, but not on $\varepsilon$.

###### Proof.

Suppose that $\inf_{X\times\Sigma}\widetilde{\varphi}=A$ and
$\sup_{X\times\Sigma}\widetilde{\varphi}=B$. Consider the following function,

$\displaystyle\alpha=\log\beta-\gamma\circ\widetilde{\varphi},$

where $\beta=|\nabla\widetilde{\varphi}|^{2}_{\Theta_{\Psi}}$ and
$\gamma:[A,B]\rightarrow\mathbb{R}$ is a smooth function to be determined
later. According to the assumption that $\widetilde{\varphi}$ lies in the
space $\mathcal{C}^{1}$, Yau’s maximum principle can be applied here. In
particular, there exists a sequence in $\\{x_{k}\\}$ in
$X\times\Sigma^{\circ}$ such that,

$\displaystyle\lim_{k\rightarrow\infty}\alpha(x_{k})=\sup_{X\times\Sigma}\alpha,\quad\lim_{k\rightarrow\infty}|\nabla\alpha(x_{k})|_{\Theta_{\Psi}}=0,\quad\limsup_{k\rightarrow\infty}\Delta\alpha(x_{k})\leq
0,$

where $\Delta=\Delta_{\Theta_{\Psi}}$. Then, for a sufficiently small
$\mathbf{e}>0$ to be determined later and all $k\gg 1$, we have

$\displaystyle\alpha(x_{k})\geq\sup_{X\times\Sigma}\alpha-\mathbf{e},\quad|\nabla\alpha(x_{k})|_{\Theta_{\Psi}}\leq\mathbf{e},\qquad\Delta\alpha(x_{k})\leq\mathbf{e}.$
(4.1)

Fixing $O=x_{k}$ satisfying (4.1), we can pick the normal coordinates around
$O$. Let $g$ and $\widetilde{g}$ denote the metric tensors corresponding to
$\Theta_{\Psi}$ and
$\Theta_{\widetilde{\Phi}}=\Theta_{\Psi}+dd^{c}\widetilde{\varphi}$. Then
there exist local holomorphic coordinates near $O$ such that,

$\displaystyle g_{i\overline{j}}(O)=\delta_{ij},\quad
g_{i\overline{j},k}(O)=0\quad\text{ and
}\quad\widetilde{g}_{i\overline{j}}(O)\text{ is diagonal}.$

By taking derivative of $\alpha$,

$\displaystyle\alpha_{p}=\frac{\beta_{p}}{\beta}-(\gamma^{\prime}\circ\widetilde{\varphi})\cdot\widetilde{\varphi}_{p}.$

Combining with condition (4.1), $|\alpha_{p}(O)|\leq\mathbf{e}$. Then, at the
point $O$, we have

$\displaystyle\alpha_{p\overline{p}}\geq\frac{\beta_{p\overline{p}}}{\beta}-[(\gamma^{\prime})^{2}+\gamma^{\prime\prime}]|\widetilde{\varphi}_{p}|^{2}-\gamma^{\prime}\widetilde{\varphi}_{p\overline{p}}-\mathbf{e}|\gamma^{\prime}||\widetilde{\varphi}_{p}|-\mathbf{e}.$
(4.2)

If we write the local potential of $\widetilde{g}_{i\overline{j}}$ as $u$ near
$O$, then the $\varepsilon$-geodesic equation is locally given by
$\det(u_{i\overline{j}})=\upsilon(\varepsilon)\det(g_{i\overline{j}})$. The
direct derivative of the equation at $O$ gives,

$\displaystyle\sum_{p}\frac{u_{p\overline{p}j}}{u_{p\overline{p}}}=\big{(}\log\upsilon(\varepsilon)\big{)}_{j}.$
(4.3)

Also, notice that,

$\displaystyle\beta_{p\overline{p}}\geq-D\beta+2\operatorname{Re}\sum_{j}u_{p\overline{p}j}\widetilde{\varphi}_{\overline{j}}+\sum_{j}|\widetilde{\varphi}_{jp}|^{2}+\widetilde{\varphi}_{p\overline{p}}^{2},$

where $-D$ is the negative lower bound of bisectional curvature of
$\Theta_{\Psi}$. Recall that we have the assumption
$C^{-1}g_{i\overline{j}}\leq u_{i\overline{j}}\leq Cg_{i\overline{j}}$ and
$|\widetilde{\varphi}_{p}|<C$, where $C$ is the constant from our assumption
at the beginning of this section and we will get rid of this constant in the
end. Together with (4.2) and (4.3), we have,

$\displaystyle\begin{split}C\mathbf{e}\geq\sum_{p}\frac{\alpha_{p\overline{p}}}{u_{p\overline{p}}}\geq(\gamma^{\prime}&-D)\sum_{p}\frac{1}{u_{p\overline{p}}}+\frac{1}{\beta}\sum_{jp}\frac{|\widetilde{\varphi}_{jp}|}{u_{p\overline{p}}}\\\
&-2\operatorname{Re}\frac{1}{\beta}\sum_{j}\big{(}\log\upsilon(\varepsilon)\big{)}_{j}\widetilde{\varphi}_{\overline{j}}\\\
&-[(\gamma^{\prime})^{2}+\gamma^{\prime\prime}]\sum_{p}\frac{|\widetilde{\varphi}_{p}|^{2}}{u_{p\overline{p}}}-n\gamma^{\prime}-C(|\gamma^{\prime}|+1)\mathbf{e}.\end{split}$
(4.4)

According to Blocki’s key observation in [6], after modified in our case, at
the point $O$, we have

$\displaystyle\frac{1}{\beta}\sum_{j,p}\frac{|\widetilde{\varphi}_{jp}|^{2}}{u_{p\overline{p}}}\geq(\gamma^{\prime})^{2}\sum_{p}\frac{|\widetilde{\varphi}_{p}|^{2}}{u_{p\overline{p}}}-2\gamma^{\prime}-\frac{2+C\mathbf{e}}{\beta}-C(1+|\gamma^{\prime}|)\mathbf{e},$

and assuming that $\beta\geq 1$, we have

$\displaystyle\frac{2}{\beta}\operatorname{Re}\sum_{j}\big{(}\log\upsilon\big{)}_{j}\varphi_{\overline{j}}\geq-2\frac{|\nabla\log\upsilon(\varepsilon)|}{\sqrt{\beta}}\geq-2(n+1)\frac{\big{|}\nabla\big{(}\upsilon(\varepsilon)^{\frac{1}{n+1}}\big{)}\big{|}}{\upsilon(\varepsilon)^{\frac{1}{n+1}}}\geq-V\sum_{p}\frac{1}{u_{p\overline{p}}}$

where $V$ is a uniform constant satisfying

$\displaystyle V\geq
2(n+1)\big{|}\nabla\big{(}\upsilon(\varepsilon)^{\frac{1}{n+1}}\big{)}\big{|}.$

Combining with (4.4),

$\displaystyle
C(1+|\gamma^{\prime}|)\mathbf{e}\geq(\gamma^{\prime}-D-V)\sum_{p}\frac{1}{u_{p\overline{p}}}-\gamma^{\prime\prime}\sum_{p}\frac{|\widetilde{\varphi}_{p}|^{2}}{u_{p\overline{p}}}-(n+2)\gamma^{\prime}-2.$
(4.5)

Now, we choose the function $\gamma$ and the small number $\mathbf{e}>0$ in
(4.5) as follows. Let $\gamma=(D+V+3)(t-A)-(B-A)^{-1}(t-A)^{2}$ and
$\mathbf{e}\leq C^{-1}(D+V+3)^{-1}$, then we have

$\displaystyle\sum_{p}\frac{1}{u_{p\overline{p}}}+\frac{2}{B-A}\sum_{p}\frac{|\widetilde{\varphi}_{p}|^{2}}{u_{p\overline{p}}}\leq
3+(n+2)(D+V+3).$

Then, it is straightforward to conclude that
$\beta(O)\leq\max\\{[(n+3)(D+V+3)]^{n+1}n(B-A),1\\}$. Noting that
$\beta\leq\exp\\{\mathbf{e}+\log\beta(O)-\gamma\circ\widetilde{\varphi}(O)+\gamma\circ\widetilde{\varphi}\\}$,
hence, $\beta$ is controlled by some uniform constant only depending on
$\|\widetilde{\varphi}\|_{L^{\infty}}$, $D$, $V$ and $n$. ∎

##  5\. A priori estimate up to $\mathcal{C}^{1,1}$

First, we deal with the uniform $\mathcal{C}^{2}$ boundary estimate on
$X\times\partial\Sigma$. The technique is to construct local barrier functions
near the boundary, which is completely parallel to [8, 9, 17]. The statement
is the following:

######  Lemma 5.1.

Let the data $(X\times\Sigma,\Theta_{\Psi},\widetilde{\varphi})$ be the same
as in Proposition 4.1. Let $\nabla$ denote the Levi-Civita connection of
$\Theta_{\Psi}$ on $X\times\Sigma$. Then

$\displaystyle\sup_{X\times\partial\Sigma}|\nabla^{2}\widetilde{\varphi}|_{\Theta_{\Psi}}\leq
C,$

where the constant $C$ only depends on
$\sup_{X\times\Sigma}|\nabla\widetilde{\varphi}|_{\Theta_{\Psi}}$ and on
$(X\times\Sigma,\Theta_{\Psi})$.

###### Proof.

Fixing a point $p\in X\times\partial\Sigma$, we pick the local holomorphic
coordinates around the point $p$ such that the coordinates system is normal in
$X$ and $\Sigma$ direction, we still pick the standard coordinate function of
the annulus, denoted by $\\{x_{1},\ldots,x_{2n},x_{2n+1}=t,x_{2n}=s\\}$ and
the corresponding holomorphic coordinates, $z_{i}=x_{2i-1}+ix_{2i}$.
Throughout the proof, we assume the metric tensor $g$ associated with
$\Theta_{\Psi}$ satisfies $m\delta_{ij}\leq g_{i\overline{j}}\leq
M\delta_{ij}$. In general, we need to prove the boundary $C^{2}$ estimate at
$p$ in tangential-tangential, tangential-normal and normal-normal directions
respectively. However, the tangential-tangential is trivial in our case and
the normal-normal estimate follows directly from the tangential-normal
estimate. Here, we briefly summarize the proof of tangential-normal estimate
by explicitly constructing the barrier functions.

Consider a small neighborhood near $p$, $\displaystyle
B^{\prime}_{\delta}(p)=(X\times\Sigma)\cap B_{\delta}(p)$, where the small
constant $\delta$ will be determined later. Firstly, we construct the
following auxiliary function in $B^{\prime}_{\delta}(p)$,

$\displaystyle v=\widetilde{\varphi}+Nt(1-t),$ (5.1)

where $N$ is a large constant to be determined. Then, it can be easily checked
that

$\displaystyle\widetilde{\Delta}v\leq
n+1-m\sum_{i}\widetilde{g}^{i\overline{i}}-N\widetilde{g}^{n+1,\overline{n+1}},$

where $\widetilde{g}$ again denotes the metric tensor associated with
$\Theta_{\widetilde{\Phi}}=\Theta_{\Psi}+dd^{c}\widetilde{\varphi}$ and
$\widetilde{\Delta}$ denotes the corresponding Laplacian. Notice that

$\displaystyle-\frac{m}{2}\sum_{i}\widetilde{g}^{i\overline{i}}-N\widetilde{g}^{n+1,\overline{n+1}}\leq-\frac{mN^{\frac{1}{n+1}}}{2(\det\widetilde{g})^{\frac{1}{n+1}}}=-\frac{mN^{\frac{1}{n+1}}}{2\upsilon(\varepsilon)^{\frac{1}{n+1}}}(\det
g)^{-\frac{1}{n+1}}.$

By taking $N=[(n+1)(2/m)]^{n+1}\max_{B_{\delta}^{\prime}(p)}(\det g)$, we have
$\displaystyle\widetilde{\Delta}v\leq-\frac{m}{2}\sum_{i}\widetilde{g}^{i\overline{i}}$.
Noting that $\widetilde{\varphi}=\widetilde{\Phi}-\Psi\geq 0$, we have $v\geq
0$ on $\partial B^{\prime}_{\delta}(p)$. Then, the barrier functions can be
constructed as follows:

$\displaystyle w=Av+B|z|^{2}\pm\frac{\partial}{\partial
x_{k}}\widetilde{\varphi},\quad\text{ for }1\leq k\leq 2n\text{ or }k=2n+2.$

By differentiating the Monge-Ampère equation $(E_{\varepsilon})$ in the local
coordinates,

$\displaystyle\pm\widetilde{\Delta}\Big{(}\frac{\partial}{\partial
x_{k}}\widetilde{\varphi}\Big{)}=\pm\big{(}\widetilde{g}^{i\overline{j}}(\widetilde{g})_{i\overline{j},k}-\widetilde{g}^{i\overline{j}}g_{i\overline{j},k}\big{)}\leq
C(1+\sum\widetilde{g}^{i\overline{i}}),$

where $A$ and $B$ are large positive constants to be determined. According to
the $\mathcal{C}^{1}$ estimate of $\widetilde{\varphi}$, we assume that
$|\partial_{k}\widetilde{\varphi}|\leq C$. By picking a very large constant
$B$ such that, on $\partial B^{\prime}_{\delta}(p)$,
$B|z|^{2}\pm\partial_{k}\widetilde{\varphi}\geq 0$, we have $w\geq 0$ on
$\partial B^{\prime}_{\delta}(p)$. Then, we choose a large constant $A$ such
that $\widetilde{\Delta}w\leq 0$ in $B^{\prime}_{\delta}(p)$. Then, by maximum
principle, $w\geq 0$ in $B^{\prime}_{\delta}(p)$. Together with the fact that
$w(p)=0$, we have $\partial_{t}w\geq 0$ at $p$, which implies the tangential-
normal estimate on the boundary. ∎

Lemma 5.1 together with Yau’s standard calculation on Laplacian estimate
implies the following interior Laplacian estimate, referring to [31].

######  Lemma 5.2.

Let $\widetilde{\varphi}$ be the solution of $(E_{\varepsilon})$ and $\Delta$,
$\widetilde{\Delta}$, the Laplacian operators of $g=\Theta_{\Psi}$ and
$\widetilde{g}=\Theta_{\widetilde{\Phi}}=\Theta_{\Psi}+dd^{c}\widetilde{\varphi}$
respectively. Then, for any constant $C$,

$\displaystyle\widetilde{\Delta}\big{(}e^{-C\widetilde{\varphi}}(n+1+\Delta\widetilde{\varphi})\big{)}\geq$
$\displaystyle\quad
e^{-C\widetilde{\varphi}}\big{(}\Delta\log\upsilon(\varepsilon)-(n+1)^{2}\inf_{i\neq
l}(R_{i\overline{i}l\overline{l}})\big{)}$ $\displaystyle-
Ce^{-C\widetilde{\varphi}}(n+1)(n+1+\Delta\widetilde{\varphi})$
$\displaystyle+(C+\inf_{i\neq
l}(R_{i\overline{i}l\overline{l}}))e^{-C\widetilde{\varphi}}(n+1+\Delta\widetilde{\varphi})^{1+\frac{1}{n}}\upsilon(\varepsilon)^{-1},$

where $R$ denotes the curvature tensor of $g$. From this, we can deduce the
estimate

$\displaystyle\sup_{X\times\Sigma}|\Delta\widetilde{\varphi}|\leq
C(1+\sup_{X\times\partial\Sigma}|\Delta\widetilde{\varphi}|),$

where $C$ only depends on $\sup_{X\times\Sigma}\widetilde{\varphi}$ and on a
negative lower bound of $\inf_{i\neq l}(R_{i\overline{i}l\overline{l}})$.

