# Mass and Expansion of Asymptotically Conical Kähler Metrics

Qi Yao

###### Abstract.

We prove an expansion theorem on scalar-flat asymptotically conical (AC)
Kähler metrics. The ADM mass is known to be defined in ALE Riemannian
manifolds. The concept of ADM mass can be generalized to AC cases. Consider an
AC Kähler manifold with asymptotic to a Ricci-flat Kähler metric cone with
complex dimension $n$. Assuming the weak decay conditions required for the
mass to be well-defined, then each scalar-flat AC Kähler metric admits an
expansion that the main term is given by the standard Kähler metric of the
metric cone and the leading error term is of $O(r^{2-2n})$ with coefficient
only depending on the ADM mass and its dimension. Besides, the mass formula by
Hein-LeBrun [12] also can be proved in our setting. As an interesting
application, a new version of the positive mass theorem will also be discussed
in the cases of the resolutions of the Ricci-flat Kähler cones.

##  1\. Introduction and Statement of Main Theorems

Schwarzschild metric is known to be a solution of Einstein’s vacuum equations.
In general space-time $\mathbb{R}^{n+1}$ $(n\geq 3)$, the Schwarzschild metric
has the following explicit expression.

$\displaystyle
g=-(1-\frac{2m}{r^{n-2}})dt^{2}+(1-\frac{2m}{r^{n-2}})dr^{2}+r^{2}g_{S^{n-1}},$

where the constant $m$ is the mass of the the Schwarzschild metric. One can
easily observe that, by restricting to each space slice $t=c$, the
Schwarzschild metric differs from the Euclidean metric by quantity of order
$O(r^{2-n})$.

Motivated by Schwarschild metrics, Arnowitt, Deser and Misner [1] first
generalized the definition of the mass (ADM mass) to asymptotically flat
metrics. Here, we apply the definition of mass to asymptotically locally
Euclidean (ALE) manifolds. Let $(X,g)$ be an complete non-compact Riemannian
manifold, we say $(X,g)$ is ALE if there is a compact subset $K\subseteq X$
such that $X-K$ has finitely many components, denoted by $X_{i,\infty}$ with
$X-K=\bigcup_{i}X_{i,\infty}$ and each $X_{i,\infty}$ is diffeomorphic to
$(\mathbb{R}^{n}-B_{R})/\Gamma_{i}$, where $B_{R}$ is a closed ball of radius
$R$ and $\Gamma$ is a finite subset of $O(n)$. The metric $g$ is asymptotic to
the Euclidean metric with rapid decay rate. Then, the ADM mass of the end
$X_{i,\infty}$ is given by

$\displaystyle
m_{i}(g)=\lim_{r\rightarrow\infty}\frac{\Gamma(\frac{n}{2})}{4(n-1)\pi^{n/2}}\int_{S_{r}/\Gamma_{i}}(g_{ij,i}-g_{ii,j})n^{j}d\mu$
(1.1)

where $S_{r}$ is the Euclidean sphere of radius $r$, $n$ is the outward
Euclidean unit normal vector and $d\mu$ is the volume form induced by standard
metric on $S_{r}$. Bartnik [2] and Chruściel [9] give the appropriate decay
condition to make the ADM mass coordinate-invariant, for Sobolev case and
Hölder case respectively. Here, we list the decay condition of Hölder case for
the metric $g$ on each end $X_{i,\infty}$,

  1. (i)

the scalar curvature $R$ of $g$ belongs to $L^{1}$.

  2. (ii)

the metric $g$ is asymptotic to the Euclidean metric $\delta_{ij}$ at the end
with decay rate $-\tau$ for some $\tau>(n-2)/2$,

$\displaystyle
g_{ij}=\delta_{ij}+O(r^{-\tau}),\qquad|\nabla(\psi^{-1})^{*}g)|_{g_{0}}=O(r^{-\tau-1}).$
(1.2)

In this paper, one of main results is that the scalar-flat ALE Kähler metric
of complex dimension $n$ is of decay rate $2-2n$, which has an expansion,
$g_{ij}=\delta_{ij}+O(r^{2-2n})$ and the leading error term is determined by
the ADM mass of $g$ (see theorem D). Furthermore, the expansion also works in
scalar-flat AC Kähler cases if we replace the Euclidean metric by the standard
metric on Kähler cone.

Rather than directly introduce the expansion theorem, we first introduce an
important tool in Kähler geometry, the ddbar lemma. The ddbar lemma is a
standard result in compact Kähler manifolds and it can be easily proved in ALE
Kähler manifolds if we assume a fast decay condition (see [14, theorem
8.4.4]). In Colon-Hein [10, theorem 3.11], the ddbar lemma is generalized to
asymptotically conical (AC) Kähler manifolds with a lower decay condition
$(\text{only need }-\tau<0)$, with additional assumption of non-negative Ricci
curvature. In the author’s previous paper [24, Proposition 3.6], the ddbar
lemma can be proved in negative line bundle over Kähler C-spaces (compact
simply-connected homogeneous Kähler manifolds) without any decay assumption at
infinity. Here, we give a new version of ddbar lemma by revising the Colon-
Hein’s result and dropping the non-negative Ricci curvature assumption.

The ddbar lemma on the complement of $K\subseteq X$ will also be discussed in
this paper. In Goto [11, section5] and Conlon-Hein [10, appendix A], ddbar
lemma is proved in the case of complex dimension $n\geq 3$ with a trivial
canonical bundle. In this paper, we derive a ddbar lemma on $X\backslash K$
without making the above assumptions, but with high decay rate error terms.

Let $(L,g_{L})$ be a compact Riemannian manifold, the Riemannian cone $C_{L}$
associated with $L$ is defined to be $L\times\mathbb{R}_{>0}$ with Riemannian
metric $g_{0}=dr^{2}+r^{2}g_{L}$. A Riemannian cone $C_{L}$ is said to be
Kähler if there exists a $g_{0}$-parallel complex structure $J_{0}$ such that
the corresponding fundamental form $\omega_{0}=g_{0}(J_{0}\cdot,\cdot)$ is
closed, in particular, $\omega_{0}=i\partial\overline{\partial}r^{2}$. If we
assume that the Kähler cone $(C_{L},g_{0},J_{0})$ is Ricci-flat with
$\dim_{\mathbb{C}}C_{L}=n$, then, according to the standard calculation in
[22, section 1.4] and [3, section 11.1], the link $L$ is Sasaki-Einstein with
$\textup{Ric}g_{L}=2(n-1)g_{L}$.

Let $(X,J,g)$ be an AC Kähler manifold asymptotic to $C_{L}$, where
$(C_{L},g_{0})$ is a Ricci-flat Kähler cone with link $L$. Throughout the
paper we always assume that the manifolds only have one end, then there exists
a compact subset $K\subseteq X$ and $B_{R}=\\{x\in C_{L},r(x)<R\\}$, such that
$\psi:X-K\rightarrow C_{L}-B_{R}$ is a diffeomorphism satisfying the following
decay conditions,

  1. (i)

the scalar curvature $R$ of $g$ belongs to $L^{1}$.

  2. (ii)

the complex structure $J$ on $X$ decays to $J_{0}$, the induced almost complex
structure from $C_{L}$.

  3. (iii)

the metric $g$ is asymptotic to the reference metric $g_{L}$ at the end with
decay rate $-\tau$ for some $\tau>0$, for $i=0,1,\ldots,k$,

$\displaystyle|\nabla^{i}((\psi^{-1})^{*}g-g_{0})|_{g_{0}}=O(r^{-\tau-i}).$
(1.3)

######  Theorem A.

(ddbar lemma) Let $X$ be AC Kähler manifolds asymptotic to a Ricci-flat Kähler
cone $C_{L}$. Let $k$ be a large positive integer, $\alpha\in(0,1)$ and
$\delta>0$,

  1. (i)

Let $\omega$ be a $d$-exact real $(1,1)$-form on $X$ satisfying the decay
condition $\omega\in\mathcal{C}^{k,\alpha}_{-\delta}$. Then, there exists a
real function $\varphi\in\mathcal{C}^{k+2,\alpha}_{2-\delta}$ such that
$\omega=dd^{c}\varphi$.

  2. (ii)

Let $\omega$ be a $d$-exact real $(1,1)$-form on the end $X\backslash K$. Then
there exists a real function
$\varphi\in\mathcal{C}^{k+2,\alpha}_{2-\delta}(X\backslash K)$ such that,

$\displaystyle\omega=dd^{c}\varphi+O(r^{-2n}).$

We generalize the definition of mass to AC Riemannian manifolds. Let
$(C_{L},g_{0})$ be a Riemannian metric cone of link $(L,g_{L})$ and $(X,g)$, a
complete Riemannian manifold asymptotic to $(C_{L},g)$ at infinity. The
definition in (1.1) requires a coordinate system at infinity, which does not
exist in general AC Riemannian manifolds. Hence, we start from a coordinate-
free expression of mass. In Lee’s book [17, Section 3.1.3], the mass is
defined to be

$\displaystyle\mathfrak{m}(g)=\frac{1}{2(2n-1)\operatorname{Vol}(L)}\lim_{r\rightarrow\infty}\int_{L(r)}(\overline{\nabla}^{j}g_{ij}-(\operatorname{tr}_{g_{0}}g)_{i})n^{i}d\operatorname{Vol}_{L(r)},$
(1.4)

where $\overline{\nabla}$ is the Levi-Civita connection with respect to
$g_{0}$ and $n$ is the outer normal vector field on $L(r)$. In Hein-LeBrun
[12], the ADM mass in ALE Kähler manifolds is a quantity determined by the
total scalar curvature and its topological data. A similar mass formula also
holds in AC Kähler manifolds asymptotic to Ricci-flat Kähler cones. Let
$\iota:H^{2}_{c}(X)\rightarrow H^{2}_{dR}(X)$ be the map induced by the
natural embedding of chain complex
$\Omega^{\bullet}_{c}\hookrightarrow\Omega^{\bullet}_{dR}$. Generally
speaking, $\iota$ is not an isomorphism, but the first Chern class of $(X,J)$
always has a pre-image under $\iota$ (see Lemma 4.2). Then, we have the
following mass formula.

######  Theorem B.

Let $(X,J,g)$ be an AC Kähler manifolds asymptotic to a Ricci-flat Kähler cone
$(C_{L},J_{0},g_{0})$ satisfying the decay condition (1.3) with
$\tau=n-1+\epsilon$. Then, we have

$\displaystyle\mathfrak{m}(g)=-\frac{2\pi\langle\iota^{-1}c_{1},[\omega]^{n-1}\rangle}{(2n-1)(n-1)!\operatorname{Vol}(L)}+\frac{1}{2(2n-1)\operatorname{Vol}(L)}\int_{X}R_{g}d\operatorname{Vol}_{g},$
(1.5)

where $[\omega]$ is the Kähler class of $(g,J)$, $c_{1}$ is the first Chern
class of $(X,J)$ and $R_{g}$ is the scalar curvature of $g$ on $X$.

An interesting application of the mass formula (1.5) is the positive mass
theorem. The positive mass theorem is first proved in AE manifolds of lower
dimension by Schoen, Yau [21]. Afterwards, Witten [23] develop a new method to
prove the positive mass theorem on spin manifolds. And in Kähler cases, Hein-
Lebrun [12] confirms the positive mass theorem on AE Kähler manifolds.
However, in ALE cases, Lebrun [15] constructs the first counter-example of
positive mass theorem. Here, we discuss a new version of positive mass theorem
on resolution spaces of Ricci-flat Kähler cones with an isolated canonical
singularity at vertex.

######  Theorem C.

(positive mass theorem) Let $(X,J)$ be a resolution space of a Ricci-flat
Kähler cone $C_{L}$ such that the only singularity $O\in C_{L}$ is canonical.
If $(X,g)$ has scalar curvature $R\geq 0$, then the mass
$\mathfrak{m}(X,g)\geq 0$, and equals zero only if $(X,J,g)$ is a crepant
resolution of $C_{L}$ with a scalar-flat Kähler metric $g$.

Now, we can introduce the following expansion theorem in AC Kähler case.

######  Theorem D.

Let $(X,J)$ be a AC Kähler manifold asymptotic to a Ricci-flat Kähler cone
$(C_{L},J_{0})$ and we assume that $k$ be a large positive integer and
$\tau=n-1+\epsilon$,

  1. (i)

Let $\omega_{1}$, $\omega_{2}$ be Kähler forms on $X$ with the corresponding
metrics satisfying decay condition (1.3) and $R_{1}$, $R_{2}$, the scalar
curvature of $\omega_{1}$ and $\omega_{2}$. If

    1. $\bullet$

$[\omega_{1}]=[\omega_{2}]$

    2. $\bullet$

$R_{1}=R_{2}$,

then $\omega_{2}=\omega_{1}+dd^{c}\varphi$ with the potential
$\varphi\in\mathcal{C}^{k+2,\alpha}_{2-2\tilde{\tau}}$, for some
$\tilde{\tau}>n-1$ depending on $(n,L,\tau)$.

  2. (ii)

Let $\omega$ be a scalar flat Kähler form on X satifying decay condition
(1.3). And the complex structure $J$ is asymptotic to $J_{0}$ satisfying

$\displaystyle
J=J_{0}+O(r^{2-2n-\epsilon^{\prime}}),\qquad\epsilon^{\prime}>0,$ (1.6)

Then, the scalar flat Kähler form $\omega$ admits the following expansion at
infinity. In particular, outside a compact set of $X$, for complex dimension
$n\geq 3$, we have

$\displaystyle\omega=\frac{1}{2}dd^{c}r^{2}+\frac{(2n-1)\mathfrak{m}(\omega)}{2(4-2n)(n-1)}dd^{c}r^{4-2n}+O(r^{-2\tilde{\tau}}),$

where $m(\omega)$ is the ADM mass of $\omega$ and $\tilde{\tau}>n-1$ only
depends on $(n,L,\epsilon^{\prime},\tau)$. And for complex dimension $n=2$, we
have

$\displaystyle\omega=\frac{1}{2}dd^{c}r^{2}+\frac{3\mathfrak{m}(\omega)}{2}dd^{c}\log
r+O(r^{-2\tilde{\tau}}),$

where $\tilde{\tau}=\min\\{\tau,3/2\\}$.

As special cases of AC Kähler metrics, all ALE Kähler metrics satisfy the
expansion theorem. According to the statement (i) of Theorem D in ALE Kähler
cases, we can define Kähler potential spaces with relatively ”good” decay
rate. Precisely, by fixing a Kähler metric $\omega$, then, we can define the
following potential space of ALE Kähler metrics,

$\displaystyle\mathcal{H}_{-\tau}([\omega])=\Big{\\{}f\in\mathcal{C}^{\infty}_{-2\tilde{\tau}}:\omega+i\partial\overline{\partial}f>0,\
\tilde{\tau}=\min\\{\tau,n-\frac{1}{2}\\}\Big{\\}}$

In fact, $\mathcal{H}_{-\tau}([\omega])$ does not contain all ALE Kähler
metrics, but it is enough for ”prescribed scalar curvature” problem, as all
ALE Kähler metrics with the same scalar curvature as $\omega$ are contained in
$\mathcal{H}_{-\tau}([\omega])$. For the statement (ii), in ALE Kähler case,
the fall-off condition of complex structure (1.6) is automatically satisfied.
In fact, for $n\geq 3$, in the asymptotic chart, the complex structure $J$
coincides with the standard one $J_{0}$, and for $n=2$, in the asymptotic
chart, $J=J_{0}+O(r^{-3})$. One can check [12, Lemma 2.3, Proposition 4.5] for
details.

This paper is a part of Ph.D thesis of the author. The author would like to
thank 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. This work is completed while the author is
supported by scholarship from University of Münster, WWU.

##  2\. Preliminary on Analysis

###  2.1. The Laplacian of 1-form on Riemanian cones

This subsection is mainly dedicated to preparing for the proof of Theorem A.
We will solve the laplacian equation of $1$-form on Riemannian cones. The fact
(lemma 2.1) was also claimed in [10, Lemma 3.7(ii)]. Here, we just give a
detailed proof for this lemma.

Let $L$ be a closed Riemannian manifold with dimension $n-1$ ($n\geq 4$) and
$C(L)=L\times\mathbb{R}_{>0}$ with standard cone metric $g_{C}=dr^{2}+r^{2}g$.
Throughout this paper, we will apply the weighted Hölder norms on Riemannian
cones. Let $T$ be a tensor field on the Riemannian cone $(C_{L},g)$ and $U$ be
an open domain in $C_{L}$, the weighted Hölder norm with order $\rho$ is
defined to be the following,

$\displaystyle||T||_{C^{k,\alpha}_{\rho}(U)}=\sum_{i=0}^{k}\sup_{x\in
U}\big{|}(r^{2}+1)^{\frac{1}{2}(-\rho+i)}\nabla^{i}T\big{|}_{g}+\sup_{x,y\in
U}(r^{2}+1)^{\frac{1}{2}(-\rho+k+\alpha)}\frac{|\nabla^{k}T(x)-\nabla^{k}T(y)|_{g}}{|x-y|^{\alpha}}$
(2.1)

where $\nabla$ is the Levi-Civita connection of $g$ and $|T(x)-T(y)|_{g}$ is
defined via parallel transport minimal geodesic from $x$ to $y$. A tensor
field $T$ is belong to $\mathcal{C}^{k,\alpha}_{\rho}(U)$ if the Hölder norm
$||T||_{\mathcal{C}^{k,\alpha}_{\rho}(U)}$ is finite. Similarly, we can define
a weighted Hölder norm on each AC Riemannian manifold as in (2.1) by fixing a
smooth radial function $\tilde{r}$, which is obtained by smoothly extending
the radial function $r$ to the whole manifold.

Let $\Delta=dd^{*}+d^{*}d$ be the Hodge-Laplacian operator on $C_{L}$ and
$\Delta_{L}=d_{L}d^{*}_{L}+d_{L}^{*}d_{L}$, the Hodge-Laplacian operator on
$L$. Let
$0=\lambda^{\prime}_{0}<\lambda^{\prime}_{1}\leq\lambda^{\prime}_{2}\leq\ldots$
(listed with multiplicity) be the increasing sequence of eigenvalues of
$\Delta_{L}$ on functions and $\kappa_{0}$ (constant), $\kappa_{1},\ldots$,
the corresponding eigenfuncions satisfying $||\kappa_{i}||_{L^{2}(L)}=1$, for
all $i\geq 0$. Viewing $\Delta_{L}$ as an operator acting on the spaces of
1-forms on $L$, we immediately get a family of eigen-1-forms,
$\Delta_{L}d_{L}\kappa_{i}=\lambda^{\prime}_{i}d_{L}\kappa_{i}$, $i\geq 1$.
The $L^{2}$-normalization of $\kappa_{i}$ implies that
$||d_{L}\kappa_{i}||_{L^{2}(L)}=(\lambda^{\prime}_{i})^{1/2}$. Besides,
according to Hodge decomposition on $L$, we also have a family of coclosed
eigen-1-forms, $\Delta_{L}\eta_{j}=\lambda^{\prime\prime}_{j}\eta_{j}$
$(d^{*}_{L}\eta_{j}=0)$, where the eigenvalues are listed as an increasing
sequence,
$0\leq\lambda^{\prime\prime}_{1}\leq\lambda^{\prime\prime}_{2}\leq\ldots$ and
$||\eta_{j}||_{L^{2}(L)}=1$. In summary, we have

  1. (i)

Exact eigenforms: $d_{L}\kappa_{i}$ with eigenvalue $\lambda^{\prime}_{i}$ and
$||d_{L}\kappa_{i}||_{L^{2}(L)}=\lambda_{i}^{1/2}$, for $i\geq 1$.

