Calabi’s Conjecture of the Kähler–Ricci Soliton Type

Abstract

In this paper, we discuss Calabi’s equation of the Kähler–Ricci soliton type on a compact Kähler manifold. This equation was introduced by Zhu as a generalization of Calabi’s conjecture. We give necessary and sufficient conditions for the unique existence of a solution for this equation on a compact Kähler manifold with a holomorphic vector field which has a zero point. We also consider the case of a nowhere vanishing holomorphic vector field, and give sufficient conditions for the unique existence of a solution for this equation.

1 Introduction

Let $(M, \omega )$ be an m-dimensional compact Kähler manifold. In Kähler geometry, the following theorem is widely known as Calabi’s conjecture:

Theorem 1.1

Let $\Omega \in 2 \pi c_1(M)$ be a real (1, 1)-form. Then there exists a unique Kähler form $\omega '$ in the Kähler class $[ \omega ]$ such that $\mathop {\mathrm {Ric}}\nolimits (\omega ) = \Omega$.

Yau [8] proved this theorem by the continuity method and Cao [1] also proved it by using some geometric flow. This theorem is deeply related to Kähler–Einstein metrics. For instance, as an immediate corollary, we have

Corollary 1.2

If $c_1(M) = 0$, then there exists a unique Ricci-flat Kähler form $\omega '$ in Kähler class $[ \omega ]$.

As a generalization of Calabi’s conjecture, Zhu [9] considered the following problem:

Problem 1.3

(Calabi’s conjecture of the Kähler–Ricci soliton type). Let $\Omega \in 2 \pi c_1(M)$ be a real (1, 1)-form and X be a holomorphic vector field on M. Then, does there exist a Kähler form $\omega '$ in the Kähler class $[ \omega ]$ such that

$$\begin{aligned} \mathop {\mathrm {Ric}}\nolimits (\omega ') - \Omega = \mathcal {L}_X \omega ' ? \end{aligned}$$

(1.1)

Here $\mathcal {L}_X$ denotes the Lie derivative along X. We call (1.1) Calabi’s equation of the Kähler–Ricci soliton type. One of the motivations for which he introduced Eq. (1.1) was to study Kähler–Ricci solitons. A Kähler form $\omega '$ is called a Kähler–Ricci soliton if it satisfies

$$\begin{aligned} \mathop {\mathrm {Ric}}\nolimits (\omega ') - \omega ' = \mathcal {L}_X \omega ' \end{aligned}$$

(1.2)

for some holomorphic vector field X. In particular, if $X = 0$, then a Kähler–Ricci soliton is nothing but a Kähler–Einstein metric. Clearly, a Kähler–Ricci soliton $\omega '$ is a solution for (1.1) when $\Omega = \omega '$. In his paper, Zhu [9] showed the following theorem:

Theorem 1.4

[9] Let $(M, \omega )$ be a compact Kähler manifold with $c_1(M) > 0$. Let $\Omega \in 2 \pi c_1(M)$ be a positive definite (1, 1)-form on M and X be a holomorphic vector field on M. Then Eq. (1.1) has a unique solution $\omega '$ in the Kähler class $[ \omega ]$ if and only if

• (i) There exists a maximal compact subgroup K of $\mathop {\mathrm {Aut}}\nolimits _0(M)$ such that it contains the one-parameter family $\{ \exp ( t \mathop {\mathrm {Im}}\nolimits X) \}_{t \in \mathbb {R}}$,

• (ii) $\mathcal {L}_X \Omega$ is a real (1, 1)-form on M.

Here $\mathop {\mathrm {Aut}}\nolimits _0(M)$ is the identity component of the group $\mathop {\mathrm {Aut}}\nolimits (M)$ of holomorphic automorphisms of M.