Lemma 5.2, together with Lemma 5.1, implies that there exists a uniform
constant $C$ only depending on $\sup_{X\times\Sigma}\Delta\widetilde{\varphi}$
such that $\varepsilon
C^{-1}g_{i\overline{j}}\leq\widetilde{g}_{i\overline{j}}\leq
Cg_{i\overline{j}}$. This is already enough to apply the standard local
regularity theory of the Monge-Ampère equation to prove
$\mathcal{C}^{k,\alpha}$ estimates for any $k\geq 2$ that depend on a positive
lower bound for $\varepsilon$. In this way the equation $(E_{\varepsilon})$
can be solved using the continuity path $(E_{s})$, $s\in[\varepsilon,1]$.
However, in order to construct an honest geodesic by letting $\varepsilon\to
0$, we require a full $\mathcal{C}^{1,1}$ estimate which is uniform in
$\varepsilon$. In [12], $\mathcal{C}^{1,1}$ regularity is proved in the
compact case. The method can also be applied in the ALE Kähler setting.

######  Proposition 5.3.

Let the data $(X\times\Sigma,\Theta_{\Psi},\widetilde{\varphi})$ be the same
as in Proposition 4.1. If $\widetilde{\varphi}$ lies in the space
$\mathcal{C}^{2}(X\times\Sigma,\Theta_{\Psi})$, then there exists a constant
$C$ such that

$\displaystyle|\nabla^{2}\widetilde{\varphi}|_{\Theta_{\Psi}}\leq C,$

where $\nabla$ again denotes the Levi-Civita connection of the metric
$\Theta_{\Psi}$ and $C$ depends only on $(X\times\Sigma,\Theta_{\Psi})$ and on
$\sup_{X\times\Sigma}|\widetilde{\varphi}|$,
$\sup_{X\times\Sigma}|\nabla\widetilde{\varphi}|_{\Theta_{\Psi}}$,
$\sup_{X\times\Sigma}|\Delta\widetilde{\varphi}|$,
$\sup_{X\times\partial\Sigma}|\nabla^{2}\widetilde{\varphi}|_{\Theta_{\Psi}}$.

###### Proof.

We again write $g$ for the metric tensor associated with $\Theta_{\Psi}$. Let
$\lambda_{1}(\nabla^{2}\widetilde{\varphi})$ be the largest eigenvalue of the
real Hessian $\nabla^{2}\widetilde{\varphi}$. By observing that there exists a
uniform constant $C$ such that
$\lambda_{1}(\nabla^{2}\widetilde{\varphi})\leq|\nabla^{2}\widetilde{\varphi}|_{g}\leq
C\lambda_{1}(\nabla^{2}\widetilde{\varphi})+C$, it suffices to prove that
$\lambda_{1}(\nabla^{2}\widetilde{\varphi})$ has a uniform upper bound.
Consider the following quantity,

$\displaystyle
Q=\log\lambda_{1}(\nabla^{2}\widetilde{\varphi})+h(|\nabla\widetilde{\varphi}|_{g}^{2})-A\widetilde{\varphi},$

where $h$ is defined to be $\displaystyle
h(s)=-\frac{1}{2}\log\big{(}1+\sup_{X\times\Sigma}|\nabla\widetilde{\varphi}|_{g}^{2}-s\big{)}$
and $A$ is a uniform large positive constant to be determined later. We can
further modify this quantity to $Q_{\mathbf{e}}=Q-\mathbf{e}r$, where
$\mathbf{e}$ is a small positive constant to be determined later. According to
the assumption that $|\nabla^{2}\widetilde{\varphi}|$ is bounded and hence so
is $Q$, the modified quantity $Q_{\mathbf{e}}$ attains its maximum at some
point $x_{\mathbf{e}}\in X\times\Sigma$. The same argument as in Lemma 2.2
implies that $\lim_{\mathbf{e}\rightarrow
0}Q(x_{\mathbf{e}})=\sup_{X\times\Sigma}Q$. In the following, we assume
$\mathbf{e}$ is small enough such that
$|Q(x_{\mathbf{e}})-\sup_{X\times\Sigma}Q|<1$ and always write
$p=x_{\mathbf{e}}$. Since $Q_{\mathbf{e}}$ might not be smooth at $p$ if the
eigenspace of $\lambda_{1}(\nabla^{2}\widetilde{\varphi})(p)$ has dimension
greater than one, a perturbation argument used in [12] can be applied to the
quantity $Q_{\mathbf{e}}$ here.

Fix normal coordinates $(z_{1},\ldots,z_{n+1})$ with respect to $g$ at $p$
such that $(\widetilde{\varphi}_{i\overline{j}})$ is diagonal at $p$. Define
the corresponding real coordinates $(x_{1},\ldots,x_{2n})$ by
$z_{i}=x_{2i-1}+ix_{2i}$. Let
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{2n}$ be the eigenvalues of
$\nabla^{2}\widetilde{\varphi}$ at $p$ and $V_{1},\ldots,V_{2n}$, the
corresponding unit eigenvectors at $p$. The eigenvectors can be extended to
vector fields with constant coefficients in a small neighborhood of $p$, also
denoted by $V_{1},\ldots,V_{2n}$, and can be represented by
$V_{\alpha}=V^{\beta}_{\alpha}{\partial_{x_{\beta}}}$ in the local
coordinates. The perturbation argument is to perturb
$\nabla^{2}\widetilde{\varphi}$ locally around $p$ and to ensure that
$\lambda_{1}>\lambda_{2}$ near $p$. Precisely, consider the following locally
defined tensor field,

$\displaystyle
P=\sum_{\alpha,\beta}\big{(}\delta_{\alpha\beta}-V^{\alpha}_{1}V^{\beta}_{1}\big{)}dx_{\alpha}\otimes
dx_{\beta}.$

Let $\lambda_{i}^{\prime}=\lambda_{i}(\nabla^{2}\widetilde{\varphi}-P)$. Then,
one can easily check that $\lambda^{\prime}_{1}(p)=\lambda_{1}(p)$ and
$\lambda^{\prime}_{i}(p)=\lambda_{i}(p)-1$ for $i\geq 2$. Hence, there exists
a neighborhood of $p$ such that
$\lambda^{\prime}_{1}>\lambda^{\prime}_{2}\geq\ldots\geq\lambda^{\prime}_{2n}$
and $\lambda^{\prime}_{1}\leq\lambda_{1}$. Consider the following perturbed
quantities,

$\displaystyle\hat{Q}=\log\lambda^{\prime}_{1}+h(|\nabla\widetilde{\varphi}|^{2}_{g})-A\widetilde{\varphi},\quad\hat{Q}_{\mathbf{e}}=\hat{Q}-\mathbf{e}r.$

Therefore, $\hat{Q}_{\mathbf{e}}$ is a smooth quantity with a local maximum at
$p$. Then, we have,

$\displaystyle|d\hat{Q}(p)|_{g}\leq C\mathbf{e},\quad\Delta\hat{Q}(p)\leq
C\mathbf{e}.$

The following inequality follows directly from the calculation in [12, Lemma
2.1]. The only information we need in the calculation is the second derivative
of the Monge-Ampère equation at $p$. We will not repeat the details here. By
assuming $\lambda_{1}^{\prime}\geq 1$ at $p$, and again writing
$\widetilde{g}$ for the metric tensor associated with
$\Theta_{\widetilde{\Phi}}=\Theta_{\Psi}+dd^{c}\widetilde{\varphi}$, we have

$\displaystyle\begin{split}\Delta\hat{Q}\geq&\
2\sum_{\alpha>1}\frac{\widetilde{g}^{i\overline{i}}|\partial_{i}(\widetilde{\varphi}_{V_{\alpha}V_{1}})|^{2}}{\lambda_{1}(\lambda_{1}-\lambda_{\alpha})}+\frac{\widetilde{g}^{i\overline{i}}\widetilde{g}^{j\overline{j}}\big{|}V_{1}\big{(}\widetilde{g}_{i\overline{j}}\big{)}\big{|}^{2}}{\lambda_{1}}-\frac{\widetilde{g}^{i\overline{i}}|\partial_{i}(\widetilde{\varphi}_{V_{1}V_{1}})|^{2}}{\lambda_{1}^{2}}\\\
&+h^{\prime}\sum_{k}\widetilde{g}^{i\overline{i}}\big{(}|\widetilde{\varphi}_{ik}|^{2}+|\widetilde{\varphi}_{i\overline{k}}|^{2}\big{)}+h^{\prime\prime}\widetilde{g}^{i\overline{i}}\big{|}\partial_{i}|\nabla\widetilde{\varphi}|^{2}_{g}\big{|}\\\
&+(A-B)\sum_{i}\widetilde{g}^{i\overline{i}}-An,\end{split}$ (5.2)

where the constant $B$ only depends on $(X\times\Sigma,g)$ and
$\sup_{X\times\Sigma}|\nabla\widetilde{\varphi}|_{g}$. To cancel the annoying
terms, we deal with the third term in (5.2),
$\displaystyle{{\lambda_{1}^{-2}}\widetilde{g}^{i\overline{i}}|\partial_{i}(\widetilde{\varphi}_{V_{1}V_{1}})|^{2}}$.
To estimate the term, we split it into the following two parts,

$\displaystyle
I_{1}=(1-2\delta)\frac{\widetilde{g}^{i\overline{i}}|\partial_{i}(\widetilde{\varphi}_{V_{1}V_{1}})|^{2}}{\lambda_{1}^{2}},$
$\displaystyle
I_{2}=2\delta\frac{\widetilde{g}^{i\overline{i}}|\partial_{i}(\widetilde{\varphi}_{V_{1}V_{1}})|^{2}}{\lambda_{1}^{2}},$

where $0<\delta<1/4$ is to be determined later. For $I_{1}$, referring to [12,
Lemma 2.2], by assuming that $\lambda_{1}^{\prime}\geq D/\delta$, where $D$
only depends on $(X\times\Sigma,g)$ and
$\sup_{X\times\Sigma}\Delta\widetilde{\varphi}$, we have

$\displaystyle
I_{1}\leq\sum_{i,j}\frac{\widetilde{g}^{i\overline{i}}\widetilde{g}^{j\overline{j}}\big{|}V_{1}\big{(}\widetilde{g}_{i\overline{j}}\big{)}\big{|}}{\lambda_{1}}+2\sum_{\alpha>1}\sum_{i}\frac{\widetilde{g}^{i\overline{i}}|\partial_{i}(\widetilde{\varphi}_{V_{\alpha}V_{1}})|^{2}}{\lambda_{1}(\lambda_{1}-\lambda_{\alpha})}+\sum_{i}\widetilde{g}^{i\overline{i}}.$
(5.3)

To estimate $I_{2}$, recall the fact that $d\hat{Q}_{\mathbf{e}}=0$ and apply
the derivative of eigenvalues referring to [12, Lemma 5.2]. Then, we have

$\displaystyle\begin{split}I_{2}&=2\delta\sum_{i}\widetilde{g}^{i\overline{i}}\big{|}A\widetilde{\varphi}_{i}+h^{\prime}\partial_{i}|\nabla\widetilde{\varphi}|^{2}_{g}-\mathbf{e}r_{i}\big{|}^{2}\\\
&\leq 8\delta
A^{2}\sum_{i}\widetilde{g}^{i\overline{i}}|\widetilde{\varphi}_{i}|^{2}+2(h^{\prime})^{2}\sum_{i}\widetilde{g}^{i\overline{i}}|\partial_{i}|\nabla\widetilde{\varphi}|^{2}_{g}|^{2}+C\mathbf{e}\sum_{i}\widetilde{g}^{i\overline{i}}.\end{split}$
(5.4)

Combining (5.2), (5.3), (5.4) and $\Delta\hat{Q}\leq C\mathbf{e}$, then, by
assuming $\lambda_{1}^{\prime}\geq D/\delta$, we have

$\displaystyle C\mathbf{e}\geq$ $\displaystyle\
h^{\prime}\sum_{k}\widetilde{g}^{i\overline{i}}\big{(}|\widetilde{\varphi}_{ik}|^{2}+|\widetilde{\varphi}_{i\overline{k}}|^{2}\big{)}+\big{(}h^{\prime\prime}-2(h^{\prime})^{2}\big{)}\widetilde{g}^{i\overline{i}}\big{|}\partial_{i}|\nabla\widetilde{\varphi}|^{2}\big{|}$
$\displaystyle-8\delta
A^{2}\widetilde{g}^{i\overline{i}}|\widetilde{\varphi}_{i}|^{2}+(A-B-C\mathbf{e})\sum_{i}\widetilde{g}^{i\overline{i}}-An.$

Notice that $h^{\prime\prime}=2(h^{\prime})^{2}$. Picking $\mathbf{e}\leq
1/C$, $A=B+2$ and
$\displaystyle\delta=\big{(}8A^{2}(\sup_{X\times\Sigma}|\nabla\widetilde{\varphi}|^{2}+1)\big{)}^{-1}$,
then we have

$\displaystyle
h^{\prime}\sum_{k}\widetilde{g}^{i\overline{i}}\big{(}|\widetilde{\varphi}_{ik}|^{2}+|\widetilde{\varphi}_{i\overline{k}}|^{2}\big{)}+\sum_{i}\widetilde{g}^{i\overline{i}}\leq
An+1.$

Recall $\widetilde{g}_{i\overline{j}}\leq Cg_{i\overline{j}}$, where $C$ only
depends on $\sup_{X\times\Sigma}\Delta\widetilde{\varphi}$. Hence, at $p$,
$\widetilde{g}^{i\overline{i}}\geq C^{-1}$. Then,

$\displaystyle\lambda_{1}(p)\leq\max\Big{\\{}\frac{D}{\delta},\big{\\{}(An+1)C-n\big{\\}}(1+\sup_{X\times\Sigma}|\nabla\widetilde{\varphi}|_{g}^{2})\Big{\\}}.$

Together with the fact that $\sup_{X\times\Sigma}Q\leq Q(p)+1$, we prove that
$\sup_{X\times\Sigma}\lambda_{1}$ is bounded by some uniform constant. ∎

##  6\. The asymptotic behavior of $\varepsilon$-geodesics

In this section, we prove Theorem B on the asymptotic behavior of
$\varepsilon$-geodesics for a fixed $\varepsilon>0$. We use the notation
introduced before Theorem B and we assume
$\psi_{0},\psi_{1}\in\mathcal{H}_{-\gamma}(\omega)$ $(\gamma>0)$. We are
really interested in the case when $-\gamma=2-2\tilde{\tau}$ due to theorem
1.1. In $\varepsilon$-geodesic equation $(E_{\varepsilon})$, the derivatives
of function $\upsilon(\varepsilon)$ decays at infinity with order
$-\varsigma$, $|\nabla^{k}\upsilon(\varepsilon)|\leq O(r^{-\varsigma-k})$ with
$\varsigma\geq\gamma$ for $k\geq 1$. Without loss of generality, we assume
$\varsigma\geq\gamma>\tau$, otherwise theorem B can be proved more easily
without iteration (step 3).

