# Modified extremal Kähler metrics and multiplier Hermitian-Einstein metrics

Yasuhiro Nakagawa and Satoshi Nakamura  Y. Nakagawa: Faculty of Advanced
Science and Technology, Kumamoto University, Kurokami 2-40-1, Chuo-ku,
Kumamoto 860-8555, Japan<EMAIL_ADDRESS>S. Nakamura:
Department of Mathematics, Tokyo Institute of Technology, 2-12-1, Ookayama,
Meguro-ku, Tokyo, 152-8551, Japan<EMAIL_ADDRESS>
###### Abstract.

Motivated by the notion of multiplier Hermitian-Einstein metric of type
$\sigma$ introduced by Mabuchi, we introduce the notion of $\sigma$-extremal
Kähler metrics on compact Kähler manifolds, which generalizes Calabi’s
extremal Kähler metrics. We characterize the existence of this metric in terms
of the coercivity of a certain functional on the space of Kähler metrics to
show that, on a Fano manifold, the existence of a multiplier Hermitian-
Einstein metric of type $\sigma$ implies the existence of a $\sigma$-extremal
Kähler metric.

###### Key words and phrases:

Extremal Kähler metrics, Multiplier Hermitian-Einstein metrics.

###### 2010 Mathematics Subject Classification:

Primary 53C25; Secondary 53C55, 58E11.

The first author is supported by JSPS Grant-in-Aid for Scientific Research (C)
No. 23K03094. The second author is supported by JSPS Grant-in-Aid for Science
Research Activity Start-up No. 21K20342 and Grant-in-Aid for Early-Career
Scientists No. 24K16917.

###### Contents

  1. 1 Introduction
  2. 2 Preliminary
  3. 3 An $h$-modified extremal Kähler metric and the properness of $K_{V}^{h}$
  4. 4 An $h$-modified extremal Kähler metrics and the geodesic stability
  5. 5 Existence for a $\sigma$-soliton and a $\sigma$-extremal Kähler metrics

##  1\. Introduction

Various generalizations of Kähler-Einstein metrics are discussed in the theory
for canonical Kähler metrics. Calabi’s extremal Kähler metric [7] is one of
the most important generalizations for a compact Kähler manifold. For a Fano
manifold, Mabuchi’s soliton [31] and a Kähler-Ricci soliton [27] are also
extensively studied and can be seen as typical examples of multiplier
Hermitian-Einstein metrics introduced by Mabuchi [32].

It is known that for a Fano manifold, the existence of Mabuchi’s soliton
implies the existence of Calabi’s extremal Kähler metric in the first Chern
class as pointed out in [25, Remark 2.22], [33, Section 9.6] and [45, Remark
5.8] for example. In this paper, we focus on this interesting relation for the
existence of different canonical Kähler metrics from view points of multiplier
Hermitian-Einstein metrics.

Let $M$ be an $n$ dimensional compact connected Kähler manifold without
boundary and $\Omega$ its Kähler class. we denote, by $\mathfrak{X}(M)$, the
Lie algebra of holomorphic vector fields on $M$ and put
$\mathfrak{X}_{0}(M):=\left\\{X\in\mathfrak{X}(M)\,|\,\mathrm{Zero}(X)\neq\varnothing\right\\}$,
where $\mathrm{Zero}(X)$ is the zero-set of $X\in\mathfrak{X}(M)$. Note that
if $M$ is a Fano manifold then $\mathfrak{X}(M)=\mathfrak{X}_{0}(M)$. For
$X\in\mathfrak{X}(M)$ and $\omega\in\Omega$, there exists a unique complex
valued smooth function $\theta_{X}^{(\omega)}\in C^{\infty}(M;\mathbb{C})$ on
$M$ such that

(1)
$i_{X}\omega=\mathbb{H}_{\omega}\left(i_{X}\omega\right)+\sqrt{-1}\,\overline{\partial}\theta_{X}^{(\omega)}\quad\text{and}\quad\int_{M}\theta_{X}^{(\omega)}\omega^{n}=0,$

where $i_{X}\omega$ is the interior product of $\omega$ with $X$ and
$\mathbb{H}_{\omega}(i_{X}\omega)$ the harmonic part of $i_{X}\omega$ with
respect to $\omega$ (cf. [26, pp. 90–95]). Then we have

$L_{X}\omega=di_{X}\omega=\sqrt{-1}\partial\overline{\partial}\theta_{X}^{(\omega)},$

where $L_{X}\omega$ is the Lie differentiation of $\omega$ with respect to
$X$. $\theta_{X}^{(\omega)}$ is called the complex potential of $X$ with
respect to $\omega$. If $X\in\mathfrak{X}_{0}(M)$, then
$\mathbb{H}_{\omega}(i_{X}\omega)=0$, hence we have
$i_{X}\omega=\sqrt{-1}\,\overline{\partial}\theta_{X}^{(\omega)}$.

Fix $V\in\mathfrak{X}_{0}(M)$ and a $V_{\mathrm{Im}}$-invariant Kähler metric
$\omega_{0}\in\Omega$, where
$V_{\mathrm{Im}}:=\frac{1}{2\sqrt{-1}}(V-\overline{V})$ is the imaginary part
of $V$. Let

$\mathcal{H}:=\set{\phi\in{C^{\infty}(M;\mathbb{R})}\,}{\,\omega_{\phi}:=\omega_{0}+\sqrt{-1}\partial\overline{\partial}\phi>0}$

be the set of all Kähler potentials, and put
$\mathcal{H}_{V}=\set{\phi\in\mathcal{H}\,}{\,L_{V_{\mathrm{Im}}}\omega_{\phi}=0}$.
For each $\phi\in\mathcal{H}_{V}$,
$\theta_{V}^{(\omega_{\phi})}\in{C^{\infty}(M;\mathbb{R})}$ is a real valued
function on $M$. In this article, if there is no fear of confusion,
$\theta_{V}^{(\omega_{\phi})}$ and $\theta_{V}^{(\phi)}$ are used
interchangeably. According to [20], two real numbers
$\min_{M}\theta_{V}^{(\phi)}$ and $\max_{M}\theta_{V}^{(\phi)}$ are
independent of the choice of $\phi\in\mathcal{H}_{V}$. Set $I=(\alpha,\beta)$
where

$-\infty\leqq\alpha<\min_{M}\theta_{V}^{(\phi)}\leqq\max_{M}\theta_{V}^{(\phi)}<\beta\leqq+\infty.$

Let $\sigma(s)$ be a real-valued smooth function on $I$. Since
$\int_{M}\exp(-\sigma(\theta_{V}^{(\phi)}))\omega_{\phi}^{n}$ is independent
of the choice of $\phi\in\mathcal{H}_{V}$, we normalize $\sigma$ by the
condition

$\int_{M}\exp(-\sigma(\theta_{V}^{(\phi)}))\omega_{\phi}^{n}=\int_{M}\omega_{\phi}^{n}.$

######  Definition 1.1.

A Kähler metric $\omega_{\phi}$ ($\phi\in\mathcal{H}_{V}$) is an
$\sigma$-extremal Kähler metric if

$S(\omega_{\phi})-\underline{S}=1-e^{-\sigma(\theta_{V}^{(\phi)})}$

where $S(\omega_{\phi})$ is the scalar curvature of $\omega_{\phi}$ and
$\underline{S}$ its average
$\int_{M}S(\omega_{\phi})\omega_{\phi}^{n}/\int_{M}\omega_{\phi}^{n}$. We note
that the constant $\underline{S}$ is independent of the choice of
$\phi\in\mathcal{H}_{V}$.

When $1-e^{-\sigma(\theta_{V}^{(\phi)})}=\theta_{V}^{(\phi)}$, a
$\sigma$-extremal Kähler metric is nothing but an extremal Kähler metric
introduced by Calabi [7]. The first result in this article is to characterize
the existence for a $\sigma$-extremal Kähler metric in terms of the properness
of a certain energy functional on the space of Kähler metrics as a
generalization of the works by Chen-Cheng [10, 11] and He [23]. Let
$\mathrm{Aut}_{0}(M)$ be the identity component of the holomorphic
automorphism group for $M$, and $G=\mathrm{Aut}_{0}(M,V)$ be its subgroup
which commutes with the action of $V$.

######  Theorem 1.2.

Let $\mathrm{Aut}_{0}(M)$ be the identity component of the holomorphic
automorphism group for $M$, and $\mathrm{Aut}_{0}(M,V)$ be its subgroup which
commutes with the action of $V$. Then, there exists a $\sigma$-extremal Kähler
metric in $\mathcal{H}_{V}$ if and only if the $\sigma$-Mabuchi functional is
coercive.

We shall prove a more general statement in Theorem 3.1 for an $h$-modified
extremal Kähler metric in the sense that
$S(\omega_{\phi})-\underline{S}=h(\theta_{V}^{(\phi)})$ for a real valued
smooth function $h$ on the interval $I$ satisfying
$\int_{M}h(\theta_{V}^{(\phi)})\omega_{\phi}^{n}=0$. Here, we note that the
integral $\int_{M}h(\theta_{V}^{(\phi)})\omega_{\phi}^{n}$ is independent of
the choice of $\phi\in\mathcal{H}_{V}$.

An $h$-modified extremal Kähler metric is a special case of a weighted
constant scalar curvature Kähler metric [1, 28, 29]. In our setting, we can
extend an argument of Chen-Cheng [11] and He [23] on the method of continuity
to show the regularity of minimizers for a Mabuchi type energy functional
corresponds to $h$-modified extremal Kähler metrics in Section 3.2. As the
result, we can establish not only the coercivity theorem (Theorem 3.1) but
also the geodesic stability theorems (Theorems 4.2 and 4.6).

The definition of the $\sigma$-extremal Kähler metric is motivated by both the
notion of the multiplier Hermitian-Einstein metrics of type $\sigma$
introduced by Mabuchi [32] and the inequality (6). Consider the case when $M$
is a Fano manifold and $\Omega=2\pi c_{1}(M)$. Assume one of the following
conditions: (i) $\dot{\sigma}\leqq{0}\leqq\ddot{\sigma}$, (ii)
$\ddot{\sigma}>0$, where $\dot{\sigma}$ and $\ddot{\sigma}$ are the first
derivative and the second derivative respectively. Fix
$\omega_{\phi}\in\Omega$. Mabuchi called a conformally Kähler metric

$\widetilde{\omega}_{\phi}:=\omega_{\phi}\exp\left(-\frac{1}{n}\sigma(\theta_{V}^{(\phi)})\right)$

a multiplier Hermitian metric of type $\sigma$. Note that the multiplier
Hermitian metric $\widetilde{\omega}_{\phi}$ can be seen as an Hermitian
metric on the holomorphic tangent bundle $TM$. Then
$\widetilde{\omega}_{\phi}$ defines the Hermitian connection

$\widetilde{\nabla}:=\nabla-\frac{\partial(\sigma(\theta_{V}^{(\phi)}))}{n}\mathrm{id}_{TM},$

where $\nabla$ is the natural connection with respect to $\omega$. The Ricci
form $\mathrm{Ric}^{\sigma}_{V}(\omega_{\phi})$ of
$(\widetilde{\omega}_{\phi},\widetilde{\nabla})$ is equal to
$\mathrm{Ric}(\omega_{\phi})+\sqrt{-1}\partial\overline{\partial}\sigma(\theta_{V}^{(\phi)})$,
where $\mathrm{Ric}(\omega_{\phi})\in{2\pi}c_{1}(M)$ is the Ricci form for
$\omega_{\phi}$ defined by
$-\sqrt{-1}\partial\overline{\partial}\log\omega_{\phi}^{n}$.

######  Definition 1.3.

The conformally Kähler metric $\widetilde{\omega}_{\phi}$ is a multiplier
Hermitian-Einstein metric of type $\sigma$ if
$\mathrm{Ric}^{\sigma}_{V}(\omega_{\phi})=\omega_{\phi}$. Then we call
$\omega_{\phi}$ itself a $\sigma$-soliton.

Define the Ricci potential $\rho_{\phi}$ for $\omega_{\phi}$ as follows.

(2)
$\mathrm{Ric}(\omega_{\phi})-\omega_{\phi}=\sqrt{-1}\partial\overline{\partial}\rho_{\phi}\quad\text{and}\quad\int_{X}(1-e^{\rho_{\phi}})\omega_{\phi}^{n}=0.$

In this terminology, $\omega_{\phi}$ is a $\sigma$-soliton if and only if
$\rho_{\phi}+\sigma(\theta_{V}^{(\phi)})=0$.

A $\sigma$-soliton gives some well-known generalizations of Kähler-Einstein
metrics.

  1. (i)

When $V=0$, a $\sigma$-soliton $\omega_{\phi}$ is nothing but a Kähler-
Einstein metric.

  2. (ii)

When $\sigma(s)=-s+C$, where $C$ is a constant, $\sigma$-soliton
$\omega_{\phi}$ is a Kähler-Ricci soliton in the sense that
$\mathrm{Ric}(\omega_{\phi})-\omega_{\phi}=L_{V}\omega_{\phi}$.