  2. (ii)

Coclosed eigenforms: $\eta_{j}$ with eigenvalue $\lambda^{\prime\prime}_{j}$
and $||\eta_{j}||_{L^{2}(L)}=1$.

Consider an 1-form $\beta$ defined on $C(L)$, then we can write
$\beta=\kappa(r,y)dr+\eta(r,y)$. By spectral decomposition of 1-form on $L$,
we obtain the Fourier series of $\kappa(r,y)$ and $\eta(r,y)$,

$\displaystyle\begin{split}\kappa(r,y)&=\sum_{i\geq 0}f_{i}(r)\kappa_{i}(y)\\\
\eta(r,y)&=\sum_{i\geq 1}g_{i}(r)d_{L}\kappa_{i}(y)+\sum_{j\geq
1}h_{j}(r)\eta_{j}(y).\end{split}$ (2.2)

Based on calculation in [8, (2.14-2.15)] or [7, (3.8)], we have an explicit
formula for the Laplacian of 1-form $\beta=\kappa dr+\eta$. In particular,

$\begin{split}\Delta\beta=\
&dr\big{(}-\kappa^{\prime\prime}-\frac{n-1}{r}\kappa^{\prime}+\frac{n-1}{r^{2}}\kappa+\frac{1}{r^{2}}\Delta_{L}\kappa+\frac{2}{r^{3}}d^{*}_{L}\eta\big{)}\\\
&-\eta^{\prime\prime}-\frac{n-3}{r}\eta^{\prime}-\frac{2}{r}d_{L}\kappa+\frac{1}{r^{2}}\Delta_{L}\eta.\end{split}$
(2.3)

Regarding the Laplacian equation of 1-form, we introduce exceptional sets $A,\
B,\ C$ defined in (2.6), (2.13), (2.15) respectively; namely, the set of
orders of homogeneous harmonic 1-forms of three different types. If we write
$U(r_{0})=\\{x\in C_{L},r(x)>r_{0}\\}$ and $\overline{U}(r_{0})$ represents
its topological closure, then we have the following lemma,

######  Lemma 2.1.

Let $\theta$ be an 1-form defined on $\overline{U}(1)$ with
$\theta\in\mathcal{C}^{k,\alpha}_{\rho}$. If $k\geq 2n+3$ and $\rho+2\notin
A\bigcup B\bigcup C$, then there exists a solution
$\beta\in\mathcal{C}^{k+2,\alpha}_{\rho+2}(\overline{U}(1))$ satisfying the
Laplacian equation, $\Delta\beta=\theta$ and $\beta$ satisfies the estimate.

$\displaystyle||\beta||_{\mathcal{C}^{k+2,\alpha}_{\rho+2}(U(2))}\leq
C||\theta||_{\mathcal{C}^{k,\alpha}_{\rho}(U(1))}.$

where $C$ only depends on $(n,L,k,\rho)$.

###### Proof.

One of key points to solve the Laplacian equation of 1-form is to observe that
the Laplacian equation is equivalent to a system of ODEs based on (2.2) and
(2.3). By spectrum decomposition of the Laplacian, $\beta=\kappa dr+\eta$ can
be written as in (2.2) and similarly, we can also represent
$\theta=\theta_{0}dr+\theta_{1}$ as,

$\displaystyle\theta_{0}=\sum_{i\geq
0}u_{i}(r)\kappa_{i}(y),\qquad\theta_{1}=\sum_{i\geq
1}v_{i}(r)d_{L}\kappa_{i}(y)+\sum_{j\geq 1}w_{j}(r)\eta_{j}(y),$

where $v_{i},\ w_{j}\in\mathcal{C}^{{k,\alpha}}_{\rho+1}$ and
$u_{i}\in\mathcal{C}^{k,\alpha}_{\rho}$. According to (2.3), by comparing the
coefficients, the Laplacian equation $\Delta\beta=\theta$ is equivalent to the
following system of ODEs,

$\displaystyle-h^{\prime\prime}_{j}-\frac{n-3}{r}h^{\prime}_{j}+\frac{\lambda^{\prime\prime}_{j}}{r^{2}}h_{j}=w_{j},\quad(j\geq
1)$ (2.4)

and

$\displaystyle-f^{\prime\prime}_{i}-\frac{n-1}{r}f^{\prime}_{i}+\frac{n-1+\lambda^{\prime}_{i}}{r^{2}}f_{i}-\frac{2\lambda_{i}^{\prime}}{r^{3}}g_{i}=u_{i},$
(2.5a)
$\displaystyle-g^{\prime\prime}_{i}-\frac{n-3}{r}g^{\prime}_{i}+\frac{\lambda^{\prime}_{i}}{r^{2}}g_{i}-\frac{2}{r}f_{i}=v_{i},\quad(i\geq
0)$ (2.5b)

To solve equation (2.4), notice that the corresponding homogeneous equation
has the solutions $r^{a_{j}^{\pm}}$, where the orders are given by

$\displaystyle A=\\{a_{j}^{\pm}-1,j\geq 1\\},\qquad
a^{\pm}_{j}=-\frac{n-4}{2}\pm\sqrt{\Big{(}\frac{n-4}{2}\Big{)}^{2}+\lambda^{\prime\prime}_{j}},$
(2.6)

where $A$ defines the first exceptional set. If we write $\displaystyle
H_{j}=\begin{pmatrix}h_{j}\\\ h^{\prime}_{j}\end{pmatrix}$, then we have the
following representation formula,

$\displaystyle
H_{j}(r)=W_{j}(r)\Big{(}W_{j}^{-1}(1)H_{j}(1)+\int_{1}^{r}W_{j}^{-1}(s)\begin{pmatrix}0\\\
w_{j}(s)\end{pmatrix}ds\Big{)}$ (2.7)

where the Wronskian $W_{\lambda}$ is given by

$\displaystyle W_{j}(r)=\begin{pmatrix}r^{a^{+}_{j}}&r^{a^{-}_{j}}\\\
a^{+}_{j}r^{a^{+}_{j}-1}&a^{-}_{j}r^{a^{-}_{j}-1}\end{pmatrix}.$

Noting that te exceptional set can be ordered nondecreasingly as $\ldots\leq
a_{2}^{-}\leq a_{1}^{-}\leq a_{1}^{+}\leq a_{2}^{+}\leq\ldots$. Without loss
of generality, assume that $\rho>1-n$ (to ensure $\rho+3>a_{j}^{-}$ for $j\geq
1$). Then, we can deduce the following explicit formula of $h_{j}$,

$\displaystyle h_{j}(r)=\hat{h}_{j}(r)+A_{+}r^{a^{+}_{j}}+A_{-}r^{a^{-}_{j}}.$
(2.8)

In case of $a_{j}^{+}>\rho+3$, $\hat{h}_{\lambda}$, $A_{+}$ and $A_{-}$ are
given as follows,

$\displaystyle\hat{h}_{j}$
$\displaystyle=\frac{1}{a^{-}_{j}-a^{+}_{j}}\bigg{(}r^{a^{+}_{j}}\int_{\infty}^{r}s^{1-a^{+}_{j}}w_{j}(s)ds-r^{a^{-}_{j}}\int_{r_{0}}^{r}s^{1-a^{-}_{j}}w_{j}(s)ds\bigg{)},$
$\displaystyle A_{-}$
$\displaystyle=\frac{1}{a^{-}_{j}-a^{+}_{j}}\big{(}-a^{+}_{j}h_{j}(1)+h^{\prime}_{j}(1)\big{)},$
(2.9) $\displaystyle A_{+}$
$\displaystyle=\frac{1}{a^{-}_{j}-a^{+}_{j}}\bigg{(}a^{-}_{j}h_{j}(1)-h^{\prime}_{j}(1)+\int_{1}^{\infty}s^{1-a^{+}_{j}}w_{j}(s)ds\bigg{)}.$

It is obvious to see that $\hat{h}_{j}(r)=O(r^{3+\rho})$. By choosing certain
values $(h_{j}(1),h^{\prime}_{j}(1))$ such that the coefficient $A_{+}$ and
$A_{-}$ are vanishing, then we have $h_{j}(r)=O(r^{3+\rho})$. In case that
$a^{+}_{j}<3+\rho$, $h_{j}(r)$ has the same expression as (2.8) with different
$\hat{h}_{j}$, $A_{+}$ as follows

$\begin{split}\hat{h}_{j}&=\frac{1}{a^{-}_{j}-a^{+}_{j}}\bigg{(}r^{a^{+}_{j}}\int_{1}^{r}s^{1-a^{+}_{j}}w_{j}(s)ds-r^{a^{-}_{j}}\int_{1}^{r}s^{1-a^{-}_{j}}w_{j}(s)ds\bigg{)},\\\
A_{+}&=\frac{1}{a^{-}_{j}-a^{+}_{j}}\big{(}a^{-}_{j}h_{j}(1)-h_{j}^{\prime}(1)\big{)}\end{split}$
(2.10)

and $A_{-}$ has the same formula as in (2.9). The reason we exclude the
exceptional set $A$ is that if $\rho+2=a_{j}^{\pm}-1$, there exists some
$\log$ terms appear in $\hat{h}_{j}$. There is only one special case remains
to check; that is when $n=4$ and $\lambda^{\prime\prime}_{j}=0$. In this case,
the solutions of the corresponding homogenous equation of (2.4) is generated
by $1$ and $\log r$. Then, by similar computation as above, the solution
$h_{j}$ can be written as,

$\displaystyle h_{j}(r)=h(1)+h^{\prime}(1)\log r+\log
r\int_{1}^{r}sw_{j}(s)ds-\int_{1}^{r}\big{(}s\log s\big{)}w_{j}(s)ds$

The assumption $\rho+3>4-n=0$ ensures that $h_{j}(r)=O(r^{{\rho}+3})$. In
conclution, if $\rho+3\notin A$, then we have
$h_{j}\in\mathcal{C}^{k+2,\alpha}_{\rho+3}$. It is also easy to see that
$h_{j}=O(r^{3+\rho})$.

To solve (2.5a) and (2.5b), we introduce an auxiliary functions,
$D_{i}=-g_{i}^{\prime\prime}-(n-1)r^{-1}g^{\prime}_{i}+{\lambda^{\prime}_{i}}{r^{-2}}g_{i}$
and $E_{i}=f-g^{\prime}$, then the equations (2.5a) and (2.5b) can be
rewritten as follows,

$\displaystyle-$ $\displaystyle
E^{\prime\prime}_{i}-\frac{n-1}{r}E^{\prime}_{i}+\frac{\lambda^{\prime}_{i}+n-1}{r^{2}}E_{i}+D^{\prime}_{i}=u_{i},$
(2.11a) $\displaystyle D_{i}-\frac{2}{r}E_{i}=v_{i}.$ (2.11b)

The system of equations can be reduces to

$\displaystyle-E^{\prime\prime}_{i}-\frac{n-3}{r}E^{\prime}_{i}+\frac{\lambda^{\prime}_{i}+n-3}{r^{2}}E_{i}=\vartheta_{i},$
(2.12)

where $\vartheta_{i}=u_{i}-v^{\prime}_{i}\in C^{k-1,\alpha}_{\rho}$. The
equation (2.12) can be solved by the same method as (2.4). Only to notice that
the exceptional set is different from $A$,

$\displaystyle B=\\{b_{i}^{\pm},i\geq 0\\},\qquad
b_{i}^{\pm}=-\frac{n-4}{2}\pm\sqrt{\Big{(}\frac{n-4}{2}\Big{)}^{2}+\lambda^{\prime}_{i}+n-3}.$
(2.13)

If $\rho+2\notin B$, by similar discussion from (2.7) to (2.10), there exists
a solution $E_{i}\in C^{k+1,\alpha}_{\rho+2}$ satisfying (2.12). It suffices
to solve $g$ and $f$. The equation (2.11a) can be rewritten as

$\displaystyle-g^{\prime\prime}_{i}-\frac{n-1}{r}g^{\prime}_{i}+\frac{\lambda^{\prime}_{i}}{r^{2}}g_{i}=\varpi_{i},$
(2.14)

where $\varpi_{i}=2r^{-1}E_{i}+v_{i}\in\mathcal{C}^{k,\alpha}_{\rho+1}$. If we
introduce another exceptional set,

$\displaystyle C=\\{c_{i}^{\pm}-1,i\geq 1\\},\qquad
c_{i}^{\pm}=-\frac{n-2}{2}\pm\sqrt{\Big{(}\frac{n-2}{2}\Big{)}^{2}+\lambda^{\prime}_{i}},$
(2.15)

then by the same method, assuming $\rho+2\notin B\bigcup C$, there exists a
solution $g_{i}\in\mathcal{C}^{k+2,\alpha}_{\rho+3}$. By the definition of
$E_{i}$, we obtain that
$f_{i}=g^{\prime}_{i}+E_{i}\in\mathcal{C}^{k+1,\alpha}_{\rho+2}$. According to
previous discussion, we have found coefficients satisfying the right decay
condition of Fourier series of $\beta$. It remains to show that the Fourier
series converges in $\mathcal{C}^{2}_{loc}$ topology.

If $\lambda_{j}>0$, $j\in J$, we have the following estimte for $w_{j}(r)$,

$\displaystyle w_{j}(r)=\int_{L}(\theta_{1}(r,y),\eta_{j}(y))_{g_{L}}dy$
$\displaystyle={(\lambda^{\prime\prime}_{j})^{-\frac{k}{2}}}\int_{L}\big{(}\Delta_{L}^{\frac{k}{2}}\theta_{1}(r,y),\eta_{j}\big{)}_{g_{L}}dy$
$\displaystyle\leq\operatorname{Vol}(L)^{\frac{1}{2}}(\lambda^{\prime\prime}_{j})^{-\frac{k}{2}}r^{\rho+1}||\theta(r,y)||_{k,\alpha;\rho}.$
(2.16)

By by expression of $\hat{h}_{j}$ in (2.9), we have

$\displaystyle|\hat{h}_{j}(r)|+r|\hat{h}_{j}^{\prime}(r)|+r^{2}|\hat{h}_{j}^{\prime\prime}(r)|\leq
C(n,\rho)r^{\rho+3}\sup|r^{-\rho-1}w_{j}(r)|$ (2.17)

Combining (2.17) with (2.16), we have the estimate

$\displaystyle|\hat{h}_{j}(r)|+r|\hat{h}_{j}^{\prime}(r)|+r^{2}|\hat{h}_{j}^{\prime\prime}(r)|\leq
C(n,L,\rho)(1+\lambda_{j})^{-\frac{k}{2}}r^{\rho+3}||\theta(r,y)||_{k,\alpha;\rho}.$
(2.18)

Applying Moser iteration to $\eta_{j}$, we obtain that

$\displaystyle||\eta_{j}||_{L^{\infty}(L)}\leq
C(n,L)(\lambda^{\prime\prime}_{j})^{\frac{n-1}{2}}||\eta_{j}||_{L^{2}(L)}=C(n,L)(\lambda^{\prime\prime})_{j}^{\frac{n-1}{2}}.$

Then, the Schauder estimates for $\Delta_{L}$ implies that
$||\eta_{j}||_{C^{2,\alpha}(L)}\leq
C(n,L)(1+\lambda^{\prime\prime}_{j})^{\frac{n+1}{2}}$; hence

$\displaystyle|\eta_{j}|+r|\nabla\eta_{j}|+r^{2}|\nabla^{2}\eta_{j}|\leq
C(n,L)(1+\lambda^{\prime\prime}_{j})^{\frac{n+1}{2}}r^{-1}.$ (2.19)

Recall the Weyl’s law for differential forms [6, Appendix by J. Dodziuk],
$\lambda_{j}\sim j^{\frac{2}{n-1}}$. The Weyl’s law together with (2.18) and
(2.19) can deduce that when $k\geq 2n+3$,

$\displaystyle|\hat{h}_{j}(r)\eta_{j}(y)|+r|\nabla(\hat{h}_{j}(r)\eta_{j}(y))|+r^{2}|\nabla^{2}(\hat{h}_{j}(r)\eta_{j}(y))|\leq
C(n,L)j^{\big{(}-1+\frac{2n-k}{n-1}\big{)}}r^{\rho+2}||\theta(r,y)||_{k,\alpha;\rho+1},$

which implies that the series $\sum_{j\in J}\hat{h}_{j}(r)\eta_{j}(y)$
converges in $\mathcal{C}^{2}_{loc}$. By choosing certain $h_{j}(1)$ and
$h^{\prime}_{j}(1)$ to ensure that $A_{+}=A_{-}=0$, $\sum_{j\in
J}{h}_{j}(r)\eta_{j}(y)$ converges in $\mathcal{C}^{2}_{loc}$. To show the
convergence of $\sum_{i\in I}{f}_{i}(r)\kappa_{i}(y)dr$ and $\sum_{i\in
I}{g}_{i}(r)d_{L}\kappa_{i}(y)$, we deduce a similar estimate as (2.18) for
$E_{i}$ at first. Recall that $\vartheta_{i}=u_{i}-v^{\prime}_{i}$, then we
have

$\displaystyle\vartheta_{i}(r)$
$\displaystyle=\int_{L}\theta_{0}(r,y)\kappa_{i}(y)dy-\lambda_{i}^{-1}\int_{L}(\partial_{r}\theta_{1}(r,y),d_{L}\kappa_{i}(y))_{g_{L}}dy$
$\displaystyle=\lambda_{i}^{-\frac{k}{2}}\int_{L}\big{(}\Delta_{L}^{\frac{k}{2}}\theta_{0}(r,y)\big{)}\kappa_{i}(y)dy-\lambda_{i}^{-\frac{k+1}{2}}\int_{L}\big{(}\partial_{r}\big{(}\Delta_{L}^{\frac{k-1}{2}}\theta_{1}(r,y)\big{)},d_{L}\kappa_{i}(y)\big{)}_{g_{L}}dy$
$\displaystyle\leq
C(n,L)\lambda_{i}^{-\frac{k}{2}}r^{\rho}||\theta||_{k,\alpha;\rho}.$ (2.20)

Recall the equation (2.12), together with (2.20), by the same method to derive
(2.18), we have,

$\displaystyle|E_{i}(r)|+r|E^{\prime}_{i}(r)|+r^{2}|E^{\prime\prime}_{i}(r)|\leq
C(n,L,\rho)(\lambda^{\prime}_{i})^{-\frac{k}{2}}r^{\rho+2}||\theta||_{k,\alpha;\rho}.$
(2.21)

Again, according to equation (2.14) and estimate (2.21), the same method shows
that

$\displaystyle\varpi_{i}(r)=\frac{2E_{i}}{r}+v_{i}\leq
C(n,L,\rho)r^{\rho+1}(\lambda^{\prime}_{i})^{-\frac{k}{2}}r^{\rho+1}||\theta||_{k,\alpha;\rho}.$

Therefore, we have

$\displaystyle|g_{i}(r)|+r|g^{\prime}_{i}(r)|+r^{2}|g^{\prime\prime}_{i}(r)|\leq
C(n,L,\rho)(1+\lambda^{\prime}_{i})^{-\frac{k}{2}}r^{\rho+3}||\theta||_{k,\alpha;\rho}.$
(2.22)

Combining (2.21), (2.22) and definition of $E_{i}$, we obtain,

$\displaystyle|f_{i}(r)|+r|f^{\prime}_{i}(r)|+r^{2}|f^{\prime\prime}_{i}(r)|\leq
C(n,L,\rho)(1+\lambda^{\prime}_{i})^{-\frac{k-1}{2}}r^{\rho+2}||\theta||_{k,\alpha;\rho}.$
(2.23)

Also, we can derive the similar estimate for $\kappa_{i}$ and
$d_{L}\kappa_{i}$ as in (2.19), by Moser iteration and Schauder estimates

$\begin{split}|\kappa_{i}|+r|\nabla\kappa_{i}|+r^{2}|\nabla^{2}\kappa_{i}|&\leq
C(n,L)(1+\lambda^{\prime}_{i})^{\frac{n+1}{2}};\\\ |d_{L}\kappa_{i}|+r|\nabla
d_{L}\kappa_{i}|+r^{2}|\nabla^{2}d_{L}\kappa_{i}|&\leq
C(n,L)(1+\lambda^{\prime}_{i})^{\frac{n+3}{2}}r^{-1}.\end{split}$ (2.24)

Then, the estmates (2.22), (2.23) and (2.24) together with Weyl’s law implies
that, when $k\geq 2n+3$, the series $\sum_{i\in I}{f}_{i}(r)\kappa_{i}(y)dr$
and $\sum_{i\in I}{g}_{i}(r)d_{L}\kappa_{i}(y)$ converge in $C^{2}_{loc}$
topology. And the convergence of series also implies that

$\displaystyle||\beta||_{L^{\infty}}\leq
C(n,L,\rho,k)r^{\rho+2}||\theta||_{k,\alpha;\rho}.$

Since the Laplacian can be viewed as an elliptic operator for vector bundle
$\wedge^{1}T^{*}X$ over $X$, according to elliptic operator theory on conical
manifolds [20, Theorem 4.12], we have Schauder-type estimate for the equation,
$\Delta\beta=\theta$.