One of the main purposes of this paper is to remove the assumption that $\Omega$ is positive definite and give a partial answer to Problem 1.3. Our first main result is as follows:

Theorem 1.5

Let $(M, \omega )$ be a compact Kähler manifold and $\Omega \in 2 \pi c_1(M)$ be a real (1, 1)-form on M. Suppose that a holomorphic vector field X has a zero point. Then Eq. (1.1) has a unique solution $\omega '$ in the Kähler class $[ \omega ]$ if and only if

• (i) There exists a maximal compact subgroup K of $\mathop {\mathrm {Aut}}\nolimits _0(M)$ such that it contains the one-parameter family $\{ \exp ( t \mathop {\mathrm {Im}}\nolimits X) \}_{t \in \mathbb {R}}$,

• (ii) $\mathcal {L}_X \Omega$ is a real (1, 1)-form on M.

As a corollary of Theorem 1.5, we have

Corollary 1.6

Let $(M, \omega )$ be a compact Kähler manifold. Let $\Omega \in 2 \pi c_1(M)$ be a real (1, 1)-form on M and X be a holomorphic vector field on M. Suppose $H^1 (M; \mathbb {R}) = 0$. Then Eq. (1.1) has a unique solution $\omega '$ in the Kähler class $[ \omega ]$ if and only if

• (i) There exists a maximal compact subgroup K of $\mathop {\mathrm {Aut}}\nolimits _0(M)$ such that it contains the one-parameter family $\{ \exp ( t \mathop {\mathrm {Im}}\nolimits X) \}_{t \in \mathbb {R}}$,

• (ii) $\mathcal {L}_X \Omega$ is a real (1, 1)-form on M.

In particular, if M is a Fano manifold, i.e., $c_1(M) > 0$, then M satisfies the condition $H^1 (M; \mathbb {R}) = 0$. Zhu used the continuity method in the proof of his theorem, but we show Theorem 1.5 by using a geometric flow.

We also consider the case of a nowhere vanishing holomorphic vector field X. This case is more complicated because the harmonic part of $i_X \omega$ does not vanish. Under the condition that X has no zero point, we show the following theorem:

Theorem 1.7

Let $(M, \omega )$ be a compact Kähler manifold and $\Omega \in 2 \pi c_1(M)$ be a real (1, 1)-form on M. Let X be a holomorphic vector field which has no zero point. Assume that both $\{ \exp ( t \mathop {\mathrm {Re}}\nolimits X) \}_{t \in \mathbb {R}}$ and $\{ \exp ( t \mathop {\mathrm {Im}}\nolimits X) \}_{t \in \mathbb {R}}$ are periodic. Moreover, suppose that $\mathcal {L}_X \Omega$ is a real (1, 1)-form on M. Then Eq. (1.1) has a unique solution $\omega '$ in the Kähler class $[ \omega ]$.

We organize this paper as follows. In Sect. 2, we review some basic facts in Kähler geometry. In Sect. 3, we show the necessity part of Theorem 1.5 (cf. [9]). In Sects. 4, 5 and 6, we introduce a geometric flow, and prove the long time existence and the convergence of the flow (cf. [1, 7]). In Sect. 7, we consider the case of a nowhere vanishing holomorphic vector field.

2 Preliminaries

Let M be an m-dimensional compact Kähler manifold and $\omega$ be a Kähler form on M. In local coordinates $( z^1, \dots , z^m)$, $\omega$ has an expression

$$\begin{aligned} \omega = \sqrt{-1}\sum _{i, \, j = 1}^m g_{i \bar{j}} \, dz^i \wedge d{\bar{z}}^j, \end{aligned}$$

where $(g_{i \bar{j}})$ is a positive definite Hermitian matrix. Recall that $g_{i \bar{j}}$ satisfy the Kähler identities

$$\begin{aligned} \partial _k g_{i \bar{j}} = \partial _i g_{k \bar{j}}, \end{aligned}$$

(2.1)

where $\partial _i = \partial / \partial z^i$ and $\partial _{\bar{j}} = \partial / \partial {\bar{z}}^j$. For arbitrary Kähler form $\omega '$ in the Kähler class $[\omega ]$, there exists a smooth real function $\varphi$ on M such that

$$\begin{aligned} \omega ' = \omega + \sqrt{-1}\partial \bar{\partial } \varphi . \end{aligned}$$