  3. (iii)

When $\sigma(s)=-\log(s+1)$ and $\theta_{V}^{(\phi)}>-1$, $\sigma$-soliton
$\omega_{\phi}$ is a Mabuchi soliton ([25, 31, 44]) in the sense that
$1-e^{\rho_{\phi}}$ is the potential function of the holomorphic vector field
$-V$ with respect to $\omega_{\phi}$.

The existence of a $\sigma$-soliton is well understood. Han-Li [21] showed
that the existence is equivalent to an algebraic stability condition for a
Fano manifold. The authors [34] established a combinatorial condition to
characterize the existence for a $\sigma$-soliton on a KSM-manifold which is a
certain toric bundle including a toric Fano manifold.

As an application of Theorem 1.2, we obtain a relation of the existence for a
$\sigma$-soliton and a $\sigma$-extremal Kähler metric, which is the second
result in this article.

######  Theorem 1.4.

Let $M$ be a Fano manifold admitting a $\sigma$-soliton in
$\Omega=2\pi{c_{1}(M)}$. Then $M$ also admits a $\sigma$-extremal Kähler
metric in $\Omega=2\pi{c_{1}(M)}$.

When $\sigma(s)=-\log(s+1)$, this relation is already known for the literature
as stated in the beginning of the paper. We note that a $\sigma$-soliton and a
$\sigma$-extremal metric are different in general. See Section 5.

### Organization

In Section 2 various energy functionals are introduced. In particular the
$h$-modified Mabuchi functional whose critical points are $h$-modified
extremal Kähler metrics is introduced. In Section 3, based on the existence /
properness principle established by Darvas-Rubinstein [15, 16], we prove
Theorem 1.2. In Section 4, based on the work of Chen-Cheng [11], we show that
the the existence of an $h$-modified extremal Kähler metric is equivalent to
some notions of geodesic stability. In Section 5, we prove Theorem 1.4 and
provide an example of an $h$-modified extremal Kähler metric related to a
Kähler-Ricci soliton.

##  2\. Preliminary

###  2.1. The metric completion of $\mathcal{H}_{V}$

Following the works of Darvas [13, 14], we introduce the completion of
$\mathcal{H}$. For $p\geqq{1}$, the length of a smooth curve $[0,1]\ni
t\mapsto c_{t}\in\mathcal{H}$ is given by
$\int_{0}^{1}\|\dot{c_{t}}\|_{L^{p}(M,\omega_{c_{t}})}dt.$ For any
$\phi_{0},\phi_{1}\in\mathcal{H}$, the distance $d_{p}(\phi_{0},\phi_{1})$ is
defined as the infimum of the length of smooth curves connecting $\phi_{0}$
and $\phi_{1}$. The pair $(\mathcal{H},d_{p})$ is in fact a metric space [13].
Let

$\mathrm{PSH}(M,\omega_{0}):=\Set{\phi\in L^{1}(M,\omega_{0})}{\phi\text{ is
u.s.c. and }\omega_{\phi}\geqq{0}}$

and

$\mathcal{E}^{p}(M,\omega_{0}):=\Set{\phi\in\mathrm{PSH}(M,\omega_{0})}{\int_{M}\omega_{\phi}^{n}=\operatorname{Vol},\quad\int_{M}|\phi|^{p}\omega_{\phi}^{n}<+\infty}.$

Darvas [13] showed that the completion of the metric space
$(\mathcal{H},d_{p})$ is identified with $(\mathcal{E}^{p},d_{p})$, where
$d_{p}(u_{0},u_{1})=\lim_{k\to\infty}d_{p}\bigl{(}u_{0}(k),u_{1}(k)\bigr{)}$
for any smooth decreasing sequences $\\{u_{i}(k)\\}_{k\in\mathbb{N}}$ in
$\mathcal{H}$ converging pointwise to $u_{i}\in\mathcal{E}^{p}$. We will need
the case when $p=1$ for our purpose. According to [16, Lemma 5.4], we can take
the completion of $(\mathcal{H}_{V},d_{1})$ which we denote by
$(\mathcal{E}_{V}^{1},d_{1})$.

###  2.2. Energy functionals

At first, we work on a compact Kähler manifold $M$. Set
$\mathrm{Vol}:=\int_{M}\omega_{0}^{n}$. For any $\phi\in\mathcal{H}$, we
define the Aubin-Yau (or Aubin-Mabuchi) functional as follows.

$E(\phi)=E_{\omega_{0}}(\phi):=\frac{\mathrm{Vol}^{-1}}{n+1}\int_{M}\phi\sum_{j=0}^{n}\omega_{0}^{j}\wedge\omega_{\phi}^{n-j}.$

We also define Aubin’s I-functional and J-functional as follows.

$\displaystyle
I(\phi)=I_{\omega_{0}}(\phi):=\mathrm{Vol}^{-1}\int_{M}\phi(\omega_{0}^{n}-\omega_{\phi}^{n}),$
$\displaystyle
J(\phi)=J_{\omega_{0}}(\phi):=\mathrm{Vol}^{-1}\int_{M}\phi\omega_{0}^{n}-E_{\omega_{0}}(\phi).$

Set $\mathbb{J}(\phi)=I(\phi)-J(\phi)$. Note that they satisfy the classical
inequality

(3) $0\leqq{I(\phi)}\leqq{(n+1)\mathbb{J}(\phi)}\leqq{nI(\phi)}.$

For $(1,1)$-form $\chi$, we define

$\mathbb{J}_{\chi}(\phi)=\mathbb{J}_{\chi,\omega_{0}}(\phi):=\mathrm{Vol}^{-1}\int_{M}\phi\sum_{j=0}^{n-1}\chi\wedge\omega_{0}^{j}\wedge\omega_{\phi}^{n-1-j}-\underline{\chi}E(\phi)$

where $\underline{\chi}=n\int_{M}\chi\wedge\omega^{n-1}/\int_{M}\omega^{n}$.
In particular, we denote

$\mathbb{J}_{-\mathrm{Ric}}(\phi):=\underline{S}E(\phi)-\mathrm{Vol}^{-1}\int_{M}\phi\sum_{j=0}^{n-1}\mathrm{Ric}(\omega_{0})\wedge\omega_{0}^{j}\wedge\omega_{\phi}^{n-1-j}.$

The entropy functional is defined as

$H(\phi)=H_{\omega_{0}}(\phi):=\mathrm{Vol}^{-1}\int_{M}\log\Bigl{(}\frac{\omega_{\phi}^{n}}{\omega_{0}^{n}}\Bigr{)}\omega_{\phi}^{n}.$

Fix a real valued function $h$ on the interval $I$ satisfying
$\int_{M}h(\theta_{V}^{(\phi)})\omega_{\phi}^{n}=0$. For any
$\phi\in\mathcal{H}_{V}$, we introduce the $h$-modified Aubin-Yau functional

$\mathbb{J}_{V}^{h}(\phi):=\mathrm{Vol}^{-1}\int_{0}^{1}dt\int_{M}\frac{d\phi_{t}}{dt}h(\theta_{V}^{(\phi_{t})})\omega_{\phi_{t}}^{n},$

where $\phi_{t}\in\mathcal{H}_{V}$ is a smooth path connecting $0$ and $\phi$,
to define

$E_{V}^{h}:=E-\mathbb{J}_{V}^{h}$

and to define the $h$-modified Mabuchi functional as

$K_{V}^{h}:=H+\mathbb{J}_{-\mathrm{Ric}}+\mathbb{J}_{V}^{h}.$

An $h$-modified extremal Kähler metric which is defined by the equation
$S(\omega_{\phi})-\underline{S}=h(\theta_{V}^{(\phi)})$ is a critical point of
$K_{V}^{h}$ since

(4) $\delta
K_{V}^{h}=-\mathrm{Vol}^{-1}\int_{M}\delta\phi\Bigl{(}S(\phi)-\underline{S}-h(\theta_{V}^{(\phi)})\Bigr{)}\omega_{\phi}^{n}.$

When $h(s)=1-e^{-\sigma(s)}$, we call the functional $K_{V}^{h}$ the
$\sigma$-Mabuchi functional.

######  Lemma 2.1.

The followings hold.

  1. (i)

When $1-h\geqq{0}$ on $I$, for any $\phi_{0}\leqq\phi_{1}$, we have
$E_{V}^{h}(\phi_{0})\leqq{E_{V}^{h}(\phi_{1})}$.

  2. (ii)

The functions $E_{V}^{h}$ and $\mathbb{J}_{V}^{h}$ are affine along any
$C^{1,1}$-geodesic connecting two points in $\mathcal{H}_{V}$.

###### Proof..

First we prove (i). Note that for any smooth path $\phi_{t}\in\mathcal{H}_{V}$
connecting $\phi_{0}$ and $\phi_{1}$,

$E_{V}^{h}(\phi_{1})-E_{V}^{h}(\phi_{0})=\mathrm{Vol}^{-1}\int_{0}^{1}dt\int_{M}\dot{\phi_{t}}(1-h(\theta_{V}^{(\phi_{t})}))\omega_{\phi_{t}}^{n}\geqq{0}.$

Thus taking $\phi_{t}:=t(\phi_{1}-\phi_{0})+\phi_{0}$, we have
$E_{V}^{h}(\phi_{0})\leqq{E_{V}^{h}(\phi_{1})}$.

Next we prove (ii). The affineness for $E$ along any $C^{1,1}$-geodesic
segment is well known. We prove $E_{V}^{h}$ is affine. The following proof is
same as [6, Proposition 10.4] and [30, Lemma 2.5]. Take a unique
$C^{1,1}$-geodesic $\phi_{t}$ connecting
$\phi_{0},\,\phi_{1}\in\mathcal{H}_{V}$ and small $\varepsilon>0$. By [9] the
geodesic $\phi_{t}$ can be approximated by smooth $\varepsilon$-geodesics
$\phi_{t}^{\varepsilon}\in\mathcal{H}_{V}$ connecting $\phi_{0},\phi_{1}$
defining

$\Big{(}\frac{\partial^{2}\phi_{t}^{\varepsilon}}{\partial\tau\partial\bar{\tau}}-\Big{|}\bar{\partial}\Big{(}\frac{\partial\phi_{t}^{\varepsilon}}{\partial\tau}\Big{)}\Big{|}^{2}_{\omega_{\phi_{t}^{\varepsilon}}}\Big{)}\omega_{\phi_{t}^{\varepsilon}}^{n}=\varepsilon\omega_{0}^{n}$

on $M\times([0,1]\times S^{1})\subset M\times\mathbb{C}$ with
$t=\mathrm{Re}(\tau)$. The argument by integration by parts in the proof of
[30, Lemma 2.5] (we need to replace
$1-\theta_{X}(\omega_{\phi_{t}^{\varepsilon}})$ by
$1-h(\theta_{V}^{\phi_{t}^{\varepsilon}})$) shows

$\displaystyle\frac{\partial^{2}}{\partial\tau\partial\bar{\tau}}E_{V}^{h}(\phi_{t}^{\varepsilon})$
$\displaystyle=$
$\displaystyle-\mathrm{Vol}^{-1}\int_{M}\Big{(}\frac{\partial^{2}\phi_{t}^{\varepsilon}}{\partial\tau\partial\bar{\tau}}-\Big{|}\bar{\partial}\Big{(}\frac{\partial\phi_{t}^{\varepsilon}}{\partial\tau}\Big{)}\Big{|}^{2}_{\omega_{\phi_{t}^{\varepsilon}}}\Big{)}(1-h(\theta_{V}^{\phi_{t}^{\varepsilon}}))\omega_{\phi_{t}^{\varepsilon}}^{n}$
$\displaystyle=$
$\displaystyle-\varepsilon\mathrm{Vol}^{-1}\int_{M}(1-h(\theta_{V}^{\phi_{t}^{\varepsilon}}))\omega_{0}^{n}.$

Since there exists a uniform constant $C$ independent of
$\phi\in\mathcal{H}_{V}$ such that $|\theta_{V}^{(\phi)}|<C$, it follows that

$\Big{|}\frac{\partial^{2}}{\partial\tau\partial\bar{\tau}}E_{V}^{h}(\phi_{t}^{\varepsilon})\Big{|}<\varepsilon
C.$

When $\varepsilon\to{0}$,
$\sqrt{-1}\partial\overline{\partial}_{\tau}E_{V}^{h}(\phi_{t}^{\varepsilon})$
converges weakly to
$\sqrt{-1}\partial\overline{\partial}_{\tau}E_{V}^{h}(\phi_{t})$ as Monge-
Ampère measures, where $\partial\overline{\partial}_{\tau}$ is
$\partial\overline{\partial}$-operator with respect to $\tau\in\mathbb{C}$.
Thus $\sqrt{-1}\partial\overline{\partial}_{\tau}E_{V}^{h}(\phi_{t})=0$. This
complete the proof. ∎

The above lemma together with the convexity of the K-energy
$K:=H+\mathbb{J}_{-\mathrm{Ric}}$ proved by Berman-Berndtsson [2] shows that
the functional $K_{V}^{h}$ is convex along $C^{1,1}$ geodesic segments in
$\mathcal{H}_{V}$, and a $h$-modified extremal metric minimizes $K_{V}^{h}$.