$\displaystyle||\beta||_{{k+2,\alpha;\rho+2}}\leq
C(n,L,\rho,k)(||\theta||_{k,\alpha;\rho}+||\beta||_{0;\rho+2})\leq
C(n,L,\rho,k)||\theta||_{k,\alpha;\rho}.$

∎

The proof of lemma 2.1 not only works for 1-forms, but also for functions in
weighted Hölder space. Also by spectral decompsition technique, we can solve
$\Delta_{0}u=f$ as follows. Recall that the increasing sequence
$0=\lambda^{\prime}_{0}<\lambda^{\prime}_{1}\leq\lambda^{\prime}_{2}\ldots$ is
eigenvalues of $\delta$ acting on functions, then the exceptional set is given
by,

$\displaystyle D=\\{d_{i}^{\pm},\text{ for }i=0,1,\ldots\\},\qquad
d^{\pm}_{i}=-\frac{n-2}{2}\pm\sqrt{\Big{(}\frac{n-2}{2}\Big{)}^{2}+\lambda^{\prime}_{i}}.$
(2.25)

######  Corollary 2.2.

Let $f$ be a function defined on $\overline{U}(1)$ with
$f\in\mathcal{C}^{k,\alpha}_{\rho-2}$. If $k\geq$ and $\rho\notin D$, then
there exists a solution $u\in\mathcal{C}^{k+2,\alpha}_{\rho}(\overline{U}(1))$
satisfying the Laplacian equation, $\Delta_{0}u=f$ and $u$ satisfies the
estimate.

$\displaystyle||u||_{\mathcal{C}^{k+2,\alpha}_{\rho}(U(2))}\leq
C||f||_{\mathcal{C}^{k,\alpha}_{\rho-2}(U(1))}.$ (2.26)

where $C$ only depends on $(n,L,k,\rho)$.

###  2.2. Laplacian on asymptotically conical manifolds

The asymptotic behavior of Laplacian on ALE manifolds has been studied in many
references. The main idea to solve the Laplacian equation on ALE manifolds is
to apply the Fredholm property of the Laplacian operator. In Lockhart [18,
Corollary 6.5], Lockhart-McOwen [19, Theorem 1.3] and Cantor [4, Theorem 6.3],
the Fredholm property is proved in Sobolev case for the elliptic operators
that are asymptotic to the Euclidean Laplacian at infinity. The Hölder case
has been discussed in [5, section 4] for real dimension 3. In Marshall [20,
Theorem 6.9], the Fredholm property of Laplacian operator has been generalized
to AC manifolds on both Sobolev and Hölder cases. In this section, we will
summarize the key results of the Hölder cases on AC manifolds and prove the
following proposition. Recall that $(C_{L},g_{0})$ is a Riemannian cone and
$(X,g)$ asymptotic to $(C_{L},g_{0})$ at infinity with a diffeomrphsim
$\psi:X_{\infty}=X-K\rightarrow(R_{0},\infty)\times L$ and
$\displaystyle|\nabla^{i}((\psi^{-1})^{*}g-g_{0})|_{g_{0}}=O(r^{-i-\tau})$,
for integers $i=0,1,\ldots,k$.

In $X_{\infty}$, let $\Delta_{0}$ and $\Delta$ be the Laplacian operators of
metrics $g_{0}$ and $g$ respectively and $D$, the exceptional set given in
(2.25).

######  Proposition 2.3.

Suppose $(X,g)$ is a complete AC manifold asymptotic to a Riemannian cone
$(C_{L},g)$ with $\dim_{\mathbb{R}}X=n\geq 4$ and
$f\in\mathcal{C}^{k,\alpha}_{\rho-2}$.

  1. (i)

Let $\rho\in(0,\infty)\backslash D$, there exists a solution
$u\in\mathcal{C}^{k+2,\alpha}_{\rho}$.

  2. (ii)

Let $\rho\in(2-n,0)$, there exists a unique solution $u\in
C^{k,\alpha}_{\rho+2}$ of $\Delta u=f$.

  3. (iii)

Let $\rho\in(-\infty,2-n)\backslash D$, there exists a unique solution
$u=Ar^{2-n}+v$, where

$\displaystyle
A=\frac{1}{(n-2)\operatorname{Vol}(L)}\int_{X}fd\operatorname{Vol}_{X}$ (2.27)

and $v\in\mathcal{C}^{k+2,\alpha}_{\tilde{\rho}}$ with
$\tilde{\rho}=\max\\{d_{1}^{-},\rho,2-n-\tau\\}$.

In the following we write
$\ker(\Delta,\delta+2)=\ker(\Delta:\mathcal{C}^{k+2,\alpha}_{\delta+2}\rightarrow\mathcal{C}^{k,\alpha}_{\delta})$
and
$\operatorname{Im}(\Delta,\delta)=\operatorname{Im}(\Delta:\mathcal{C}^{k+2,\alpha}_{\delta+2}\rightarrow\mathcal{C}^{k,\alpha}_{\delta})$.
The key observation is that the estimate (2.26) can be modified to obtain a
similar estimate for elliptic operators asymptotic to $\Delta_{0}$ on AC
Riemannian manifolds; namely, the scale broken estimate referring to [20,
Theorem 6.7]. Precisely, fixing $X_{2R}=\\{x\in X,r(x)>2R\\}$ with $R>R_{0}$,
for all $u\in\mathcal{C}^{k+2,\alpha}_{\rho}$, we have

$\displaystyle||u||_{\mathcal{C}^{k+2,\alpha}_{\rho}}\leq C(||\Delta
u||_{\mathcal{C}^{k,\alpha}_{\rho-2}}+||u||_{\mathcal{C}^{0}(X_{2R})}).$
(2.28)

It can be proved that the Laplacian operator
$\Delta:\mathcal{C}^{k+2,\alpha}_{\rho}\rightarrow\mathcal{C}^{k,\alpha}_{\rho-2}$
on $X$ is Fredholm if $\rho\notin D$. To sketch the proof of Fredholm
property, we observe that the estimate (2.28) implies that the set
$\operatorname{Im}(\Delta,\rho)$ is closed. To see this, let $\\{f_{i}\\}$ be
a Cauchy sequence in $C^{k,\alpha}_{\rho-2}$ contained in
$\operatorname{Im}(\Delta,\rho)$. Notice that there exists a closed subspace
$\mathcal{B}_{\rho}$ such that
$\mathcal{C}^{k+2,\alpha}_{\rho}=\mathcal{B}_{\rho}\oplus\ker(\Delta,\rho)$
(because $\ker(\Delta,\rho)$ is finite-dimensional), then we have a bounded
sequence of preimages $\\{u_{i}\\}$ in $\mathcal{B}_{\rho}$ with $\Delta
u_{i}=f_{i}$. According to Arzelà-Ascoli and (2.28), by taking a subsequence
of $\\{u_{i}\\}$, we obtain a Cauchy sequence in $\mathcal{C}^{0}(X_{2R})$.
Hence, $\\{u_{i}\\}$ is also a Cauchy sequence by passing to the subsequence.
Let $f$, $u$ be the convergence of $\\{f_{i}\\}$ and $\\{u_{i}\\}$
respectively, then $f=\Delta u$ is belong to $\operatorname{Im}(\Delta,\rho)$.
Based on self-adjointness of $\Delta$, the Laplacian
$\Delta:\mathcal{C}^{k+2,\alpha}_{\delta+2}\rightarrow\mathcal{C}^{k,\alpha}_{\delta}$

$\displaystyle\operatorname{Im}(\Delta,\delta)=\\{f\in\mathcal{C}^{k,\alpha}_{\rho},(f,h)_{L^{2}}=0\text{
for all }h\in\ker(\Delta,-n-\delta)\\}$ (2.29)

Then, the Fredholm property of $\Delta$ follows from the fact that
$\ker(\Delta,\rho)$ is finite-dimensional.

###### Proof of Proposition 2.3.

By the maximal principle of harmonic functions, we can show that
$\ker(\Delta,\rho)=0$ if $\rho<0$. In the cases (i) and (ii), by Fredholm
property of $\Delta$ and (2.29), when $\rho\in(-n,-2)$,
$\operatorname{Im}(\Delta,\rho)$ is annihilated by $\ker(\Delta,-n-\rho)=0$.
Hence, $\operatorname{Im}(\Delta,\rho)=\mathcal{C}^{k,\alpha}_{\rho}$ for
$\rho\in(-n,-2)$. The uniqueness in the case (ii) is directly from the fact
$\ker(\Delta,\rho)=0$ for $\rho\in(-n,-2)$. .

For the case (iii), notice that $\Delta
r^{2-n}=\Delta_{0}r^{2-n}+(\Delta-\Delta_{0})r^{2-n}=O(r^{-n-\tau})$ and

$\displaystyle\int_{X}\Delta r^{2-n}d\operatorname{Vol}_{X}$
$\displaystyle=-\lim_{r\rightarrow\infty}\int_{L(r)}\langle\nabla
r^{2-n},n\rangle_{g}d\operatorname{Vol}_{L(r)}$
$\displaystyle=\lim_{r\rightarrow\infty}(n-2)r^{1-n}\int_{L(r)}g(\partial
r,\partial r)d\operatorname{Vol}_{L(r)}$
$\displaystyle=\lim_{r\rightarrow\infty}(n-2)\operatorname{Vol}(L)+O(r^{-\tau})=(n-2)\operatorname{Vol}(L).$

Consider the function $f-A\Delta r^{2-n}$, where the constant $A$ is chosen to
be (2.27). Observing that $f-A\Delta r^{2-n}=O(r^{\max\\{-n-\tau,\rho-2\\}})$
and $\int_{X}f-A\Delta r^{2-n}$ =0. Without loss of generality, we assume
$\max\\{-n-\tau,\rho-2\\}\in(d_{1}^{-}-2,-n)$. Then (2.29) implies that the
only annihilating functions are constant functions. The integration is
vanishing indicates that $f-A\Delta r^{2-n}$ is in
$\operatorname{Im}(\Delta,\tilde{\rho}-2)$ with
$\tilde{\rho}=\max\\{d^{-}_{1},\rho,2-n-\tau\\}$; hence, we have solution
$u=Ar^{2-n}+v$ with $v\in\mathcal{C}^{k+2,\alpha}_{\tilde{\rho}}$. ∎

##  3\. Proof of $dd^{c}$ lemma

###  3.1. Proof of Theorem A (i)

Recall that $(X,J,g)$ is an AC Kähler manifold asymptotic to a Ricci-flat
Kähler cone $(C_{L},J_{0},g_{0})$ at infinity with a diiffeomorphism
$\psi:X_{\infty}=X-K\rightarrow C_{L}-B_{R_{0}}$. In the end $X_{\infty}$,
there are two sets of differential operators. In particular, we write $d$,
$\partial$, $\overline{\partial}$ and $d^{*}$, $\partial^{*}$,
$\overline{\partial}^{*}$ as the differential opertators and the dual
operators with respect to $(g,J)$, which also induces the Laplacian operators
$\Delta$ in the asymptotic chart. Also let $d_{0}$, $\partial_{0}$,
$\overline{\partial}_{0}$, $d_{0}^{*}$, $\partial_{0}^{*}$,
$\overline{\partial}_{0}^{*}$ and $\Delta_{0}$ be the Laplacian operators with
respect to $(g_{0},J_{0})$. Let $\omega\in\mathcal{C}^{k,\alpha}_{-\delta}$ be
an exact real $(1,1)$-form on $X$. Then, by [10, Theorem 3.11], there exists a
real 1-form $\alpha\in\mathcal{C}^{k+1,\alpha}_{1-\delta}$ such that
$\omega=d\alpha$. Decompose $\alpha$ to $(1,0)$-form and $(0,1)$-form with
respect to $J$, $\alpha^{1,0}$ and $\alpha^{0,1}$. Consider the following
Laplacian equation for $\phi$,

$\displaystyle\frac{1}{2}\Delta\phi=\overline{\partial}^{*}\alpha^{0,1},$
(3.1)

Based on the Proposition 2.3, there exists a solution
$\phi\in\mathcal{C}^{k+2,\alpha}_{2-\delta}$ for (3.1). Define
$\xi=\alpha^{0,1}-\overline{\partial}\phi$. Due to the low decay rate, it is
not necessary that $\xi=0$. The idea of the proof is to reduce the decay rate
of $\xi$ and to find $\hat{\phi}$ such that
$\overline{\partial}\hat{\phi}=\alpha^{0,1}$. It is observed that

$\displaystyle\overline{\partial}^{*}\xi=0,\qquad\overline{\partial}\xi=0.$
(3.2)

Hence, $\xi$ is a $J$-harmonic (0,1)-form in
$\mathcal{C}^{k+1,\alpha}_{1-\delta}$. If $1-\delta$ is nonnegative, we can
reduce the order of $\xi$ as follows. Assuming that
$\theta=\Delta_{0}\xi=(\Delta_{0}-\Delta)\xi=O(r^{-1-\delta-\tau})$, according
to lemma 2.1, there exists a solution $\beta=O(r^{1-\delta-\tau})$ such that
$\Delta_{0}\beta=\theta$ in the asymptotic chart of $X$. Hence, we can write
$\xi$ in the asymptotic chart of $X$ as

$\displaystyle\xi=\xi_{0}+\beta,$

where $\xi_{0}$ is a harmonic 1-form with respect to $\Delta_{0}$ of decay
rate $1-\delta$ and $\beta$ is a 1-form of decay rate $1-\delta-\tau$. Notice
that $\xi_{0}$ is not necessary a $(0,1)$-form with respect to $J$, but the
$(1,0)$ part of $\xi_{0}$ has the decay rate $1-\delta-\tau$. Then, according
to [8, Lemma 2.27] or referring to [13, Lemma B.1], the assumption that $0\leq
1-\delta<1$ implies there exists a harmonic function $\psi_{0}$ with respect
to $(J_{0},g_{0})$ such that $d\psi_{0}=\xi_{0}$ with
$\psi_{0}=\mathcal{C}^{k+2,\alpha}_{2-\delta}$ and
$\partial\psi_{0}=\mathcal{C}^{k+1,\alpha}_{1-\delta-\tau}$,
$\overline{\partial}\psi_{0}=\mathcal{C}^{k+1,\alpha}_{1-\delta}$. Extend
$\psi_{0}$ to a smooth function $\tilde{\psi}_{0}$ on $X$. According to
proposition 2.3, by solving equation $\Delta b=-\Delta\tilde{\psi}_{0}$, there
exists a harmonic function $\psi$ with respect to $(J,g)$ such that,

$\displaystyle\psi=\tilde{\psi_{0}}+b.$

Since in asymptotic chart of $X$,
$-\Delta\tilde{\psi}_{0}=(\Delta_{0}-\Delta)\tilde{\psi}_{0}=O(r^{-\delta-\tau})$,
we have $b$ is a function of class $\mathcal{C}^{k+2,\alpha}_{2-\delta-\tau}$.
Now, consider a new (0,1)-form given by

$\displaystyle\xi_{1}=\xi-\overline{\partial}\psi,$ (3.3)

which also satisfies conditon (3.2)
$\overline{\partial}\xi_{1}=\overline{\partial}^{*}\xi_{1}=0$. And $\xi_{1}$
has the following decay rate,

$\displaystyle\xi_{1}$ $\displaystyle=\xi_{0}+\beta-\overline{\partial}\psi$
$\displaystyle=\beta+\partial\tilde{\psi}_{0}+\overline{\partial}\tilde{\psi}_{0}-\overline{\partial}\psi$
$\displaystyle=\beta+\partial\tilde{\psi}_{0}-\overline{\partial}b,$

where $\beta$, $\partial\psi_{0}\in\mathcal{C}^{k+1,\alpha}_{1-\delta-\tau}$.
Hence, $\xi_{1}\in\mathcal{C}^{k+1,\alpha}_{1-\delta-\tau}$. Repeating this
process until the decay rate of 1-form to be negative, precisely, we can find
$\hat{\xi}=\xi-\overline{\partial}\hat{\psi}$ such that
$\hat{\psi}\in\mathcal{C}^{k+2,\alpha}_{2-\delta}$ and
$\hat{\xi}\in\mathcal{C}^{k+1,\alpha}_{1-\delta-n\tau}$ with
$1-\delta-n\tau<0$. To simplify our notation, we replace $\xi$ with the 1-form
of negative decay rate. In the following proof, we assume that
$\xi\in\mathcal{C}^{k+1,\alpha}_{1-\delta}$ ($1-\delta<0$) with
$\overline{\partial}^{*}\xi=\overline{\partial}\xi=0$. We introduce the
following lemma.

######  Lemma 3.1.

Let $k$ be a large positive integer and $1-\delta<0$,

  1. (i)

Let $\xi$ is a real 1-form on $X$ of class
$\mathcal{C}^{k+1,\alpha}_{1-\delta}$. Assuming that $\xi$ is a harmonic
1-form on $X$, then, $\xi\in\mathcal{C}^{k+1,\alpha}_{-2n+2}$.