The Ricci form $\mathop {\mathrm {Ric}}\nolimits (\omega )$ of $\omega$ is given by

$$\begin{aligned} \mathop {\mathrm {Ric}}\nolimits (\omega ) = \sqrt{-1}\sum _{i, \, j = 1}^m R_{i \bar{j}} \, dz^i \wedge d{\bar{z}}^j = - \sqrt{-1}\partial \bar{\partial }\log \det (g_{i \bar{j}}) \end{aligned}$$

and it represents $2 \pi c_1(M)$.

Let $\eta (M)$ be the space of holomorphic vector fields on M. For each holomorphic vector field X, there exists a unique function $\theta _X(\omega )$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathcal {L}_X \omega = \sqrt{-1}\partial \bar{\partial }\theta _X(\omega ), \\ \int _M \theta _X(\omega ) \, \frac{\omega ^m}{m!} = 0. \end{array}\right. } \end{aligned}$$

(2.2)

Put $\alpha _X := i_X \omega - \sqrt{-1}\bar{\partial }\theta _X(\omega )$. Then $\alpha _X$ is a harmonic (0, 1)-form with respect to $\omega$. The following propositions are widely known, but we give proofs for the reader’s convenience.

Proposition 2.1

Let $\omega _\varphi = \omega + \sqrt{-1}\partial \bar{\partial }\varphi$ be a Kähler form on M in the Kähler class $[\omega ]$. Then

$$\begin{aligned} \theta _X(\omega _\varphi ) = \theta _X(\omega ) + X(\varphi ). \end{aligned}$$

(2.3)

Proof

Let $\omega _s = \omega + s \sqrt{-1}\partial \bar{\partial }\varphi$. From the definition of $\theta _X$, we have

$$\begin{aligned} \sqrt{-1}\partial \bar{\partial }\theta _X(\omega _s) = \sqrt{-1}\partial \bar{\partial }\theta _X(\omega ) + s \sqrt{-1}\partial \bar{\partial }X(\varphi ) \end{aligned}$$

(2.4)

and hence

$$\begin{aligned} \theta _X(\omega _s) = \theta _X(\omega ) + s X(\varphi ) + c_s \end{aligned}$$

(2.5)

for some constants $c_s$. Clearly, $c_0 = 0$.

We now compute

$$\begin{aligned} 0&\equiv \frac{d}{ds} \int _M (\theta _X(\omega ) + s X(\varphi ) + c_s) \, \frac{\omega _s^m}{m!} \nonumber \\&= \int _M \left( X(\varphi ) + \frac{d c_s}{ds} + \left( \theta _X(\omega ) + s X(\varphi ) + c_s \right) \triangle _{\omega _s} \varphi \right) \, \frac{\omega _s^m}{m!} \nonumber \\&= \int _M X(\varphi ) \, \frac{\omega _s^m}{m!} + \int _M \varphi \mathcal {L}_X \left( \frac{\omega _s^m}{m!} \right) + \frac{d c_s}{ds} \int _M \frac{\omega _s^m}{m!} \nonumber \\&= \frac{d c_s}{ds} \int _M \frac{\omega _s^m}{m!}, \end{aligned}$$

(2.6)

where $\triangle _{\omega _s} = g_s^{i \bar{j}} \partial _i \partial _{\bar{j}}$ denotes the complex Laplacian with respect to $\omega _s$. Thus we conclude $c_s \equiv 0$. $\square$

Proposition 2.2

$\alpha _X$ is independent of the choice of $\omega '$ in the Kähler class $[\omega ]$.

Proof

From Proposition 2.1, it follows that

$$\begin{aligned} i_X \omega _\varphi - \sqrt{-1}\bar{\partial }\theta _X(\omega _\varphi ) = i_X \omega - \sqrt{-1}\bar{\partial }\theta _X(\omega ) + \sqrt{-1}\left( i_X \partial \bar{\partial }\varphi - \bar{\partial }(X(\varphi ))\right) . \end{aligned}$$

(2.7)

Since X is holomorphic, we have $i_X \partial \bar{\partial }\varphi - \bar{\partial }(X(\varphi )) = 0$. $\square$

Proposition 2.3

[4] $\alpha _X \equiv 0$ if and only if X has a zero point.