Berman-Darvas-Lu [4] showed that the K-energy
$K:=H+\mathbb{J}_{-\mathrm{Ric}}$ on $\mathcal{H}$ can be extended to a
functional $\mathcal{E}^{1}\to\mathbb{R}\cup\\{+\infty\\}$ using the same
formula and such a K-energy in $\mathcal{E}^{1}$ is the greatest $d_{1}$-lsc
extension of $K$ on $\mathcal{H}$. Moreover they showed the K-energy is convex
along finite energy geodesics in $\mathcal{E}^{1}$. These fact and the
following lemma show that the functional $K_{V}^{h}$ can be extended to a
functional $\mathcal{E}^{1}_{V}\to\mathbb{R}\cup\\{+\infty\\}$ and the
extended functional $K_{V}^{h}$ on $\mathcal{E}^{1}_{V}$ is the greatest
$d_{1}$-lsc extension of $K_{V}^{h}$ on $\mathcal{H}_{V}$. Moreover,
$K_{V}^{h}$ is convex along finite energy geodesics in $\mathcal{E}^{1}_{V}$,
and a $h$-modified extremal Kähler metric is a minimizer on
$\mathcal{E}^{1}_{V}$.

######  Lemma 2.2.

The function $\mathbb{J}_{V}^{h}$ on $\mathcal{H}_{V}$ has a unique
$d_{1}$-continuous extension to $\mathcal{E}^{1}_{V}$.

###### Proof..

Note that there exists a uniform constant $C$ independent of
$\phi\in\mathcal{H}_{V}$ by [46] such that $|\theta_{V}^{(\phi)}|<C$. Take any
smooth curve $\phi_{t}$ in $\mathcal{H}_{V}$ connecting $\phi_{0}$ and
$\phi_{1}$. Observe

$|\mathbb{J}_{V}^{h}(\phi_{0})-\mathbb{J}_{V}^{h}(\phi_{1})|\leqq{C\int_{0}^{1}dt\int_{M}|\dot{\phi_{t}}|\omega_{\phi_{t}}^{n}}$

to show
$|\mathbb{J}_{V}^{h}(\phi_{0})-\mathbb{J}_{V}^{h}(\phi_{1})|\leqq{C}d_{1}(\phi_{0},\phi_{1})$
by taking infimum over all smooth curves. ∎

Consider the case when $M$ is Fano and $\Omega=2\pi c_{1}(M)$. We define

$L(\phi):=-\log\Bigl{(}\mathrm{Vol}^{-1}\int_{M}e^{\rho_{0}-\phi}\omega_{0}^{n}\Bigr{)}$

to introduce the $h$-modified Ding functional as

$D_{V}^{h}:=-E_{V}^{h}+L.$

A $\sigma$-soliton is a critical point of $D_{V}^{h}$ with
$h(s)=1-e^{-\sigma(s)}$ since

$\delta
D_{V}^{h}=-\mathrm{Vol}^{-1}\int_{M}\delta\phi\Bigl{(}1-h(\theta_{V}^{\phi})-e^{\rho_{\phi}}\Bigr{)}\omega_{\phi}^{n}.$

We call the functional $D_{V}^{h}$ with $h(s)=1-e^{-\sigma(s)}$ the
$\sigma$-Ding functional.

The inequality in the following lemma motivated to introduce the notion of an
$\sigma$-extremal Kähler metric. This is well-known for the literature in the
case when $V=0$ and in the case when $h(s)=s$. See [33, Section 9.6] for
example.

######  Lemma 2.3.

Let $M$ be a Fano manifold with $\Omega=2\pi c_{1}(M)$. For any
$\phi\in\mathcal{H}_{V}$, we have

(5)
$K_{V}^{h}(\phi)=\mathrm{Vol}^{-1}\int_{M}\log\Bigl{(}\frac{\omega_{\phi}^{n}}{e^{\rho_{0}}\omega_{0}^{n}}\Bigl{)}\omega_{\phi}^{n}-E_{V}^{h}(\phi)+\mathrm{Vol}^{-1}\int_{M}\phi\omega_{\phi}^{n}+\mathrm{Vol}^{-1}\int_{M}\rho_{0}\omega_{0}^{n}.$

As a consequence, for any $\phi\in\mathcal{H}_{V}$ we have

(6)
$K_{V}^{h}(\phi)-\mathrm{Vol}^{-1}\int_{M}\rho_{0}\omega_{0}^{n}\geqq{D_{V}^{h}(\phi)}.$

###### Proof..

It is easy to see that the derivative of the RHS in (5) equals to the RHS in
(4). We note that the value of the RHS in (5) is $0$ at $\phi=0$ to show the
equality. Next we prove the inequality (6). We can assume
$\int_{M}e^{\rho_{0}-\phi}\omega_{0}^{n}=\operatorname{Vol}$. This is because
$K_{V}^{h}$ and $D_{V}^{h}$ are invariant under the translation
$\phi\mapsto\phi+\text{constant}$. Observe that Jensen’s inequality for the
convex function $x\log x$ yields

$\mathrm{Vol}^{-1}\int_{M}\log\Bigl{(}\frac{\omega_{\phi}^{n}}{e^{\rho_{0}-\phi}\omega_{0}^{n}}\Bigl{)}\omega_{\phi}^{n}=\operatorname{Vol}^{-1}\int_{M}\frac{\omega_{\phi}^{n}}{e^{\rho_{0}-\phi}\omega_{0}^{n}}\log\Bigl{(}\frac{\omega_{\phi}^{n}}{e^{\rho_{0}-\phi}\omega_{0}^{n}}\Bigl{)}e^{\rho_{0}-\phi}\omega_{0}^{n}\geqq{0}.$

Therefore

$K_{V}^{h}(\phi)-\operatorname{Vol}^{-1}\int_{M}\rho_{0}\omega_{0}^{n}\geqq{-E_{V}^{h}(\phi)}=D_{V}^{h}(\phi).$

∎

###  2.3. The notion of the coercivity

Set $G:=\mathrm{Aut}_{0}(M,V)$, $\mathcal{H}_{0}:=\mathcal{H}\cap E^{-1}(0)$
and $\mathcal{H}_{V,0}:=\mathcal{H}_{V}\cap E^{-1}(0)$. For any Kähler metric
$\omega_{\phi}=\omega_{0}+\sqrt{-1}\partial\overline{\partial}\phi$ and
$\sigma\in\mathrm{Aut}_{0}(M)$, we define a Kähler potential
$\sigma[\phi]\in\mathcal{H}_{0}$ by
$\sigma^{*}\omega_{\phi}=\omega_{0}+\sqrt{-1}\partial\overline{\partial}\sigma[\phi]$.
Then for any $\phi,\psi\in\mathcal{H}_{0}$ the $d_{1}$-distance relative to
$G$ is defined by
$d_{1,G}(\phi,\psi):=\inf_{\sigma_{1},\sigma_{2}\in{G}}d_{1}(\sigma_{1}[\phi],\sigma_{2}[\psi])$
which is also equal to $\inf_{\sigma\in G}d_{1}(\phi,\sigma[\psi])$. This
distance can be defined on $\mathcal{H}_{V,0}$, since the $G$-action preserves
$\mathcal{H}_{V,0}$ and the inclusion
$\mathcal{H}_{V,0}\subset\mathcal{H}_{0}$ is totally geodesic. For later use,
we also define $\mathcal{E}^{1}_{V,0}:=\mathcal{E}^{1}_{V}\cap E^{-1}(0)$
which is the completion of $\mathcal{H}_{V,0}$.

######  Definition 2.4.

A $G$-invariant functional $F$ on $\mathcal{H}_{V,0}$ is coercive if there
exist constants $C,D>0$ such that for all $\phi\in\mathcal{H}_{V,0}$,

$F(\phi)\geqq{Cd_{1,G}(0,\phi)-D}.$

###  2.4. Futaki type invariants

Differentiating the functional $K_{V}^{h}$, we get a Futaki type invariant for
$M$. For $X\in\mathfrak{X}_{0}(M)$ and $\omega\in\mathcal{H}_{V}$, we put

$F_{V}^{h}(X):=-\int_{M}\theta_{X}^{(\omega)}\Big{(}S(\omega)-\underline{S}-h(\theta_{V}^{(\omega)})\Big{)}\omega^{n}.$

Since $\int_{M}\left(S(\omega)-\underline{S}\right)\omega^{n}=0$ and
$\int_{M}h(\theta_{V}^{(\omega)})\omega^{n}=0$, there exist
$\rho_{\omega},\widetilde{h}_{V}^{(\omega)}\in{C^{\infty}(M;\mathbb{R})}$
satisfying

$\displaystyle
S(\omega)-\underline{S}=\square_{\omega}\rho_{\omega}:=\sum_{i,j=1}^{n}g^{\overline{\jmath}i}\frac{\partial^{2}\rho_{\omega}}{\partial{z^{i}}\partial\overline{z^{\jmath}}},$
$\displaystyle
h(\theta_{V}^{(\omega)})=\square_{\omega}\widetilde{h}_{V}^{(\omega)},$

where
$\omega=\sqrt{-1}\sum_{i,j=1}^{n}g_{i\overline{\jmath}}dz^{i}\wedge\overline{dz^{\jmath}}$
and $\left(g^{\overline{\jmath}i}\right)_{i,j=1}^{n}$ is the inverse matrix of
$\left(g_{i\overline{\jmath}}\right)_{i,j=1}^{n}$. Then we have

$F_{V}^{h}(X)=\int_{M}X\left(\rho_{\omega}-\widetilde{h}_{V}^{(\omega)}\right)\omega^{n}.$

######  Proposition 2.5.

The value $F_{V}^{h}(X)$ is independent of the choice of
$\omega\in\mathcal{H}_{V}$. In particular, if there exists an $h$-modified
extremal Kähler metrics in $\mathcal{H}_{V}$, the invariant $F_{V}^{h}$
vanishes identically.

###### Proof..

Since
$-\int_{M}\theta_{X}^{(\omega)}\left(S(\omega)-\underline{S}\right)\omega^{n}=\int_{M}X\left(\rho_{\omega}\right)\omega^{n}$
is the original Futaki invariant ([17], see also [18]), this is independent of
the choice of $\omega\in\mathcal{H}_{V}$.

For an arbitrary smooth path
$\left\\{\omega_{\phi_{t}}\,|\,a\leqq{t}\leqq{b}\right\\}$ in
$\mathcal{H}_{V}$, by a simple calculation, we obtain

$\frac{d}{dt}\theta_{X}^{(\omega_{\phi_{t}})}=X\\!\left(\dot{\phi}_{t}\right),\qquad\frac{d}{dt}h(\theta_{V}^{(\omega_{\phi_{t}})})=V\\!\left(\dot{\phi}_{t}\right)h^{\prime}(\theta_{V}^{(\omega_{\phi_{t}})}),\qquad\frac{d}{dt}\omega_{\phi_{t}}^{n}=\left(\square_{\omega_{\phi_{t}}}\dot{\phi}_{t}\right)\omega_{\phi_{t}}^{n}.$

In view of these equalities, we can conclude that
$\frac{d}{dt}\int_{M}\theta_{X}^{(\omega_{\phi_{t}})}h(\theta_{V}^{(\omega_{\phi_{t}})})\omega_{\phi_{t}}^{n}=0$.
Hence, $\int_{M}\theta_{X}^{(\omega)}h(\theta_{V}^{(\omega)})\omega^{n}$ is
independent of the choice of $\omega\in\mathcal{H}_{V}$. This completes the
proof. ∎

We note that we can take $V=0$ provided the original Futaki invariant
vanishes.