  2. (ii)

Let $\xi$ is a harmonic (0,1)-form on $X$ of class
$\mathcal{C}^{k+1,\alpha}_{1-\delta}$ with $\overline{\partial}\xi=0$, then
$\xi\in\mathcal{C}^{k+1,\alpha}_{-2n+1}$

###### Proof.

Let $\xi$ be a real harmonic 1-form on $X$. According to lemma 2.1, $\xi$ can
be represented outside a compact set of $X$ as follows,

$\displaystyle\xi=\xi_{0}+\beta$ (3.4)

where $\xi_{0}$ is a harmonic form with respect to $\Delta_{0}$ of the same
decay rate as $\xi$ and $\beta$ is a 1-form of decay rate at most
$1-\delta-\tau$. By assumption that $X$ is asymptotic to the conical manifold
$C_{L}$, $\xi_{0}$ can be expressed on basis of spectral decomposition with
respect to Laplacian operator on link $L$,

$\displaystyle\xi_{0}$ $\displaystyle=\sum_{i\geq
0}[f_{i}(r)\kappa_{i}(y)dr+g_{i}(r)d_{L}\kappa_{i}(y)]+\sum_{j\in
J}h_{j}(r)\eta_{j}(y)$ $\displaystyle=\sum_{i\geq
0}[d(g_{i}(r)\kappa_{i}(y))+E_{i}(r)\kappa_{i}(y)dr]+\sum_{j\geq
1}h_{j}(r)\eta_{j}(y).$

where $E_{i}(r)=f_{i}(r)-g^{\prime}_{i}(r)$. Then, the function $g_{i}$,
$E_{i}$ satisfy the following equations, ($\dim_{\mathbb{R}}X=2n$)

$\displaystyle-$ $\displaystyle
E^{\prime\prime}_{i}-\frac{2n-3}{r}E^{\prime}_{i}+\frac{\lambda^{\prime}_{i}+2n-3}{r^{2}}E_{i}=0,$
(3.5a) $\displaystyle-$ $\displaystyle
g_{i}^{\prime\prime}-\frac{2n-1}{r}g_{i}^{\prime}+\frac{\lambda^{\prime}_{i}}{r^{2}}g_{i}=\frac{2}{r}E_{i}.$
(3.5b)

By solving the equations (3.5a) and (3.5b), the harmonic 1-forms with respect
to $\Delta_{0}$ of negative decay rate at infinity have the following two
types,

  1. (I)

Let $\lambda_{i}$ be all eigenvalues of $\Delta_{L}$ for functions on $L$,

$\displaystyle d(r^{c_{i}^{-}}\kappa_{i}),\quad\text{with
}c^{-}_{i}=-(n-1)-\sqrt{(n-1)^{2}+\lambda^{\prime}_{i}}$

  2. (II)

Let $\lambda_{i}$ be all positive eigenvalues of $\Delta_{L}$ for functions on
$L$,

$\displaystyle
B_{i}r^{b_{i}^{-}}\kappa_{i}dr+2d(r^{b_{i}^{-}+1}\kappa_{i}),\quad\text{with
}b_{i}^{-}=-(n-2)-\sqrt{(n-2)^{2}+\lambda^{\prime}_{i}+2n-3},$

and $B_{i}$ is a constant given by
$B_{i}=-(b_{i}^{-})^{2}-2nb_{i}^{-}+\lambda_{i}+1-2n$.

For the coclosed part of decomposition, consider the following equation,

$\displaystyle-h^{\prime\prime}_{j}-\frac{2n-3}{r}h^{\prime}_{j}+\frac{\lambda_{j}}{r^{2}}h_{j}=0,$
(3.6)

By solving (3.6), we have the third type of harmonic 1-form,

  1. (III)

Let $\lambda_{j}$ be the eigenvalues of $\Delta_{L}$ for the co-closed 1-forms
on $L$,

$\displaystyle r^{a^{-}_{j}}\eta_{j},\quad\text{with
}a_{j}^{-}=-(n-2)-\sqrt{(n-2)^{2}+\lambda^{\prime\prime}_{j}}$

To estimate the eigenvalues, we notice that $L$ is a Sasakian-Einstein
manifold with Ricci curvature $\textup{Ric}_{g_{L}}=2(n-1)g_{L}$. By
Lichnerowicz-Obata first eigenvalue theorem and [13, Lemma B.2], let
$\lambda^{\prime}_{1}$ and $\lambda^{\prime\prime}_{1}$ be the first positive
eigenvalue of closed and coclosed 1-form, we have the following estimate,

$\displaystyle\lambda^{\prime}_{1}(L)\geq
2n-1,\qquad\lambda^{\prime\prime}_{1}(L)\geq 4n-4.$

Then, we obtain the greatest possibe decay rate of harmonic 1-forms for all
these three types. For type (I), the forms decays to rate at most $-2n+1$ at
infinity. For type (II), the forms decays to rate at most $-2n+2$. And the
forms of type (III) decays to rate at most $-2n+1$. Hence, we obtain
$\xi_{0}=O(r^{-2n+2})$. If $1-\delta-\tau\geq-2n+2$, then, by expression
(3.4), $\xi$ is of decay rate $1-\delta-\tau$. Again, noticing that $\beta$ is
the solution of $\Delta_{0}\beta=(\Delta_{0}-\Delta)\xi$, there exists a
$d$-closed solution $\beta$ of decay rate $1-\delta-2\tau$. Repeating the
process, we can finally obtain that $\xi=O(r^{2-2n})$.

For the second statement, let $\xi$ be a harmonic $(0,1)$-form on $X$ with
$\overline{\partial}\xi=0$. Notice that $\xi$ also can be written as in (3.4),
$\xi=\xi_{0}+\beta$. According to (i), we can assume $\xi$, $\xi_{0}$ of rate
$-d\leq 2-2n$ and $\beta$ of rate $-d-\tau$. $\xi$ can be decomposed to
$(1,0)$ and $(0,1)$ form with respect to $J_{0}$,

$\displaystyle\xi=\xi^{1,0}_{J_{0}}+\xi^{0,1}_{J_{0}},\quad,\xi^{1,0}_{J_{0}}=\frac{\xi-
iJ_{0}\xi}{2},\ \xi^{0,1}_{J_{0}}=\frac{\xi+iJ_{0}\xi}{2}.$

Since $\xi$ is a $(0,1)$-form with respect to $J$, $\xi-
iJ_{0}\xi=i(J-J_{0})\xi=O(r^{-d-\tau})$. The $(1,0)$ part of $\xi$ with
respect to $J_{0}$ has rate $-d-\tau$, so does $\xi_{0}$. Then, the highest
order terms of $\xi_{0}$ is a homogeneous harmonic $(0,1)$-form with respect
to $J_{0}$, denoted by $\xi_{0}^{h}$. The fact $\overline{\partial}\xi=0$
implies that $\overline{\partial}_{0}\xi_{0}^{h}=O(r^{-d-1-\tau})$, If we
assume the rate of $\xi^{h}_{0}$ is greater than $1-2n$, based on the
classification of harmonic 1-forms, $\xi^{h}_{0}$ must be of type (II). Then,
$\xi_{0}^{h}$ can be written as,

$\displaystyle\xi^{h}_{0}=C(2b^{-}_{i}+6-4n)r^{b^{-}_{i}}\kappa_{i}\overline{\partial}_{0}r+2r^{b^{-}_{i}+1}\overline{\partial}_{0}\kappa_{i},$

where $i\geq 1$ and $b_{i}^{-}\geq 1-2n$. By taking differentiation, we have

$\displaystyle\overline{\partial}_{0}\xi^{h}_{0}=C(4b_{i}^{-}+4-4n)r^{b_{i}^{-}}\overline{\partial}_{0}\kappa_{i}\wedge\overline{\partial}_{0}r.$

It is easy to check that $\overline{\partial}_{0}\xi^{h}_{0}$ is a
nonvanishing homogeneous form of rate $-d-1$, which contradicts against
$\overline{\partial}_{0}\xi_{0}^{h}=O(r^{b^{-}_{i}-1-\tau})$. Therefore, we
have $\xi=O(r^{1-2n})$. ∎

According to lemma 3.1, we reduce the proposition A to the case of fast decay
rate. Let $\hat{\omega}=2\operatorname{Re}\partial\xi$ be a closed real (1,1)
form of decay rate at most $-2n+1$. Also notice that
$\operatorname{tr}_{\omega}\hat{\omega}=2\operatorname{Re}\overline{\partial}^{*}\xi=0$.
Recall that there exists a natural radial function defined in $X$ and let
$B_{R}=\\{x\in X|r(x)\leq R\\}$, for some very large positive real number $R$.
Then, it is easy to check that

$\displaystyle
2(n-2)!\int_{B_{R}}|\hat{\omega}|^{2}d\operatorname{Vol}=\int_{B_{R}}d(2\operatorname{Re}\xi\wedge\hat{\omega}\wedge\omega^{n-2})=\int_{\partial
B_{R}}2\operatorname{Re}\xi\wedge\hat{\omega}\wedge\omega^{n-2}.$ (3.7)

The decay conditions of $\hat{\omega}$ and $\xi$ imply that the integration in
(3.7) tends to zero as $R$ goes to infinity. Hence, we have $\hat{\omega}=0$.
In conclusion, the exact real (1,1)-form $\omega$ can be expressed as

$\displaystyle\omega=2\operatorname{Re}\partial\alpha^{0,1}=2\operatorname{Re}\partial(\xi+\overline{\partial}\phi)=\hat{\omega}+2i\partial\overline{\partial}\operatorname{Im}\phi=dd^{c}\operatorname{Im}\phi,$

where $\phi\in\mathcal{C}^{k+2,\alpha}_{2-\delta}$. Therefore, we complete the
proof of proposition A.

###  3.2. Proof of Theorem A (ii): an obstruction of ddbar lemma

Let $(X,J)$ be an AC Kähler manifold asymptotic to a Ricci-flat Kähler cone
$(C_{L},J_{0})$ and $K$ be the compact set in $X$ such that $X\backslash
K\cong C_{L}\backslash\overline{B(R_{0})}$. We apply the notation as in
section 3.1. Let $\omega$ be the real exact (1,1)-form on $X\backslash K$ with
$\omega=d\alpha$, where $\alpha$ is a real 1-form on $X\backslash K$. Then,
$\alpha$ can split into the sum of a (0,1)-form and a (1,0)-form with respect
to $J$, $\alpha=\alpha^{1,0}+\alpha^{0,1}$. Here
$\overline{\partial}^{*}\alpha^{0,1}$ is a complex function defined on
$X\backslash K$ of class $\mathcal{C}^{k,\alpha}_{-\delta}$. By extending
$\overline{\partial}^{*}\alpha^{0,1}$ to a function
$f\in\mathcal{C}^{k,\alpha}_{-\delta}$ on the whole manifold $X$. Then the
Laplacian equation (3.1) can be rewritten as $\Delta\phi=2f$. According to
Proposition 2.3, we have a solution
$\phi\in\mathcal{C}^{k+2,\alpha}_{2-\delta}$. Hence, we can define a
(0,1)-form on $X\backslash K$, $\xi=\alpha^{0,1}-\overline{\partial}\phi$. It
is also easy to check $\overline{\partial}\xi=0$,
$\overline{\partial}^{*}\xi=0$. Based on the proof of Theorem A (i), we can
reduce the rate of $\xi$ at infinity as in (3.3); i.e. we can find a harmonic
function $\psi\in\mathcal{C}^{k+2,\alpha}_{2-\delta}$ such that
$\hat{\xi}=\xi-\overline{\partial}\psi$ is a (0,1)-form of class
$\mathcal{C}^{k+1,\alpha}_{1-\delta-m\tau}$ with $1-\delta-m\tau<0$ and
$\overline{\partial}\hat{\xi}=0$, $\overline{\partial}^{*}\hat{\xi}=0$. Then,

$\displaystyle\omega=2\operatorname{Re}(\partial\alpha^{0,1})=2\operatorname{Re}(\partial\hat{\xi}+\partial\overline{\partial}\phi+\partial\overline{\partial}\psi)=2i\partial\overline{\partial}\operatorname{Im}(\phi+\psi)+2\operatorname{Re}(\partial\hat{\xi}).$
(3.8)

The above expression of $\omega$ (3.8) implies that the obstruction of ddbar
lemma is the harmonic 1-form $\hat{\xi}$. Unfortunately, the obstruction term,
$\operatorname{Re}(\partial\hat{\xi})$, cannot be vanishing in general, see
counter example 3.3. However, the decay rate of the term can be reduced to
enough high order. According to Lemma 3.1, the decay rate of $\hat{\xi}$ is at
most $1-2n$. Hence, $\omega=i\partial\overline{\partial}f+O(r^{-2n})$ for some
real function $f\in\mathcal{C}^{k+2,\alpha}_{2-\delta}$, which completes the
proof of theorem A (ii).

######  Remark 3.2.

Fixing $X\backslash K$ as above, we assume that $J_{0}=J$ in this remark. An
interesting fact we observed in the proof of theorem A (ii) is that the
obstruction of ddbar lemma is described by the class
$\mathcal{H}^{0,1}_{\overline{\partial},\overline{\partial}^{*}}/\overline{\partial}\mathcal{H}$,
where $\overline{\partial}\mathcal{H}$ is the $\overline{\partial}$-image of
complex valued harmonic functions on $X\backslash K$ asymptotic to zero at
infinity and

$\displaystyle\mathcal{H}^{0,1}_{\overline{\partial},\overline{\partial}^{*}}=\\{\xi\in\Omega^{0,1}(X\backslash
K)|\ \overline{\partial}\xi=0,\ \overline{\partial}^{*}\xi=0\text{ and
}\xi\text{ decays to 0 at infinty }\\}.$

The fact directly follows from 3.8 and the classification of harmonic 1-forms
on metric cones as discussed in the proof of Lemma 3.1.

######  Example 3.3.

In this example, we discuss the ddbar lemma on
$\mathbb{C}^{n}\backslash\\{0\\}$. For $n\geq 3$, any exact real $(1,1)$-form
$\omega$ satisfies the ddbar lemma; i.e. $\omega=dd^{c}f$ for a real function
function $f$. In the higher dimension cases, One refers to [11, Lemma 5.5] or
[10, Corollary A.3] for details. However, in the case of $n=2$, the ddbar
lemma doesn’t hold. According to remark 3.2, the obstruction of ddbar lemma is
given by
$\mathcal{H}^{0,1}_{\overline{\partial},\overline{\partial}^{*}}/\overline{\partial}\mathcal{H}$.
Then, we can calculate the dimension of the spaces,
$\mathcal{H}^{0,1}_{\overline{\partial},\overline{\partial}^{*}}$ and
$\overline{\partial}\mathcal{H}$ for each growth rate. For $k\geq 0$, we have

$\displaystyle\dim_{\mathbb{C}}(\mathcal{H}^{0,1}_{\overline{\partial},\overline{\partial}^{*}})_{-2-k}=k(k+1),\quad\dim_{\mathbb{C}}(\overline{\partial}\mathcal{H})_{-2-k}=k^{2}.$

Let $\omega$ be an exact real $(1,1)$-form of decay rate $-\delta$, then there
exists a real function $f$ of decay rate $2-\delta$ such that,

$\displaystyle\omega=dd^{c}f+\theta_{-4}+\theta_{-5}+O(r^{-6}).$

And we can write down $\theta_{-4}$ and $\theta_{-5}$ explicitly,

$\displaystyle\theta_{-4}=\operatorname{Re}(C_{-3}\partial\xi_{-3})=\operatorname{Re}\bigg{\\{}C_{-3}\partial\bigg{(}\frac{\bar{z}_{2}d\bar{z}_{1}-\bar{z}_{1}d\bar{z}_{2}}{(|z_{1}|^{2}+|z_{2}|^{2})^{2}}\bigg{)}\bigg{\\}},$

where $C_{-3}$ is a complex constant and the form $\xi_{-3}$ is a
representative of
$\mathcal{H}^{0,1}_{\overline{\partial},\overline{\partial}^{*}}/\overline{\partial}\mathcal{H}$
of decay rate $-3$. Similarly, $\theta_{-5}$ can be written as follows,

$\displaystyle\theta_{-5}$
$\displaystyle=\operatorname{Re}(C_{-4,1}\xi_{-4,1}+C_{-4,2}\xi_{-4,2})$
$\displaystyle=\operatorname{Re}\bigg{\\{}C_{-4,1}\partial\bigg{(}\frac{\bar{z}_{1}\bar{z}_{2}d\bar{z}_{1}-\bar{z}^{2}_{1}d\bar{z}_{2}}{(|z_{1}|^{2}+|z_{2}|^{2})^{3}}\bigg{)}+C_{-4,2}\partial\bigg{(}\frac{\bar{z}^{2}_{2}d\bar{z}_{1}-\bar{z}_{1}\bar{z}_{2}d\bar{z}_{2}}{(|z_{1}|^{2}+|z_{2}|^{2})^{3}}\bigg{)}\bigg{\\}},$

where $C_{-4,1}$ and $C_{-4,2}$ are complex constant. In Joyce’s book [14,
Theorem 8.9.2], the statement of ddbar lemma on $\mathbb{C}^{2}/\\{0\\}$,
which states that $\omega=dd^{c}f+\theta_{-4}$, is wrong, as we can find
infinite error terms.

##  4\. The Mass formula on AC Kähler Manifolds

In this section, we dedicate to generalize the Hein-LeBrun’s mass [12, Theorem
C] formula to AC Kähler manifolds asymptotic to Ricci-flat Kähler cones. In
section 4.1, we introduce some basic set-ups on the manifolds. By introducing
a system of coframes at infinity, we calculate the Chern connection forms and
the curvature forms of canonical line bundle. In section 4.2, the generalized
definition of mass will be given in the AC Riemannian manifolds. And we
complete the proof of mass formula in AC Kähler manifolds asymptotic to Ricci-
flat Kähler cones.

###  4.1. AC Kähler manifolds Asymptotic to Locally Calabi-Yau Cones

Let $(C_{L},J_{0})$ be a Kähler cone with Ricci-flat metric $g_{0}$. The
associated link $L$ has positive Ricci curvature
$\textup{Ric}(g_{L})=(2n-2)g_{L}$. According to Myers’s Theorem, the
fundamental group $\pi_{1}$ is finite. Hence, there exists a universal
covering $\widetilde{L}$ with finite deck transformation $\Gamma$ acting on
$\widetilde{L}$. Let $(C_{\widetilde{L}},\widetilde{g}_{0})$ be the metric
cone of $\widetilde{L}$, then $\widetilde{C}_{L}$ is a finite covering of
$C_{L}$ with $\widetilde{C}_{{L}}/\Gamma\cong C_{L}$. By pulling back the
complex structure on $C_{L}$, the covering space
$(\widetilde{C}_{L},\widetilde{J}_{0})$ is also a Kähler cone with Ricci-flat
metric $\widetilde{g}_{0}$. Since
$(\widetilde{C},\widetilde{J}_{0},\widetilde{g}_{0})$ is Kähler Ricci-flat
with trivial fundamental group, then there exists a nowhere vanishing
holomorphic section of canonical bundle $K_{\widetilde{C}_{L}}$,
$\widetilde{\Omega}_{0}$ and
$\tilde{\omega}_{0}^{n}=i^{n^{2}}\widetilde{\Omega}_{0}\wedge\overline{\widetilde{\Omega}}_{0}$.
$\widetilde{\Omega}_{0}$ induces a multi-valued nonvanishing holomorphic
section, $\Omega_{0}$, of $K_{C_{L}}$ on $C_{L}$ through the covering map;
i.e., there exists a group representation $\kappa:\Gamma\rightarrow U(1)$ such
that

$\displaystyle\gamma^{*}\widetilde{\Omega}_{0}=\kappa(\gamma)\widetilde{\Omega}_{0},\qquad\gamma\in\Gamma.$
(4.1)

In other words, the multi-valued section $\Omega_{0}$ has $|\Gamma|$ branches
and each branch differs by a constant factor $\kappa(\gamma)$,
$\gamma\in\Gamma$.

Let $(C_{L},J_{0},g_{0})$ be a Ricci-flat Kähler cone with the link $L$. Let
$(X,J,g)$ be an AC Kähler manifold asymptotic to $C_{L}$ together with the
diffeomorphism $\Psi:X_{\infty}=X-K\rightarrow C_{L}-B_{R}$, where
$B_{R}=\\{x\in C_{L},r(x)\leq R\\}$. We also assume that $g$ decays to $g_{0}$
in the end $X_{\infty}$ with rate $-\tau=1-n-\epsilon$ and $J$ decays to
$J_{0}$ at infinity.