Proof

Suppose $\alpha _X \equiv 0$. Let $p \in M$ be a point at which $\theta _X(\omega )$ attains its maximum. Then X vanishes at p. Conversely, suppose X vanishes at $q \in M$. Since $\alpha _X$ is harmonic, $\bar{\partial }^* \alpha _X = 0$ and $\partial \alpha _X = 0$. Thus we have

$$\begin{aligned} 0 \le \int _M | \alpha _X |^2_\omega \, \frac{\omega ^m}{m!}&= \int _M (i_X \omega - \sqrt{-1}\bar{\partial }\theta _X(\omega ), \alpha _X)_\omega \, \frac{\omega ^m}{m!} \nonumber \\&= \int _M (i_X \omega , \alpha _X)_\omega \, \frac{\omega ^m}{m!} \nonumber \\&= \int _M \bar{\alpha }_X (X) \, \frac{\omega ^m}{m!}. \end{aligned}$$

(2.8)

Furthermore, $\bar{\alpha }_X ( X )$ is a holomorphic function on M. Since M is compact and $X_q = 0$, it follows that $\bar{\alpha }_X ( X ) \equiv 0$. Therefore, $\alpha _X \equiv 0$. $\square$

As a corollary of Proposition 2.3, we have

Corollary 2.4

Suppose $H^1 (M; \mathbb {R}) = 0$. Then, for arbitrary holomorphic vector field X, $\alpha _X \equiv 0$.

3 Calabi’s Equation of the Kähler–Ricci Soliton Type

Let $\Omega \in 2 \pi c_1(M)$ be a real (1, 1)-form on M and X be a holomorphic vector field on M. In this section, we assume a Kähler form $\omega$ is a solution for Calabi’s equation of the Kähler–Ricci soliton type:

$$\begin{aligned} \mathop {\mathrm {Ric}}\nolimits (\omega ) - \Omega = \mathcal {L}_X \omega . \end{aligned}$$

(3.1)

The aim of this section is to derive the necessary conditions for the existence of the solution $\omega$, which was obtained by Zhu ([9]).

Since $\mathrm {Ric}(\omega )$ and $\Omega$ are real (1, 1)-forms on M, we can see $\mathcal {L}_X \omega$ is a real (1, 1)-form. Therefore, $\mathop {\mathrm {Im}}\nolimits X$ is a Killing vector field, that is, $\mathop {\mathrm {Im}}\nolimits X$ generates a one-parameter group of isometries of $(M, \omega )$. Thus, there exists a maximal compact subgroup K of $\mathrm {Aut}_0(M)$ such that it contains the one-parameter group $\{ \exp ( t \mathop {\mathrm {Im}}\nolimits X) \}_{t \in \mathbb {R}}$.

Moreover, we can see the following:

Proposition 3.1

[9] Assume that there exists a solution $\omega$ for (3.1). Then $\mathcal {L}_X \Omega$ is a real (1, 1)-form on M.

Proof

First note that $\theta _X(\omega )$ is a real-valued function. We have

$$\begin{aligned} \mathcal {L}_{\mathop {\mathrm {Re}}\nolimits X} \mathop {\mathrm {Ric}}\nolimits (\omega )&= \frac{d}{dt} \Big |_{t=0} \left( \exp (t \mathop {\mathrm {Re}}\nolimits X) \right) ^* \mathop {\mathrm {Ric}}\nolimits (\omega ) \nonumber \\&=- \sqrt{-1}\partial \bar{\partial }\triangle _\omega \theta _X(\omega ), \end{aligned}$$

(3.2)

and

$$\begin{aligned} \mathcal {L}_{\mathop {\mathrm {Im}}\nolimits X} \mathop {\mathrm {Ric}}\nolimits (\omega ) = 0. \end{aligned}$$

(3.3)

Hence $\mathcal {L}_X \mathop {\mathrm {Ric}}\nolimits (\omega )$ is a real (1, 1)-form.