We consider the case when $M$ is Fano and $\Omega=2\pi{c_{1}(M)}$. For
$\omega\in\Omega=2\pi{c_{1}(M)}$ and
$X\in\mathfrak{X}_{0}(M)=\mathfrak{X}(M)$, we define
$\widetilde{\theta}_{V}^{(\omega)}\in{C^{\infty}(M;\mathbb{C})}$ by

$i_{X}\omega=\sqrt{-1}\,\overline{\partial}\widetilde{\theta}_{X}^{(\omega)}\quad\text{and}\quad\int_{M}\widetilde{\theta}_{X}^{(\omega)}e^{\rho_{\omega}}\omega^{n}=0,$

where $\rho_{\omega}$ is the Ricci potential for $\omega$. Then we have
$\square_{\omega}\widetilde{\theta}_{X}^{(\omega)}+X\left(\rho_{\omega}\right)+\widetilde{\theta}_{X}^{(\omega)}=0$
(cf. Tian-Zhu [42, (2.1)]), hence we obtain

$\int_{M}\widetilde{\theta}_{X}^{(\omega)}\omega^{n}=-\int_{M}X\left(\rho_{\omega}\right)\omega^{n}$

and
$\widetilde{\theta}_{X}^{(\omega)}=\theta_{V}^{(\omega)}-\mathrm{Vol}^{-1}\int_{M}X\left(\rho_{\omega}\right)\omega^{n}$.
In view of these equalities, we conclude that

$\displaystyle F_{V}^{h}(X)$
$\displaystyle=\int_{M}X\left(\rho_{\omega}\right)\omega^{n}+\int_{M}\theta_{X}^{(\omega)}h(\theta_{V}^{(\omega)})\omega^{n}$
$\displaystyle=-\int_{M}\widetilde{\theta}_{X}^{(\omega)}\omega^{n}+\int_{M}\widetilde{\theta}_{X}^{(\omega)}h(\theta_{V}^{(\omega)})\omega^{n}$
$\displaystyle=-\int_{M}\widetilde{\theta}_{X}^{(\omega)}\left(1-e^{\rho_{\omega}}\right)\omega^{n}+\int_{M}\widetilde{\theta}_{X}^{(\omega)}h(\theta_{V}^{(\omega)})\omega^{n}$
$\displaystyle=-\int_{M}\widetilde{\theta}_{X}^{(\omega)}\left(1-e^{\rho_{\omega}}-h(\theta_{V}^{(\omega)})\right)\omega^{n}$
$\displaystyle=-\int_{M}{\theta}_{X}^{(\omega)}\left(1-e^{\rho_{\omega}}-h(\theta_{V}^{(\omega)})\right)\omega^{n}.$

Therefore, we can see that this invariant $F_{V}^{h}(X)$ is the derivative of
the $h$-modified Ding functional $D_{V}^{h}$ along the ray generated by $X$.
Moreover, assuming $h(s)=1-e^{-\sigma(s)}$, since
$\int_{M}\theta_{X}^{(\omega)}e^{\rho_{\omega}}\omega^{n}=\int_{M}X\left(\rho_{\omega}\right)\omega^{n}$,
we can conclude

$\displaystyle F_{V}^{h}(X)$
$\displaystyle=-\int_{M}\theta_{X}^{(\omega)}\left(e^{-\sigma(\theta_{V}^{(\omega)})}-e^{\rho_{\omega}}\right)\omega^{n}$
$\displaystyle=\int_{M}\theta_{X}^{(\omega)}e^{\rho_{\omega}}\omega^{n}-\int_{M}\theta_{X}^{(\omega)}e^{-\sigma(\theta_{V}^{(\omega)})}\omega^{n}$
$\displaystyle=\int_{M}X\left(\rho_{\omega}\right)\omega^{n}-\int_{M}\theta_{X}^{(\omega)}e^{-\sigma(\theta_{V}^{(\omega)})}\omega^{n}$
$\displaystyle=-\int_{M}\widetilde{\theta}_{X}^{(\omega)}e^{-\sigma(\theta_{V}^{(\omega)})}\omega^{n}$
$\displaystyle=\int_{M}\left(\square_{\omega}\widetilde{\theta}_{X}^{(\omega)}+X\left(\rho_{\omega}\right)\right)e^{-\sigma(\theta_{V}^{(\omega)})}\omega^{n}$
$\displaystyle=\int_{M}X\left(\rho_{\omega}+\sigma(\theta_{V}^{(\omega)})\right)e^{-\sigma(\theta_{V}^{(\omega)})}\omega^{n}.$

This is nothing but an obstruction for the existence of multiplier Hermitian-
Einstein metrics of type $\sigma$, which is introduced by Futaki [19].

##  3\. An $h$-modified extremal Kähler metric and the properness of
$K_{V}^{h}$

######  Theorem 3.1.

There exists an $h$-modified extremal Kähler metric in $\mathcal{H}_{V,0}$ if
and only if the $h$-modified Mabuchi functional $K_{V}^{h}$ is
$\mathrm{Aut}_{0}(M,V)$-invariant and coercive. Moreover the condition of
$\mathrm{Aut}_{0}(M,V)$-invariance can be removed.

###### Proof..

Our proof is due to the existence / properness principle established by
Darvas-Rubinstein [16] (see also Darvas’s survey [15]). We apply [15, Theorem
4.7] to the data $\mathcal{R}=\mathcal{H}_{V,0}$, $d=d_{1}$, $F=K_{V}^{h}$ and
$G=\mathrm{Aut_{0}}(M,V)$. It is enough to check the above data enjoy the
properties (P1)–(P6) in [15, Section 4.2].

(P1):

This is stated in the preliminary.

(P2):

This is due to Proposition 3.2 stated below.

(P3):

This is due to Theorem 3.8 stated below.

(P4):

This is due to [16, Lemma 5.9].

(P5):

This is due to Theorem 3.9 stated below.

(P6):

This is the cocycle condition for $K_{V}^{h}$.

We argue that the condition of $G$-invariance can be removed. For
$X\in\mathrm{Lie}(\mathrm{Aut}_{0}(M,V))$, let $g_{t}$ be a one-parameter
subgroup generated by the flow of the real part
$X_{\mathrm{Re}}:=\frac{1}{2}\left(X+\overline{X}\right)$. For any
$V_{\mathrm{Im}}$-invariant metric $\omega$, we write
$g_{t}^{*}\omega=\omega_{0}+\sqrt{-1}\partial\overline{\partial}\phi_{t}$.
Note that $g_{t}^{*}\omega$ is also $V_{\mathrm{Im}}$-invariant. Since

$\frac{d}{dt}g_{t}^{*}\omega|_{t=0}=L_{X_{\mathrm{Re}}}\omega=\sqrt{-1}\partial\overline{\partial}\mathrm{Re}(\theta_{X}^{(\omega)})$

where $\theta_{X}^{(\omega)}$ is the complex potential of $X$ with respect to
$\omega$, then we have

$\displaystyle\frac{d}{dt}K_{V}^{h}(\phi_{t})$ $\displaystyle=$
$\displaystyle-\int_{M}g_{t}^{*}\Big{[}\frac{d\phi}{dt}|_{t=0}\Big{(}S(\omega)-\underline{S}-h(\theta_{V}^{(\omega)}\Big{)}\omega^{n}\Big{]}$
$\displaystyle=$
$\displaystyle-\int_{M}\mathrm{Re}(\theta_{X}^{(\omega)})\Big{(}S(\omega)-\underline{S}-h(\theta_{V}^{(\omega)})\Big{)}\omega^{n}$

which is independent of $t$. Thus if $K_{V}^{h}$ is coercive, in particular
bounded from below, then $K_{V}^{h}(\phi_{t})$ is constant in $t$ which states
that $K_{V}^{h}$ is $G$-invariant. This completes the proof. ∎

###  3.1. Compactness properties for sequences

We argue the property (P2) in the proof of Theorem 3.1.

######  Proposition 3.2.

If $\\{\phi_{i}\\}_{i}\subset\mathcal{E}^{1}_{V,0}$ satisfies
$\lim_{i\to\infty}K^{h}_{V}(\phi_{i})=\inf_{\mathcal{E}^{1}_{V,0}}K_{V}^{h}$
and $d_{1}(0,\phi_{i})<C$ for any $i$ for some $C>0$ then there exists a
minimizer $\phi_{*}$ of $K_{V}^{1}$ in $\mathcal{E}^{1}_{V,0}$ and a
subsequence $\\{\phi_{i_{k}}\\}_{k}$ which converges to $\phi_{*}$ in
$d_{1}$-topology.

###### Proof..

Recall $K_{V}^{h}=H+\mathbb{J}_{-\mathrm{Ric}}+\mathbb{J}_{V}^{h}$. Both
functionals $\mathbb{J}_{-\mathrm{Ric}}$ and $\mathbb{J}_{V}^{h}$ are
controlled by the distance $d_{1}$ in general. Thus $H(\phi_{i})$ is uniformly
bounded. The quantity $|\sup_{M}\phi_{i}|$ is also uniformly bounded since
there exists a constant $C>0$ such that for any
$\phi\in\mathrm{PSH}(M,\omega_{0})$, we have
$\sup_{M}\phi\leqq\mathrm{Vol}^{-1}\int_{M}\phi\omega_{0}^{n}+C\leqq\sup_{M}\phi+C$
(see for example [41, p. 49]). Then a compactness result in [16, Theorem 5.6]
shows that there exists $\phi_{*}\in\mathcal{E}^{1}_{V,0}$ and a subsequence
$\\{\phi_{i_{k}}\\}_{k}$ satisfying $d_{1}(\phi_{i_{k}},\phi_{*})\to 0$. Since
$K_{V}^{h}$ is $d_{1}$-lower semicontinuous, the limit $\phi_{*}$ is noting
but a minimizer. This completes the proof. ∎

The following stronger proposition than above one will be applied to establish
a geodesic stability in Section 4.

######  Proposition 3.3.

If $\\{\phi_{i}\\}_{i}\subset\mathcal{E}^{1}_{V,0}$ satisfies
$K^{h}_{V}(\phi_{i})<C$ and $d_{1}(0,\phi_{i})<C$ for any $i$ for some $C>0$
then there exists $\phi\in\mathcal{H}^{1}_{V,0}$ and a subsequence
$\\{\phi_{i_{k}}\\}_{k}$ which converges to $\phi$ in $d_{1}$-topology.

###### Proof..

By the same argument as above, we can use a compactness result in [16, Theorem
5.6] to conclude the proposition. ∎

###  3.2. Regularity of minimizers

We argue the property (P3) in the proof of Theorem 3.1.

####  3.2.1. A continuity path

Let $T$ be the compact torus in $G:=\mathrm{Aut}_{0}(M,V)$ defined by the
closure of the set of flows generated by $\mathrm{Im}V$. For a $T$-invariant
Kähler metrics $\omega$ and $\alpha$, we call $\omega$ a $\alpha$-twisted
$h$-modified extremal metric if
$S(\omega)-h(\theta_{V}^{(\omega)})-\Lambda_{\omega}\alpha=\text{const}$. The
following theorem which generalizes Hashimoto’s result [22], gives an openness
property for a continuity path used to prove the regularity of minimizers.
(The following argument is essentially due to He [24, pp. 10–12], where [24]
is a preprint version of [23].)

######  Theorem 3.4.

Let $\omega$ and $\alpha$ be $T$-invariant Kähler metrics satisfying
$\mathrm{tr}_{\omega}\alpha=\text{const}$. Then there exists a constant
$r(\omega,\alpha)\in\mathbb{R}$ depending only on $\omega$ and $\alpha$ such
that for any $r\geqq{r(\omega,\alpha)}$, there exists $\phi\in\mathcal{H}_{V}$
satisfying
$S(\phi)-h(\theta^{(\phi)}_{V})-\Lambda_{\omega_{\phi}}(r\alpha)=\text{const}$.

###### Proof..

At first, we take $T$-invariant Kähler metrics $\omega$ and $\alpha$ such that
$\Lambda_{\omega}\alpha$ is not necessarily constant. Set
$\tilde{S}(\omega)=S(\omega)-h(\theta_{V}^{(\omega)})-\Lambda_{\omega}\alpha$.
Let $\psi$ be a variation of the potential function for $\omega$, that is,
$\delta\omega=\sqrt{-1}\partial\overline{\partial}\psi$. We follow Hashimoto’s
computation [22] to obtain the linearization of the LHS of the above equation
as follows.

$\displaystyle\mathcal{L}(\psi)$ $\displaystyle=$
$\displaystyle-\mathcal{D}^{*}_{\omega}\mathcal{D}_{\omega}\psi+(\partial\tilde{S}(\omega),\overline{\partial}\psi)_{\omega}+(\alpha,\sqrt{-1}\partial\overline{\partial}\psi)_{\omega}+(\partial\Lambda_{\omega}\alpha,\overline{\partial}\psi)_{\omega}$
$\displaystyle=$
$\displaystyle-\mathcal{D}^{*}_{\omega}\mathcal{D}_{\omega}\psi+(\partial\tilde{S}(\omega),\overline{\partial}\psi)_{\omega}+F_{\omega,\alpha}(\psi)$

where $(\cdot,\cdot)_{\omega}$ is the point wise inner product on the space of
differential forms defined by $\omega$,
$\mathcal{D}_{\omega}:C^{\infty}(M;\mathbb{R})\to
C^{\infty}(T^{1,0}M\otimes\Omega^{1,0}(M))$ is an operator defined by
$\mathcal{D}_{\omega}\psi=\overline{\partial}(\mathrm{grad}^{1,0}_{\omega}\psi)$
and $\mathcal{D}_{\omega}^{*}$ is its formal adjoint with respect to the inner
product. Here we put
$F_{\omega,\alpha}(\psi):=(\alpha,\sqrt{-1}\partial\overline{\partial}\psi)_{\omega}+(\partial\Lambda_{\omega}\alpha,\overline{\partial}\psi)_{\omega}$.
The kernel of $\mathcal{D}^{*}_{\omega}\mathcal{D}_{\omega}$ which is equal to
the kernel of $\mathcal{D}_{\omega}$, is the set of functions whose complex
gradient is holomorphic (see for example [39, Section 4]). The operator
$F_{\omega,\alpha}$ is complex self-adjoint and second order elliptic and its
kernel is the set of constant functions ([22, Lemma 1]). To apply the inverse
function theorem we first observe the following lemma:

######  Lemma 3.5.

Let $\omega$ and $\alpha$ be $T$-invariant Kähler metrics satisfying
$\Lambda_{\omega}\alpha=\text{const}$. Then, for each $m\in\mathbb{N}$ there
exists $\phi_{1},\dots,\phi_{m}\in{C^{\infty}(M;\mathbb{R})}$ such that

$\omega_{m}:=\omega+\sqrt{-1}\partial\overline{\partial}(r^{-1}\phi_{1}+\cdots+r^{-m}\phi_{m})$

satisfies

$S(\omega_{m})-h(\theta^{(\omega_{m})}_{V})-r\Lambda_{\omega_{m}}\alpha=\text{const}+r^{-m}f_{m,r}$

for a function $f_{m,r}$ with average $0$ with respect to $\omega_{m}$ which
is bounded in $C^{\infty}(M;\mathbb{R})$ for any sufficiently large $r$.