In order to make analysis in the setting, we introduce a local frame system
for Kähler cones. Recall that the link $L$ is a Sasakian manifold with metric
$g_{L}$. Let $\partial r$ be the normal vector field in radial direction,
there exists a canonical vector field over $L$ defined by $\xi=J_{0}(r\partial
r)$, namely, the Reeb vector field on $L$. And the contact 1-form $\eta$ is
the dual of $\xi$ with respect to the metric $g_{L}$. The contact 1-form
$\eta$ defines a subbundle of tangent bundle on the link $L$, namely
$D=\ker\eta\rightarrow L$. Restricting $J_{0}$ on the subbundle $D$ defines an
almost complex structure on $D$, which guarantees the existence of a local
basis $(m_{i},\overline{m}_{i})$ for $D$, $i=1,\ldots,n-1$, where $m_{i}$ is
an (1,0) type vector with respect to $J_{0}|_{D}$. For any point in $L$, there
exists a neighborhood $U$ and local basis $(\xi;m_{i},\overline{m}_{i})$ for
the tangent bundle $T_{L}(U)$. Let $m_{0}=r\partial r-i\xi$, then
$(m_{i},\overline{m}_{i})_{i=0}^{n-1}$ defines a basis on $T_{C_{L}}(U)$.
According to the fact $r\partial r$ is a global real holomorphic vector field
on $C_{L}$, the scaling map of $C_{L}$ is holomorphic. Hence, the local basis
$(m_{i},\overline{m}_{i})_{i=0}^{n-1}$ is defined in $U\times\mathbb{R}_{>0}$
by pulling forward the scaling map. Let
$\\{v_{i},\overline{v}_{i}\\}_{i=0}^{n-1}$ be the normalizing basis by
defining $v_{i}=r^{-1}m_{i}$ and $v_{i}$, $\overline{v}_{i}$. are (1,0), (0,1)
type vectors according to $J_{0}$. Let 1-forms, $\mu^{i}$,
$\overline{\mu}^{i}$ be the dual basis of $v_{i}$, $\overline{v}_{i}$, then
$J_{0}$ can be written explicitly as

$\displaystyle J_{0}=i\mu^{i}\otimes v_{i}-i\bar{\mu}^{i}\otimes\bar{v}_{i}.$

Let $\overline{\nabla}$ be the Levi-Civita connection of $(C_{L},g_{0})$. One
can easily check that

$\displaystyle\overline{\nabla}_{\partial r}v_{i}=\overline{\nabla}_{\partial
r}\bar{v}_{i}=0,\qquad\overline{\nabla}_{\partial
r}\mu^{i}=\overline{\nabla}_{\partial r}\bar{\mu}^{i}=0$ (4.2)

and $\overline{\nabla}J_{0}=0$ also implies that $\overline{\nabla}_{T}v_{i}$
and $\overline{\nabla}_{T}\bar{v}_{i}$ are vectors of type $(1,0)$ and $(0,1)$
respectively for any tangent vector fields $T$ of $C_{L}$.

In the end $X_{\infty}$, the frames $\\{\mu^{i},\bar{\mu}^{i}\\}$ and
$\\{v_{i},\bar{v}_{i}\\}$ are still defined by diffeomorphism $\Psi$.
Precisely, there exists a finite open covering on link $L=\bigcup U_{k}$ such
that $X_{\infty}\subseteq\bigcup V_{k}$, where
$V_{k}=\Psi^{-1}(U_{k}\times\mathbb{R}_{>R})$. And on each open set $V_{k}$,
we have local frames $\\{\mu^{i},\bar{\mu}^{i}\\}$ and
$\\{v_{i},\bar{v}_{i}\\}$. We call the data
$(V_{k},\\{\mu^{i},\bar{\mu}^{i}\\},\\{v_{i},\bar{v}_{i}\\})$ a system of
asymptotic local frames on $X$. A quick calculation shows that $J-J_{0}$ has
decay rate at least $-\tau$. Precisely, if we write $\nabla$ as Levi-Civita
connection of $g$, then by $J$ (resp. $J_{0}$) is $\nabla$-parellel (resp.
$\nabla_{0}$-parellel),

$\displaystyle\overline{\nabla}(J-J_{0})=-(\nabla-\overline{\nabla})J.$ (4.3)

If we write the difference of two Levi-Civita connection as
$A=\nabla-\overline{\nabla}$, $A$ can be viewed as a tensor on $X$. The tensor
$A$ can be represented with respect to the system of asymptotic local frame
$\\{v_{i},\bar{v}_{i}\\}$, for instance,
$(\nabla-\overline{\nabla})_{v_{j}}v_{i}=A^{k}_{ji}v_{k}+A^{\bar{k}}_{ji}\bar{v}_{k}$.
Noticing that $|A|=O(r^{-\tau-1})$, if we also write $J$ based on a system of
asymptotic local frames, then (4.2) implies that $|\overline{\nabla}_{\partial
r}(J-J_{0})|=O(r^{-\tau-1})$. Hence, along each ray, the derivative of each
coefficient of $J-J_{0}$ has decay rate $-\tau$. Therefore, we have
$J-J_{0}=O(r^{-\tau})$. Explicitly, $J$ can be written as the following
tensor,

$\displaystyle J=J_{0}+i\mathcal{J}^{i}_{\bar{j}}\bar{\mu}^{j}$
$\displaystyle\otimes v_{i}+i\mathcal{K}^{i}_{j}\mu^{j}\otimes v_{i}$
$\displaystyle-i\overline{\mathcal{J}^{i}_{\bar{j}}}\mu^{j}\otimes\bar{v}_{i}-i\overline{\mathcal{K}^{i}_{j}}\bar{\mu}^{j}\otimes\bar{v}_{i},$

with $\mathcal{J}^{i}_{\bar{j}}$, $\mathcal{K}^{i}_{j}=O(r^{-\tau})$.
According to $J^{2}=-1$ and comparing the terms of decay rate $-\tau$, we have

$\displaystyle\mathcal{K}^{i}_{j}=0\quad\mod O(r^{-2\tau}).$

Therefore,

$\displaystyle J=J_{0}+i\mathcal{J}^{i}_{\bar{j}}\bar{\mu}^{j}$
$\displaystyle\otimes
v_{i}-i\overline{\mathcal{J}^{i}_{\bar{j}}}\mu^{j}\otimes\bar{v}_{i}+O(r^{-2\tau})$
(4.4)

According to (4.3), we can derive the relationship between $A$ and
$\overline{\nabla}\mathcal{J}$,

$\displaystyle-(\nabla-\overline{\nabla})J_{0}=$
$\displaystyle-2iA^{\bar{j}}_{i\bar{k}}\mu^{i}\otimes\bar{\mu}^{k}\otimes\bar{v}_{j}-2iA^{\bar{j}}_{ik}\mu^{i}\otimes{\mu}^{k}\otimes\bar{v}_{j}$
$\displaystyle+2iA^{j}_{\bar{i}\bar{k}}\bar{\mu}^{i}\otimes\bar{\mu}^{k}\otimes{v}_{j}+2iA^{j}_{\bar{i}k}\bar{\mu}^{i}\otimes{\mu}^{k}\otimes{v}_{j}$

and

$\displaystyle\overline{\nabla}(J-J_{0})=i(\overline{\nabla}\mathcal{J})^{j}_{\bar{k},i}+i(\overline{\nabla}\mathcal{J})^{j}_{\bar{k},\bar{i}}-i(\overline{\nabla}{\mathcal{J}})^{\bar{j}}_{k,i}-i(\overline{\nabla}{\mathcal{J}})^{\bar{j}}_{k,\bar{i}}.$

The relation
$\overline{\nabla}(J-J_{0})=-(\nabla-\overline{\nabla})J_{0}+O(r^{-2\tau-1})$
implies that,

$\displaystyle(\overline{\nabla}\mathcal{J})^{j}_{\bar{k},i}=2A^{j}_{i\bar{k}}+O(r^{-2\tau-1}),\qquad(\overline{\nabla}\mathcal{J})^{j}_{\bar{k},\bar{i}}=2A^{j}_{\bar{i}\bar{k}}+O(r^{-2\tau-1})$
(4.5)

The fact that $A$ is symmetric implies that

$\displaystyle(\overline{\nabla}\mathcal{J})^{j}_{\bar{k},\bar{i}}-(\overline{\nabla}\mathcal{J})^{j}_{\bar{i},\bar{k}}=O(r^{-2\tau-1})$
(4.6)

Recall that, on the cone $C_{L}$, there exists a nowhere vanishing multi-
valued holomorphic volume form $\Omega_{0}$. By restricting to local charts
$V_{k}$, $\Omega_{0}$ can be expressed as a single-valued form for each
$\gamma\in\Gamma$ with respect to the asymptotic local frame of $V_{k}$,

$\displaystyle\big{(}\Omega_{0}|_{V_{k}}\big{)}_{\gamma}=\kappa(\gamma)f\mu^{0}\wedge\cdots\wedge\mu^{n-1},$

where $\kappa(\gamma)$ is defined in (4.1). And the condition
$\overline{\nabla}_{\partial r}\Omega_{0}=0$, together with (4.2), implies
that $f$ is bounded on $V_{k}$. It is possible to define a smooth $(n,0)$ form
$\Omega$ on $X_{\infty}$ with respect to $J$ which tends to $\Omega_{0}$ at
infinity with decay rate $-\tau$. Let $\varpi^{i}=(\mu^{i}-iJ\mu^{i})/2$ be
the $(1,0)$ part of $\mu^{i}$ with respect to $J$,

$\displaystyle\big{(}\Omega|_{V_{k}}\big{)}_{\gamma}$
$\displaystyle=\kappa(\gamma)f\varpi^{0}\wedge\ldots\wedge\varpi^{n-1}$
$\displaystyle=\big{(}\Omega_{0}|_{V_{k}}\big{)}_{\gamma}+\frac{\kappa(\gamma)}{2}\sum_{i=0}^{n-1}(-1)^{i-1}f\mathcal{J}^{i}_{\bar{j}}\bar{\mu}^{j}\wedge\mu^{0}\wedge\ldots\wedge\widehat{\mu^{i}}\wedge\ldots\wedge\mu^{n-1}+O(r^{-2\tau})$
(4.7)

It’s easy to check that $(\Omega|_{V_{k}},V_{k})$ can be patched together to
obtain a multi-valued form $\Omega$ on the end, as $\Omega|_{V_{k}}$ can be
viewed as $(n,0)$ projection of $\Omega_{0}$ in each $(V_{k},J)$. The fall-off
condition of $\Omega$ to $\Omega_{0}$ indicates that $\Omega$ is nowhere
vanishing if the radial function $r$ large enough. Without loss of generality,
we assume that $\Omega$ is nowhere vanishing on $X_{\infty}$. We can rewrite
the formula (4.7) on $V_{k}$ as follows,

$\displaystyle\Omega=\Omega_{0}+i\langle
J-J_{0},\Omega_{0}\rangle+O(r^{-2\tau}).$ (4.8)

where $\langle J-J_{0},\Omega_{0}\rangle$ means the contraction of tensor
fields by viewing $J-J_{0}$ and $\Omega_{0}$ as tensor fields on $V_{k}$. Let
$h$ be the standard metric on canonical bundle of $X$ defined to be
$h(\mathcal{S})=|\mathcal{S}\wedge\overline{\mathcal{S}}|/\omega^{n}$ for any
section $\mathcal{S}$ of canonical bundle on $X$. Let $D_{h}$ be the Chern
connection of $h$, then there exists a connection 1-form $\theta$ defined on
$X_{\infty}$ as $D_{h}\mathcal{S}=\theta\otimes\mathcal{S}$. Locally, we can
take $\mathcal{S}=\Omega_{\gamma}$, where $\Omega_{\gamma}$ is a single-valued
local section of canonical bundle. Recall that $(0,1)$ part of the covariant
derivative $D^{0,1}_{h}=\bar{\partial}$, then
$D^{(1,0)}\Omega_{\gamma}=\bar{\partial}\Omega_{\gamma}=d\Omega_{\gamma}$. By
writing $D^{0,1}_{h}\Omega_{\gamma}=\alpha\otimes\Omega_{\gamma}$, in the
following lemma, we derive an explicit formula of $\alpha$, in which we can
see that $\alpha$ is independent of the choice of $\gamma$ modeling the higher
decay terms.

######  Lemma 4.1.

Let $\alpha$ be a $(0,1)$ part of connection 1-form satisfying
$D_{h}^{0,1}\Omega_{\gamma}=\alpha\otimes\Omega_{\gamma}$, then in each
asymptotic local frame $\\{V_{k},\mu^{j}\\}$,

$\displaystyle\alpha|_{V_{k}}=-\sum_{i,j}A^{i}_{i\bar{j}}\bar{\mu}^{j}+O(r^{-2\tau-1})$

###### Proof.

Observe that the operator $d$ can be represented as
$d=\mu^{i}\wedge\overline{\nabla}_{v_{i}}+\bar{\mu}^{i}\wedge\overline{\nabla}_{\overline{v}_{i}}$.
Then, $\Omega_{0}$ is a holomorphic volume form on $(C_{L},J_{0})$ implies
that

$\displaystyle
0=d\Omega_{0}=\bar{\mu}^{i}\wedge\overline{\nabla}_{\bar{v}_{i}}\Omega_{0}.$

Hence, $\overline{\nabla}_{\bar{v}_{i}}\Omega_{0}=0$. Together with (4.8), we
have

$\displaystyle d\Omega$ $\displaystyle=d\Omega_{0}+id\langle
J-J_{0},\Omega_{0}\rangle+O(r^{-2\tau-1})$
$\displaystyle=\frac{i}{2}\bar{\mu}^{i}\wedge\langle\overline{\nabla}_{\bar{v}_{i}}(J-J_{0}),\Omega_{0}\rangle$
(4.9)
$\displaystyle\qquad+\frac{i}{2}\mu^{i}\wedge\big{(}\langle\overline{\nabla}_{v_{i}}(J-J_{0}),\Omega_{0}\rangle+\langle
J-J_{0},\overline{\nabla}_{v_{i}}\Omega_{0}\rangle\big{)}+O(r^{-2\tau-1}).$

To simplify the expression in (4.9), we apply (4.3) or (4.5) and (4.6). Then,
we have

$\begin{split}i\bar{\mu}^{i}\wedge\langle\overline{\nabla}_{\bar{v}_{i}}(J-J_{0}),\Omega_{0}\rangle&=i\bar{\mu}^{i}\wedge\langle(\nabla-\overline{\nabla})_{\bar{v}_{i}}J_{0},\Omega_{0}\rangle+O(r^{-2\tau-1})\\\
&=\kappa(\gamma)\sum_{i,j,k}(-1)^{i}2fA_{\bar{i}\bar{j}}^{k}\bar{\mu}^{i}\wedge\bar{\mu}^{j}\wedge\mu^{0}\wedge\ldots\wedge\widehat{\mu^{k}}\wedge\ldots\wedge\mu^{n}+O(r^{-2\tau-1})\\\
&=O(r^{-2\tau-1}).\end{split}$ (4.10)

Let $\overline{\Gamma}$ be the Christoffel symbol of $\overline{\nabla}$, then
a quick calculation shows that

$\displaystyle i\mu^{i}\wedge\langle
J-J_{0},\overline{\nabla}_{v_{i}}\Omega_{0}\rangle=\kappa(\gamma)\sum_{i,j,k}\mathcal{K}^{i}_{\bar{j}}(f_{i}-f\overline{\Gamma}^{k}_{ik})\bar{\mu}^{{j}}\wedge\mu^{0}\wedge\ldots\wedge\mu^{n-1},$
(4.11)

and

$\displaystyle
i\mu^{i}\wedge\langle\overline{\nabla}_{v_{i}}(J-J_{0}),\Omega_{0}\rangle$
$\displaystyle=i\mu^{i}\wedge\langle(\nabla-\overline{\nabla})_{v_{i}}J_{0},\Omega_{0}\rangle+O(r^{-2\tau-1})$
$\displaystyle=-2\kappa(\gamma)\sum_{i,j}fA_{i\bar{j}}^{i}\bar{\mu}^{j}\wedge\mu^{0}\wedge\ldots\wedge\mu^{n-1}+O(r^{-2\tau-1}).$
(4.12)

Then, (4.10)-(4.12) imply that $D^{0,1}_{h}\Omega=\alpha\otimes\Omega$ with

$\displaystyle\alpha|_{V_{k}}=\frac{1}{2}\sum_{i,j,k}\big{(}\mathcal{K}^{i}_{\bar{j}}(\log
f)_{i}-\mathcal{K}^{i}_{\bar{j}}\overline{\Gamma}^{k}_{ik}-2A^{i}_{i\bar{j}}\big{)}\bar{\varpi}^{j}+O(r^{-2\tau-1})$
(4.13)

The formula (4.13) can be simplified further by considering the holomorphic
volume form condition $\nabla_{\bar{v}_{i}}\Omega_{0}=0$,

$\displaystyle\nabla_{\bar{v}_{i}}\Omega_{0}=\kappa(\gamma)((\log
f)_{\bar{i}}-\sum_{k}\overline{\Gamma}^{k}_{\bar{i}k})f\mu^{1}\wedge\ldots\wedge\mu^{n}$
(4.14)

The base Kähler metric $\omega_{0}=g_{0}(J_{0}\cdot,\cdot)$ can be written as
$\omega_{0}=i(g_{0})_{i\bar{j}}\mu^{i}\wedge\bar{\mu}^{j}$. The relation
$i^{n^{2}}\Omega_{0}\wedge\overline{\Omega}_{0}=\frac{1}{n!}\omega_{0}^{n}$
implies that $|f|^{2}=\det((g_{0})_{i\bar{j}})$, then

$\displaystyle(\log|f|^{2})_{i}$
$\displaystyle=(g_{0})^{j\bar{k}}(g_{0})_{j\bar{k},i}$
$\displaystyle=(g_{0})^{j\bar{k}}\big{(}\overline{\Gamma}^{p}_{ij}(g_{0})_{p\bar{k}}+\overline{\Gamma}^{\bar{q}}_{i\bar{k}}(g_{0})_{j\bar{q}}\big{)}$
$\displaystyle=\sum_{j}\overline{\Gamma}^{j}_{ij}+\sum_{k}\overline{\Gamma}^{\bar{k}}_{i\bar{k}}.$

Therefore, combining with (4.14),

$\displaystyle(\log f)_{i}-\sum_{k}\overline{\Gamma}^{k}_{ik}$
$\displaystyle=(\log|f|^{2})_{i}-(\log\bar{f})_{i}-\sum_{k}\overline{\Gamma}^{k}_{ik}$
$\displaystyle=(\log|f|^{2})_{i}-\sum_{k}\overline{\Gamma}^{\bar{k}}_{i\bar{k}}-\sum_{k}\overline{\Gamma}^{k}_{ik}=0$

we complete the proof of the lemma. ∎

Therefore, after fixing a Ricci-flat Kähler cone $(C_{L},J_{0},g_{0})$, (4.5)
implies that $\alpha$ only depends on the complex structure $J$ after modeling
higher decay terms. By assuming that $D^{1,0}\Omega=\beta\otimes\Omega$, we
have

$\displaystyle\partial h(\Omega)$ $\displaystyle=\langle
D^{1,0}\Omega,\Omega\rangle+\langle\Omega,\overline{D^{0,1}\Omega}\rangle$
$\displaystyle=h(\Omega)\beta+h(\Omega)\overline{\alpha}.$

Hence, we have $\beta=-\overline{\alpha}+\partial\log h(\Omega)$. Let
$\theta_{\omega}$ be the real part of $-i\theta$, we obtain the following
expression,

$\displaystyle\theta_{\omega}=2\operatorname{Im}\alpha-\frac{1}{2}d^{C}\log\frac{\omega^{n}}{|\Omega_{0}\wedge\bar{\Omega}_{0}|}+O(r^{-2\tau-1}).$
(4.15)

Recall that, in $X_{\infty}$, the Ricci form is given by
$\rho=d\theta_{\omega}$ and the first Chern class is $c_{1}=[\rho/\pi]$. By
adding a copy of link $L$ to the end of $X$, we have the following long exact
sequence,

$\displaystyle\cdots\rightarrow H_{dR}^{1}(L)\rightarrow
H^{2}_{c}(X)\xrightarrow{\iota}H_{dR}^{2}(X)\rightarrow
H_{dR}^{2}(L)\rightarrow\cdots$ (4.16)

Noting that $L$ is the link of a Ricci-flat Kähler cone $C_{L}$, It is known
that $L$ is Sasakian-Einstein with $Ric_{g_{L}}=2(n-1)g_{L}$. According to the
standard Bochner technique, we have $H^{1}_{dR}(X)$ vanishes.