We also write $\varphi_{\varepsilon}=\Phi_{\varepsilon}-\Psi$, so that the
solution is given by
$\Theta+dd^{c}\Phi_{\varepsilon}=\Theta_{\Psi}+dd^{c}\varphi_{\varepsilon}$
with $\varphi_{\varepsilon}=0$ on $X\times\partial\Sigma$. In Aleyasin [1], a
rough idea is given to prove the asymptotic behavior of
$\varepsilon$-geodesics by constructing barrier functions in the (strictly
easier) special case where the asymptotic coordinates are $J$-holomorphic and
the decay rate of the ALE Kähler metric to the Euclidean metric is high
enough. However, even in this special case, the details are more involved than
what is suggested in [1]. Here we give a complete proof in the general
setting.

### Step 1: Differentiating the Monge-Ampère equation.

The Monge-Ampère equation can be written explicitly in the asymptotic
coordinates of $X\times\Sigma$. As the complex structure $J$ of $X$ does not
coincide with the Euclidean complex structure $J_{0}$ of the asymptotic
coordinates in general, we will use real coordinates for clarity. By passing
to the universal covering of the end, we are able to work with the global
coordinates. Precisely, let $\\{z_{1},\ldots,z_{n}\\}$ be the asymptotic
complex coordinates of $\mathbb{C}^{n}\backslash B_{R}$ and $w=t+is$ the
complex coordinate of $\Sigma$. The corresponding real coordinates are
$\\{x_{1},\ldots,x_{2n},$ $x_{2n+1}=t,x_{2n+2}={s}\\}$, where
$z_{k}=x_{2k-1}+ix_{2k}$ for $k=1,\ldots,n$. From now on:

  * $\bullet$

Latin indices $i,j,\ldots$ will denote the real coordinates from $1$ to
$2n+2$.

  * $\bullet$

Greek indices $\alpha,\beta,\ldots$ will denote the real coordinates from $1$
to $2n$.

  * $\bullet$

The bold Greek indices $\boldsymbol{\mu},\boldsymbol{\nu}$ will denote the
real coordinates from $2n+1$ to $2n+2$.

In these coordinates, we write the Riemannian metric tensors corresponding to
$\Theta_{\Psi}$ and $\Theta_{\Psi}+dd^{c}\varphi_{\varepsilon}$ as $g_{ij}$
and $(g_{\varphi_{\varepsilon}})_{ij}$, respectively.

Throughout this section, we work in the asymptotic chart of $X$. This allows
us to use the Euclidean metric on $(\mathbb{R}^{2n}\setminus
B_{R})\times\Sigma$ as a reference metric to measure derivatives. This is
helpful because it enables us to write down equations with a good structure.
Let $|\cdot|_{0}$ denote the Euclidean length, $\nabla_{0}$ the Euclidean
Levi-Civita connection and $\nabla_{0,X}$ ($\nabla_{0,\Sigma}$) the component
of $\nabla_{0}$ acting only in the space (time) directions on
$(\mathbb{R}^{2n}\setminus B_{R})\times\Sigma$.

Then, the equation $(E_{\varepsilon})$ can be written as

$\displaystyle\sqrt{\det{\big{(}(g_{{\varphi_{\varepsilon}}})_{ij}\big{)}}}=\upsilon(\varepsilon)\sqrt{\det(g_{ij})}.$
(6.1)

Recall that $\upsilon$ satisfies conditions in (1.8). By differentiating the
log of both sides by $D_{\alpha}=\partial/\partial_{x_{\alpha}}$, we have

$\displaystyle
g_{\varphi_{\varepsilon}}^{ij}D_{\alpha}(g_{\varphi_{\varepsilon}})_{ij}=g^{ij}D_{\alpha}g_{ij}+D_{\alpha}\log\upsilon(\varepsilon).$
(6.2)

The first goal is to rewrite the equation (6.2) to be an elliptic equation in
terms of $D_{\alpha}{\varphi_{\varepsilon}}$. Let $e_{1},\ldots,e_{2n+2}$
represent the real coordinate vector fields of $x_{1},\ldots,x_{2n+2}$. Notice
that
$(g_{\varphi_{\varepsilon}})_{ij}=g_{ij}+dd^{c}{\varphi_{\varepsilon}}(e_{i},Je_{j})$.
We compute $D_{\alpha}$ of the second term:

$\displaystyle\begin{split}D_{\alpha}[dd^{c}{\varphi_{\varepsilon}}(e_{i},Je_{j})]=&-d\circ
J\circ
d(D_{\alpha}{\varphi_{\varepsilon}})(e_{i},Je_{j})-d\circ(D_{\alpha}J)\circ
d{\varphi_{\varepsilon}}(e_{i},Je_{j})\\\ &-d\circ J\circ
d{\varphi_{\varepsilon}}(e_{i},(D_{\alpha}J)e_{j}).\end{split}$ (6.3)

Observe that $D_{\alpha}J$ is completely horizontal because $J$ preserves the
product structure of the tangent bundle $T((\mathbb{R}^{2n}\setminus
B_{R})\times\Sigma)$ and $J|_{T\Sigma}$ is constant. Thus,

$D_{\alpha}J=(D_{\alpha}J)^{\beta}_{\xi}(e_{\xi}^{*}\otimes
e_{\beta}),\quad(D_{\alpha}J)_{\xi}^{\boldsymbol{\mu}}=0,\quad(D_{\alpha}J)^{\beta}_{\boldsymbol{\nu}}=0,\quad(D_{\alpha}J)^{\boldsymbol{\mu}}_{\boldsymbol{\nu}}=0,$
(6.4)

where the coefficients $(D_{\alpha}J)^{\beta}_{\xi}$ depend only on
$x_{1},\ldots,x_{2n}$ and not on $x_{2n+1},x_{2n+2}$. In the same way, we can
also see that

$|\nabla_{0,X}^{m}(D_{\alpha}J)|_{0}=O(r^{-\tau-1-m})\;\,(\text{all}\;m\geq
0),\;\,\nabla_{0,\Sigma}^{m}(D_{\alpha}J)=0\;\,(\text{all}\;m\geq 1).$ (6.5)

Moreover, it is obvious that

$\Delta_{g_{\varphi_{\varepsilon}}}(D_{\alpha}{\varphi_{\varepsilon}})=\operatorname{tr}_{g_{{\varphi_{\varepsilon}}}}(dd^{c}(D_{\alpha}{\varphi_{\varepsilon}})(\cdot,J\cdot))=g^{ij}_{{\varphi_{\varepsilon}}}dd^{c}(D_{\alpha}{\varphi_{\varepsilon}})(e_{i},Je_{j}).$
(6.6)

Then, (6.3)–(6.4) imply that

$\displaystyle\begin{split}g_{\varphi_{\varepsilon}}^{ij}D_{\alpha}[dd^{c}{\varphi_{\varepsilon}}(e_{i},Je_{j})]&=\Delta_{g_{\varphi_{\varepsilon}}}(D_{\alpha}{\varphi_{\varepsilon}})+\mathbf{O}(r^{-\tau-1})\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0}\nabla_{0,X}{\varphi_{\varepsilon}}\\\
&+\mathbf{O}(r^{-\tau-2})\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0,X}\varphi_{\varepsilon},\end{split}$
(6.7)

where $\circledast$ denotes a contraction and $\mathbf{O}$ denotes the
following behavior of a tensor $T$:

$T=\mathbf{O}(r^{-\rho}):\Longleftrightarrow|\nabla_{0,X}^{m}T|_{0}=O(r^{-\rho-m})\;\,(\text{all}\;m\geq
0),\;\,\nabla_{0,\Sigma}^{m}T=0\;\,(\text{all}\;m\geq 1).$

Then, abbreviating the estimates

$\begin{split}|\nabla_{0,X}^{m}(D_{\alpha}g_{ij})|_{0}=&\;O(r^{-\tau-1-m})\;\,(\text{all}\;m\geq
0),\\\
|\nabla_{0,X}^{m}\nabla_{0,\Sigma}(D_{\alpha}g_{ij})|_{0}=O(r^{-2\tau-1-m})\;\,&(\text{all}\;m\geq
0),\;\,\nabla_{0,\Sigma}^{m}(D_{\alpha}g_{ij})=0\;\,(\text{all}\;m\geq
2),\end{split}$

by $D_{\alpha}g_{ij}=\widehat{\mathbf{O}}(r^{-\tau-1})$ and

$\displaystyle|\nabla^{m}_{0,X}\nabla^{k}_{0,\Sigma}D_{\alpha}\log\upsilon(\varepsilon)|_{0}=O(r^{-\varsigma-1-m})\
(\text{all }m,k\geq 0),$

by
$D_{\alpha}\log\upsilon(\varepsilon)=\widehat{\widehat{\mathbf{O}}}(r^{-\varsigma-1})$
the equation (6.2) can be rewritten as

$\displaystyle\begin{split}\Delta_{g_{\varphi_{\varepsilon}}}(D_{\alpha}{\varphi_{\varepsilon}})&=\mathbf{O}(r^{-\tau-1})\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0}\nabla_{0,X}{\varphi_{\varepsilon}}\\\
&+\mathbf{O}(r^{-\tau-2})\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0,X}\varphi_{\varepsilon}\\\
&+D_{\alpha}g_{ij}\cdot(g_{\varphi_{\varepsilon}}^{ij}-g^{ij}),\quad
D_{\alpha}g_{ij}=\widehat{\mathbf{O}}(r^{-\tau-1}),\\\
&+\widehat{\widehat{\mathbf{O}}}(r^{-\varsigma-1})\end{split}$ (6.8)

We will later use this formula in full but for now it is enough to take
absolute values. Using the fact that
$\Lambda^{-1}\varepsilon\delta_{ij}\leq(g_{{\varphi_{\varepsilon}}})_{ij}\leq\Lambda\delta_{ij}$,
and according to the uniform estimates of
$|\nabla_{0}{\varphi_{\varepsilon}}|_{0}$ and
$|\nabla_{0}^{2}{\varphi_{\varepsilon}}|_{0}$ from Theorem A, the formula
(6.8) implies that

$\displaystyle|\Delta_{g_{\varphi_{\varepsilon}}}(D_{\alpha}{\varphi_{\varepsilon}})|\leq
C\varepsilon^{-1}r^{-\tau-1}$ (6.9)

for some constant
$C=C(\|{\varphi_{\varepsilon}}\|_{\mathcal{C}^{2}(X\times\Sigma,\Theta_{\Psi})},\Lambda,g,J)$
bounded above independently of $\varepsilon$.

### Step 2: Barrier estimate of the first derivatives.

The next target is to construct the upper barrier and lower barrier functions
to control $|D_{\alpha}{\varphi_{\varepsilon}}|$. Consider a smooth cutoff
function $\chi:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ satisfying
$\chi(x)=0$ for $x\leq 1$, $\chi(x)=1$ for $x\geq 2$ and
$|\chi^{\prime}(x)|\leq 4$, $|\chi^{\prime\prime}(x)|\leq 4$ for $1\leq x\leq
2$. The function $D_{\alpha}{\varphi_{\varepsilon}}$ can be extended smoothly
to $X\times\Sigma$ by defining

$\displaystyle h=\chi_{R_{0}}\cdot D_{\alpha}{\varphi_{\varepsilon}},$ (6.10)

where $R_{0}$ is a large positive constant to be determined later such that
$\\{r(p)\geq R_{0}/2\\}$ is contained in the asymptotic chart of $X$ and
$\chi_{R_{0}}(p)=\chi(r(p)/R_{0})$. From (6.9),

$\displaystyle\begin{split}|\Delta_{g_{\varphi_{\varepsilon}}}h|\leq\begin{cases}C\varepsilon^{-1}r^{-\tau-1},&\text{for}\;\,r\geq
2R_{0},\\\
4\Lambda\varepsilon^{-1}\big{(}R_{0}^{-2}|\nabla_{0,X}{\varphi_{\varepsilon}}|_{0}+R_{0}^{-1}|\nabla_{0,X}^{2}{\varphi_{\varepsilon}}|_{0}\big{)}+C\varepsilon^{-1}R_{0}^{-\tau-1},&\text{for}\;\,R_{0}\leq
r\leq 2R_{0},\\\ 0,&\text{for}\;\,r\leq R_{0}.\end{cases}\end{split}$ (6.11)

Then, we can pick a barrier function as follows:

$\displaystyle
u_{1}=E\Big{\\{}\big{(}1-\chi_{\frac{R_{0}}{2}}\big{)}\Big{(}\frac{R_{0}}{2}\Big{)}^{-\tau-1}t(t-1)+\chi_{\frac{R_{0}}{2}}t(t-1)r^{-\tau-1}\Big{\\}}\leq
0,$ (6.12)

where the constant $E$ is to be determined later. The barrier function $u_{1}$
is defined in $X\times\Sigma$ with $u_{1}=0$ on $X\times\partial\Sigma$. We
also have

$\displaystyle\Delta_{g_{\varphi_{\varepsilon}}}u_{1}$
$\displaystyle=\frac{1}{2}\operatorname{tr}_{g_{\varphi_{\varepsilon}}}(dd^{c}u_{1}(\cdot,J\cdot))=\frac{1}{2}\left\\{\sum_{1\leq\alpha,\beta\leq
2n}g^{\alpha\beta}_{{\varphi_{\varepsilon}}}(u_{1,\alpha\beta}+u_{1,J\alpha,J\beta})\right.$
$\displaystyle\left.+\sum_{\begin{subarray}{c}1\leq\alpha\leq 2n,\\\
2n+1\leq\boldsymbol{\mu}\leq
2n+2\end{subarray}}g^{\boldsymbol{\mu}\alpha}_{{\varphi_{\varepsilon}}}(u_{1,\alpha\boldsymbol{\mu}}+u_{1,J\alpha,J{\boldsymbol{\mu}}})+\sum_{2n+1\leq\boldsymbol{\mu},\boldsymbol{\nu}\leq
2n+2}g_{\varphi_{\varepsilon}}^{\boldsymbol{\mu}\boldsymbol{\nu}}(u_{1,\boldsymbol{\mu}\boldsymbol{\nu}}+u_{1,J\boldsymbol{\nu},J\boldsymbol{\nu}})\right\\}.$

Using the estimate
$\Lambda^{-1}\varepsilon\delta_{ij}\leq(g_{{\varphi_{\varepsilon}}})_{ij}\leq\Lambda\delta_{ij}$,
we obtain that

$\displaystyle\Delta_{g_{\varphi_{\varepsilon}}}u_{1}\geq\begin{cases}E(\Lambda^{-1}r^{-\tau-1}-\Lambda\varepsilon^{-1}r^{-\tau-2}-\Lambda\varepsilon^{-1}r^{-\tau-3})&\text{for}\;\,r\geq
R_{0}/2,\\\ E\Lambda^{-1}R_{0}^{-\tau-1}&\text{for}\;\,r\leq
R_{0}/2.\end{cases}$ (6.13)

By taking

$\displaystyle R_{0}\geq 4\Lambda^{2}\varepsilon^{-1},\quad
E=8R_{0}^{\tau}C\;\,\text{with}\;\,C=C(\|{\varphi_{\varepsilon}}\|_{\mathcal{C}^{2}(X\times\Sigma,\Theta_{\Psi})},\Lambda,g,J),$
(6.14)

and comparing with the inequality (6.11), we have
$\Delta_{g_{\varphi_{\varepsilon}}}u_{1}\geq\Delta_{g_{\varphi_{\varepsilon}}}h$.
Together with the fact that $u_{1}=h=0$ on $X\times\partial\Sigma$, Lemma 2.2
implies that $h\geq u_{1}$ in $X\times\Sigma$. The same method shows the upper
bound $h\leq-u_{1}$, which, together with the lower bound, implies that for
each spatial index $1\leq\alpha\leq 2n$,

$\displaystyle|D_{\alpha}{\varphi_{\varepsilon}}|\leq
C\big{(}\|{\varphi_{\varepsilon}}\|_{C^{2}(X\times\Sigma,\Theta_{\Psi})},\Lambda,g,J,\varepsilon^{-1}\big{)}t(1-t)r^{-\tau-1}\;\,\text{on}\;\,\\{r\geq
2R_{0}\\}\times\Sigma.$ (6.15)

### Step 3: Barrier estimate of the second derivatives.

Now, it comes to deal with the asymptotic behavior of the second derivative.