###### Proof of Lemma 3.5.

The proof is done by the induction.

By [22, Lemma 2], there exists a $T$-invariant function $\phi_{1}\in
C^{\infty}(M;\mathbb{R})$ such that
$-(S(\omega)-\underline{S}-h(\theta^{(\omega)}_{V}))=F_{\omega,\alpha}(\phi_{1})$,
that is,

$(S(\omega)-\underline{S}-h(\theta^{(\omega)}_{V}))+(\alpha,\sqrt{-1}\partial\overline{\partial}\phi_{1})_{\omega}=0.$

Since $F_{\omega,\alpha}$ is an elliptic operator, the uniform bounds of
derivatives of $\phi_{1}$ is guaranteed. For large $r>0$, consider the Kähler
metric
$\omega_{1}:=\omega+r^{-1}\sqrt{-1}\partial\overline{\partial}\phi_{1}$. Note
that $\theta^{(\omega_{1})}_{V}=\theta^{(\omega)}_{V}+V(r^{-1}\phi_{1})$. The
Taylor expansion of the function $h$ and the uniform bound of
$|\theta^{(\omega_{1})}_{V}|$ show
$h(\theta^{(\omega_{1})}_{V})=h(\theta^{(\omega)}_{V})+\mathcal{O}(r^{-1})$.
Together with a computation in [22, section 3.1], we have

$\displaystyle
S(\omega_{1})-h(\theta^{(\omega_{1})}_{V})-r\Lambda_{\omega_{1}}\alpha$
$\displaystyle=$
$\displaystyle\text{const}+(S(\omega)-\underline{S}-h(\theta^{(\omega)}_{V}))+(\alpha,\sqrt{-1}\partial\overline{\partial}\phi_{1})_{\omega}+\mathcal{O}(r^{-1})$
$\displaystyle=$ $\displaystyle\text{const}+\mathcal{O}(r^{-1}).$

Therefore there exists a uniformly bounded smooth function $f_{1,r}$ such that

$S(\omega_{1})-h(\theta^{(\omega_{1})}_{V})-r\Lambda_{\omega_{1}}\alpha=\text{const}+r^{-1}f_{1,r}\quad\text{and}\quad\int_{M}f_{1,r}\omega_{1}^{n}=0.$

Assume we have functions $\phi_{1},\dots,\phi_{m-1}$ such that the statement
in the lemma holds. There exists a $T$-invariant function $\phi_{m}$
satisfying

$(S(\omega_{m-1})-\underline{S}-h(\theta_{V}^{(\omega_{m-1})})-r\Lambda_{\omega_{m-1}}\alpha+rc)+r^{1-m}(\alpha,\sqrt{-1}\partial\overline{\partial}\phi_{m})_{\omega_{m-1}}=0$

where $c$ is the average of $\Lambda_{\omega_{m-1}}\alpha$ with respect to
$\omega_{m-1}$. By the same argument as above, for large $r>0$, consider the
Kähler metric
$\omega_{m}=\omega_{m-1}+r^{-m}\sqrt{-1}\partial\overline{\partial}\phi_{m}$
to compute

$\displaystyle
S(\omega_{m})-h(\theta^{(\omega_{m})}_{V})-r\Lambda_{\omega_{m}}\alpha$
$\displaystyle=$
$\displaystyle\text{const}+(S(\omega_{m-1})-\underline{S}-h(\theta_{V}^{(\omega_{m-1})})-r\Lambda_{\omega_{m-1}}\alpha+rc)$
$\displaystyle+$ $\displaystyle
r^{1-m}(\alpha,\sqrt{-1}\partial\overline{\partial}\phi_{m})_{\omega_{m-1}}+\mathcal{O}(r^{-m})$
$\displaystyle=$ $\displaystyle\text{const}+\mathcal{O}(r^{-m}).$

Therefore there exists a uniformly bounded smooth function $f_{m,r}$ such that

$S(\omega_{m})-h(\theta^{(\omega_{m})}_{V})-r\Lambda_{\omega_{m}}\alpha=\text{const}+r^{-1}f_{m,r}\quad\text{and}\quad\int_{M}f_{m,r}\omega_{m}^{n}=0.$

This completes the proof of Lemma 3.5. ∎

Take a smooth function $h_{m,r}$ satisfying
$\square_{\omega_{m}}h_{m,r}=f_{m,r}$ and $\int_{M}h_{m,r}\omega_{m}^{n}=0$.
Since $\omega_{m}=\omega+\mathcal{O}(r^{-1})$, the elliptic theory gives the
uniform bound of $h_{m,r}$. Define
$\alpha_{m}=\alpha+r^{-m-1}\sqrt{-1}\partial\overline{\partial}h_{m,r}$ to
satisfy

$S(\omega_{m})-h(\theta^{\omega_{m}}_{V})-r\Lambda\omega_{m}\alpha_{m}=\text{const}.$

In order to apply the inverse function theorem for Banach spaces, let
$W^{k,2}_{0,T}$ be the $T$-invariant Sobolev space on $M$ such that the
average of each function is $0$ with respect to $\omega$ and let
$\Omega_{T}^{1,1}$ be the set of $T$-invariant real $(1,1)$-forms on $M$
completed by the $W^{2,k}$-Sobolev norm. Here $k$ is sufficiently large and
all Sobolev norms are defined by $\omega$. Consider Banach spaces
$B_{1}:=\Omega_{T}^{1,1}\times W^{k+4,2}_{0,T}$ and
$B_{2}:=\Omega_{T}^{1,1}\times W^{k,2}_{0,T}$ and take an open set
$U:=\Set{(\eta,\phi)\in{B_{1}}}{\omega_{m}+\sqrt{-1}\partial\overline{\partial}\phi>0}$.
For $r>0$ we define a map $T_{r}:U\to B_{2}$ as follows.

$T_{r}(\eta,\phi)=(\alpha_{m}+\eta,S(\omega_{m,\phi})-h(\theta^{\omega_{m,\phi}}_{V})-r\Lambda_{\omega_{m,\phi}}(\alpha_{m}+\eta))$

where $\omega_{m,\phi}=\omega_{m}+\sqrt{-1}\partial\overline{\partial}\phi$.
The point $0\in U$ corresponds to the $r\alpha_{m}$-twisted $h$-extremal
metric $\omega_{m}$. Note that $T_{r}(0,0)=(\alpha_{m},\underline{S}-rc)$
where $c$ is the average of $\Lambda_{\omega_{m}}\alpha_{m}$ which is a
topological constant. We need to find $(\eta,\phi)\in U$ satisfying
$T_{r}(\eta,\phi)=(\alpha,\underline{S}-rc)$ by applying the inverse function
theorem. We wright $\Lambda_{m}$ for $\Lambda_{\omega_{m}}$,
$\mathcal{D}_{m}^{*}\mathcal{D}_{m}$ for
$\mathcal{D}_{\omega_{m,}}^{*}\mathcal{D}_{\omega_{m}}$ and $F_{m}$ for
$F_{\omega_{m},\alpha_{m}}$. The linearization of $T_{r}$ at $0\in U$ is given
by

$DT_{r}|_{0}(\tilde{\eta},\tilde{\phi})=(\tilde{\eta},\tilde{\phi})\begin{pmatrix}1&-r\Lambda_{m}\\\
0&-\mathcal{D}^{*}_{m}\mathcal{D}_{m}+rF_{m}\end{pmatrix}.$

This linearized operator is exactly same as Hashimoto’s one for a twisted CSCK
metric [22]. Therefore the property $\alpha_{m}=\alpha+\mathcal{O}(r_{-m-1})$
and an inverse function theorem argument by Hashimoto [22, section3.2] show
theorem 3.4. ∎

Similar to [22, Corollary 1], we have the following which shows the openness
property for a continuity path used to prove the regularity of minimizer.

######  Corollary 3.6.

Let $\omega$ is an $\alpha$-twisted $h$-modified extremal metric such that
$\omega$ and $\alpha$ are $T$-invariant Kähler metrics. If $\tilde{\alpha}$ is
a $T$-invariant Kähler metric which is close to $\alpha$ in the
$C^{\infty}$-norm, then there exists an $\tilde{\alpha}$-twisted $h$-modified
extremal metric in $[\omega]$ which is $T$-invariant.

####  3.2.2. Proof of the regularity

######  Lemma 3.7.

Suppose there exists a minimizer $\phi_{*}$ of $K_{V}^{h}$ in
$\mathcal{E}^{1}_{V}$. Then there exists a smooth $h$-modified extremal Kähler
metric in $\mathcal{H}_{V}$.

###### Proof..

Consider the continuity path

(7)
$t(S(\phi)-\underline{S}-h(\theta^{\phi}_{V}))-(1-t)(\Lambda_{\omega_{\phi}}\omega_{0}-n)=0$

where $\omega_{0}$ and $\omega_{\phi}$ are $T$-invariant and $t\in[0,1]$.
Theorem 3.4 and Corollary 3.6 show the set of $t$ such that a solution of (7)
exists is open in $[0,1]$. A solution $\phi(t)$ of the equation (7) is a
minimizer of the functional

$\tilde{K}_{t}:=tK^{h}_{V}+(1-t)\mathbb{J}.$

This is because $K^{h}_{V}$ is convex and $\mathbb{J}$ is affine along
$C^{1,1}$ geodesic segments in $\mathcal{H}_{V}$.

By the assumption, $K_{V}^{h}$ is bounded from below. The inequality (3) and
an estimate in [16, Proposition 5.5] shows

(8) $\mathbb{J}(\phi)\geqq{C^{-1}d_{1}(0,\phi)-C}$

for any $\phi\in\mathcal{H}_{0}.$ Thus
$\tilde{K}_{t}(\phi)\geqq{C^{-1}(1-t)d_{1}(0,\phi)-C}$ for any
$\phi\in\mathcal{H}_{0}$. Let $\phi(t)\in\mathcal{H}_{V,0}$ is a solution of
(7). Since $\phi(t)$ is a minimizer of $\tilde{K}_{t}$, we have
$d_{1}(0,\phi(t))<C((1-t)^{-1}+1)$. Since $|\mathbb{J}_{-\mathrm{Ric}}(\phi)|$
and $|\mathbb{J}^{h}_{V}(\phi)|$ are uniformly bounded by $Cd_{1}(0,\phi)$, we
have a uniform entropy bound

$H(\phi)=\mathrm{Vol}^{-1}\int_{M}\log\Big{(}\frac{\omega_{\phi}^{n}}{\omega_{0}^{n}}\Big{)}\omega_{\phi}^{n}<C((1-t)^{-1}+1).$

A priori estimate by Chen-Cheng [10, Theorem 1.2] shows the equation (7) has a
smooth solution for any $t\in[0,1)$.

Take a sequence $t_{i}\to 1$, and take the solution $\phi_{i}$ of the equation
(7) for $t=t_{i}$ with the normalization $E(\phi_{i})=0$. An argument in [11,
Corollary 4.5] shows $\phi_{i}$ minimizes $tK_{V}^{h}+(1-t)\mathbb{J}$ on the
completion $\mathcal{E}_{V}^{1}$, which is based on the fact that
$tK_{V}^{h}+(1-t)\mathbb{J}$ is convex along finite energy geodesic segments
(cf. [4, Theorem 4.7]). Since

$t_{i}K_{V}^{h}(\phi_{*})+(1-t_{i})\mathbb{J}(\phi_{i})\leqq
t_{i}K_{V}^{h}(\phi_{i})+(1-t_{i})\mathbb{J}(\phi_{i})\leqq
t_{i}K_{V}^{h}(\phi_{*})+(1-t_{i})\mathbb{J}(\phi_{*}),$

we have $\mathbb{J}(\phi_{i})\leq\mathbb{J}(\phi_{*})$ for any $i$. Together
with (8), $\sup_{i}d_{1}(0,\phi_{i})<+\infty$. By the same argument as above,
we have a uniform entropy bound $\sup_{i}H(\phi_{i})<+\infty$. A priori
estimate by Chen-Cheng [10, Theorem 1.2] again shows $\phi_{i}$ converges to a
smooth $h$-extremal Kähler metric $\phi_{\infty}$. This completes the proof of
the Lemma. ∎

######  Theorem 3.8.

Let $\phi_{*}$ be the minimizer of $K_{V}^{h}$ in $\mathcal{E}_{V}^{1}$. Then
$\phi_{*}$ is smooth and defines a smooth $h$-modified extremal Kähler metric.