######  Lemma 4.2.

Let $[\varrho]\in H^{2}_{dR}(X)$ where $\varrho$ is a closed 2-form with decay
rate $-\delta$, $\delta>2$. Then, there exists a unique preimage of
$[\varrho]$ in $H^{2}_{c}(X)$, denoted by $\iota^{-1}[\varrho]$.

###### Proof.

The uniqueness of preimage directly follows from $H^{1}_{dR}(L)=0$. Now, we
describe the mapping $r:H^{2}_{dR}(X)\rightarrow H^{2}_{dR}(L)$ in (4.16).
Notice that, topologically, $X_{\infty}\cong L\times(R_{0},\infty)$, then, the
closed 2-form $\varrho$ can be written as $\varrho=\eta_{2}+dr\wedge\eta_{1}$.
According to the fall-off condition of $\varrho$ with decay rate $-\delta$,
the norm of $\eta_{1}$, $\eta_{2}$ on $L$ satisfy

$\displaystyle|\eta_{1}|_{g_{L}}=r|\eta_{1}|_{g_{0}}=O(r^{1-\delta}),\qquad|\eta_{2}|_{g_{L}}=r^{2}|\eta_{2}|_{g_{0}}=O(r^{2-\delta})$

By adding a copy of link $L$ at infinity, then in topological space
$\overline{X}_{\infty}=L\times(R_{0},\infty]$, $\eta_{1}$ and $\eta_{2}$ can
extend to infinity by defining zero forms at infinity. Hence, we obtain an
extension of $\varrho$ in the space $\overline{X}_{\infty}$, denoted by
$\overline{\varrho}$, with $\overline{\varrho}|_{\infty}=0$. Then, the class
$[\varrho]$ under the mapping $r$ is given as follows,

$\displaystyle r([\varrho])=\big{[}\overline{\varrho}|_{\infty}\big{]}=0$

Then, the long exact sequence (4.16) implies that we have a preimage of
$[\varrho]$ in $H_{c}^{2}(X)$. ∎

Based on the proof of lemma 4.2, we can explicitly construct a closed form
with compact support represents $\iota^{-1}[\varrho]$. Recall that
$\varrho=\eta_{2}+dr\wedge\eta_{1}$ in $X_{\infty}$. We integrate the form
$dr\wedge\eta_{1}$ in $r$ direction; in particular
$\tilde{\eta}_{1}(r)=-\int_{\infty}^{r}dr\wedge\eta_{1}$. The 1-form
$\tilde{\eta}_{1}$ is well-defined as $|\eta_{1}|_{g_{L}}$ has decay rate
$1-\delta<-1$. Then
$\varrho=\eta_{2}-d_{L}\tilde{\eta}_{1}+d\tilde{\eta}_{1}(r)$. Let
$\tilde{\eta}_{2}=\eta_{2}-d_{L}\tilde{\eta}_{1}$, then the closedness of
$\tilde{\eta}_{2}$ implies that

$\displaystyle
d\tilde{\eta}_{2}=d_{L}\tilde{\eta}_{2}+dr\wedge\partial_{r}\tilde{\eta}_{2}=0\quad\Rightarrow\quad
d_{L}\tilde{\eta}_{2}=0,\quad\partial_{r}\tilde{\eta}_{2}=0.$

Observing that $\lim_{r\rightarrow\infty}\tilde{\eta}_{2}=0$, together with
$\partial_{r}\tilde{\eta}_{2}=0$, we obtain that $\tilde{\eta}_{2}=0$. Hence,
in $X_{\infty}$, $\varrho=d\tilde{\eta}_{1}$. Let $\chi$ be a smooth cut-off
function defined in $X_{\infty}$ such that

$\displaystyle\begin{split}f(x)=\begin{cases}1,\quad\text{ if }|x|\geq
3R_{0};\\\ 0,\quad\text{ if }|x|\leq 2R_{0}.\end{cases}\end{split}$ (4.17)

Then, $\varrho-d(\chi\tilde{\eta}_{1})$ represents the class
$\iota^{-1}[\varrho]$. Back to the case of the first Chern class
$c_{1}=[\rho/2\pi]$, the expresstion (4.15) shows that the Ricci form decays
at infinity with order $-\tau-2<-2$. By lemma 4.2, we have $\iota^{-1}c_{1}\in
H^{2}_{c}(X)$ and $(\rho-d(f\theta_{\omega}))/2\pi$ represents this class.

###  4.2. The Mass Formula

To give a reasonable definition of mass on AC Kähler manifolds, we will invoke
an expression of mass without applying Euclidean coordinates. As mentioned in
[17, Section 3.1.3], the ADM mass can be interpreted as the integral of
linearization of scalar curvature at the Euclidean background metric. This
idea can be generalized to the AC Kähler cases if we replace the background
Euclidean metric with the Riemannian cone metric. If we perturb the cone
metric $g_{0}$ and consider $g(t)=(1-t)g_{0}+tg_{1}$ where $g_{1}$ is another
Riemannian metric defined on $C_{L}$ with the corresponding scalar curvature
$R(t)\in L^{1}(C_{L})$ and bounded near $0$, according to Appendix A, lemma
A.2 for any real local frame on $X_{\infty}$

$\displaystyle\frac{d}{dt}\Big{|}_{t=0}R(t)=\overline{\nabla}^{i}\overline{\nabla}^{j}g_{ij}-\overline{\Delta}\operatorname{tr}_{g_{0}}g$

Notice that

$\displaystyle\int_{C_{L}}\frac{d}{dt}\Big{|}_{t=0}R(t)=\lim_{r\rightarrow\infty}\int_{L(r)}(\overline{\nabla}^{j}g_{ij}-(\operatorname{tr}_{g_{0}}g)_{i})n^{i}d\operatorname{Vol}_{L(r)},$

where $n$ is a outer normal vector field of $L(r)$ and the integrand of the
right-hand side is independent of the choice of real frame. Let $(X,g)$ be an
AC Riemannian manifold asymptotic to metric cone $(C_{L},g_{0})$, we give the
abstract definition of mass on $(X,g)$ by

$\displaystyle\mathfrak{m}(g)=\frac{1}{2(2n-1)\operatorname{Vol}(L)}\lim_{r\rightarrow\infty}\int_{L(r)}(\overline{\nabla}^{j}g_{ij}-(\operatorname{tr}_{g_{0}}g)_{i})n^{i}d\operatorname{Vol}_{L(r)}$
(4.18)

Let $(X,g,J)$ be an AC Kähler manifold asymptotic to a Ricci-flat Kähler cone
$(C_{L},g_{0},J_{0})$ of complex dimension $n$ with the metric $g$ asymptotic
to $g_{0}$ in the end $X_{\infty}$, the remaining part of this section is
aimed to derive a mass formula on $(X,g,J)$ which generalize the Hein-LeBrun’s
mass formula [12]. Observe that

$\displaystyle-\frac{1}{2}(\operatorname{tr}_{g_{0}}g)_{i}=-\frac{1}{2}\Big{(}\log\frac{\det
g}{\det g_{0}}\Big{)}_{i}+O(r^{-2\tau-1})$ (4.19)

and according to the calculation in Appendix (A.3)

$\displaystyle\overline{\nabla}^{j}g_{ij}-\frac{1}{2}(\operatorname{tr}_{g_{0}}g)_{i}=\frac{1}{2}g_{0}^{jk}(\overline{\nabla}_{k}g_{ij}+\overline{\nabla}_{j}g_{ik}-\overline{\nabla}_{i}g_{jk})=g_{0}^{jk}A_{jk,i},$
(4.20)

where $A_{jk,i}=g_{il}A^{l}_{jk}$. Also observe that the formulas (4.19) and
(4.20) can relate with the terms of $\theta_{\omega}$ in (4.15). Then, we have
the following mass formula,

######  Proposition 4.3.

Let $(X,g,J)$ be an AC Kähler manifold asymptotic to a locally Calabi-Yau cone
$(C_{L},g_{0},J_{0})$ of complex dimension $n$ with
$|\overline{\nabla}^{i}(g-g_{0})|_{g_{0}}=O(r^{-\tau-i})$ for $i=0,1,\ldots$,
where $\tau=1-n-\epsilon$. Let $\omega$ represent the Kähler form of $g$ on
$X$. Let $\theta_{\omega}$ be the connection 1-form as in (4.15). Then, the
mass has the following formula,

$\displaystyle\mathfrak{m}(g)=\frac{1}{(2n-1)(n-1)!\operatorname{Vol}(L)}\lim_{r\rightarrow\infty}\int_{L(r)}\theta_{\omega}\wedge\omega^{n-1}.$
(4.21)

###### Proof.

According to the observation from (4.19) and (4.20), the mass formula can be
rewritten on asymptotic local coframe $\\{\mu_{i},\bar{\mu}_{i}\\}$,

$\displaystyle\mathfrak{m}(g)=\frac{1}{2(2n-1)\operatorname{Vol}(L)}\lim_{r\rightarrow\infty}\bigg{\\{}\int_{L(r)}*2\operatorname{Re}\big{(}A_{,\bar{i}}\bar{\mu}^{i}\big{)}-\int_{L(r)}*\bigg{(}d\log\frac{\sqrt{detg}}{\sqrt{\det{g_{0}}}}\bigg{)}+O(r^{-2\epsilon})\bigg{\\}}$

where $*$ is the Hodge star operator of on $(X,g)$ and
$A_{,i}=2g_{0}^{j\bar{k}}A_{j\bar{k},i}$. Also notice that

$\displaystyle*\bigg{(}d\log\frac{\sqrt{\det
g}}{\sqrt{\det{g_{0}}}}\bigg{)}=-\Big{(}Jd\log\frac{\omega^{n}}{|\Omega_{0}\wedge\Omega_{0}|}\Big{)}\wedge\frac{\omega^{n-1}}{(n-1)!}=d^{C}\log\frac{\omega^{n}}{|\Omega_{0}\wedge\Omega_{0}|}\wedge\frac{\omega^{n-1}}{(n-1)!}.$

and

$\displaystyle\operatorname{Im}\alpha\wedge\frac{\omega^{n-1}}{(n-1)!}=(J\operatorname{Re}\alpha)\wedge\frac{\omega^{n-1}}{(n-1)!}=*(\operatorname{Re}\alpha)$

Comparing with the explicit expression of $\theta$ in (4.15), then the mass
formula can be rewritten as

$\displaystyle\mathfrak{m}(g)=\frac{1}{(2n-1)\operatorname{Vol}(L)}\lim_{r\rightarrow\infty}\bigg{\\{}\int_{L(r)}*\operatorname{Re}\big{(}A_{,\bar{i}}\mu^{\bar{i}}-2\alpha\big{)}+\frac{1}{(n-1)!}\int_{L(r)}\theta_{\omega}\wedge\omega^{n-1}\bigg{\\}}$
(4.22)

It suffices to prove the difference $A_{,\bar{i}}\mu^{\bar{i}}-2\alpha$ does
not contribute to the integral. If we write the metric $g$ in terms of frames
$\\{\mu^{i},\mu^{\bar{i}}\\}$,

$\displaystyle
g=g_{ij}\mu^{i}\otimes\mu^{j}+g_{i\bar{j}}\mu^{i}\otimes\mu^{\bar{j}}+\overline{g_{i\bar{j}}}\mu^{\bar{i}}\otimes\mu^{j}+\overline{g_{ij}}\mu^{\bar{i}}\otimes\mu^{\bar{j}}.$

Noting that $\omega=g(J\cdot,\cdot)$ and inserting (4.4) into the expression
of $g$, we obtain that,

$\displaystyle\omega=ig_{i\bar{j}}\mu^{i}\wedge\mu^{\bar{j}}+i(g_{i\bar{k}}\mathcal{J}^{\bar{k}}_{j}-g_{j\bar{k}}\mathcal{J}^{\bar{k}}_{i})\mu^{i}\wedge\mu^{j}-i(g_{k\bar{i}}\mathcal{J}^{k}_{\bar{j}}-g_{k\bar{j}}\mathcal{J}^{k}_{\bar{i}})\mu^{\bar{i}}\wedge\mu^{\bar{j}}+O(r^{-2\tau}).$

The (0,2)-part of $\omega$ with respect to $J_{0}$ is given by
$\omega^{0,2}=-i(g_{k\bar{i}}\mathcal{J}^{k}_{\bar{j}}-g_{k\bar{j}}\mathcal{J}^{k}_{\bar{i}})\mu^{\bar{i}}\wedge\mu^{\bar{j}}$.Now
we calculate $d^{*}\omega^{0,2}$. Notice that $d^{*}=[d^{C},\Lambda]$, then

$\begin{split}d^{*}\omega^{0,2}&=i\Lambda(J\circ
d(g_{k\bar{i}}\mathcal{J}^{k}_{\bar{j}}-g_{k\bar{j}}\mathcal{J}^{k}_{\bar{i}})\mu^{\bar{i}}\wedge\mu^{\bar{j}})+O(r^{-2\tau-1})\\\
&=(\overline{\nabla}\mathcal{J})^{k}_{\bar{j},l}(g_{0})_{k\bar{i}}(g_{0})^{l\bar{i}}\mu^{\bar{j}}-(\overline{\nabla}\mathcal{J})^{k}_{\bar{i},l}(g_{0})_{k\bar{j}}(g_{0})^{l\bar{i}}\mu^{\bar{j}}+O(r^{-2\tau-1})\\\
&=2A^{l}_{l\bar{j}}\mu^{\bar{j}}-2(g_{0})^{i\bar{k}}A_{i\bar{k},\bar{j}}\mu^{\bar{j}}+O(r^{-2\tau-1})\\\
&=2\alpha-A_{,\bar{j}}\mu^{\bar{j}}+O(r^{-2\tau-1}).\end{split}$ (4.23)

Inserting (4.23) to the first term of (4.4), we have

$\displaystyle\int_{L(r)}*\operatorname{Re}\big{(}A_{,\bar{i}}\mu^{\bar{i}}-2\alpha\big{)}$
$\displaystyle=\frac{1}{2}\int_{L(r)}*\big{(}d^{*}\omega^{0,2}+d^{*}\overline{\omega^{0,2}}\big{)}+O(r^{-2\epsilon})$
$\displaystyle=\frac{1}{2}\int_{L(r)}d*(\omega^{0,2}+\overline{\omega^{0,2}})+O(r^{-2\epsilon})=O(r^{-2\epsilon}),$

hence, the proof is completed. ∎

######  Theorem 4.4.

Let $(X,g,J)$ satisfy the same condition as in proposition 4.3, then we have
the following mass formula

$\displaystyle\mathfrak{m}(g)=-\frac{2\pi\langle\iota^{-1}c_{1},[\omega]^{n-1}\rangle}{(2n-1)(n-1)!\operatorname{Vol}(L)}+\frac{1}{2(2n-1)\operatorname{Vol}(L)}\int_{X}R_{g}d\operatorname{Vol}_{g}$
(4.24)

where $c_{1}$ is the first Chern class of $(X,J)$, $[\omega]$ is the Kähler
class of $g$, $R_{g}$ is the scalar curvature of $g$ and $(\cdot,\cdot)$ is
the duality pairing between $H^{2}_{c}(X)$ and $H^{2n-2}(X)$.

###### Proof.

In proposition 4.3, we have already proved a mass formula (4.4), where
$\theta_{\omega}$ is a 1-form defined on $X_{\infty}$ with
$d\theta_{\omega}=\rho$. Consider a smooth cut-off function $f$ as in (4.17)
which is $\equiv 0$ on $X-X_{\infty}$ and $\equiv 1$ if the radius is of large
enough. According to the discussion after lemma 4.2, the first Chern class
admits a representative with compact support
$\iota^{-1}c_{1}=[\rho_{0}/2\pi]$, where $\rho_{0}=\rho-d(f\theta_{\omega})$
is compactly supported. Let $X_{r\leq R}=\\{x\in X,r(x)\leq R\\}$ with
$R>3R_{0}$, then $\rho_{0}$ has compact support on $X_{r<R}$. We have

$\displaystyle\frac{(n-1)!}{2}\int_{X_{r\leq R}}R_{g}d\operatorname{Vol}_{g}$
$\displaystyle=\int_{X_{r\leq R}}\rho\wedge\omega^{n-1}$
$\displaystyle=\int_{X_{r\leq
R}}\big{(}\rho_{0}+d(f\theta_{\omega})\big{)}\wedge\omega^{n-1}$
$\displaystyle={2\pi}\langle\iota^{-1}c_{1},[\omega]^{n-1}\rangle+\int_{L(R)}\theta_{\omega}\wedge\omega^{n-1}$

Then, according to (4.4), we have

$\displaystyle\mathfrak{m}(g)$
$\displaystyle=\frac{1}{(2n-1)(n-1)!\operatorname{Vol}(L)}\lim_{r\rightarrow\infty}\int_{L(r)}\theta_{\omega}\wedge\omega^{n-1}$
$\displaystyle=-\frac{2\pi\langle\iota^{-1}c_{1},[\omega]^{n-1}\rangle}{(2n-1)(n-1)!\operatorname{Vol}(L)}+\frac{1}{2(2n-1)\operatorname{Vol}(L)}\int_{X}R_{g}d\operatorname{Vol}_{g}$

which completes the proof of mass formula ∎

##  5\. Expansion of AC Kähler Metrics

###  5.1. Proof of Theorem D (i)

According to the conditions given in theorem D (i), $\omega_{1}$ and
$\omega_{2}$ are two Kähler forms in the same Kähler class and decay to the
model Kähler form $\omega_{0}$ with rate $-\tau$ ($\tau=n-1+\epsilon$) in
$X_{\infty}$. We also assume that their scalar curvature are identically
equal. Firstly, we obtain an expression of scalar curvature $R(\omega)$ with
respect to some Kähler metric $\omega$ satisfying the fall-off condition,
based on the formula (4.15). Noting that, in (4.15), the leading term of
$\operatorname{Im}\alpha$ only depends on the complex structure $J$ by lemma
4.1. In particular, let $A_{J}=2\operatorname{Im}\alpha+O(r^{-2\tau-1})$ in
(4.15), then

$\displaystyle
R(\omega)=\operatorname{tr}_{\omega}\Big{(}dA_{J}-\frac{1}{2}dd^{C}\log\frac{\omega^{n}}{|\Omega_{0}\wedge\overline{\Omega}_{0}|}\Big{)}+O(r^{-2-2\tau})$
(5.1)

Theorem A implies that there exists a function
$\varphi\in\mathcal{C}^{k+2,\alpha}_{2-2\tau}$ such that
$\omega_{2}=\omega_{1}+dd^{C}\varphi$. Based on the condition $R_{1}=R_{2}$,
we have

$\displaystyle 0$
$\displaystyle=\operatorname{tr}_{\omega_{2}}\rho(\omega_{2})-\operatorname{tr}_{\omega_{1}}\rho(\omega_{1})$
$\displaystyle=\operatorname{tr}_{\omega_{1}}(\rho({\omega_{2}})-\rho({\omega_{1}}))+(\operatorname{tr}_{\omega_{2}}-\operatorname{tr}_{\omega_{1}})\rho({\omega_{2}})$
(5.2)

By inserting (4.15), we rewrite the first term of (5.2),

$\displaystyle\operatorname{tr}_{\omega_{1}}(\rho({\omega_{2}})-\rho({\omega_{1}}))$
$\displaystyle=\operatorname{tr}_{\omega_{1}}\bigg{(}dA_{J}-\frac{1}{2}dd^{C}\log\frac{(\omega_{1}+dd^{C}\varphi)^{n}}{|\Omega\wedge\overline{\Omega}|}-dA_{J}+\frac{1}{2}dd^{C}\log\frac{\omega_{1}^{n}}{|\Omega\wedge\overline{\Omega}|}\bigg{)}$
$\displaystyle=-\frac{1}{2}(\operatorname{tr}_{\omega_{1}}dd^{C})^{2}\varphi+O(r^{-2\tau-2})$
$\displaystyle=-\frac{1}{2}\Delta_{\omega_{1}}^{2}\varphi+O(r^{-2\tau-2}).$