For a preliminary estimate, we go back to the full formula (6.8) for
$\Delta_{g_{\varphi_{\varepsilon}}}(D_{\alpha}\varphi_{\varepsilon})$. For
every $\mathbf{a}\in(0,1)$, the Euclidean $\mathcal{C}^{0,\mathbf{a}}$ norm of
the right-hand side on a restricted unit ball
$\hat{B}_{1}(p)=B_{1}(p)\cap((\mathbb{R}^{2n}\setminus B_{R})\times\Sigma)$
with $r(p)=r\geq 2R$ is still bounded by
$C(\|{\varphi_{\varepsilon}}\|_{C^{2}(X\times\Sigma,\Theta_{\Psi})},\Lambda,g,J,\varepsilon^{-1})r^{-\tau-1}$
thanks to the Evans-Krylov estimates applied to $\varphi_{\varepsilon}$ in the
interior and the estimates of [8, Sections 2.1–2.2] at the boundary. (The
precise dependence of this constant on the ellipticity, and hence on
$\varepsilon^{-1}$, is not clear but also not needed.) Likewise, the
$\mathcal{C}^{0,\mathbf{a}}$ norm of the coefficient tensor of the PDE,
$g_{\varphi_{\varepsilon}}^{-1}$, is bounded by
$C(\|{\varphi_{\varepsilon}}\|_{C^{2}(X\times\Sigma,\Theta_{\Psi})},\Lambda,g,J,\varepsilon^{-1})$.
Applying the classic interior and boundary Schauder estimates to (6.8), we
thus obtain from (6.15) that

$\displaystyle\|D_{\alpha}{\varphi_{\varepsilon}}\|_{\mathcal{C}^{2,\mathbf{a}}(\hat{B}_{1}(p))}\leq
C\big{(}\|\varphi_{\varepsilon}\|_{C^{2}(X\times\Sigma,\Theta_{\Psi})},\Lambda,g,J,\varepsilon^{-1}\big{)}r^{-\tau-1}.$
(6.16)

These estimates will now be used to start a bootstrap to obtain some decay for
$D_{\beta}D_{\alpha}\varphi_{\varepsilon}$ using the same barrier method as in
Step 2. Differentiate the equation (6.8) again by $D_{\beta}=\partial/\partial
x_{\beta}$ for $1\leq\beta\leq 2n$. This yields

$\displaystyle\begin{split}&\Delta_{g_{\varphi_{\varepsilon}}}(D_{\beta}D_{\alpha}{\varphi_{\varepsilon}})=g_{\varphi_{\varepsilon}}^{-1}\circledast
D_{\beta}g_{\varphi_{\varepsilon}}\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0}^{2}\nabla_{0,X}\varphi_{\varepsilon}\\\
&+\mathbf{O}(r^{-\tau-2})\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0}\nabla_{0,X}\varphi_{\varepsilon}\\\
&+\mathbf{O}(r^{-\tau-1})\circledast g_{\varphi_{\varepsilon}}^{-1}\circledast
D_{\beta}g_{\varphi_{\varepsilon}}\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0}\nabla_{0,X}\varphi_{\varepsilon}\\\
&+\mathbf{O}(r^{-\tau-1})\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0}\nabla_{0,X}^{2}\varphi_{\varepsilon}\\\
&+\mathbf{O}(r^{-\tau-3})\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0,X}\varphi_{\varepsilon}\\\
&+\mathbf{O}(r^{-\tau-2})\circledast g_{\varphi_{\varepsilon}}^{-1}\circledast
D_{\beta}g_{\varphi_{\varepsilon}}\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0,X}\varphi_{\varepsilon}\\\
&+\mathbf{O}(r^{-\tau-2})\circledast
g_{\varphi_{\varepsilon}}^{-1}\circledast\nabla_{0,X}^{2}\varphi_{\varepsilon}\\\
&+D_{\beta}D_{\alpha}g_{ij}\cdot(g_{\varphi_{\varepsilon}}^{ij}-g^{ij}),\quad
D_{\beta}D_{\alpha}g_{ij}=\widehat{\mathbf{O}}(r^{-\tau-2}),\\\
&+\widehat{\mathbf{O}}(r^{-\tau-1})\circledast(g_{\varphi_{\varepsilon}}^{-1}\circledast
D_{\beta}g_{\varphi_{\varepsilon}}\circledast
g_{\varphi_{\varepsilon}}^{-1}-g^{-1}\circledast D_{\beta}g\circledast
g^{-1})\\\ &+\widehat{\widehat{\mathbf{O}}}(r^{-\varsigma-2})\end{split}$
(6.17)

As before, we have that $\Lambda^{-1}g^{-1}\leq
g_{\varphi_{\varepsilon}}^{-1}\leq\varepsilon^{-1}\Lambda g^{-1}$, and we also
have

$|D_{\beta}g_{\varphi_{\varepsilon}}|_{0}\leq|D_{\beta}g|_{0}+|\nabla_{0,X}\nabla_{0}^{2}\varphi_{\varepsilon}|_{0}=O(r^{-\tau-1})$
(6.18)

thanks to the preliminary estimate (6.16). Similarly, all derivatives of
$\varphi_{\varepsilon}$ on the right-hand side of (6.17) are at worst of order
$3$, with at least one purely spatial derivative, and hence can be bounded by
$O(r^{-\tau-1})$ thanks to (6.16). In this way, we obtain that

$\displaystyle\Delta_{g_{\varphi_{\varepsilon}}}(D_{\beta}D_{\alpha}{\varphi_{\varepsilon}})=\widehat{\mathbf{O}}(r^{-\tau-2})\circledast(g_{\varphi_{\varepsilon}}^{-1}-g^{-1})+O(r^{-2\tau-2}).$
(6.19)

The majority of terms on the right-hand side actually decay faster than
$O(r^{-2\tau-2})$, and the only term that might decay more slowly is
$\widehat{\mathbf{O}}(r^{-\tau-2})\circledast(g_{\varphi_{\varepsilon}}^{-1}-g^{-1})$.
So far, we can only bound this by $O(r^{-\tau-2})$. However, by applying the
same method as in the weighted estimate of the first derivative in Step 2, we
can then construct the following barrier function for
$D_{\beta}D_{\alpha}{\varphi_{\varepsilon}}$:

$\displaystyle
u_{2}=E^{\prime}\Big{\\{}\big{(}1-\chi_{\frac{R_{0}}{2}}\big{)}\Big{(}\frac{R_{0}}{2}\Big{)}^{-\tau-2}t(t-1)+\chi_{\frac{R_{0}}{2}}t(t-1)r^{-\tau-2}\Big{\\}},$
(6.20)

where $R_{0}$ is the same constant as in (6.12) and $E^{\prime}$ is another
uniform constant depending on $R_{0}$,
$\|{\varphi_{\varepsilon}}\|_{C^{2}(X\times\Sigma,\Theta_{\Psi})}$, $\Lambda$,
$g$, $J$ and on the constant of (6.16). Hence, we get the weighted estimate
for $D_{\beta}D_{\alpha}{\varphi_{\varepsilon}}$:

$\displaystyle|D_{\beta}D_{\alpha}{\varphi_{\varepsilon}}|\leq
C\big{(}\|\varphi_{\varepsilon}\|_{C^{2}(X\times\Sigma,\Theta_{\Psi})},\Lambda,g,J,\varepsilon^{-1}\big{)}r^{-\tau-2}.$
(6.21)

According to the full formula (6.17) for
$\Delta_{g_{\varphi_{\varepsilon}}}(D_{\beta}D_{\alpha}\varphi_{\varepsilon})$
and (6.16), in the restricted unit ball $\hat{B}_{1}(p)$, the
$\mathcal{C}^{0,\mathbf{a}}$ norm of all terms on the right hand side of
(6.17) are bounded by
$C(\|{\varphi_{\varepsilon}}\|_{C^{2}},\Lambda,g,J,\varepsilon^{-1})r^{-\tau-2}$.
Applying the classic interior and boundary Schauder estimates to (6.17), we
thus obtain from (6.15) that

$\displaystyle\|D_{\alpha}D_{\beta}{\varphi_{\varepsilon}}\|_{\mathcal{C}^{2,\mathbf{a}}(\hat{B}_{1}(p))}\leq
C\big{(}\|\varphi_{\varepsilon}\|_{C^{2}(X\times\Sigma,\Theta_{\Psi})},\Lambda,g,J,\varepsilon^{-1}\big{)}r^{-\tau-2}.$
(6.22)

### Step 4: Iterative improvement of the barrier estimates.

In this step, we improve the decay order of the estimates we obtain in (6.16)
and (6.22) by an iteration argument. Recall that from Steps 2–3 we have the
following weighted estimates to start the iteration process (see (6.16) and
(6.22)):

$\displaystyle\begin{split}\|\nabla_{0,X}\varphi_{\varepsilon}\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}+\|\nabla_{0}\nabla_{0,X}\varphi_{\varepsilon}\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}&\\\
+\;\|\nabla_{0}^{2}\nabla_{0,X}\varphi_{\varepsilon}\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}&=O(r^{-\tau-1}),\\\
\|\nabla^{2}_{0,X}\varphi_{\varepsilon}\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}+\|\nabla_{0}\nabla_{0,X}^{2}\varphi_{\varepsilon}\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}&=O(r^{-\tau-2}).\end{split}$
(6.23)

To complete the iteration argument, we need to improve the decay of the term
$g_{\varphi_{\varepsilon}}^{-1}-g^{-1}$. More precisely, this term occurs in a
combination $(g_{\varphi_{\varepsilon}}^{ij}-g^{ij})D_{\alpha}g_{ij}$ in the
first derivative estimate (Step 2), and in combinations
$(g_{\varphi_{\varepsilon}}^{ij}-g^{ij})D_{\beta}D_{\alpha}g_{ij}$ and
$[g_{\varphi_{\varepsilon}}^{ik}D_{\beta}(g_{\varphi_{\varepsilon}})_{kl}g_{\varphi_{\varepsilon}}^{jl}-g^{ik}D_{\beta}g_{kl}g^{jl}]D_{\alpha}g_{ij}$
(to get optimal decay rate of $D_{\alpha}D_{\beta}\varphi_{\varepsilon}$, we
need to analyze this term) in the second derivative estimate (Step 3). We will
now analyze these combinations more carefully. All constants in this step may
depend on
$\|\varphi_{\varepsilon}\|_{C^{2}(X\times\Sigma,\Theta_{\Psi})},\Lambda,g,J,\varepsilon^{-1}$.
Let $\varphi$ be a continuous function defined in $(X\backslash
B_{R})\times\Sigma$ with at most polynomial growth rate at infinity, for
simplicity, we introduce the notation $(\varphi)^{\sharp}$ to denote the decay
rate of $\varphi$ and $(D_{X}\varphi)^{\sharp}$, $(D_{X}^{2}\varphi)^{\sharp}$
to denote the decay rate of
$\|\nabla_{0,X}\varphi\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}+\|\nabla_{0}\nabla_{0,X}\varphi\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}+\|\nabla_{0}^{2}\nabla_{0,X}\varphi\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}$,
$\|\nabla^{2}_{0,X}\varphi\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}+\|\nabla_{0}\nabla_{0,X}^{2}\varphi\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}$
respectively.