###### Proof..

By [23, Lemma 3.8], we first take a sequence $\phi_{i}\in\mathcal{H}_{V}$ such
that $\lim_{i\to\infty}d_{1}(\phi_{i},\phi_{*})=0$ and
$\lim_{i\to\infty}H(\phi_{i})=H(\phi)$. Then
$\lim_{i\to\infty}K_{V}^{h}(\phi_{i})=K_{V}^{h}(\phi_{*})$ by the
$d_{1}$-continuity. For each $i$, consider the continuity path

(9)
$t(S(\phi)-h(\theta_{V}^{\phi})-\underline{S})-(1-t)(\Lambda_{\omega_{\phi}}\omega_{\phi_{i}}-n)=0$

for $t\in[0,1]$. Theorem 3.4 and Corollary 3.6 show the set of $t$ such that a
solution of (9) exists is open in $[0,1]$. Consider the Aubin’s functionals
with a reference metric $\omega_{\phi_{i}}$. Namely we consider

$\displaystyle
I_{i}(\phi):=\mathrm{Vol}^{-1}\int_{M}\phi(\omega_{\phi_{i}}^{n}-\omega_{\phi}^{n}),$
$\displaystyle
J_{i}(\phi):=\mathrm{Vol}^{-1}\int_{M}\phi\omega_{\phi_{i}}^{n}-\frac{\mathrm{Vol}^{-1}}{n+1}\int_{M}\phi\sum_{j=0}^{n}\omega_{\phi_{i}}^{n}\wedge\omega_{\phi}^{n}$

and $\mathbb{J}_{i}:=I_{i}-J_{i}$. By the same argument as in Lemma 3.7, there
exists a unique smooth solution $\phi_{i}^{t}\in\mathcal{H}_{V,0}$ for any
$t\in[0,1)$ which is a minimizer of
$\tilde{K}_{t}^{i}:=tK_{V}^{h}+(1-t)\mathbb{J}_{i}$ on $\mathcal{E}_{V}$. Thus
for any $\phi\in\mathcal{E}_{V}$,

(10) $tK_{V}^{h}(\phi_{i}^{t})+(1-t)\mathbb{J}_{i}(\phi_{i}^{t})\leqq
tK_{V}^{h}(\phi)+(1-t)\mathbb{J}_{i}(\phi).$

Since it is easy to see that $\phi_{i}$ minimizes $\mathbb{J}_{i}$, we have
$0=\mathbb{J}_{i}(\phi_{i})\leqq\mathbb{J}(\phi_{i}^{t}).$ Together with (10),
we have

(11) $K_{V}^{h}(\phi_{i}^{t})\leqq{K_{V}^{h}(\phi_{i})}.$

Since $\phi_{*}$ minimizes $K_{V}^{h}$, the inequality (10) gives
$\mathbb{J}_{i}(\phi_{i}^{t})\leqq\mathbb{J}_{i}(\phi_{*}).$ Note the
classical inequality $I_{i}\leqq(n+1)\mathbb{J}_{i}\leqq{nI_{i}}$ and an
inequality $I_{i}(\phi_{*})\leqq{Cd_{1}(\phi_{i},\phi_{*})}$ proved by
Darvas([13]). Since $\lim_{i\to\infty}d_{1}(\phi_{i},\phi_{*})=0$, the above
inequalities shows $\lim_{i\to\infty}I_{i}(\phi_{i}^{t})=0$ which is uniform
in $t$. By [3, Theorem 1.8], we have
$I_{\omega_{0}}(\phi_{i}^{t})\leqq{C(I_{\omega_{0}}(\phi_{i})+I_{i}(\phi_{i}^{t}))}\leq{C}$
which is uniform in $t$. By [16, Proposition 5], we have the uniform distance
bound $d_{1}(0,\phi_{i}^{t})\leqq{CI_{\omega_{0}}(\phi_{i}^{t})+C}\leqq{C}$.
The inequality (11) and this distance bound show that the entropy
$H(\phi_{i}^{t})$ has uniform upper bound in $t\in[0,1)$. By a priori estimate
by Chen-Cheng [10, Theorem 1.2], when $t\to 1$, the potential $\phi_{i}^{t}$
converges to $u_{i}\in\mathcal{H}_{V,0}$ satisfying the h-extremal Kähler
metric equation $S(u_{i})-\underline{S}-h(\theta_{V}^{(u_{i})})=0$.

Note that $\lim_{i\to\infty}I_{i}(u_{i})=0$ and $d_{1}(0,u_{i})$ is uniformly
bounded. We can take a subsequence $\\{u_{i}\\}_{i}$ such that this converges
to a smooth h-modified extremal Kähler metric $u\in\mathcal{H}_{V,0}$. By an
argument in [11, Lemma 5.7], the difference between $u$ and $\phi_{*}$ is
constant. This completes the proof. ∎

###  3.3. Bando-Mabuchi type uniqueness theorem

We argue the property (P5) in the proof of Theorem 3.1.

######  Theorem 3.9.

For any two $h$-modified extremal Kähler metrics
$\omega_{\phi_{0}},\omega_{\phi_{1}}$ in $\mathcal{H}_{V}$, there exists
$g\in\mathrm{Aut}_{0}(X,V)$ such that
$\omega_{\phi_{0}}=g^{*}\omega_{\phi_{1}}$.

###### Proof..

We follow an argument of Berman-Berndtsson [2]. Let $T$ be the compact torus
defined by the closure of the set of flows generated by $\mathrm{Im}V$, and
let $\mu$ be a $T$-invariant smooth volume form on $M$ with
$\int_{M}d\mu=\operatorname{Vol}$. For any $\phi\in\mathcal{H}_{V}$, define

$F_{\mu}(\phi):=\int_{M}\phi d\mu-E(\phi):=I_{\mu}(\phi)-E(\phi).$

Consider the functional $K_{V}^{h}+sF_{\mu}$ for small $s>0$ to define the
derivative $F_{V}(\phi,s):=d(K_{V}^{h}+sF_{\mu})|_{\phi}$ at
$\phi\in\mathcal{H}_{V}$.

The space of Kähler metrics $\mathcal{H}_{V}$ has a natural $L^{2}$-metric
(called the Mabuchi metric) defined by
$\int_{M}v_{0}v_{1}\omega_{\phi}^{n}/\operatorname{Vol}$ for tangent vectors
$v_{0},v_{1}$ in $T_{\phi}\mathcal{H}_{V}$. This induces the connection $D$ on
the space of $1$-forms on $\mathcal{H}_{V}$. We regard the differentials
$dK_{V}^{h}$ and $dF_{\mu}$ of the functionals $K_{V}^{h}$ and $F_{\mu}$ as
$1$-forms on $\mathcal{H}_{V}$ and consider the equation

(12) $-D_{v_{0}}dK_{V}^{h}|_{\phi_{0}}=dF_{\mu}|_{\phi_{0}}$

for unknown $v_{0}\in T_{\phi_{0}}\mathcal{H}_{V}$.

We put

$\mathfrak{g}_{\omega_{0},V}:=\Set{v\in\mathrm{Lie}(\mathrm{Aut}_{0}(M,V))\,}{\,i_{v}\omega_{0}=\sqrt{-1}\,\overline{\partial}{f}\text{
where }f\in C^{\infty}(M;\mathbb{R})}$

and $\Gamma_{\omega_{0},V}:=\exp(\mathfrak{g}_{\omega_{0},V})$. In view of an
argument in [2, Proposition 4.7], we can assume that the $h$-extremal Kähler
metric $\phi_{0}$ minimizes $F_{\mu}$ on the orbit
$(\Gamma_{\omega_{0},V})\cdot[\phi_{0}]\subset\mathcal{H}_{V}$ by the
replacing $\omega_{\phi_{0}}$ by $g_{0}^{*}\omega_{\phi_{0}}$ for some
$g_{0}\in\Gamma_{\omega_{0},V}$. Since the Hessian of $\mathbb{J}_{V}^{H}$
vanishes on $T_{\phi_{0}}\mathcal{H}_{V}$, an argument in [2, Proposition 4.3,
Proof of Theorem 4.4 and pp. 1192-1193] shows that the equation (12) has a
$T$-invariant solution $v_{0}$. This implies

(13) $F_{V}(\phi_{0}+sv_{0},s)=\mathcal{O}(s^{2}),$

since for a smooth curve $\psi_{s}$ of continuous function,

$\frac{s}{ds}\Big{|}_{0}\langle
F_{V}(\phi_{0}+sv_{0},s),\psi_{s}\rangle=\langle
D_{v_{0}}dK_{V}^{h}|_{\phi_{0}},\psi_{0}\rangle+\langle
dK_{V}^{h}|_{\phi_{0}},D_{v_{0}\psi_{s}}\rangle+\langle
dF_{\mu}|_{\phi_{0}},\psi_{0}\rangle=0$

where $\langle\cdot,\cdot\rangle$ is the pairing for $1$-forms. Similarly,
under replacing $\omega_{\phi_{1}}$ by $g_{1}^{*}\omega_{\phi_{1}}$ for some
$g_{1}\in\mathrm{Aut}_{0}(X,V)$, there exists $T$-invariant $v_{1}\in
T_{\phi_{1}}\mathcal{H}_{V}$ satisfying

(14) $F_{V}(\phi_{1}+sv_{1},s)=\mathcal{O}(s^{2}).$

Consider the $C^{1,1}$-geodesic $\phi_{t}^{s}$ connecting $\phi_{0}+sv_{0}$
and $\phi_{1}+sv_{1}$. Note that $\phi_{t}^{s}$ is also $T$-invariant.
Finally, an argument in [2, Proposition 4.1 and Proof of Theorem 4.4] together
with the convexity of $t\mapsto K_{V}^{h}(\phi_{t}^{s})$, the linearity of
$t\mapsto E(\phi_{t}^{s})$ and the properties (13) and (14) show that there
exists a uniform constant $C>0$ such that
$d_{2}(\phi_{0}^{s},\phi_{1}^{s})^{2}\leqq{Cs}$, which implies
$\omega_{\phi_{0}}=\omega_{\phi_{1}}$. ∎

##  4\. An $h$-modified extremal Kähler metrics and the geodesic stability

We argue the characterization for the existence of an $h$-modified extremal
Kähler metric in terms of geodesic stability conditions which has independent
interest to study of the metric.

###  4.1. Uniform geodesic stability

We first characterize the existence of an $h$-modified extremal Kähler metric
as a uniform geodesic stability condition. Let $G:=\mathrm{Aut}_{0}(M,V)$.

######  Definition 4.1.

Fix $\psi_{0}\in\mathcal{H}_{V,0}$ with $K_{V}^{h}(\psi_{0})<+\infty$. We say
$(M,\Omega,V)$ is uniform geodesic stable at $\psi_{0}$ if there exists $C>0$
such that for any geodesic ray $[0,+\infty)\ni
t\to\psi_{t}\in\mathcal{E}^{1}_{V,0}$ starting at $\psi_{0}$, we have

$\lim_{t\to+\infty}\frac{K^{h}_{V}(\psi_{t})-K^{h}_{V}(\psi_{0})}{t}\geqq{C\limsup_{t\to+\infty}\frac{d_{1,G}(\psi_{t},\psi_{0})}{t}}.$

######  Theorem 4.2.

Fix $\psi_{0}\in\mathcal{E}_{V,0}^{1}$ with $K_{V}^{h}(\psi_{0})<+\infty$.
There exists an $h$-modified extremal Kähler metric in $\mathcal{H}_{V,0}$ if
and only if the $h$-modified Mabuchi functional $K^{h}_{V}$ is $G$-invariant
and $(M,\Omega,V)$ is uniform geodesic stable at
$\psi_{0}\in\mathcal{E}_{V,0}^{1}$ with $K_{V}^{h}(\psi_{0})<+\infty$.
Moreover, the condition of $G$-invariance can be removed.

###### Proof..

Following [15, Theorem 4.7], it suffice to establish properties (P1), (P3),
(P4), (P5), (P6) and (P2)*. The last property (P2)* is due to Proposition 3.3.
The other properties are explained in the proof of Theorem 3.1. We argue that
the condition of $G$-invariance can be removed in the next lemma. This
completes the proof. ∎

Note that the limit in the left hand side of the above inequality exists
because of the convexity of $K^{h}_{V}$.

######  Lemma 4.3.

Fix $\psi_{0}\in\mathcal{H}_{V,0}$ with $K_{V}^{h}(\psi_{0})<+\infty$. If for
any geodesic ray $[0,+\infty)\ni{t}\to\psi_{t}\in\mathcal{E}^{1}_{V,0}$
starting at $\psi_{0}$, we have

$\lim_{t\to+\infty}\frac{K^{h}_{V}(\psi_{t})-K^{h}_{V}(\psi_{0})}{t}\geqq{0},$

then the $h$-modified Mabuchi functional $K_{V}^{h}$ is $G$-invariant.

###### Proof..

We follow the proof of [11, Lemma 7.1]. Take any $\phi\in\mathcal{H}_{V,0}$
and any one parameter subgroup
$[0,+\infty)\ni{t}\mapsto{g_{t}}\in\mathrm{Aut}_{0}(M,V)$ generated by
$X_{\mathrm{Re}}$ where $X\in\mathrm{Aut}_{0}(M,V)$. Define
$\phi_{t}:=g_{t}[\phi]\in\mathcal{H}_{V,0}$. It suffice to show
$\frac{d}{dt}K_{V}^{h}(\phi_{t})=0$. Since the same argument as in the proof
of Theorem 3.1 shows $\frac{d}{dt}K_{V}^{h}(\phi_{t})$ is independent of $t$,
we can assume $\frac{d}{dt}K_{V}^{h}(\phi_{t})(=:a)<0$ without loss of
generality.