By writing the second term of (5.2) in locally asymptotic frame, it is
observed that the coefficients of $\rho(\omega_{2})$ belongs to
$\mathcal{C}^{k-2,\alpha}_{-\tau-2}$ and
$(\operatorname{tr}_{\omega_{2}}-\operatorname{tr}_{\omega_{1}})\in\mathcal{C}^{k,\alpha}_{-\tau}$,
which implies that

$\displaystyle(\operatorname{tr}_{\omega_{2}}-\operatorname{tr}_{\omega_{1}})\rho(\omega_{2})\in\mathcal{C}^{k-2,\alpha}_{-2\tau-2}.$

Hence, we can deduce that $\varphi$ satisfies the equation,
$\Delta_{\omega_{1}}^{2}\varphi=f$ with
$f\in\mathcal{C}^{k-2,\alpha}_{-2-2\tau}$ on $X$ with $-2-2\tau<-2n$. By
solving the Laplacian on asymptotic conical manifolds, proposition 2.3 (iii)
imples

$\displaystyle\Delta_{\omega_{1}}\varphi=Cr^{2-2n}+\phi,\qquad\phi\in\mathcal{C}^{k,\alpha}_{-2\tilde{\tau}}\text{
with }\tilde{\tau}=\min\\{\tau,-d^{-}_{1}/2\\},$ (5.3)

where $d_{1}^{-}$ is the second negative exceptional weight in $D$ defined in
2.25 with $d_{1}^{-}<d_{0}^{-}=2-2n$. By proposition 4.3, combining with
(4.15), we have

$\displaystyle\mathfrak{m}(\omega_{2})-\mathfrak{m}(\omega_{1})=\lim_{r\rightarrow\infty}\frac{1}{(2n-1)(n-1)!\operatorname{Vol}(L)}$
$\displaystyle\bigg{[}\int_{L(r)}\theta_{\omega_{2}}\wedge\omega_{2}^{n-1}-\int_{L(r)}\theta_{\omega_{1}}\wedge\omega_{1}^{n-1}\bigg{]}$
$\displaystyle=\lim_{r\rightarrow\infty}C(n,L)$
$\displaystyle\bigg{[}\int_{L(r)}\theta_{\omega_{2}}\wedge(\omega_{2}^{n-1}-\omega_{1}^{n-1})$
$\displaystyle\qquad-\int_{L(r)}(\theta_{\omega_{2}}-\theta_{\omega_{1}})\wedge\omega_{1}^{n-1}\bigg{]}.$
$\displaystyle=\lim_{r\rightarrow\infty}C(n,L)$
$\displaystyle\bigg{[}\int_{L(r)}\theta_{\omega_{2}}\wedge(\omega_{2}^{n-1}-\omega_{1}^{n-1})$
$\displaystyle\qquad-\frac{1}{2}\int_{L(r)}d^{C}\log\frac{(\omega_{1}+dd^{C}\varphi)^{n}}{\omega_{1}^{n}}\wedge\omega_{1}^{n-1}\bigg{]},$
(5.4)

where $C(n,L)$ is a constant given in proposition 4.3. Since
$\varphi\in\mathcal{C}^{k,\alpha}_{-\tau}$ and by (4.15), the coefficients of
$\theta_{\omega_{2}}$ with respect to a local asymptotic frame are of class
$\mathcal{C}^{k-1,\alpha}_{-\tau-1}$, then we have

$\displaystyle\int_{L(r)}\theta_{\omega_{2}}\wedge(\omega_{2}^{n-1}-\omega_{1}^{n-1})=\int_{L(r)}\theta_{\omega_{2}}\wedge
dd^{C}\varphi\wedge(\omega_{1}^{n-2}+\omega_{1}^{n-3}\wedge\omega_{2}+\ldots+\omega^{n-2}_{2})=O(r^{-2\epsilon}),$
(5.5)

and

$\displaystyle\int_{L(r)}d^{C}\log\frac{(\omega_{1}+dd^{C}\varphi)^{n}}{\omega_{1}^{n}}\wedge\omega_{1}^{n-1}=\int_{L(r)}d^{C}(\Delta_{\omega_{1}}\varphi)\wedge\omega_{1}^{n-1}+O(r^{-2\epsilon}).$
(5.6)

By inserting (5.5) and (5.6), the difference of mass is

$\displaystyle\mathfrak{m}(\omega_{2})-\mathfrak{m}(\omega_{1})=-\lim_{r\rightarrow\infty}C(n,L)\int_{L(r)}d^{C}(\Delta_{\omega_{1}}\varphi)\wedge\omega_{1}^{n-1}$

Notice that the standard volume form on the link $L$ of the model cone $C_{L}$
is given by $d\operatorname{Vol}_{g_{L}}=\eta\wedge\omega_{0}^{n-1}|_{L(1)}$,
where $\eta=r^{-1}J_{0}dr$ is the contact 1-form of the Sasakian manifold $L$.
Then,

$\begin{split}-\int_{L(r)}d^{C}(\Delta_{\omega_{1}}\varphi)\wedge\omega_{1}^{n-1}&=-C\int_{L(r)}Jd(r^{2-2n})\wedge\omega_{1}^{n-1}\\\
&=C\int_{L(r)}(2n-2)r^{1-2n}(r\eta)\wedge\omega_{0}^{n-1}+O(r^{-\tau})\\\
&=C(2n-2){\operatorname{Vol}(L)}+O(r^{-\tau}),\end{split}$ (5.7)

where $C$ is the constant in (5.3). According to the mass formula in theorem
4.4, the mass only depends on the Kähler class, the first Chern class and
scalar curvature; hence, $\mathfrak{m}(\omega_{1})=\mathfrak{m}(\omega_{2})$.
Then, we have $C=0$, which completes the proof of theorem D (i).

###  5.2.  Proof of Theorem D (ii)

For the proof theorem D (ii), in the case of $\dim_{\mathbb{C}}X_{\infty}=2$,
noticing that the link of Ricci-flat Kähler cone is an Sasakian Einstein
manifold of real dimensional 3, $L$ has constant sectional curvature. Thus,
$C_{L}$ is biholomorphic to $\mathbb{C}^{2}/\Gamma$ for some finite subgroup
$\Gamma\subseteq U(2)$. Hence, the discussion can be reduced to ALE cases.
Since $H^{2}(X_{\infty})=H^{2}(S^{3}/\Gamma)=0$, $\omega-\omega_{euc}$ is
exact real $(1,1)$ form on $X_{\infty}$ with decay rate $-\tau$. According to
Theorem A (ii), there exists $\varphi\in\mathcal{C}^{k+2,\alpha}_{2-\tau}$
such that

$\displaystyle\omega-\omega_{euc}=dd^{c}\varphi+O(r^{-4}).$ (5.8)

If $\omega$ is a scalar-flat Kähler metric, then by formula of scalar
curvature, in the end $X_{\infty}$,

$\displaystyle 0=\operatorname{tr}_{\omega}\rho$
$\displaystyle=\operatorname{tr}_{\omega}dA_{J}-\frac{1}{2}\operatorname{tr}_{\omega}dd^{c}\log\frac{\omega^{n}}{\omega_{0}^{n}}+O(r^{-2\tau-2})$
$\displaystyle=\operatorname{tr}_{\omega}dA_{J}-\frac{1}{2}\Delta_{\omega}^{2}\varphi+O(r^{\max\\{-2\tau-2,-6\\}}).$
(5.9)

Recall that, in complex dimension $2$, the complex structure of ALE manifolds
decays to the standard complex structure in Euclidean space as
$J-J_{0}=O(r^{-3})$ (see [12, Proposition 4.5]). Since
$A_{J}=2\operatorname{Im}\alpha$, where $\alpha$ is given in Lemma 4.1, then
its differential has $dA_{J}=O(r^{-5})$. Then, the equation (5.2) can be
rewritten as,

$\displaystyle\Delta^{2}_{\omega}\varphi=f,\quad\text{ for
}f\in\mathcal{C}^{k-2,\alpha}_{-2\tilde{\tau}-2},$ (5.10)

where $\tilde{\tau}=\min\\{\tau,3/2\\}$. Also, by Proposition 2.3 (iii),
$\Delta_{\omega}\varphi=Cr^{2-2n}+\phi$ with
$\phi\in\mathcal{C}^{k,\alpha}_{-2\tilde{\tau}}$. According to Proposition
4.3, we have

$\displaystyle\mathfrak{m}(\omega)$
$\displaystyle=\lim_{r\rightarrow\infty}\frac{1}{3\operatorname{Vol}(L)}\int_{L(r)}\theta_{\omega}\wedge\omega^{n-1}$
$\displaystyle=\lim_{r\rightarrow\infty}\frac{1}{3\operatorname{Vol}(L)}\int_{L(r)}\Big{(}A_{J}-\frac{1}{2}d^{C}\log\frac{(\omega_{0}+dd^{C}\varphi)^{n}}{\omega_{0}^{n}}\Big{)}\wedge\omega^{n-1}$
$\displaystyle=-\lim_{r\rightarrow\infty}\frac{1}{3\operatorname{Vol}(L)}\int_{L(r)}d^{C}(\Delta_{\omega}\varphi)\wedge\omega^{n-1}$

Then, repeating the calculation in (5.7), we have the constant $C$,

$\displaystyle C=3\mathfrak{m}(\omega).$

Notice that $\displaystyle\Delta_{0}\log r=\frac{2}{r^{2}}$, we have the
solution

$\displaystyle\varphi=\frac{3\mathfrak{m}(\omega)}{2}\log
r+O(r^{-2\tilde{\tau}+2}).$

In the case of $\dim_{\mathbb{C}}X_{\infty}\geq 3$, the asymptotic condition
of the Kähler form on $X_{\infty}$, $\omega-\omega_{0}=O(r^{-\tau})$ with
$\tau=n-1+\epsilon>2$, implies that $\omega-\omega_{0}$ is an exact form. One
can check it by identifying $L\times[T_{0},\infty)$ with $X_{\infty}$. Define
a new parameter $t=\log r$ on cylinder $L\times[T_{0},\infty)$. Consider the
cylinder metric $g_{cyl}=dt^{2}+g_{L}$, then the relation with the conical
metric is $g_{cyl}=e^{-2t}g_{0}$. Therefore,
$|\omega-\omega_{0}|_{g_{cyl}}=O(e^{2-\tau})$. Now, let
$\omega-\omega_{0}=\eta_{2}+\eta_{1}\wedge dt$. If we integrate the form in
$t$ direction, $\eta=-\int_{\infty}^{t}\eta_{1}\wedge dt$, then

$\displaystyle d\eta$ $\displaystyle=d_{L}\eta-\partial_{t}\eta\wedge dt$
$\displaystyle=-\int_{\infty}^{t}d_{L}\eta_{1}\wedge dt+\eta_{1}\wedge dt$
$\displaystyle=\int^{t}_{\infty}\partial_{t}\eta_{2}\wedge dt+\eta_{1}\wedge
dt=\eta_{2}+\eta_{1}\wedge dt,$

where the third equality is from the closedness of $\omega-\omega_{0}$.
According to Theorem A (ii), there exists a real function
$\varphi\in\mathcal{C}^{k+2,\alpha}_{2-\tau}$ such that
$\omega-\omega_{0}=i\partial\overline{\partial}\varphi+O(r^{-2n})$. Based on
the formula (5.1) and $R(\omega)=0$, we have the equation
$\Delta^{2}_{\omega}\varphi=f$, for
$f\in\mathcal{C}^{k-2,\alpha}_{-2\tau^{\prime}-2}$, where the decay rate of
$f$ is determined by the decay rate of $dA_{J}$ (by assumption (1.6)), the
dimension $n$ and $\tau$; in particular,
$\tau^{\prime}=\min\\{n,\tau,n-1+\epsilon^{\prime}\\}$. Also, by proposition
2.3 (ii), $\Delta_{\omega}\varphi=Cr^{2-2n}+\phi\ (**)$ with
$\phi\in\mathcal{C}^{k,\alpha}_{-2\tilde{\tau}}$,
$\tilde{\tau}=\min\\{\tau^{\prime},-d^{-}_{1}/2\\}$. Combining proposition
4.3, (4.15) and (5.7), we can obtain a formula for the constant $C$ in $(**)$,

$\displaystyle C=\frac{2n-1}{n-1}m(\omega).$

Notice that $\Delta_{0}r^{4-2n}=2(4-2n)r^{2-2n}.$ Therefore, we have the
expansion of the solution of $(**)$,

$\displaystyle\varphi=\frac{C}{2(4-2n)}r^{4-2n}+O(r^{2-2\tilde{\tau}})$

Noting the the standard Kähler form on $C_{L}$ is $\omega_{0}=dd^{c}r^{2}/2$,
the Kähler form $\omega$ admits the following expansion,

$\displaystyle\omega=\frac{1}{2}dd^{c}r^{2}+\frac{(2n-1)\mathfrak{m}(\omega)}{2(4-2n)(n-1)}dd^{c}r^{4-2n}+O(r^{-2\tilde{\tau}}),$

which completes the proof.

##  6\. Positive Mass Theorem on Resolution Spaces of Calabi-Yau Cones

We conclude the paper by proving the positive mass theorem based on the
generalized ADM mass defined in (4.18).

Let $(C_{L},J_{0},g_{0})$ be a Ricci-flat Kähler cone with only one
singularity at the vertex $O$. Recall the resolution of the Ricci-flat Kähler
cone is a smooth complex manifold $X$ with a proper map $\pi:X\rightarrow
C_{L}$ such that the map $\pi:X\backslash E\rightarrow C_{L}\backslash\\{O\\}$
is biholomorphic, where $E=\pi^{-1}(O)$ is the exceptional divisor. The
positive mass theorem does not always hold for resolution spaces of Ricci-flat
Kähler cone cones, for instance, the counter example constructed by LeBrun
[16]. So it is reasonable to consider a special class of resolution spaces. In
particular, we say the isolated singularity $O\in C_{L}$ is canonical if the
resolution spaces $(X,\pi)$ satisfies,

$\displaystyle K_{X}=\pi^{*}K_{C_{L}}+\sum_{i}a_{i}E_{i},$ (6.1)

where $E_{i}$ are the irreducible hypersurfaces and $a_{i}\geq 0$. Then, we
can prove the following positive mass theorem on canonical resolution spaces
of Calabi-Yau cones.

######  Theorem 6.1.

Let $(C_{L},J_{0},g_{0})$ be a Ricci-flat Kähler cone such that the only
singularity $O\in C_{L}$ is canonical. Suppose that $X$ admits an
asymptotically conical Kähler metrics $g$ with decay rate $-\tau$,
$\tau=n-1+\epsilon$ $(\epsilon>0)$. If $(X,g)$ has scalar curvature $R\geq 0$,
then the mass $\mathfrak{m}(X,g)\geq 0$, and equals zero only if $(X,J,g)$ is
a crepant resolution of $C_{L}$ with a scalar-flat Kähler metric $g$.

###### Proof.

According to the assumption that $s\geq 0$, the integral of scalar curvature
in the mass formula (4.24) is nonnegative, and equals zero only if $g$ is
scalar-flat. It suffices to prove the first term of the mass formula is also
nonnegative.

Firstly, recalling the map $\iota:H_{c}^{2}(X)\rightarrow H^{2}(X)$ induced by
inclusion of cohomology classes, we prove that $[\omega]\in H^{2}(X)$ has a
preimage in $H_{c}^{2}(X)$. For $\dim_{\mathbb{C}}X=2$, the Calabi-Yau cone
must be $\mathbb{C}^{2}/\Gamma$ with $\Gamma$ is a finite subgroup of $U(2)$.
The fact $H^{1}(S^{3}/\Gamma)=H^{2}(S^{3}/\Gamma)=0$ implies that $\iota$ is
an isomorphism. For $\dim_{\mathbb{C}}X\geq 3$, since we have the decay
condition $\omega-\frac{1}{2}dd^{C}r^{2}=O(r^{-\tau})$ with $\tau>2$ in
$X_{\infty}$, the lemma 4.2 implies that $\omega-\frac{1}{2}dd^{C}r^{2}$ is an
exact form in $X_{\infty}$, written by $d\theta$. Let $f$ be a smooth function
identically equal to $1$ near infinity and vanishing in a compact set of $X$,
for instance, the function defined in (4.17), then
$\omega-d(f(d^{C}r^{2}/4+\theta))$ has a compact support.

Notice that $c_{1}(X)=-c_{1}(K_{X})$, then,

$\displaystyle-\langle\iota^{-1}c_{1},[\omega]^{n-1}\rangle=-\langle\iota^{-1}c_{1},(\iota^{-1}[\omega])^{n-1}\rangle=\langle
c_{1}(K_{X}),(\iota^{-1}[\omega])^{n-1}\rangle.$

According to (6.1) and Poincaré duality, we have,

$\displaystyle\langle
c_{1}(K_{X}),(\iota^{-1}[\omega])^{n-1}\rangle=\sum_{i}a_{i}\int_{E_{i}}\omega^{n-1}\geq
0.$

The above inequality only if $a_{i}=0$ for all $i$, which completes the proof.

∎

######  Example 6.2.

Let $D$ be a compact Fano manifold with $\dim_{\mathbb{C}}D=n-1$ and
$K_{D}^{\times}$, the space by shrinking the zero section of the canonical
line line bundle of $D$. In this example, we consider the standard resolution
of $(K_{D}^{\alpha})^{\times}$, where $\alpha$ is a positive fractional
number. Firstly, we check that $K_{D}^{\times}$ is a Calabi-Yau cone. Let $U$
be a local chart in $D$ as well as a local trivialization of $K_{D}$. We
assume that $U\times\mathbb{C}\subseteq K_{D}$ has a holomorphic coordinate
system $\\{z_{1},\ldots,z_{n-1},u\\}$. There exists a local holomorphic
$(n,0)$ form on $U\times\mathbb{C}$, $\Omega_{U}=dz_{1}\wedge\ldots\wedge
dz_{n-1}\wedge du$ and $\Omega_{U}$ can be extended naturally to be a global
nonvanishing holomorphic $(n,0)$ form on $K_{D}$. To see this, let $V$ be
another local chart of $D$ with $K_{D}|_{V}=V\times\mathbb{C}$ and
$\\{w_{1},\ldots,w_{n-1},v\\}$, its holomorphic coordinates. Recall that the
transition function of $K_{D}$ is given by $v=u\cdot J(\partial z/\partial
w)$, then,

$\displaystyle dw_{1}\wedge\ldots\wedge dw_{n-1}\wedge dv=J(\partial
z/\partial w)dw_{1}\wedge\ldots\wedge dw_{n-1}\wedge
du=dz_{1}\wedge\ldots\wedge dz_{1}\wedge du.$

Hence, we have a global nonvanishing holomorphic $(n,0)$ form, $\Omega$ on
$K_{D}$. By restricting $\Omega$ to the space away from the zero section of
$K_{D}$, we show that $K_{D}^{\times}$ is a Calabi-Yau cone. Similarly, we can
find a global holomorphic $(n,0)$ form, $\Omega$, on $K_{D}^{\alpha}$. Let $U$
be a local chart in $D$ with $(K_{D})^{\alpha}|_{U}=U\times\mathbb{C}$ and
$\\{z_{1},\ldots,z_{n-1},t\\}$ be a holomorphic coordinates of
$\\{U\times\mathbb{C}\\}$. Then, locally, $\Omega$ can be written as

$\displaystyle\Omega|_{U}=\frac{1}{\alpha}t^{\frac{1-\alpha}{\alpha}}dz_{1}\wedge\ldots\wedge
dz_{n-1}\wedge dt.$

Also, by restricting $\Omega$ away from the zero section of $K_{D}^{\alpha}$,
we obtain a nonvanishing holomorphic (possibly multi-valued) $(n,0)$ form on
$(K_{D}^{\alpha})^{\times}$. Note that $\Omega$ is multi-valued if the order
$(1-\alpha)/\alpha$ is not an integer. Therefore, for the resolution map by
shrinking zero section,
$\pi:K^{\alpha}_{D}\rightarrow(K^{\alpha}_{D})^{\times}$, we have the
following adjunction type formula,

$\displaystyle
K_{K_{D}^{\alpha}}=\pi^{*}K_{(K_{D}^{\alpha})^{\times}}+\frac{1-\alpha}{\alpha}D$

Hence, $K^{\alpha}_{D}$ satisfies the positive mass theorem if and only if
$\alpha\leq 1$.