The metric tensor $(g_{\varphi_{\varepsilon}})_{ij}$ and its inverse can be
written as $(2n+2)\times(2n+2)$-matrices

$\displaystyle\widetilde{P}=\begin{pmatrix}P&\eta^{t}\\\
\eta&\mathfrak{p}\end{pmatrix},\quad(\widetilde{P})^{-1}=\begin{pmatrix}Q&\xi^{t}\\\
\xi&\mathfrak{q}\end{pmatrix},$

where $P$, $Q$ are $2n\times 2n$-matrices, $\mathfrak{p}$, $\mathfrak{q}$ are
$2\times 2$-matrices and $\eta$, $\xi$ are $2\times 2n$-matrices. By direct
calculation, we have

$\displaystyle Q=P^{-1}-P^{-1}\eta^{t}\xi,\quad\xi=-\mathfrak{p}^{-1}\eta
Q,\quad\mathfrak{q}=(I_{2}-\xi\eta^{t})\mathfrak{p}^{-1}.$ (6.24)

The fact that $\Lambda^{-1}\varepsilon I_{2n+2}\leq\widetilde{P}\leq\Lambda
I_{2n+2}$ implies that $|\xi|\leq C|\eta|$. The weighted estimate (6.23),
together with the fall-off condition of the metric $g$, implies that
$|\eta|=O(r^{(D_{X}\varphi_{\varepsilon})^{\sharp}})$. Then, from (6.24), we
have

$\displaystyle
Q=P^{-1}+O(|\eta|^{2}),\quad\xi=O(|\eta|),\quad\mathfrak{q}=\mathfrak{p}^{-1}+O(|\eta|^{2}).$
(6.25)

Similarly, let $\widetilde{P}^{\prime}$ denote the matrix of $g$ in asymptotic
coordinates. If we write

$\displaystyle\widetilde{P}^{\prime}=\begin{pmatrix}P^{\prime}&(\eta^{\prime})^{t}\\\
\eta^{\prime}&\mathfrak{p}^{\prime}\end{pmatrix},\quad(\widetilde{P}^{\prime})^{-1}=\begin{pmatrix}Q^{\prime}&(\xi^{\prime})^{t}\\\
\xi^{\prime}&\mathfrak{q}^{\prime}\end{pmatrix},$

then we have

$\displaystyle
Q^{\prime}=(P^{\prime})^{-1}+O(|\eta^{\prime}|^{2}),\quad\xi^{\prime}=O(|\eta^{\prime}|),\quad\mathfrak{q}^{\prime}=(\mathfrak{p}^{\prime})^{-1}+O(|\eta^{\prime}|^{2}),$
(6.26)

where $|\eta^{\prime}|=O(r^{-\gamma-1})$. According to the estimate (6.23),
$|P-P^{\prime}|=O(r^{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp}})$ and hence
$|P^{-1}-(P^{\prime})^{-1}|=O(r^{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp}})$
as well because $P,P^{\prime}$ are uniformly bounded. Moreover,
$\mathfrak{p},\mathfrak{q},\mathfrak{p}^{\prime},\mathfrak{q}^{\prime}$ are
all uniformly equivalent to $I_{2}$ but there is no reason for
$\mathfrak{p}-\mathfrak{p}^{\prime}$ to decay. Then (6.25) and (6.26) imply
that

$\displaystyle\begin{split}|Q-Q^{\prime}|&=O(r^{\max\\{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp},2(D_{X}\varphi_{\varepsilon})^{\sharp},-2\gamma-2\\}}),\\\
|\xi-\xi^{\prime}|&=O(r^{\max\\{(D_{X}\varphi_{\varepsilon})^{\sharp},-\gamma-1\\}}),\\\
|\mathfrak{p}-\mathfrak{p}^{\prime}|&=O(1).\end{split}$ (6.27)

Then, by calculating blockwise and using that
$D_{\alpha}g_{\boldsymbol{\mu}\boldsymbol{\nu}}=0$, we have

$\displaystyle\begin{split}(g_{\varphi_{\varepsilon}}^{ij}-g^{ij})D_{\alpha}g_{ij}&=|Q-Q^{\prime}|O(r^{-\tau-1})+|\xi-\xi^{\prime}|O(r^{-\gamma-2}),\\\
(g_{\varphi_{\varepsilon}}^{ij}-g^{ij})D_{\beta}D_{\alpha}g_{ij}&=|Q-Q^{\prime}|O(r^{-\tau-2})+|\xi-\xi^{\prime}|O(r^{-\gamma-3}).\end{split}$
(6.28)

By inserting (6.27), (6.28) into (6.8), we have

$\displaystyle\begin{split}|D_{\alpha}\varphi_{\varepsilon}|&=O(r^{\max\\{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp}-\tau-1,(D_{X}\varphi_{\varepsilon})^{\sharp}-\tau-1,-2\gamma-3,-\varsigma-1\\}}).\end{split}$
(6.29)

For the last but one term of (6.17),

$\displaystyle\begin{split}D_{\alpha}g_{ij}\big{(}g_{\varphi_{\varepsilon}}^{ik}g^{jl}_{\varphi_{\varepsilon}}D_{\beta}(g_{\varphi_{\varepsilon}})_{kl}-g^{ik}g^{jl}D_{\beta}g_{kl}\big{)}&=D_{\alpha}g_{ij}g^{ik}_{\varphi_{\varepsilon}}g^{jl}_{\varphi_{\varepsilon}}(D_{\beta}(g_{\varphi_{\varepsilon}})_{kl}-D_{\beta}g_{kl})\\\
&+D_{\alpha}g_{ij}D_{\beta}g_{kl}g_{\varphi_{\varepsilon}}^{jl}(g^{ik}_{\varphi_{\varepsilon}}-g^{ik})\\\
&+D_{\alpha}g_{ij}D_{\beta}g_{kl}g^{ik}(g^{jl}_{\varphi_{\varepsilon}}-g^{jl}).\end{split}$
(6.30)

For the first term of right-hand side of (6.30), by using
$|D_{\beta}(g_{\varphi_{\varepsilon}})_{kl}-D_{\beta}g_{kl}|\leq|\nabla_{0}^{2}\nabla_{0,X}\varphi_{\varepsilon}|$,
we obtain that the decay rate of the first term is given by
$(D_{X}\varphi_{\varepsilon})^{\sharp}-\tau-1$. For the second and third
terms, we need to analyze $(g^{ik}_{\varphi_{\varepsilon}}-g^{ik})$. Similar
to (6.28), we have

$\displaystyle
D_{\alpha}g_{ij}D_{\beta}g_{kl}g^{ik}(g^{jl}_{\varphi_{\varepsilon}}-g^{jl})=|Q-Q^{\prime}|O(r^{-2\tau-2})+|\xi-\xi^{\prime}|O(r^{-\tau-\gamma-3})+O(r^{-2\gamma-4}).$
(6.31)

By inserting (6.27) into (6.31), we have

$\displaystyle\begin{split}D_{\alpha}g_{ij}\big{(}g_{\varphi_{\varepsilon}}^{ik}g^{jl}_{\varphi_{\varepsilon}}&D_{\beta}(g_{\varphi_{\varepsilon}})_{kl}-g^{ik}g^{jl}D_{\beta}g_{kl}\big{)}=O(r^{\max\\{(D_{X}\varphi_{\varepsilon})^{\sharp}-\tau-1,(D_{X}^{2}\varphi_{\varepsilon})^{\sharp}-2\tau-2,-2\gamma-4\\}}).\end{split}$
(6.32)

Then, inserting (6.27), (6.28) into (6.17), we have

$\displaystyle|D_{\alpha}D_{\beta}\varphi_{\varepsilon}|$
$\displaystyle=O(r^{\max\\{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp}-\tau-1,(D_{X}\varphi_{\varepsilon})^{\sharp}-\tau-1,-2\gamma-4,-\varsigma-2\\}}).$

We can go one step further by applying Schauder estimates to (6.8) and (6.17)
and to obtain $\mathcal{C}^{2,\mathbf{a}}$ estimates for
$D_{\alpha}\varphi_{\varepsilon}$ and
$D_{\alpha}D_{\beta}\varphi_{\varepsilon}$ in $\hat{B}_{1}(p)$. Indeed, those
terms on the right-hand side of the PDEs (6.8), (6.17) that were known to
decay pointwise with rate
$\max\\{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp},(D_{X}\varphi_{\varepsilon})^{\sharp}\\}-\tau-1$
already after Step 3 are actually also decaying at rate
$\max\\{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp},(D_{X}\varphi_{\varepsilon})^{\sharp}\\}-\tau-1$
in $\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))$ norm. This is clear from
(6.23). So we just need to find the decay rates of the most difficult terms,
$(g_{\varphi_{\varepsilon}}^{ij}-g^{ij})D_{\alpha}g_{ij}$ in (6.8) and
$(g_{\varphi_{\varepsilon}}^{ij}-g^{ij})D_{\beta}D_{\alpha}g_{ij}$,
$D_{\alpha}g_{ij}(g_{\varphi_{\varepsilon}}^{ik}g^{jl}_{\varphi_{\varepsilon}}D_{\beta}(g_{\varphi_{\varepsilon}})_{kl}-g^{ik}g^{jl}D_{\beta}g_{kl})$
in (6.17) in $\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))$ norm as well. For
this, we need to go back and also estimate the
$\mathcal{C}^{0,\mathbf{a}}$-norm of $Q-Q^{\prime}$ and $\xi-\xi^{\prime}$ in
$\hat{B}_{1}(p)$, as follows. By using (6.23), we have that

$\displaystyle[P^{-1}-(P^{\prime})^{-1}]_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}=O(r^{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp}}),\quad[\xi]_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}=O(r^{(D_{X}\varphi_{\varepsilon})^{\sharp}}).$

Then, based on (6.25), we have that

$\displaystyle\begin{split}[Q-Q^{\prime}]_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}&=O(r^{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp},2(D_{X}\varphi_{\varepsilon})^{\sharp},-2\gamma-2}),\\\
[\xi-\xi^{\prime}]_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}&=O(r^{\max\\{(D_{X}\varphi_{\varepsilon})^{\sharp},-\gamma-1\\}}).\end{split}$
(6.33)

Then we can proceed as in (6.28) and (6.32), obtaining that the decay rates of
$[\Delta_{\varphi_{\varepsilon}}D_{\alpha}\varphi_{\varepsilon}]_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}_{1}(p))}$
and
$[\Delta_{\varphi_{\varepsilon}}D_{\alpha}D_{\beta}\varphi_{\varepsilon}]_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}(p))}$
are
$\max\\{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp}-\tau-1,(D_{X}\varphi_{\varepsilon})^{\sharp}-\tau-1,-2\gamma-3,-\varsigma-2\\}$
and
$\max\\{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp}-\tau-1,(D_{X}\varphi_{\varepsilon})^{\sharp}-\tau-1,-2\gamma-4,-\varsigma-2\\}$
respectively. According to the classic interior and boundary Schauder
estimates, we improve (6.29) to $\mathcal{C}^{2,\mathbf{a}}(\hat{B}_{1}(p))$
norm,

$\displaystyle\begin{split}\|D_{\alpha}\varphi_{\varepsilon}\|_{\mathcal{C}^{2,\mathbf{a}}(\hat{B}_{1}(p))}&=O(r^{\max\\{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp}-\tau-1,(D_{X}\varphi_{\varepsilon})^{\sharp}-\tau-1,-2\gamma-3,-\varsigma-1\\}}),\\\
\|D_{\alpha}D_{\beta}\varphi_{\varepsilon}\|_{\mathcal{C}^{2,\mathbf{a}}(\hat{B}_{1}(p))}&=O(r^{\max\\{(D_{X}^{2}\varphi_{\varepsilon})^{\sharp}-\tau-1,(D_{X}\varphi_{\varepsilon})^{\sharp}-\tau-1,-2\gamma-4,-\varsigma-2\\}}).\end{split}$
(6.34)

Inserting (6.23) into (6.34), and using (6.34) again to improve (6.23), we
finally obtain the following estimates:

$\displaystyle\|D_{\alpha}\varphi_{\varepsilon}\|_{\mathcal{C}^{2,\mathbf{a}}(\hat{B}_{1}(p))}=O(r^{\max\\{-2\gamma-3,-\varsigma-1\\}}),\quad\|D_{\alpha}D_{\beta}\varphi_{\varepsilon}\|_{\mathcal{C}^{2,\mathbf{a}}(\hat{B}_{1}(p))}=O(r^{\max\\{-2\gamma-4,-\varsigma-2\\}}).$
(6.35)

Note that according to (6.35), because $\Psi$ was chosen to be linear in $t$,
the decay rate of $\varphi_{\varepsilon}$ is faster than the decay rate of the
boundary data $\psi_{0},\psi_{1}$.

### Step 5: Proof of Theorem B

In Step 4, we have obtained the optimal decay rates in the cases of $k=1,2$
(even though it is not required in the proof of Theorem B). In this step, we
give optimal estimates for $k\geq 3$ and complete the proof of Theorem B.

For the higher order derivatives, by differentiating the Monge-Ampère equation
(6.2) $m$ times, similar to (6.8) and (6.17) and writing
$D_{K}=D_{\kappa_{1}}\cdots D_{\kappa_{m}}$ ($1\leq\kappa_{i}\leq 2n$, for
$i=1,\ldots,m$), instead of giving a full formula as (6.8) and (6.17), we
write a simplified formula of
$\Delta_{\varphi_{\varepsilon}}D_{K}\varphi_{\varepsilon}$:

$\displaystyle\begin{split}|\Delta_{\varphi_{\varepsilon}}D_{K}\varphi_{\varepsilon}|&\leq\sum_{i=1}^{m}O(r^{-\tau-2-m+i})|\nabla_{0,X}^{i}\varphi_{\varepsilon}|+\sum_{i=1}^{m}O(r^{-\tau-1-m+i})|\nabla_{0}\nabla_{0,X}^{i}\varphi_{\varepsilon}|\\\
&+\sum_{i=1}^{m-1}O(r^{-\tau-m+i})|\nabla_{0}^{2}\nabla_{0,X}^{i}\varphi_{\varepsilon}|+\sum_{i=1}^{m}|\nabla_{0,X}^{i}g_{jl}||\nabla_{0,X}^{m-i}(g^{jl}_{\varphi_{\varepsilon}}-g^{jl})|\\\
&+O(r^{-\varsigma-m}).\end{split}$ (6.36)

Applying induction on $m$, according to iteration process (step 4), we can
assume for $k\leq m-1$

$\displaystyle||\nabla^{k}_{0,X}\varphi_{\varepsilon}||_{\hat{B}_{1}(p)}=O(r^{\max\\{-2\gamma-2,-\varsigma\\}-k}).$
(6.37)

To find the optimal decay rates, the most difficult term is
$\sum_{i=1}^{m}|\nabla_{0,X}^{i}g_{jl}||\nabla_{0,X}^{m-i}(g^{jl}_{\varphi_{\varepsilon}}-g^{jl})|$.
Notice that by (6.31) and (6.34), we have

$\displaystyle\big{|}D_{K_{1}}{g_{jl}}D_{K_{2}}g_{ik}(g^{ij}_{\varphi_{\varepsilon}}-g^{ij})\big{|}=O(r^{-2\gamma-2-k_{1}-k_{2}}),$

where $K_{1}$, $K_{2}$ are $k_{1}$-, $k_{2}$-multi-indices respectively. Then,
we apply induction on $k$ to find the decay rate of
$|D_{K_{1}}{g_{jl}}D_{K_{2}}g_{ik}D_{K}(g^{ij}_{\varphi_{\varepsilon}}-g^{ij})|$,
where $K$ is a $k$-multi-index. Applying one derivative to
$(g^{ij}_{\varphi_{\varepsilon}}-g^{ij})$, by (6.30), we can prove that

$\displaystyle|D_{K_{1}}{g_{jl}}D_{K_{2}}g_{ik}D_{K}(g^{ij}_{\varphi_{\varepsilon}}-g^{ij})|=O(r^{-2\gamma-2-k_{1}-k_{2}-k}).$
(6.38)

Then, by (6.30) and (6.38), we have

$\displaystyle|\nabla_{0,X}^{i}g_{jl}||\nabla_{0,X}^{m-i}(g^{jl}_{\varphi_{\varepsilon}}-g^{jl})|$
$\displaystyle\leq|\nabla^{i}_{0,X}g_{jl}|\Big{\\{}\big{|}\nabla_{0,X}^{m-i-1}\big{[}g^{jk}_{\varphi_{\varepsilon}}g^{sl}_{\varphi_{\varepsilon}}(\nabla_{0,X}(g_{\varphi_{\varepsilon}})_{ks}-\nabla_{0,X}g_{ks})\big{]}\big{|}$
$\displaystyle+2\big{|}\nabla^{m-i-1}\big{[}g_{\varphi_{\varepsilon}}^{jk}(g^{ls}_{\varphi_{\varepsilon}}-g^{ik})\nabla_{0,X}g_{ks}\big{]}\big{|}\Big{\\}}$
$\displaystyle=O(r^{-2\gamma-2-m}).$