The same argument as in the proof of [11, Lemma 7.1] shows that there exists a
constant $C>0$ such that for a sequence $t_{k}\nearrow+\infty$ and for the
unit speed finite energy geodesic segments
$\rho_{k}(s):[0,d_{1}(\psi_{0},\phi_{t_{k}})]\to\mathcal{E}_{V,0}^{h}$
connecting $\psi_{0}$ and $\phi_{t_{k}}$, we have

$K_{V}^{h}(\rho_{k}(s))\leqq\mathrm{max}\\{K_{V}^{h}(\psi_{0}),K_{V}^{h}(\phi)\\}+\frac{sat_{k}}{Ct_{k}+d_{1}(\psi_{0},\phi)}$

for any $s\in[0,d_{1}(\psi_{0},\phi_{t_{k}})]$. Here we used the lower semi
continuity of $K_{V}^{h}$ and the convexity of $K_{V}^{h}$ along $\rho_{k}$.
Thus for each fixed $s$, the functional $K_{V}^{h}(\rho_{k}(s))$ is bounded
above which is uniform in $k$. As discussed in the proof of Proposition 3.2,
for each fixed $s$, a compactness result in [16, Theorem 5.6] can be applied
to show that a subsequence
$\\{\rho_{k_{l}}(s)\\}_{l}\subset\mathcal{E}_{V,0}^{1}$ converges in
$d_{1}$-distance to an element in $\mathcal{E}_{V,0}^{1}$. The same argument
which is used Cantor’s diagonal sequence argument as in the proof of [11,
Lemma 6.5] concludes that there exists a subsequence $\\{k_{l}\\}_{l}$ such
that for any $s\geqq{0}$, $\rho_{k_{l}}(s)$ converges in $d_{1}$-distance to a
unit speed locally finite energy geodesic ray $\rho_{\infty}(s)$ starting at
$\psi_{0}$. By the lower semicontinuity of $K_{V}^{h}$,

$K_{V}^{h}(\rho_{\infty}(s))\leqq\liminf_{l}K_{V}^{h}(\rho_{k_{l}}(s))\leqq\mathrm{max}\\{K_{V}^{h}(\phi),K_{V}^{h}(\psi_{0})\\}+\frac{sa}{C}$

to conclude

$\lim_{s\to+\infty}\frac{K_{V}^{h}(\rho_{\infty}(s)-K_{V}^{h}(\psi_{0})}{s}\leqq\frac{a}{C}<0$

which is a contradiction. ∎

###  4.2. Geodesic stability

Next we characterizes the existence of an $h$-modified extremal Kähler metric
as a geodesic stability condition.

######  Definition 4.4.

Fix $\psi_{0}\in\mathcal{E}_{V,0}^{1}$ with $K_{V}^{h}(\psi_{0})<+\infty$. We
say $(M,\Omega,V)$ is geodesic stable at $\psi_{0}$ if for any locally finite
energy geodesic ray $\psi_{t}:[0,+\infty)\to\mathcal{E}_{V,0}^{1}$ starting at
$\psi_{0}$ with unit speed,

$\lim_{t\to+\infty}\frac{K_{V}^{h}(\psi_{t})-K_{V}^{h}(\psi_{0})}{t}\geqq{0}$

and the equality holds if and only if $\rho$ is parallel to another geodesic
ray $\tilde{\psi_{t}}:[0,+\infty)\to\mathcal{E}_{V,0}^{1}$ which is generated
from a holomorphic vector field in $\mathrm{Lie}(G)$. Here we say that
$\psi_{t}$ is parallel to $\tilde{\psi_{t}}$ if
$\sup_{t>0}d_{1}(\psi_{t},\tilde{\psi_{t}})<+\infty$.

######  Proposition 4.5.

If $(M,\Omega,V)$ is geodesic stable at $\psi_{0}\in\mathcal{E}_{V,0}^{1}$
with $K_{V}^{h}(\psi_{0})<+\infty$, then the $h$-modified Mabuchi functional
is bounded from below.

###### Proof..

We follow the proof of [11, Proposition 7.3]. In order to imply a
contradiction, suppose that there exists a sequence
$\\{\tilde{\phi_{i}}\\}\subset\mathcal{E}_{V,0}^{1}$ satisfying
$K_{V}^{h}(\tilde{\phi_{i}})\to-\infty$. Take $g_{i}\in
G:=\mathrm{Aut}_{0}(M,V)$ to define
$\phi_{i}:=g_{i}[\tilde{\phi_{i}}]\in\mathcal{E}_{V,0}^{1}$ with the property
$d_{1,G}(\psi_{0},\tilde{\phi_{i}})\leqq{d_{1}(\psi_{0},\phi_{i})}\leqq{d_{1,G}(\psi_{0},\tilde{\phi_{i}})+1}$.
In view of Lemma 4.3, $K_{V}^{h}(\phi_{i})\to-\infty$.

If $\sup_{i}d_{1}(\psi_{0},\phi_{i})<+\infty$, then the same compactness
argument as in Proposition 3.2 shows a subsequence $\\{\phi_{i_{k}}\\}_{k}$
converges to $\phi_{\infty}\in\mathcal{E}_{V,0}^{1}$ in $d_{1}$-distance. It
follows from the lower semicontinuity of $K_{V}^{h}$ that
$K_{V}^{h}(\phi_{\infty})\leqq\liminf_{fty}K_{V}^{h}(\phi_{i_{k}})=-\infty$
which is a contradiction.

Thus we focus on the case where $\sup_{i}d_{1}(\psi_{0},\phi_{i})=+\infty$. We
can assume $d_{1}(\psi_{0},\phi_{i})\to+\infty$. Let
$\rho_{i}:[0,d_{1}(\psi_{0},\phi_{i})]\to\mathcal{E}_{V,0}^{1}$ be the unit
speed geodesic segment connecting $\psi_{0}$ and $\phi_{i}$. By the convexity
of $K_{V}^{h}$ along $\rho_{i}$, we have
$K_{V}^{h}(\rho_{i}(t))\leqq\mathrm{max}\\{K_{V}^{h}(\psi_{0}),K_{V}^{h}(\phi_{i})\\}$
for $t\in[0,d_{1}(\psi_{0},\phi_{i})]$. Since $K_{V}^{h}(\rho_{i}(t))$ is
bounded from above for each fixed $t>0$, the same argument as in [11, Lemma
6.5] again shows that the subsequence of the geodesic segments converges to a
unit speed locally finite geodesic ray
$\rho_{\infty}:[0,+\infty)\to\mathcal{E}_{V,0}^{1}$ starting at $\psi_{0}$. By
the lower semicontinuity of $K_{V}^{h}$, the functional $K_{V}^{h}$ is still
bounded above on $\rho_{\infty}$. Then the geodesic stability condition
implies that $\rho_{\infty}$ is parallel to a geodesic ray generated from a
holomorphic vector field. This concludes
$d_{1,G}(\rho_{\infty}(0),\rho_{\infty}(t))$ is bounded uniformly in $t$.

On the other hand, by [11, Lemma 7.4], we have

$d_{1,G}(\psi_{0},\rho_{i}(t))\geqq{d_{1}(\psi_{0},\rho_{i}(t))-1}=t-1$

for $t\in[0,d_{1}(\psi_{0},\phi_{i})]$ to conclude

$d_{1,G}(\psi_{0},\rho_{\infty}(t))\geqq{d_{1,G}(\psi_{0},\rho_{i}(t))-d_{1,G}(\rho_{i}(t),\rho_{\infty}(t))}\geqq{t-1-d_{1}(\rho_{i}(t),\rho_{\infty}(t))}.$

Taking $i\to\infty$, we have a contradiction to boundedness of
$d_{1,G}(\rho_{\infty}(0),\rho_{\infty}(t))$. ∎

######  Theorem 4.6.

Fix $\psi_{0}\in\mathcal{E}_{V,0}^{1}$ with $K_{V}^{h}(\psi_{0})<+\infty$.
There exists an $h$-modified extremal Kähler metrics in $\mathcal{H}_{V,0}$ if
and only if $(M,\Omega,V)$ is geodesic stable at
$\psi_{0}\in\mathcal{E}_{V,0}^{1}$.

###### Proof..

In order to show the “only if” part, we follow an argument in [11, pp.
992–993]. Assume that the reference metric $\omega_{0}$ is an $h$-modified
extremal Kähler metric. By Theorem 3.1, the functional $K_{V}^{h}$ is
coercive.

Let $\rho:[0,+\infty)\to\mathcal{E}_{V,0}^{1}$ be a unit speed geodesic ray
starting at $\psi_{0}$. If $K_{V}^{h}$ is unbounded from above along $\rho$,
then the convexity of $K_{V}^{h}$ shows
$\lim_{t\to+\infty}K_{V}^{h}(\rho(t))/t>0$. Thus we argue that $\rho$ is
parallel to a geodesic ray coming from a holomorphic vector field, provided
that $K_{V}^{h}$ is bounded from above on $\rho$.

Let $\\{t_{k}\\}_{k}\subset\mathbb{R}$ be a sequence such that $t_{k}>0$ and
$t_{k}\to+\infty$. Let
$r_{k}:[0,d_{1}(0,\rho(t_{k}))]]\to\mathcal{E}_{V,0}^{1}$ be the unit speed
finite energy geodesic segment connecting $0$ and $\rho(t_{k})$. We use the
fact that the convexity of $K_{V}^{h}$ along $r_{k}$ and that a $h$-modified
extremal Kähler metric minimizes $K_{V}^{h}$ to conclude for any
$t\in[0,d_{1}(0,\rho(t_{k}))]$,

$K_{V}^{h}(r_{k}(t))\leqq\Big{(}1-\frac{t}{d_{1}(0,\rho(t_{k}))}\Big{)}\inf_{\mathcal{E}_{V,0}^{1}}K_{V}^{h}+\frac{t}{d_{1}(0,\rho(t_{k}))}\sup_{t\geqq{0}}K_{V}^{h}(\rho(t)).$

Since $K_{V}^{h}$ is uniformly bounded from above, we can apply the same
argument as in [11, Lemma 6.5] to show that for a subsequence
$\\{k_{l}\\}_{l}$ and for any $t\geqq{0}$, the ray $r_{k_{l}}(t)$ converges to
a locally finite energy geodesic ray $r_{\infty}(t)$ with unit speed. Then by
the above inequality and the lower semicontinuity of $K_{V}^{h}$, we have

$K_{V}^{h}(r_{\infty}(t))\leqq\liminf_{k_{l}}K_{V}^{h}(r_{k_{l}}(t))\leqq\inf_{\mathcal{E}_{V,0}^{1}}K_{V}^{h},$

for any $t>0$. Since $r_{\infty}(t)$ is thus an $h$-modified extremal Kähler
metric for any $t\geqq{0}$, Theorem 3.9 shows the ray $r_{\infty}$ is
generated by a holomorphic vector field. Moreover by [11, Lemma 7.6], we know
that $\rho$ and $r_{\infty}$ are parallel. This completes the proof of the
“only if” part.

In order to show the “if” part, we follow an argument in [11, p. 992]. Recall
the continuity path (7) to construct an $h$-modified extremal Kähler metric.
By Proposition 4.5 and a argument in the proof of Lemma 3.7, there exists a
solution $\tilde{\phi}_{t}$ of the equation (7) for any $t<1$. Take a sequence
$t_{i}\nearrow 1$ and define $\tilde{\phi}_{i}$ as the solution of (7) at
$t=t_{i}$.

If $\sup_{i}d_{1,G}(\psi_{0},\tilde{\phi}_{i})<+\infty$, then Proposition 4.7
stated below and Lemma 4.3 show the existence of an $h$-modified extremal
Kähler metric.