##  Appendix A Linearization of Scalar Curvature and Mass on AE manifolds

The goal of this appendix is to complete the detailed calculation of
linearization of scalar curvature. Based on the calculation, we will see that,
on AE manifolds, the ADM mass can be interpreted as the integral of
linearization of scalar curvature over the sphere at infinity.

Let $X$ be a Riemannian manifold with a base metric $\overline{g}$. If we
assume that $g$ is another metric over $X$, then the linearization of scalar
curvature $DR(g)$ with respect to $(X,\overline{g})$ is defined to be

$\displaystyle DR(g)=\frac{d}{dt}\Big{|}_{t=0}R(t)$

where $R(t)$ is the scalar curvature of $g(t)=\overline{g}+t(g-\overline{g})$.
Let $\\{\xi_{i},1\leq i\leq n\\}$ be a system of local frame on $X$. Let
$\nabla$ (resp. $\overline{\nabla}$) be the Levi-Civita connection of $g$
(resp. $\overline{g}$) and $\Gamma$ (resp. $\overline{\Gamma}$) be the
Christoffel symbol of $\nabla$ (resp. $\overline{\nabla}$). Also the
difference of two connections is denoted by $A=\nabla-\overline{\nabla}$.
Then, the Ricci curvature of $g$ and $\overline{g}$ has the following
relation,

######  Lemma A.1.

Let $R_{ij}$ and $\overline{R}_{ij}$ be the Ricci curvature of $g$ and
$\overline{g}$ with respect to the local frame system $\\{\xi_{i},1\leq i\leq
n\\}$ and its dual $\\{\omega^{i},1\leq i\leq n\\}$, then we have

$\displaystyle
R_{ij}=\overline{R}_{ij}+(\overline{\nabla}_{k}A^{k}_{ij}-\overline{\nabla}_{i}A^{k}_{kj})+(A^{k}_{kl}A^{l}_{ij}-A^{k}_{jl}A^{l}_{ik})+A^{k}_{lj}\omega^{l}([\xi_{k},\xi_{i}]).$

###### Proof.

Notice that
$R_{ij}=\omega^{k}(R(\xi_{k},\xi_{i})\xi_{j})=\omega^{k}(\nabla_{\xi_{k}}\nabla_{\xi_{i}}\xi_{j}-\nabla_{\xi_{i}}\nabla_{\xi_{k}}\xi_{j})$
and $\overline{R}_{ij}$ admits a similar formula. Then, we have

$\displaystyle\begin{split}R_{ij}-\overline{R}_{ij}=&\
\omega^{k}(\nabla_{\xi_{k}}\nabla_{\xi_{i}}\xi_{j}-\nabla_{\xi_{i}}\nabla_{\xi_{k}}\xi_{j})-\omega^{k}(\overline{\nabla}_{\xi_{k}}\overline{\nabla}_{\xi_{i}}\xi_{j}-\overline{\nabla}_{\xi_{i}}\overline{\nabla}_{\xi_{k}}\xi_{j})\\\
=&\
\omega^{k}\Big{[}\overline{\nabla}_{\xi_{k}}(\nabla_{\xi_{i}}-\overline{\nabla}_{\xi_{i}})\xi_{j}+(\nabla_{\xi_{k}}-\overline{\nabla}_{\xi_{k}})\overline{\nabla}_{\xi_{i}}\xi_{j}+(\nabla_{\xi_{k}}-\overline{\nabla}_{\xi_{k}})(\nabla_{\xi_{i}}-\nabla_{\xi_{i}})\xi_{j}\Big{]}\\\
&-\omega^{k}\Big{[}\overline{\nabla}_{\xi_{i}}(\nabla_{\xi_{k}}-\overline{\nabla}_{\xi_{k}})\xi_{j}+(\nabla_{\xi_{i}}-\overline{\nabla}_{\xi_{i}})\overline{\nabla}_{\xi_{k}}\xi_{j}+(\nabla_{\xi_{i}}-\overline{\nabla}_{\xi_{i}})(\nabla_{\xi_{k}}-\nabla_{\xi_{k}})\xi_{j}\Big{]}\\\
=&\
\big{(}\xi_{k}(A_{ij}^{k})+A^{l}_{ij}\overline{\Gamma}^{k}_{lk}+A^{k}_{lk}\overline{\Gamma}^{l}_{ij}+A^{l}_{ij}A^{k}_{lk}\big{)}-\big{(}\xi_{i}(A_{kj}^{k})+A^{l}_{kj}\overline{\Gamma}^{k}_{il}+A^{k}_{li}\overline{\Gamma}^{l}_{kj}+A^{l}_{kj}A^{k}_{li}\big{)}\\\
=&\
\xi_{k}(A_{ij}^{k})-\xi_{i}(A_{kj}^{k})+A^{l}_{ij}\overline{\Gamma}^{k}_{lk}+A^{k}_{lk}\overline{\Gamma}^{l}_{ij}-A^{l}_{kj}\overline{\Gamma}^{k}_{il}-A^{k}_{li}\overline{\Gamma}^{l}_{kj}\\\
&\ +A^{l}_{ij}A^{k}_{lk}-A^{l}_{kj}A^{k}_{li}.\end{split}$ (A.1)

Also we can compute $\overline{\nabla}A$, then

$\displaystyle\begin{split}\overline{\nabla}_{k}A^{k}_{ij}-\overline{\nabla}_{i}A^{k}_{kj}=&\
\xi_{k}(A_{ij}^{k})+A^{l}_{ij}\overline{\Gamma}_{lk}^{k}-A^{k}_{lj}\overline{\Gamma}_{ki}^{l}-A^{k}_{il}\overline{\Gamma}^{l}_{kj}\\\
&-\xi_{i}(A_{kj}^{k})-A^{l}_{kj}\overline{\Gamma}^{k}_{il}+A^{k}_{lj}\overline{\Gamma}^{l}_{ik}+A^{k}_{lk}\overline{\Gamma}^{l}_{ij}\\\
=&\
\xi_{k}(A_{ij}^{k})-\xi_{i}(A_{kj}^{k})+A^{l}_{ij}\overline{\Gamma}^{k}_{lk}+A^{k}_{lk}\overline{\Gamma}^{l}_{ij}-A^{l}_{kj}\overline{\Gamma}^{k}_{il}-A^{k}_{li}\overline{\Gamma}^{l}_{kj}\\\
&+A^{k}_{lj}\overline{\Gamma}^{l}_{ik}-A^{k}_{lj}\overline{\Gamma}^{l}_{ki}.\end{split}$
(A.2)

Notice that $A$ is a symmetric tensor, but $\overline{\Gamma}$ is not and

$\displaystyle\overline{\Gamma}^{l}_{ik}-\overline{\Gamma}^{l}_{ki}=\omega^{l}([\xi_{i},\xi_{k}]).$

By plugging (A.2) into (A.1), we have

$\displaystyle
R_{ij}-\overline{R}_{ij}=\overline{\nabla}_{k}A^{k}_{ij}-\overline{\nabla}_{i}A^{k}_{kj}-A^{k}_{lj}\omega^{l}([\xi_{i},\xi_{k}])+A^{l}_{ij}A^{k}_{lk}-A^{l}_{kj}A^{k}_{li},$

which completes the proof. ∎

######  Lemma A.2.

Let $DR(g)$ be the linearization of scalar curvature with respect to
$(X,\overline{g})$. Then, on the local frame system $\\{\xi_{i},1\leq i\leq
n\\}$,

$\displaystyle
DR(g)=\langle\textup{Ric}_{\overline{g}},g-\overline{g}\rangle_{\overline{g}}+\overline{\nabla}^{j}\overline{\nabla}^{i}g_{ij}-\overline{\Delta}\operatorname{tr}_{\overline{g}}g.$

###### Proof.

Firstly, we derive a formula for $A$. Notice that

$\displaystyle\overline{\nabla}_{i}g_{jk}$
$\displaystyle=\xi_{i}(g_{jk})-g(\overline{\nabla}_{\xi_{i}}\xi_{j},\xi_{k})-g(\xi_{j},\overline{\nabla}_{\xi_{i}}\xi_{k})$
$\displaystyle=g\big{(}(\nabla_{\xi_{i}}-\overline{\nabla}_{\xi_{i}})\xi_{j},\xi_{k}\big{)}+g\big{(}\xi_{j},(\nabla_{\xi_{i}}-\overline{\nabla}_{\xi_{i}})\xi_{k}\big{)}$
$\displaystyle=g_{lk}A^{l}_{ij}+g_{jl}A^{l}_{ik}.$

By permutation of the indice $i,j,k$, we have

$\displaystyle
A^{k}_{ij}=\frac{1}{2}g^{kl}(\overline{\nabla}_{i}g_{jl}+\overline{\nabla}_{j}g_{il}-\overline{\nabla}_{l}g_{ij})$
(A.3)

Let $A(t)=\nabla_{t}-\overline{\nabla}$ and $\nabla_{t}$ be the Levi-Civita
connection of $g(t)=\overline{g}+t(g-\overline{g})$, then

$\displaystyle
R(t)=g^{ij}(t)\big{(}\overline{R}_{ij}+\overline{\nabla}_{k}A^{k}_{ij}(t)-\overline{\nabla}_{i}A^{k}_{kj}(t)-A^{k}_{lj}(t)\omega^{l}([\xi_{i},\xi_{k}])+A^{l}_{ij}(t)A^{k}_{lk}(t)-A^{l}_{kj}(t)A^{k}_{li}(t)\big{)}$

Notice that $A(0)=0$, we have

$\displaystyle\begin{split}\frac{d}{dt}\Big{|}_{t=0}R(t)=&\overline{R}_{ij}\frac{d}{dt}\Big{|}_{t=0}g^{ij}(t)\\\
&+\overline{g}^{ij}\frac{d}{dt}\big{|}_{t=0}\big{(}\overline{\nabla}_{k}A^{k}_{ij}(t)-\overline{\nabla}_{i}A^{k}_{kj}(t)\big{)}-\omega^{l}([\xi_{i},\xi_{k}])\overline{g}^{ij}\frac{d}{dt}\Big{|}_{t=0}A^{k}_{lj}(t).\end{split}$
(A.4)

Then, we compute each term in (A.4) separately.

$\displaystyle\frac{d}{dt}\Big{|}_{t=0}A^{k}_{lj}(t)$
$\displaystyle=\frac{1}{2}\overline{g}^{km}(\overline{\nabla}_{l}\dot{g}_{jm}+\overline{\nabla}_{j}\dot{g}_{ml}-\overline{\nabla}_{m}\dot{g}_{jl})(0)$
$\displaystyle=\frac{1}{2}\overline{g}^{km}(\overline{\nabla}_{l}{g}_{jm}+\overline{\nabla}_{j}{g}_{ml}-\overline{\nabla}_{m}{g}_{jl})$

Inserting into the third term of (A.4),

$\displaystyle\omega^{l}([\xi_{i},\xi_{k}])\overline{g}^{ij}\frac{d}{dt}\Big{|}_{t=0}A^{k}_{lj}(t)$
$\displaystyle=\frac{1}{2}\overline{g}^{ij}\overline{g}^{km}(\overline{\nabla}_{[i,k]}{g}_{jm}+\overline{\nabla}_{j}{g}_{m[i,k]}-\overline{\nabla}_{m}{g}_{j[i,k]})=0$

The above identity is vanishing because of the symmetry of indices; in
particular, by permuting $(ij)$ and $(km)$, we can check that each term in
above identity is vanishing. Notice that

$\displaystyle\overline{g}^{ij}\frac{d}{dt}\Big{|}_{t=0}\overline{\nabla}_{i}A^{k}_{kj}(t)$
$\displaystyle=\frac{1}{2}\overline{g}^{ij}\overline{\nabla}_{i}\big{(}\overline{g}^{kl}(\overline{\nabla}_{j}g_{kl}+\overline{\nabla}_{k}g_{lj}-\overline{\nabla}_{l}g_{jk})\big{)}$
$\displaystyle=\frac{1}{2}\overline{g}^{ij}\overline{\nabla}_{i}\overline{\nabla}_{j}\big{(}\overline{g}^{kl}g_{kl}\big{)}$
$\displaystyle=\frac{1}{2}\Delta\operatorname{tr}_{\overline{g}}g$

and

$\displaystyle\overline{g}^{ij}\frac{d}{dt}\Big{|}_{t=0}\overline{\nabla}_{k}A^{k}_{ij}(t)$
$\displaystyle=\frac{1}{2}\overline{g}^{ij}\overline{\nabla}_{k}\big{(}\overline{g}^{kl}(\overline{\nabla}_{j}g_{il}+\overline{\nabla}_{i}g_{lj}-\overline{\nabla}_{l}g_{ji})\big{)}$
$\displaystyle=\overline{g}^{ij}\overline{\nabla}_{k}\big{(}\overline{g}^{kl}\overline{\nabla}_{j}g_{il}\big{)}-\frac{1}{2}\overline{g}^{ij}\overline{\nabla}_{k}\big{(}\overline{g}^{kl}\overline{\nabla}_{l}g_{ij}\big{)}$
$\displaystyle=\overline{\nabla}^{j}\overline{\nabla}^{i}g_{ij}-\frac{1}{2}\overline{\Delta}\operatorname{tr}_{\overline{g}}g,$

combining with
$d/dt|_{t=0}g^{ij}(t)=\overline{g}^{ik}\overline{g}^{jl}(g-\overline{g})_{kl}$
and A.4, then we have,

$\displaystyle\frac{d}{dt}\Big{|}_{t=0}R(t)=\langle\textup{Ric}_{\overline{g}},g-\overline{g}\rangle_{\overline{g}}+\overline{\nabla}^{j}\overline{\nabla}^{i}g_{ij}-\overline{\Delta}\operatorname{tr}_{\overline{g}}g,$

which completes the proof. ∎

Based on lemma A.2, we can quickly derive the ADM mass on AE manifolds from
the integral of linearization of scalar curvature. Let $g$ be a Riemannian
metric in $\mathbb{R}^{n}$ and $DR(g)$ be the linearization of scalar
curvature with respect to the Euclidean metric on $\mathbb{R}^{n}$. Then, the
integral of $DR(g)$ is given by

$\displaystyle\int_{\mathbb{R}^{n}}DR(g)d\operatorname{Vol}_{euc}$
$\displaystyle=\lim_{r\rightarrow\infty}\int_{S^{n-1}(r)}\big{(}\overline{\nabla}^{j}g_{ij}-(\operatorname{tr}_{g_{0}}g)\big{)}n^{i}d\operatorname{Vol}_{S^{n-1}(r)}$
$\displaystyle=\lim_{r\rightarrow\infty}\int_{S^{n-1}(r)}(g_{ij,j}-g_{jj,i})n^{i}d\operatorname{Vol}_{S^{n-1}(r)}.$
(A.5)

Notice that the formula (A.5) only depends on the information of metrics at
infinity. Hence, up to a constant factor, (A.5) can be defined to be the mass
on AE manifolds.

## References

  * [1] R. Arnowitt, S. Deser, and C. W. Misner.  Coordinate invariance and energy expressions in general relativity.  Phys. Rev., 122:997–1006, May 1961.
  * [2] Robert Bartnik.  The mass of an asymptotically flat manifold.  Communications on pure and applied mathematics, 39(5):661–693, 1986\.
  * [3] Charles Boyer and Krazysztof Galicki.  Sasakian geometry.  Oxford Univ. Press, 2008.
  * [4] Murray Cantor.  Elliptic operators and the decomposition of tensor fields.  Bulletin of the American Mathematical Society, 5(3):235–262, 1981\.
  * [5] Alice Chaljub-Simon and Yvonne Choquet-Bruhat.  Problémes elliptiques du second ordre sur une variété euclidienne á l’infini.  In Annales de la Faculté des sciences de Toulouse: Mathématiques, volume 1, pages 9–25, 1979.
  * [6] Isaac Chavel.  Eigenvalues in Riemannian geometry.  Academic press, 1984.
  * [7] Jeff Cheeger et al.  Spectral geometry of singular riemannian spaces.  Journal of differential geometry, 18(4):575–657, 1983.
  * [8] Jeff Cheeger and Gang Tian.  On the cone structure at infinity of ricci flat manifolds with euclidean volume growth and quadratic curvature decay.  Inventiones mathematicae, 118(1):493–571, 1994.
  * [9] P. Chrv[ciel.  Boundary conditions at spatial infinity from a hamiltonian point of view.  arXiv: General Relativity and Quantum Cosmology, pages 49–59, 1985\.
  * [10] Ronan J Conlon, Hans-Joachim Hein, et al.  Asymptotically conical calabi–yau manifolds, i.  Duke Mathematical Journal, 162(15):2855–2902, 2013.
  * [11] Ryushi Goto.  Calabi-yau structures and einstein-sasakian structures on crepant resolutions of isolated singularities.  Journal of the Mathematical Society of Japan, 64(3):1005–1052, 2012\.
  * [12] Hans-Joachim Hein and Claude LeBrun.  Mass in kähler geometry.  Communications in Mathematical Physics, 347(1):183–221, 2016.
  * [13] Hans-Joachim Hein and Song Sun.  Calabi-yau manifolds with isolated conical singularities.  Publications mathématiques de l’IHÉS, 126(1):73–130, 2017\.
  * [14] Dominic D Joyce.  Compact manifolds with special holonomy.  Oxford University Press on Demand, 2000.
  * [15] C. Lebrun.  Counter-examples to the generalized positive action conjecture.  Communications in Mathematical Physics, 118:591–596, 1988.
  * [16] Claude LeBrun.  Counter-examples to the generalized positive action conjecture.  Communications in Mathematical Physics, 118(4):591 – 596, 1988\.
  * [17] Dan A Lee.  Geometric relativity, volume 201.  American Mathematical Soc., 2019.
  * [18] R. Lockhart.  Fredholm properties of a class of elliptic operators on non-compact manifolds.  Duke Mathematical Journal, 48:289–312, 1981.
  * [19] Robert B. Lockhart and Robert C. Mc Owen.  Elliptic differential operators on noncompact manifolds.  Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Ser. 4, 12(3):409–447, 1985.
  * [20] Stephen P Marshal.  Deformations of special Lagrangian submanifolds.  PhD thesis, University of Oxford, 2002.
  * [21] Richard M Schoen and Shing-Tung Yau.  Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity.  Proceedings of the National Academy of Sciences, 76(3):1024–1025, 1979.
  * [22] James Sparks.  Sasaki-einstein manifolds.  arXiv preprint arXiv:1004.2461, 2010.
  * [23] Edward Witten.  A new proof of the positive energy theorem.  Communications in Mathematical Physics, 80(3):381–402, 1981.
  * [24] Qi Yao.  Invariant scalar-flat kähler metrics on line bundles over generalized flag varieties, 2020.