Combining with (6.37), we have that the right-hand side of (6.36) is
$O(r^{-2\tau+2-k})$. Using the construction of barrier functions in Step 2–3,
we obtain that $|D_{K}\varphi_{\varepsilon}|\leq Cr^{-2\tau+2-m}$. To apply
Schauder estimates to the $m$-th derivative of Monge-Ampère equation, we also
need to know the decay rate of
$[\Delta_{\varphi_{\varepsilon}}D_{K}\varphi_{\varepsilon}]_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}(p))}$:

$\displaystyle[\Delta_{\varphi_{\varepsilon}}D_{K}\varphi_{\varepsilon}]_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}(p))}$
$\displaystyle\leq\sum_{i=1}^{m}O(r^{-\tau-2-m+i})\|\nabla_{0,X}^{i}\varphi_{\varepsilon}\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}(p))}$
$\displaystyle+\sum_{i=1}^{m}O(r^{-\tau-1-m+i})\|\nabla_{0}\nabla_{0,X}^{i}\varphi_{\varepsilon}\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}(p))}$
$\displaystyle+\sum_{i=1}^{m-1}O(r^{-\tau-m+i})\|\nabla_{0}^{2}\nabla_{0,X}^{i}\varphi_{\varepsilon}\|_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}(p))}$
$\displaystyle+\sum_{i=1}^{m}\big{\|}|\nabla_{0,X}^{i}g_{jl}||\nabla_{0,X}^{m-i}(g^{jl}_{\varphi_{\varepsilon}}-g^{jl})|\big{\|}_{\mathcal{C}^{0,\mathbf{a}}(\hat{B}(p))}+O(r^{-\varsigma-m})$
$\displaystyle=O(r^{{\max\\{-2\gamma-2,-\varsigma\\}-m}})$

Hence, we have
$\|D_{K}\varphi_{\varepsilon}\|_{\mathcal{C}^{2,\mathbf{a}}(\hat{B}_{1}(p))}\leq
Cr^{{{\max\\{-2\gamma-2,-\varsigma\\}-m}}}$, for $m\geq 1$. To prove that
$\varphi_{\varepsilon}$ is in $\mathcal{H}_{-\gamma}$, by integrating
$({\varphi_{\varepsilon}})_{r}=O(r^{\max\\{-2\gamma-2,-\varsigma\\}-1})$ in
the radial direction from infinity to $r=R$, we obtain a function
$\hat{\varphi}_{\varepsilon}$ defined in $X\setminus B_{R}$ with decay rate
$\max\\{-2\gamma-2,-\varsigma\\}$. Then,

$\displaystyle\varphi_{\varepsilon}-\hat{\varphi}_{\varepsilon}=c(\theta,t),$
(6.39)

where $c(\theta,t)$ is a function in $X\setminus B_{R}$ independent of radius
$r$ and $\theta$ be viewed as a variable on the link. It suffices to prove
that $c(\theta,t)$ is independent of $\theta$. By taking derivative of (6.39),
we have $|\nabla_{0,X}c(\theta,t)|=O(r^{\max\\{-2\gamma-2,-\varsigma\\}-1})$.
In the case that $c(\theta,t)$ is not constant with respect to $\theta$,
$|\nabla_{0,X}c(\theta,t)|\sim r^{-1}$, which contradicts to the fact that
$\max\\{-2\gamma-2,-\varsigma\\}<-1$. Hence we proved that
$\varphi_{\varepsilon}=c(t)+O(r^{-\max\\{-2\gamma-2,-\varsigma\\}})$. We
conclude that, for $\Phi_{\varepsilon}=\varphi_{\varepsilon}+\Psi$,

$\displaystyle\sup_{(\mathbb{R}^{2n}\backslash
B_{R})\times\Sigma}\left(|\nabla^{k}_{0,X}\Phi_{\varepsilon}|+|\nabla^{k}_{0,X}\dot{\Phi}_{\varepsilon}|+|\nabla^{k}_{0,X}\ddot{\Phi}_{\varepsilon}|\right)\leq
C(k,\varepsilon^{-1})r^{-\gamma-k}\quad\text{for all}\ k\geq 1.$

In conclusion, we have proved Theorem B.

##  7\. Convexity of the Mabuchi $K$-energy

According to Theorem 1.1 (assuming $\tau=\tilde{\tau}$), we can restrict
ourselves to the space

$\displaystyle\mathcal{H}_{-2\tau+2}=\\{\varphi\in\hat{\mathcal{C}}^{\infty}_{-2\tau+2}:\omega_{\varphi}=\omega+dd^{c}\varphi>0\\},\quad\tau>n-1,$

and the function $\upsilon(\varepsilon)$ is constructed by (1.9) and (1.10).
In the previous section, we proved that for any two given boundary data
$\psi_{0},\psi_{1}\in\mathcal{H}_{-2\tau+2}$, there exists a solution of the
$\varepsilon$-geodesic equation $(E_{\varepsilon})$ in the same space
$\mathcal{H}_{-2\tau+2}$.

The derivative of the Mabuchi $K$-energy can be defined as follows: for
$\psi\in T_{\varphi}\mathcal{H}_{-2\tau+2}$,

$\displaystyle\delta_{\psi}\mathcal{K}(\varphi)=-\int_{X}\psi
R(\omega_{\varphi})\frac{\omega_{\varphi}^{n}}{n!}.$

The integral converges because $-2-2\tau<-2n$, equivalently, $\tau>n-1$. In
the following proposition, the second derivative of Mabuchi $K$-energy will be
calculated in $M=X_{R}=\\{x\in X:r(x)\leq R\\}$ containing boundary terms, and
it will be clear that these boundary terms go to zero as $R\to\infty$.
Precisely, we consider Mabuchi $K$-energy restricted in $M$,

$\displaystyle\delta_{\psi}\mathcal{K}_{M}(\varphi)=-\int_{M}\psi
R(\omega_{\varphi})\omega_{\varphi}^{n}.$ (7.1)

The calculation of the second variation of $\mathcal{K}_{M}$ is due to my
advisor Bianca Santoro in one of her unpublished notes, several years before I
started this project. The limiting case $R\to\infty$ was previously stated by
Aleyasin [1] without details concerning the vanishing of boundary terms.

To simplify the notation, in the following proposition, we write
$R_{\varphi}=R(\omega_{\varphi})$,
$\textup{Ric}_{\varphi}=\textup{Ric}(\omega_{\varphi})$,
$\Delta=g^{i\overline{k}}_{\varphi}\partial_{i}\partial_{\overline{k}}$,
$|\cdot|=|\cdot|_{\omega_{\varphi}}$, $\nabla=\nabla_{\omega_{\varphi}}$ and
$\mathcal{D}f=\nabla_{i}\nabla_{k}fdz^{i}dz^{k}$, where
$\nabla_{i}\nabla_{k}f=f_{,ik}$ is a covariant derivative of $f$ with respect
to $\omega_{\varphi}$. Recall that $\mathcal{D}$ is called the Lichnerowicz
operator, and $\mathcal{D}f=0$ if and only if ${\rm grad}^{1,0}f$ is a
holomorphic type $(1,0)$ vector field.

######  Proposition 7.1 (Santoro).

Along a path of potentials $\varphi(t)\in\mathcal{H}_{-2\tau+2}$,

$\displaystyle\begin{split}\frac{d^{2}\mathcal{K}_{M}}{dt^{2}}&=-\int_{M}[\ddot{\varphi}-\frac{1}{2}|\nabla\dot{\varphi}|^{2}]R_{\varphi}\omega_{\varphi}^{n}+\int_{M}|\mathcal{D}\dot{\varphi}|^{2}\omega_{\varphi}^{n}\\\
&-\frac{n(n-1)}{2}\int_{\partial
M}\dot{\varphi}d^{c}\dot{\varphi}\wedge\textup{Ric}_{\varphi}\wedge\omega_{\varphi}^{n-2}+ni\int_{\partial
M}\dot{\varphi}g_{\varphi}^{k\overline{l}}(\textup{Ric}_{\varphi})_{i\overline{l}}{\dot{\varphi}}_{k}dz^{i}\wedge\omega_{\varphi}^{n-1}\\\
&-ni\int_{\partial
M}\dot{\varphi}g_{\varphi}^{i\overline{j}}{\dot{\varphi}}_{,ik\overline{j}}dz^{k}\wedge\omega_{\varphi}^{n-1}-ni\int_{\partial
M}g_{\varphi}^{k\overline{l}}{\dot{\varphi}}_{\overline{l}}{\dot{\varphi}}_{,ki}dz^{i}\wedge\omega_{\varphi}^{n-1}.\end{split}$
(7.2)

Furthermore, by taking $R\to\infty$ in (7.2), we have

$\displaystyle\frac{d^{2}\mathcal{K}}{dt^{2}}=\int_{X}\big{[}\ddot{\varphi}-\frac{1}{2}|\nabla\dot{\varphi}|^{2}\big{]}R_{\varphi}\frac{\omega_{\varphi}^{n}}{n!}+\int_{X}|\mathcal{D}\dot{\varphi}|^{2}\frac{\omega_{\varphi}^{n}}{n!}.$
(7.3)

###### Proof.

By taking the second derivative of Mabuchi $K$-energy in $M$, we have

$\displaystyle\frac{d^{2}\mathcal{K}_{M}}{dt^{2}}=$
$\displaystyle\frac{d}{dt}\left[-\int_{M}R_{\varphi}\dot{\varphi}\,\omega_{\varphi}^{n}\right]$
$\displaystyle=$
$\displaystyle-\int_{M}\ddot{\varphi}\,R_{\varphi}\,\omega_{\varphi}^{n}-n\int_{M}\dot{\varphi}\frac{d}{dt}(\textup{Ric}_{\varphi})\wedge\omega_{\varphi}^{n-1}$
(7.4)
$\displaystyle-n(n-1)\int_{M}\dot{\varphi}\textup{Ric}_{\varphi}\wedge\omega_{\varphi}^{n-2}\wedge(i\partial\overline{\partial}\dot{\varphi}).$

The second term of (7.4) needs one integration by parts, and we get

$\displaystyle-n\int_{M}\dot{\varphi}\frac{d}{dt}(\textup{Ric}_{\varphi})\wedge\omega_{\varphi}^{n-1}$
$\displaystyle=-n\int_{M}\dot{\varphi}\left[-i\partial\overline{\partial}\left(\frac{d}{dt}(\log\omega_{\varphi}^{n})\right)\wedge\omega_{\varphi}^{n-1}\right]$
$\displaystyle=n\int_{M}\dot{\varphi}\left[i\partial\overline{\partial}\left(\frac{n\omega_{\varphi}^{n-1}\wedge
i\partial\overline{\partial}\dot{\varphi}}{\omega_{\varphi}^{n}}\right)\wedge\omega_{\varphi}^{n-1}\right]$
$\displaystyle=\int_{M}\dot{\varphi}(\Delta^{2}\dot{\varphi})\omega_{\varphi}^{n}.$

Now, to the term
$\int_{M}\dot{\varphi}\textup{Ric}_{\varphi}\wedge\omega_{\varphi}^{n-2}\wedge
i\partial\overline{\partial}\dot{\varphi}$. For simplicity, $\dot{\varphi}=u$,

$\displaystyle\int_{M}\dot{\varphi}\textup{Ric}_{\varphi}\wedge\omega_{\varphi}^{n-2}\wedge
i\partial\overline{\partial}\dot{\varphi}$ $\displaystyle=-i\int_{M}\partial
u\wedge\overline{\partial}u\wedge\textup{Ric}_{\varphi}\wedge\omega_{\varphi}^{n-2}+\frac{1}{2}\int_{\partial
M}ud^{c}u\wedge\textup{Ric}_{\varphi}\wedge\omega_{\varphi}^{n-2}$
$\displaystyle=-i\int_{M}\partial
u\wedge\overline{\partial}u\wedge\mathring{\textup{Ric}}_{\varphi}\wedge\omega^{n-2}_{\varphi}-\frac{i}{n}\int_{M}\partial
u\wedge\overline{\partial}u\wedge R_{\varphi}\omega^{n-1}_{\varphi}$
$\displaystyle+\frac{1}{2}\int_{\partial
M}ud^{c}u\wedge\textup{Ric}_{\varphi}\wedge\omega_{\varphi}^{n-2}$
$\displaystyle=-i\int_{M}\partial
u\wedge\overline{\partial}u\wedge\mathring{\textup{Ric}}_{\varphi}\wedge\omega^{n-2}_{\varphi}-\frac{1}{2n^{2}}\int_{M}|\nabla
u|^{2}R_{\varphi}\omega^{n}_{\varphi}$
$\displaystyle+\frac{1}{2}\int_{\partial
M}ud^{c}u\wedge\textup{Ric}_{\varphi}\wedge\omega_{\varphi}^{n-2},$

where $\mathring{\textup{Ric}}$ is the traceless part of Ricci. If $\psi$ is
any primitive $(1,1)$-form, then

$\displaystyle*\psi=\frac{-1}{(n-2)!}\psi\wedge\omega^{n-2},\;\,\text{and
hence}\;\,n(n-1)\mathring{\textup{Ric}}\wedge\omega_{\varphi}^{n-2}=-n!({*\mathring{\textup{Ric}}}).$

Hence,

$\displaystyle n(n-1)\int_{M}i\partial
u\wedge\overline{\partial}u\wedge\mathring{\textup{Ric}}_{\varphi}\wedge\omega_{\varphi}^{n-2}$
$\displaystyle=-\int_{M}n!(*\mathring{\textup{Ric}}_{\varphi})\wedge(i\partial
u\wedge\overline{\partial}u)$
$\displaystyle=-\int_{M}\langle\mathring{\textup{Ric}}_{\varphi},i\partial
u\wedge\overline{\partial}u\rangle\,\omega_{\varphi}^{n}$
$\displaystyle=-\int_{M}\langle\textup{Ric}_{\varphi},i\partial
u\wedge\overline{\partial}u\rangle\,\omega_{\varphi}^{n}+\int_{M}\langle\tfrac{1}{n}R_{\varphi}\,\omega_{\varphi},i\partial
u\wedge\overline{\partial}u\rangle\,\omega_{\varphi}^{n}.$

Note that

$\displaystyle\int_{M}\langle\tfrac{1}{n}R_{\varphi}\,\omega_{\varphi},i\partial
u\wedge\overline{\partial}u\rangle\,\omega_{\varphi}^{n}$
$\displaystyle=\int_{M}\langle(n-1)!R_{\varphi}\,\omega_{\varphi},i\partial
u\wedge\overline{\partial}u\rangle\,\frac{\omega_{\varphi}^{n}}{n!}$
$\displaystyle=\int_{M}(i\partial
u\wedge\overline{\partial}u)\wedge*[(n-1)!R_{\varphi}\omega_{\varphi}]$
$\displaystyle=\int_{M}R_{\varphi}(i\partial
u\wedge\overline{\partial}u)\wedge\omega_{\varphi}^{n-1}$
$\displaystyle=\frac{1}{2n}\int_{M}|\nabla
u|^{2}R_{\varphi}\,\omega_{\varphi}^{n}.$