We consider the case where
$\sup_{i}d_{1,G}(\psi_{0},\tilde{\phi}_{i})=+\infty$ to show contradiction. We
follow the same argument as the proof of Proposition 4.5. Take $g_{i}\in G$
such that $\phi_{i}:=g_{t}[\tilde{\phi}_{i}]$ satisfies
$d_{1,G}(\psi_{0},\tilde{\phi}_{i})\leqq{d_{1}(\psi_{0},\phi_{i})}\leqq
d_{1,G}(\psi_{0},\tilde{\phi}_{i})+1$. Since an argument in the proof of Lemma
3.7 shows the functional $K_{V}^{h}$ is bounded along the continuity path
$\eqref{eq-t}$, $\sup_{i}K_{V}^{h}(\tilde{\phi}_{i})<+\infty$. By Lemma 4.3,
$\sup_{i}K_{V}^{h}(\phi_{i})<+\infty$. By the same argument as the proof of
Proposition 4.5, the unit speed geodesic segment $\rho_{i}$ connecting
$\psi_{0}$ and $\phi_{i}$ yields the limiting geodesic ray $\rho_{\infty}$
starting at $\psi_{0}$. The geodesic ray $\rho_{\infty}$ decreases $K_{V}^{h}$
since each geodesic ray $\rho_{i}$ decreases $\tilde{K}_{t}$ in Lemma 3.7.
Since $(M,\Omega,V)$ is geodesic stable, $\rho_{\infty}$ is parallel to a
geodesic ray coming from a holomorphic vector field. Thus $\rho_{\infty}$ is
$d_{1,G}$-bounded. On the other hand, an argument in the proof of Proposition
4.5 shows $\rho_{\infty}$ is not $d_{1,G}$-bounded. Therefore the situation
where $\sup_{i}d_{1,G}(\psi_{0},\tilde{\phi}_{i})=+\infty$ does not occur.
This completes the proof. ∎

######  Proposition 4.7.

Suppose the $h$-modified Mabuchi functional $K_{V}^{h}$ is bounded from below.
Take a sequence $t_{i}\nearrow 1$ and let $\tilde{\phi}_{i}$ be a solution the
continuity path (7) at $t=t_{i}$ satisfying
$\tilde{\phi_{i}}\in\mathcal{H}_{V,0}$. Suppose also
$\sup_{i}d_{1,G}(0,\tilde{\phi_{i}})<+\infty$. Let
$\phi_{i}\in\mathcal{H}_{V,0}$ be in the $G$-orbit of $\tilde{\phi_{i}}$
satisfying $\sup_{i}d_{1}(0,\phi_{i})<+\infty$. Then a subsequence of
$\\{\phi_{i}\\}_{i}$ converges to a smooth $h$-modified extremal Kähler metric
in $C^{1,\alpha}$-topology for any $0<\alpha<1$.

###### Proof..

We follow an argument in [11, pp. 973–978]. Recall that $\tilde{\phi}_{i}$ is
a minimizer of the functional
$\tilde{K}_{t_{i}}:=t_{i}K_{V}^{h}+(1-t_{i})\mathbb{J}$. When $t_{i}\to 1$, we
observe

(15)
$\tilde{K}_{t_{i}}(\tilde{\phi}_{i})=\inf_{\mathcal{H}_{V}}\tilde{K}_{t_{i}}\to\inf_{\mathcal{H}_{V}}K_{V}^{h},\quad
K_{V}^{h}(\tilde{\phi}_{i})\to\inf_{\mathcal{H}_{V}}K_{V}^{h}\quad\text{and}\quad(1-t_{i})\mathbb{J}(\tilde{\phi}_{i})\to{0}$

(cf. [11, Lemma 4.15]). Indeed for any $\varepsilon>0$, we can take
$\phi^{\varepsilon}\in\mathcal{H}_{V}$ satisfying
$K_{V}^{h}(\phi^{\varepsilon})\leqq\inf_{\mathcal{H}_{V}}K_{V}^{h}+\varepsilon$
to obtain

$\tilde{K}_{t_{i}}(\tilde{\phi}_{i})\leqq{t_{i}K_{V}^{h}(\phi^{\varepsilon})+(1-t_{i})\mathbb{J}(\phi^{\varepsilon})}.$

This yields

$\limsup_{i\to\infty}\tilde{K}_{t_{i}}(\tilde{\phi}_{i})\leqq{K_{V}^{h}(\phi^{\varepsilon})}\leqq\inf_{\mathcal{H}_{V}}K_{V}^{h}+\varepsilon.$

Since $\mathbb{J}\geqq{0}$, together with the above inequality, we have

$t_{i}\inf_{\mathcal{H}_{V}}K_{V}^{h}\leqq{t_{i}K_{V}^{h}(\tilde{\phi}_{i})}\leqq{K_{t_{i}}(\tilde{\phi}_{i})}\leqq\inf_{\mathcal{H}_{V}}K_{V}^{h}+\varepsilon$

which shows (15).

We next observe

(16)
$\sup_{i}H(\phi_{i})=\sup_{i}\int_{M}\log\frac{\omega_{\phi_{i}}^{n}}{\omega_{0}^{n}}\omega_{\phi_{i}}^{n}<+\infty$

(cf. [11, Lemma 4.18]). Indeed by Lemma 4.3,
$\sup_{i}K_{V}^{h}(\tilde{\phi}_{i})=\sup_{i}K_{V}^{h}(\phi_{i})$ which is
finite by the inequality (15). Since the functionals
$\mathbb{J}_{-\mathrm{Ric}}$ and $\mathbb{J}_{V}^{h}$ in the definition of
$K_{V}^{h}$ are controlled by the $d_{1}$-distance, then the estimate (16)
follows.

Let $g_{i}\in\mathrm{Aut}_{0}(M,V)$ be the automorphism such that
$\phi_{i}=g_{i}[\tilde{\phi}_{i}]$. Define
$\omega_{i}=g_{i}^{*}\omega_{0}=\omega_{0}+\sqrt{-1}\partial\overline{\partial}h_{i}$
with $\sup_{M}h_{i}=0$. Also define
$\theta_{i}=g_{i}^{*}\theta_{V}^{(\tilde{\phi}_{i})}=\theta_{V}^{(\phi_{i})}$.
A direct computation shows that $\phi_{i}$ satisfies

(17)
$\omega_{\phi_{i}}^{n}=e^{F_{i}}\omega_{0}^{n}\quad\text{and}\quad\square_{\omega_{\phi_{i}}}F_{i}=\Big{(}\underline{S}-\frac{1-t_{i}}{t_{i}}n+h(\theta_{i})\Big{)}+\Lambda_{\omega_{\phi_{i}}}\Big{(}\mathrm{Ric}(\omega_{0})-\frac{1-t_{i}}{t_{i}}\omega_{i}\Big{)}$

(cf. [11, Lemma 4.14]).

We are now in position to prove that a subsequence of $\\{\phi_{i}\\}_{i}$
converges to a smooth $h$-modified extremal Kähler metric in
$C^{1,\alpha}$-topology for any $0<\alpha<1$. Our equation (7) is different
from Chen-Cheng’s one by the term $h(\theta_{i})$. Since this term is uniform,
we can apply estimates established by Chen-Cheng [11]. Define

$f_{i}=\frac{1-t_{i}}{t_{i}}h_{i}.$

By [11, Lemma 4.20], there exists a constant $C=C(\omega_{0})>0$ such that for
any $p>1$, there exists $\varepsilon_{p}>0$ satisfying
$\|e^{-f_{i}}\|_{L^{p}(\omega_{0}^{n})}\leqq{C}$ when
$t_{i}\in(1-\varepsilon_{p},1)$ By [11, Proposition 4.21] (see also [11,
Theorem 3.3]), there exists a constant
$C=C(p,\omega_{0},\sup_{i}H(\phi_{i}))>0$ such that

$\|F_{i}+f_{i}\|_{W^{1,2p}}+\|n+\square_{\omega_{0}}\phi_{i}\|_{L^{p}(\omega_{0}^{n})}\leqq{C}.$

By taking a subsequence, we get $\mathrm{Im}V$-invariant functions
$\phi_{*}\in W^{2,p}$ and $F_{*}\in W^{1,p}$ for any $1<p<+\infty$ such that

  * •

$\phi_{i}\to\phi_{*}$ in $C^{1,\alpha}$ for any $0<\alpha<1$ and
$\sqrt{-1}\partial\overline{\partial}\phi_{i}\to\sqrt{-1}\partial\overline{\partial}\phi_{*}$
weekly in $L^{p}$,

  * •

$F_{i}+f_{i}\to F_{*}$ in $C^{\alpha}$ for any $0<\alpha<1$ and
$\nabla(F_{i}+f_{i})\to\nabla F_{*}$ weekly in $L^{p}$,

  * •

$\omega_{\phi_{i}}^{k}\to\omega_{\phi_{*}}^{k}$ weekly in $L^{p}$ for any
$1\leqq{k}\leqq{n}$.

An argument in [11, Proof of Proposition 4.23] shows that $\phi_{*}$ is a weak
solution in the following sense:

$\omega_{\phi_{*}}^{n}=e^{F_{*}}\omega_{0}^{n},$

and for any $u\in{C^{\infty}(M;\mathbb{R})}$,

$-\int_{M}d^{c}F_{*}\wedge{du}\wedge\frac{\omega_{\phi_{*}}^{n-1}}{(n-1)!}=-\int_{M}u\Big{(}\underline{S}+h(\theta_{V}^{(\phi_{*})})\Big{)}\frac{\omega_{\phi_{*}}^{n}}{n!}+\int_{M}u\mathrm{Ric}(\omega_{0})\wedge\frac{\omega_{\phi_{*}}^{n}}{n!}$

where $d^{c}:=\frac{\sqrt{-1}}{2}(\partial-\overline{\partial})$. By [12,
Theorem 1.1], there exists a constant $C>1$ such that
$C^{-1}\omega_{0}\leqq\omega_{\phi_{*}}\leqq{C\omega_{0}}$ and
$\omega_{\phi_{*}}\in W^{3,p}$. Therefore the standard elliptic regularity
argument shows $\phi_{*}$ is a smooth $h$-modified extremal Kähler metric.
This completes the proof. ∎

##  5\. Existence for a $\sigma$-soliton and a $\sigma$-extremal Kähler
metrics

In this section we work on a Fano manifold $M$ with $\Omega=2\pi{c_{1}(M)}$ to
prove Theorem 1.4. Let $G:=\mathrm{Aut}_{0}(M,V)$.

######  Theorem 5.1.

There exists a $\sigma$-soliton in $2\pi c_{1}(M)$ if and only if the
$\sigma$-Ding functional is $G$-invariant and coercive. Moreover the condition
of $G$-invariance can be removed.

###### Proof..

We check that the data $\mathcal{R}=\mathcal{H}_{V,0}$, $d=d_{1}$,
$F=D_{V}^{h}$ with $h(s)=1-e^{-\sigma(s)}$ and $G=\mathrm{Aut_{0}}(M,V)$ enjoy
the properties (P1)–(P6) in [15, Section 4.2] to apply [15, Theorem 4.7].

(P1):

This is due to [6] and Lemma 2.1.

(P2):

This can be proved by applying the monotonicity of $E_{V}^{h}$ with
$h(s)=1-e^{-\sigma(s)}$ stated in Lemma 2.1 and [16, Lemmas 5.15, 5.20 and
5.29] where we replace $\mathrm{AM}_{X}$ and $F^{X}$ in [16] by $E_{V}^{h}$
and $D_{V}^{h}$ with $h(s)=1-e^{-\sigma(s)}$ respectively.

(P3):

: This is due to [37, Theorem 1.1] for regularity argument by an application
of a geometric flow which is based on [38, 40].

(P4):

This is due to [16, Lemma 5.9].

(P5):

This is due to (P3) and [32, Theorem C].

(P6):

This follows from the cocycle condition for $D_{V}^{h}$.

Finally, we follow the same argument as in the proof of Theorem 3.1 to
conclude that $D_{V}^{h}$ is $G$-invariant provided the coercivity of
$D_{V}^{h}$. ∎

Before proving Theorem 1.4, we note that a $\sigma$-soliton
$\omega_{\sigma\text{-sol}}$ and a $\sigma$-extremal metric
$\omega_{\sigma\text{-ext}}$ are different in general. Indeed, if
$\omega_{\sigma\text{-sol}}=\omega_{\sigma\text{-ext}}=:\omega$, or
equivalently $S(\omega)-n=1-e^{\rho_{\omega}}$, then an argument in [36,
Section 3.2] shows $\omega$ is a Kähler-Einstein metric.

###### Proof of Theorem 1.4.

, By the inequality (6), if the $\sigma$-Ding functional is coercive then so
is the $\sigma$-Mabuchi functional. Therefore Theorems 1.2 and 5.1 show that
the existence of a $\sigma$-soliton implies the existence of a
$\sigma$-extremal Kähler metric. ∎

######  Example 5.2.

Let $M$ be a toric Fano manifold and $\Omega=2\pi{c_{1}(M)}$. Then, by [43,
Lemma 2.1], there exists a unique holomorphic vector field
$V_{0}\in\mathfrak{X}_{0}(M)=\mathfrak{X}(M)$ satisfying $F_{V_{0}}^{h}(X)=0$
for any $X\in\mathfrak{X}(M)$, where $h(s)=1-e^{s+C}$ and
$C:=\log\left(\mathrm{Vol}/\int_{M}e^{\theta_{V_{0}}^{(\omega)}}\omega^{n}\right)$
for some $\omega\in\Omega=2\pi{c_{1}(M)}$. Here, we note that this constant
$C$ does not depend on the choice of $\omega\in\Omega=2\pi{c_{1}(M)}$. In view
of Wang-Zhu’s result [43, Theorem 1.1], $M$ always admits a Kähler-Ricci
soliton, i.e., there exists $\omega_{0}\in\Omega=2\pi{c_{1}(M)}$ satisfying
$\mathrm{Ric}(\omega_{0})-\omega_{0}=L_{V_{0}}\omega_{0}$. Therefore, by
Theorem 1.4, we have a $\sigma$-extremal metric
$\omega_{1}\in\Omega=2\pi{c_{1}(M)}$ for $\sigma(s)=-s-C$, i.e., $\omega_{1}$
satisfies

$S(\omega_{1})-n=1-\frac{\mathrm{Vol}}{\int_{M}e^{\theta_{V_{0}}^{(\omega_{1})}}\omega_{1}^{n}}e^{\theta_{V_{0}}^{(\omega_{1})}}.$

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