Thus, we get that

$\displaystyle\begin{split}\frac{d^{2}\mathcal{K}_{M}}{dt^{2}}=&-\int_{M}[\ddot{\varphi}-\frac{1}{2}|\nabla\dot{\varphi}|^{2}]R_{\varphi}\,\omega_{\varphi}^{n}-\int_{M}\langle\textup{Ric}_{\varphi},i\partial
u\wedge\overline{\partial}u\rangle\,\omega_{\varphi}^{n}\\\
&+\int_{M}u(\Delta^{2}u)\,\omega_{\varphi}^{n}-\frac{n(n-1)}{2}\int_{\partial
M}ud^{c}u\wedge\textup{Ric}_{\varphi}\wedge\omega_{\varphi}^{n-2}.\end{split}$
(7.5)

######  Lemma 7.2.

Let $f$ be a smooth function defined on $M$. Then we have that

$\displaystyle\Delta^{2}f=\mathcal{D}^{*}\mathcal{D}f-g_{\varphi}^{i\overline{k}}g_{\varphi}^{j\overline{l}}(\textup{Ric}_{\varphi})_{i\overline{l}}f_{j\overline{k}}-g_{\varphi}^{i\overline{k}}g_{\varphi}^{j\overline{l}}(\nabla_{\overline{k}}(\textup{Ric}_{\varphi})_{i\overline{l}})f_{j}.$

Hence,

$\displaystyle\begin{split}\int_{M}u(\Delta^{2}u)\omega_{\varphi}^{n}&-\int_{M}\langle\textup{Ric}_{\varphi},i\partial
u\wedge\overline{\partial}u\rangle\omega_{\varphi}^{n}\\\
&=\int_{M}|\mathcal{D}u|^{2}\omega_{\varphi}^{n}+ni\int_{\partial
M}ug_{\varphi}^{k\overline{l}}(\textup{Ric}_{\varphi})_{i\overline{l}}u_{k}dz^{i}\wedge\omega_{\varphi}^{n-1}\\\
&-ni\int_{\partial
M}ug_{\varphi}^{i\overline{j}}\nabla_{\overline{j}}\nabla_{k}\nabla_{i}udz^{k}\wedge\omega_{\varphi}^{n-1}-ni\int_{\partial
M}g_{\varphi}^{k\overline{l}}u_{\overline{l}}u_{,ki}dz^{i}\wedge\omega_{\varphi}^{n-1}.\end{split}$
(7.6)

###### Proof.

Notice that

$\displaystyle\nabla_{j}\nabla_{\overline{k}}\nabla_{i}f=\nabla_{\overline{k}}\nabla_{j}\nabla_{i}f-\tensor{R}{{}_{i}^{m}{}_{j}{}_{\overline{k}}}f$

Then, we have

$\displaystyle\Delta^{2}f$
$\displaystyle=g^{i\overline{j}}_{\varphi}g^{k\overline{l}}_{\varphi}\nabla_{\overline{l}}\nabla_{k}\nabla_{\overline{j}}\nabla_{i}f$
$\displaystyle=g^{i\overline{j}}_{\varphi}g^{k\overline{l}}_{\varphi}\nabla_{\overline{l}}\big{(}\nabla_{\overline{j}}\nabla_{k}\nabla_{i}f-\tensor{R}{{}_{k}^{m}{}_{i}{}_{\overline{j}}}f_{m}\big{)}$
$\displaystyle=\mathcal{D}^{*}\mathcal{D}f-g_{\varphi}^{k\overline{l}}g_{\varphi}^{m\overline{j}}\textup{Ric}_{k\overline{j}}f_{m\overline{l}}-g_{\varphi}^{k\overline{l}}g_{\varphi}^{m\overline{j}}(\nabla_{\overline{l}}\textup{Ric}_{k\overline{j}})f_{m}.$

Here
$\mathcal{D}^{*}\mathcal{D}=g^{i\overline{j}}g^{k\overline{l}}\nabla_{\overline{l}}\nabla_{\overline{j}}\nabla_{k}\nabla_{i}$.
Then, we have

$\displaystyle\int_{M}u\mathcal{D}^{*}\mathcal{D}u\omega_{\varphi}^{n}=\int_{M}g_{\varphi}^{k\overline{l}}\big{(}ug_{\varphi}^{i\overline{j}}\nabla_{\overline{j}}\nabla_{k}\nabla_{i}u\big{)}_{\overline{l}}\omega_{\varphi}^{n}-\int_{M}\big{(}g_{\varphi}^{k\overline{l}}u_{\overline{l}}g_{\varphi}^{i\overline{j}}\nabla_{\overline{j}}\nabla_{k}\nabla_{i}u\big{)}\omega_{\varphi}^{n}.$

The Stokes’ theorem can be applied to the first term in the above formula by
observing that if we write
$\mathfrak{h}=ih_{k}dz^{k}=i(ug_{\varphi}^{i\overline{j}}\nabla_{\overline{j}}\nabla_{k}\nabla_{i}u)dz^{k}$,
then
$g_{\varphi}^{k\overline{l}}(h_{k})_{\overline{l}}\omega_{\varphi}^{n}=n\overline{\partial}\mathfrak{h}\wedge\omega_{\varphi}^{n-1}$.
Hence,

$\displaystyle\int_{M}g_{\varphi}^{k\overline{l}}\big{(}ug_{\varphi}^{i\overline{j}}\nabla_{\overline{j}}\nabla_{k}\nabla_{i}u\big{)}_{\overline{l}}\omega_{\varphi}^{n}=\int_{\partial
M}n\mathfrak{h}\wedge\omega_{\varphi}^{n-1}.$

Similarly,

$\displaystyle-\int_{M}\big{(}g_{\varphi}^{k\overline{l}}u_{\overline{l}}g_{\varphi}^{i\overline{j}}\nabla_{\overline{j}}\nabla_{k}\nabla_{i}u\big{)}\omega_{\varphi}^{n}$
$\displaystyle=-\int_{M}g_{\varphi}^{i\overline{j}}\big{(}g_{\varphi}^{k\overline{l}}u_{\overline{l}}\nabla_{k}\nabla_{i}u\big{)}_{\overline{j}}\omega_{\varphi}^{n}+\int_{M}|\mathcal{D}u|^{2}\omega_{\varphi}^{n}$
$\displaystyle=\int_{M}|\mathcal{D}u|^{2}\omega_{\varphi}^{n}-ni\int_{\partial
M}g_{\varphi}^{k\overline{l}}u_{\overline{l}}u_{,ki}dz^{i}\wedge\omega_{\varphi}^{n-1}.$

We have

$\displaystyle\int_{M}u\Delta^{2}u$
$\displaystyle=\int_{M}|\mathcal{D}u|^{2}-\int_{M}ug_{\varphi}^{k\overline{l}}g_{\varphi}^{m\overline{j}}(\textup{Ric}_{\varphi})_{k\overline{j}}u_{m\overline{l}}-\int_{M}ug_{\varphi}^{k\overline{l}}g_{\varphi}^{m\overline{j}}\big{(}\nabla_{\overline{l}}(\textup{Ric}_{\varphi})_{k\overline{j}}\big{)}u_{m}$
$\displaystyle-ni\int_{\partial
M}ug_{\varphi}^{i\overline{j}}\nabla_{\overline{j}}\nabla_{k}\nabla_{i}udz^{k}\wedge\omega_{\varphi}^{n-1}-ni\int_{\partial
M}g_{\varphi}^{k\overline{l}}u_{\overline{l}}u_{,ki}dz^{i}\wedge\omega_{\varphi}^{n-1}.$

Notice that

$\displaystyle-\int_{M}\langle\textup{Ric}_{\varphi},i\partial
u\wedge\overline{\partial}u\rangle\omega_{\varphi}^{n}=-\int_{M}g_{\varphi}^{i\overline{j}}g_{\varphi}^{k\overline{l}}(\textup{Ric}_{\varphi})_{i\overline{l}}u_{k}u_{\overline{j}}\omega_{\varphi}^{n},$

and integrating by parts,

$\displaystyle-\int_{M}g_{\varphi}^{i\overline{j}}g^{k\overline{l}}_{\varphi}(\textup{Ric}_{\varphi})_{i\overline{l}}u_{k}u_{\overline{j}}\omega_{\varphi}^{n}$
$\displaystyle=\int_{M}g^{i\overline{j}}_{\varphi}\big{(}g^{k\overline{l}}_{\varphi}(\textup{Ric}_{\varphi})_{i\overline{l}}u_{k}u\big{)}_{\overline{j}}\omega_{\varphi}^{n}+\int_{M}ug^{i\overline{j}}_{\varphi}g^{k\overline{l}}_{\varphi}(\textup{Ric}_{\varphi})_{i\overline{l}}u_{k\overline{j}}\omega_{\varphi}^{n}$
$\displaystyle+\int_{M}ug_{\varphi}^{i\overline{j}}g_{\varphi}^{k\overline{l}}\big{(}\nabla_{\overline{j}}(\textup{Ric}_{\varphi})_{i\overline{l}}\big{)}u_{k}\omega_{\varphi}^{n}$
$\displaystyle=ni\int_{\partial
M}ug_{\varphi}^{k\overline{l}}(\textup{Ric}_{\varphi})_{i\overline{l}}u_{k}dz^{i}\wedge\omega_{\varphi}^{n-1}+\int_{M}ug^{i\overline{j}}_{\varphi}g^{k\overline{l}}_{\varphi}(\textup{Ric}_{\varphi})_{i\overline{l}}u_{k\overline{j}}\omega_{\varphi}^{n}$
$\displaystyle+\int_{M}ug_{\varphi}^{i\overline{j}}g_{\varphi}^{k\overline{l}}\big{(}\nabla_{\overline{j}}(\textup{Ric}_{\varphi})_{i\overline{l}}\big{)}u_{k}\omega_{\varphi}^{n}.$

Hence, we proved that

$\displaystyle\int_{M}u(\Delta^{2}u)\omega_{\varphi}^{n}$
$\displaystyle-\int_{M}\langle\textup{Ric}_{\varphi},i\partial
u\wedge\overline{\partial}u\rangle\omega_{\varphi}^{n}=\int_{M}|\mathcal{D}u|^{2}\omega_{\varphi}^{n}+ni\int_{\partial
M}ug_{\varphi}^{k\overline{l}}(\textup{Ric}_{\varphi})_{i\overline{l}}u_{k}dz^{i}\wedge\omega_{\varphi}^{n-1}$
$\displaystyle-ni\int_{\partial
M}ug_{\varphi}^{i\overline{j}}\nabla_{\overline{j}}\nabla_{k}\nabla_{i}udz^{k}\wedge\omega_{\varphi}^{n-1}-ni\int_{\partial
M}g_{\varphi}^{k\overline{l}}u_{\overline{l}}u_{,ki}dz^{i}\wedge\omega_{\varphi}^{n-1},$

which completes the proof of the lemma. ∎

The integration formula (7.6) in this lemma, together with (7.5), completes
the proof of (7.2). It suffices to show that all boundary terms in this
formula vanish as $R\to\infty$. According to Theorem B, we can check that the
decay rates of the integrands integrated on $\partial M$ are at most
$-2\tau-1<-2n+1$. This completes the proof. ∎

######  Theorem 7.3.

Assume that $\omega$ is an ALE Kähler metric on $X$ such that the Ricci
curvature of $\omega$ is non-positive, $\textup{Ric}(\omega)\leq 0$. Then,
along each $\varepsilon$-geodesic in $\mathcal{H}_{-2\tau+2}(\omega)$,
$\varphi(t)$, the Mabuchi $K$-energy is convex.

###### Proof.

The proof is parallel to Chen [9]. Here, we just do the calculation in the ALE
setting. Define $\displaystyle
f=\ddot{\varphi}-\frac{1}{2}|\nabla\dot{\varphi}|_{\omega_{\varphi}}^{2}$.
Then the $\varepsilon$-geodesic equation can be written as

$\displaystyle\varepsilon\frac{\omega^{n}}{\omega^{n}_{\varphi}}=f.$

According to (7.3), together with the observation,
$\textup{Ric}(\omega_{\varphi})=\textup{Ric}(\omega)+dd^{c}\log f$, we have

$\displaystyle\frac{d^{2}\mathcal{K}}{dt^{2}}$
$\displaystyle=\int_{X}\big{|}\mathcal{D}\dot{\varphi}(t)\big{|}^{2}_{\omega_{\varphi}}\omega_{\varphi}^{n}-\int_{X}fR(\omega_{\varphi})\,\omega_{\varphi}^{n}$
$\displaystyle=\int_{X}\big{|}\mathcal{D}\dot{\varphi}(t)\big{|}^{2}_{\omega_{\varphi}}\omega_{\varphi}^{n}-\int_{X}f\operatorname{tr}_{\omega_{\varphi}}\textup{Ric}(\omega)\,\omega_{\varphi}^{n}-\int_{X}f\Delta_{\omega_{\varphi}}\log
f\,\omega^{n}_{\varphi}$
$\displaystyle=\int_{X}\big{|}\mathcal{D}\dot{\varphi}(t)\big{|}^{2}_{\omega_{\varphi}}\omega_{\varphi}^{n}-\int_{X}f\operatorname{tr}_{\omega_{\varphi}}\textup{Ric}(\omega)\,\omega_{\varphi}^{n}+\int_{X}\frac{|\nabla
f|^{2}_{\omega_{\varphi}}}{f}\,\omega_{\varphi}^{n}\geq 0.$

We have the last equality because $f|\nabla\log
f|_{\omega_{\varphi}}=O(r^{-2\tau-1})$ and $-2\tau-1<-(2n-1)$, so that the
relevant boundary integral vanishes. Hence, we have proved the convexity of
the Mabuchi $K$-energy. ∎

######  Remark 7.4.

A quick corollary of Theorem 7.3 is that assuming $\textup{Ric}(\omega)\leq
0$, the scalar-flat Kähler metric if it exists, is unique in
$\mathcal{H}_{-2\tau+2}(\omega)$. The proof is also parallel to Chen [9].
However, if there exists a scalar-flat Kähler metric in
$\mathcal{H}_{-2\tau+2}(\omega)$, the condition, $\textup{Ric}(\omega)\leq 0$,
implies $\textup{Ric}(\omega)=0$. Hence, the uniqueness of scalar-flat ALE
Kähler metric can be reduced to the uniqueness result of Ricci-flat ALE Kähler
metric (which can be found in many reference [20, 28, 13]). A short proof is
given as follows. Let $\omega_{0}$ be a scalar-flat Kähler metric in
$\mathcal{H}_{-2\tau+2}(\omega)$. The fact, $\omega_{0}=\omega+O(r^{-2\tau})$,
implies that the ADM mass of $\omega$ and $\omega_{0}$ are equal,
$\mathfrak{m}(\omega)=\mathfrak{m}(\omega_{1})$. According to the mass formula
by Hein-LeBrun [19],

$\displaystyle\mathfrak{m}(\omega)=A(n,c_{1}(X),[\omega])+B(n)\int_{X}R(\omega)\frac{\omega^{n}}{n!},$

where $A(n,c_{1}(X),[\omega])$ is a constant only determined by the dimension
$n$, the first Chern class of $X$ and the cohomology class of $\omega$ and
$B(n)$ only depends on dimension $n$. The fact,
$\mathfrak{m}(\omega)=\mathfrak{m}(\omega_{1})$, together with the mass
formula, implies that

$\displaystyle\int_{X}R(\omega)=\int_{X}R(\omega_{1})=0.$

The assumption that $\textup{Ric}(\omega)\leq 0$ implies that
$\textup{Ric}(\omega)=0$. Then, by a simple argument, we can prove that all
scalar-flat ALE Kähler metrics in $[\omega]$ is actually Ricci-flat. The
expansion of scalar-flat Kähler metrics (Theorem 1.1) implies that the Ricci
form, $\textup{Ric}(\omega_{1})$, decays to zero at infinite with decay rate
faster than $-2n$. The ddbar lemma implies that there exist
$f\in\mathcal{C}^{\infty}_{2-2n}$ such that

$\displaystyle\textup{Ric}(\omega_{1})=dd^{c}f.$

Taking trace with respect to $g$, we have that $\Delta f=0$. By solving the
Laplacian equation (for instance, see [30, Propsition 2.3]), there is a unique
solution in the space $\mathcal{C}^{\infty}_{-\delta}$ (for
$-\delta\in(-\infty,0)\backslash D$). Hence, $f\equiv 0$, which implies that
$\omega$ is Ricci-flat.

##  8\. Nonexistence of non-positive (or non-negative) Ricci curvature

Consider the standard family of negative line bundles, $\mathcal{O}(-k)$, over
$\mathbb{C}\mathbb{P}^{n-1}$ together with their natural projections
$\pi:\mathcal{O}(-k)\rightarrow\mathbb{C}\mathbb{P}^{n-1}$. The total spaces
of $\mathcal{O}(-k)$ are fundamental examples of ALE Kähler manifolds by
viewing $\mathcal{O}(-k)$ as a resolution space of
$\mathbb{C}^{n}/\mathbb{Z}_{k}$. Let $\omega$ be any ALE Kähler metric on
$\mathcal{O}(-k)$ asymptotic to the Euclidean metric with decay rate $-\tau$
($\tau>0$). In the following, we shall prove the nonexistence of a sign of the
Ricci curvature of $\omega$ in the case $k\neq n$. When $k=n$, there always
exists a Ricci-flat ALE Kähler metric in each compactly supported ALE Kähler
class, see [21, 22, 27].

######  Theorem 8.1.

Let $\mathcal{O}(-k)$ be the standard negative line bundle over
$\mathbb{C}\mathbb{P}^{n-1}$ with $n\geq 2$ and $k\neq n$. Let $\omega$ be an
ALE Kähler metric on $\mathcal{O}(-k)$ with decay rate $-\tau$ $(\tau>0)$.
Then, the Ricci form of $\omega$, $\rho$, is of mixed type, i.e., neither
$\rho\geq 0$ nor $\rho\leq 0$ is true.

###### Proof.

Notice that for each integer $k\geq 1$, there is a compactification of
$\mathcal{O}(-k)$ by adding a divisor at infinity,
$D_{\infty}\cong\mathbb{C}\mathbb{P}^{n-1}$. We denote the compactified
manifold as $M_{k}$ and the natural embedding $j:\mathcal{O}(-k)\rightarrow
M_{k}$ is holomorphic. $M_{k}$ is a $\mathbb{C}\mathbb{P}^{1}$-bundle over
$\mathbb{C}\mathbb{P}^{n-1}$. Denote $D_{0}$ as the divisor corresponding to
the base manifold,
$\mathbb{C}\mathbb{P}^{n-1}\subset\mathcal{O}(-k)\hookrightarrow M_{k}$. Then,
the normal line bundles of $D_{0}$ and $D_{\infty}$ are given by

$\displaystyle N_{D_{0}/M_{k}}=\mathcal{O}(-k),\quad
N_{D_{\infty}/M_{k}}=\mathcal{O}(k).$ (8.1)

The following facts on the geometry of $M_{k}$ can be checked by viewing
$M_{k}$ as a smooth toric variety. $M_{k}$ can be described by $2n$ coordinate
charts with coordinates $\\{U_{i};\ u_{i}^{1},\ldots,$ $u_{i}^{n-1},u_{i}\\}$,
$\\{V_{i};\ v_{i}^{1},\ldots,v_{i}^{n-1},v_{i}\\}$ ($0\leq i\leq n-1$), where
the coordinates are related by

$\displaystyle(u^{1}_{i},\ldots,u^{n-1}_{i},u_{i})$
$\displaystyle=\left(\frac{1}{u^{i}_{0}},\frac{u_{0}^{1}}{u^{i}_{0}},\ldots,\widehat{\frac{u^{i}_{0}}{u^{i}_{0}}},\ldots,\frac{u^{n-1}_{0}}{u^{i}_{0}},(u^{i}_{0})^{k}u^{n}_{1}\right),\quad
1\leq i\leq n-1,$ $\displaystyle(v^{1}_{i},\ldots,v^{n-1}_{i},v_{i})$
$\displaystyle=\left(u^{1}_{i},\ldots,u^{n-1}_{i},\frac{1}{u_{i}}\right),\quad
0\leq i\leq n-1.$

The divisor classes of $M_{k}$ are generated by the class of
$D_{0}\cong\mathbb{C}\mathbb{P}^{n-1}$, the zero section of
$\mathcal{O}(-k)\subset M_{k}$, and the class of $D_{f}$, the total space of
the restriction of the $\mathbb{C}\mathbb{P}^{1}$-bundle $M_{k}\to D_{0}$ to a
linear subspace of $D_{0}$. Restricting $D_{0}$ and $D_{f}$ to $U_{0}$, we can
write

$\displaystyle D_{0}=\overline{(u_{0}=0)},\qquad
D_{f}=\overline{(u_{0}^{1}=0)}.$

The divisor at infinity, $D_{\infty}$, can be represented by
$(u_{0}=\infty)=(v_{0}=0)$ and $D_{\infty}$ can be represented in terms of
$D_{0}$ and $D_{f}$ as follows,

$\displaystyle D_{\infty}=D_{0}+kD_{f}$ (8.2)

By viewing $D_{0}$, $D_{f}$ and $D_{\infty}$ as smooth complex hypersurfaces
of $M_{k}$, the Poincaré duals of $D_{0}$, $D_{f}$ and $D_{\infty}$ have
natural explicit representatives denoted by $\rho_{0}$, $\rho_{f}$,
$\rho_{\infty}$ respectively. For instance, in $U_{0}$,

$\displaystyle\rho_{0}|_{U_{0}}$
$\displaystyle=\frac{1}{n\pi}i\partial\overline{\partial}\log\frac{(1+\sum_{j}|u_{0}^{j}|^{2})^{k}|u_{0}|^{2}+1}{(1+\sum_{j}|u_{0}^{j}|^{2})^{k}|u_{0}|^{2}},$
(8.3) $\displaystyle\rho_{f}|_{U_{0}}$
$\displaystyle=\frac{1}{n\pi}i\partial\overline{\partial}\log\big{(}1+\sum_{j}|u_{0}^{j}|^{2}\big{)},$
(8.4) $\displaystyle\rho_{\infty}|_{U_{0}}$
$\displaystyle=\frac{1}{n\pi}i\partial\overline{\partial}\log\big{[}\big{(}1+\sum_{j}|u_{0}^{j}|^{2}\big{)}^{k}|u_{0}|^{2}+1\big{]}.$
(8.5)

### Step 1: Extension of the ALE Ricci form to $M_{k}$.

Recall that the diffeomorphism
$\Phi:(\mathbb{C}^{n})^{*}/\mathbb{Z}_{k}\rightarrow\mathcal{O}(-k)\setminus
D_{0}$ gives a holomorphic asymptotic chart of $\mathcal{O}(-k)$. The
diffeomorphism $\Phi$ can be explicitly written as

$\displaystyle{\Phi}:(\mathbb{C}^{n})^{*}\rightarrow\mathcal{O}(-k)\setminus
D_{0},\quad{\Phi}(z_{1},\ldots,z_{n})\big{|}_{U_{0}}=\Big{(}\frac{z_{2}}{z_{1}},\ldots,\frac{z_{n}}{z_{1}},{z_{1}^{k}}\Big{)}.$

In the coordinate chart $\\{U_{0};\ u_{0}^{1},\ldots,u_{0}^{n-1},u_{0}\\}$, we
have $r^{2k}=(1+\sum_{j}|u_{0}^{j}|^{2})^{k}|u_{0}|^{2}$. By the asymptotic
condition of $\omega$, in an asymptotic chart of $\mathcal{O}(-k)$,
$\log(\omega^{n}/\omega_{0}^{n})$ can be viewed as a function of decay order
$O(r^{-\tau})$, where $\omega_{0}$ is the standard Euclidean metric on the
asymptotic chart. Thus, the Ricci form satisfies

$\displaystyle\rho=-i\partial\overline{\partial}\log\frac{\omega^{n}}{\omega^{n}_{0}}=O(r^{-\tau-2}).$

The adjunction formula tells us that as line bundles over $\mathcal{O}(-k)$,

$\displaystyle K_{\mathcal{O}(-k)}=\frac{n-k}{k}[D_{0}].$

Since $\rho$ is the curvature form of a Hermitian metric on
$K_{\mathcal{O}(-k)}^{-1}$ and $\rho_{0}$ is the curvature form of a Hermitian
metric on $[D_{0}]$, it follows that

$\displaystyle\rho+\frac{n-k}{k}\rho_{0}\;\,\text{is globally
$i\partial\overline{\partial}$-exact}.$

By restricting $\rho_{0}$ in (8.3) to the asymptotic chart of
$\mathcal{O}(-k)$, we have

$\displaystyle\rho_{0}=-i\partial\overline{\partial}\log(1+r^{-2k}).$

Hence, by Theorem 1.1, $\rho$ can be written as

$\displaystyle\rho=-\frac{n-k}{k}\rho_{0}+i\partial\overline{\partial}f\quad\text{
for
}f\in\mathcal{C}^{\infty}_{-\tau^{\prime}}(\mathcal{O}(-k)),\quad\tau^{\prime}=\min\\{2k,\tau\\}>0.$

Since $\rho$ cannot be extended smoothly to $M_{k}$, we define a smooth cut-
off function $\chi$,

$\displaystyle\chi(t)=\begin{cases}1,\quad&0\leq t\leq 1,\\\ 0,\quad&t\geq
2,\\\ \text{smooth},\quad&1<t<2,\end{cases}$

and we define $\chi_{R}(t)=\chi(t/R)$. Applying the cutoff function, we can
extend $\rho$ to be

$\displaystyle\rho_{R}=\begin{cases}\displaystyle-\frac{n-k}{k}\rho_{0}+i\partial\overline{\partial}\big{(}\chi_{R}f\big{)},\quad&\text{in
}M_{k}\setminus D_{\infty},\\\ \vspace{-2.5mm}\\\
\displaystyle-\frac{n-k}{k}\rho_{0},\quad&\text{on }D_{\infty}.\end{cases}$

### Step 2: Integral argument for $n=2$.

Recall that the intersection numbers between $D_{\infty}$, $D_{0}$ and $D_{f}$
are given by

$\displaystyle(D_{0})\cdot(D_{0})=-k,\quad(D_{0})\cdot(D_{f})=1,\quad(D_{f})\cdot(D_{f})=0,\quad(D_{0})\cdot(D_{\infty})=0.$
(8.6)

In particular, if we integrate $\rho$ over $D_{0}$, then

$\displaystyle\int_{D_{0}}{\rho}=\int_{D_{0}}\rho_{R}=\int_{M_{k}}{\rho_{R}}\wedge\rho_{0}=-\frac{2-k}{k}\int_{M_{k}}\rho_{0}^{2}=2-k.$
(8.7)

On the other hand, we have $\rho_{R}\to\rho$ pointwise and
$\rho_{R}=O(r^{-\tau^{\prime}-2})$ uniformly as $R\to\infty$. Hence, by the
dominated convergence theorem,

$\displaystyle\int_{\\{u^{1}_{0}=0\\}}\rho=\lim_{R\rightarrow\infty}\int_{\\{u^{1}_{0}=0\\}}\rho_{R}=\int_{M_{k}}{\rho_{R}}\wedge\rho_{f}=-\frac{2-k}{k}\int_{M_{k}}\rho_{0}\wedge\rho_{f}=\frac{k-2}{k}.$
(8.8)

Now assume that $\rho$ is seminegative (or semi-positive). Then the left-hand
sides of both (8.7) and (8.8) are non-positive (or non-negative). However, the
right-hand sides have opposite signs because $k\neq 2$. This is a
contradiction.

### Step 3: Integral argument for $n\geq 3$.

In higher dimension, the difficulty is to calculate the intersection numbers
of divisors. However, in the case of $M_{k}$, we can apply the formula of
intersection numbers on toric varieties [16, Chapter VII.6], or, more
explicitly, take integral of formulas of Poincaré dual (8.3)–(8.5). Notice
that

$\displaystyle\int_{D_{0}}\rho_{0}^{n-1}=(-k)^{n-1}.$ (8.9)

Then, we have

$\displaystyle\int_{D_{0}}{\rho}^{n-1}=\int_{D_{0}}\rho_{R}^{n-1}=\int_{M_{k}}\rho_{R}^{n-1}\wedge\rho_{0}=\Big{(}{-\frac{n-k}{k}}\Big{)}^{n-1}(-k)^{n-1}.$

On the other hand, we have $\rho_{R}\to\rho$ pointwise and
$\rho_{R}=O(r^{-\tau^{\prime}-2})$ uniformly as $R\to\infty$. Hence, by the
dominated convergence theorem,

$\displaystyle\begin{split}\int_{\\{u^{1}_{0}=0\\}}\rho^{n-1}&=\lim_{R\rightarrow\infty}\int_{\\{u^{1}_{0}=0\\}}\rho_{R}^{n-1}=\lim_{R\rightarrow\infty}\int_{D_{f}}\rho_{R}^{n-1}=\lim_{R\to\infty}\int_{M_{k}}\rho_{R}^{n-1}\wedge\rho_{f}\\\
&=\Big{(}{-\frac{n-k}{k}\Big{)}^{n-1}}\int_{M_{k}}\rho_{0}^{n-1}\wedge\rho_{f}=\Big{(}{-\frac{n-k}{k}}\Big{)}^{n-1}(-k)^{n-2},\end{split}$

where the last equality can be observed from (8.2) and (8.9):

$\displaystyle\int_{M_{k}}\rho_{0}^{n-1}\wedge\rho_{f}=\int_{M_{k}}\rho_{0}^{n-1}\wedge\frac{1}{k}(\rho_{\infty}-\rho_{0})=0-\frac{1}{k}\int_{M_{k}}\rho_{0}^{n}=(-k)^{n-2}$

because $\rho_{0}|_{D_{\infty}}=0$. By the same argument as in dimension $2$,
we complete the proof. ∎

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