Furthermore, we have

$$\begin{aligned} X(\theta _X(\omega )) = g_{i \bar{j}} X^i \overline{X^j} - \sqrt{-1}\overline{\alpha }_X (X). \end{aligned}$$

(3.4)

Since X is holomorphic and $\partial \alpha _X = 0$, it follows that $\bar{\partial }\left( \bar{\alpha }_X (X)\right) = 0$. Thus we obtain

$$\begin{aligned} \sqrt{-1}\partial \bar{\partial }X\left( \theta _X(\omega )\right) = \sqrt{-1}\partial \bar{\partial }(g_{i \bar{j}} X^i \overline{X^j}). \end{aligned}$$

(3.5)

Hence $\mathcal {L}_X ( \mathcal {L}_X \omega ) = \sqrt{-1}\partial \bar{\partial }X(\theta _X(\omega ))$ is a real (1, 1)-form, and we conclude $\mathcal {L}_X \Omega$ is a real (1, 1)-form. $\square$

Consequently, we complete the proof of the necessity part of Theorem 1.5.

4 A Geometric Flow of the Kähler–Ricci Soliton Type

In order to show that Calabi’s equation of the Kähler–Ricci soliton type has a solution, in this section, we introduce a geometric flow. We also show the short-time existence of the flow.

Let X be a holomorphic vector field on M. We assume that there exists a maximal compact subgroup $K \subset \mathrm {Aut}_0(M)$ such that $\{ \exp ( t\mathop {\mathrm {Im}}\nolimits X) \}_{t \in \mathbb {R}} \subset K$. By changing $\omega$ if necessary, we may assume that $\omega$ is a K-invariant Kähler form. Let $\Omega \in 2 \pi c_1(M)$ be a real (1, 1)-form such that $\mathcal {L}_X \Omega$ is a real (1, 1)-form. Since $\Omega \in 2 \pi c_1(M)$, there exists a real-valued function f on M such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathop {\mathrm {Ric}}\nolimits (\omega ) - \Omega = \sqrt{-1}\partial \bar{\partial }f, \\ \int _M e^f \, \frac{\omega ^m}{m!} = \int _M \frac{\omega ^m}{m!}. \end{array}\right. } \end{aligned}$$

(4.1)

Now we consider the following flow:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d}{dt} \omega _t = - \mathop {\mathrm {Ric}}\nolimits (\omega _t) + \Omega + \mathcal {L}_X \omega _t, \\ \omega _0 = \omega . \end{array}\right. } \end{aligned}$$

(4.2)

By the definition of this flow, we can see the following lemma:

Lemma 4.1

The flow (4.2) preserves its de Rham cohomology class.

Therefore, the flow (4.2) is equivalent to the following parabolic complex Monge–Ampère equation:

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{\varphi _t} = \log \frac{\omega _t^m}{\omega ^m} - f + \theta _X(\omega ) + X(\varphi _t), \\ \varphi _0 = 0. \end{array}\right. } \end{aligned}$$

(4.3)

First we consider the short-time existence.

Theorem 4.2

There exists a positive constant $T > 0$ such that a unique solution $\varphi _t$ for (4.3) exists for $0 \le t < T$.

Proof

Let $\Omega _t = \left( \exp (-t\mathop {\mathrm {Re}}\nolimits X)\right) ^* \Omega$. We consider the following flow:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d}{dt} \tilde{\omega }_t = - \mathop {\mathrm {Ric}}\nolimits (\tilde{\omega }_t) + \Omega _t,\\ \tilde{\omega }_0 = \omega . \end{array}\right. } \end{aligned}$$

(4.4)

Equation (4.4) has a unique short-time solution. We now fix $s \in \mathbb {R}$. Since $\mathcal {L}_{\mathop {\mathrm {Im}}\nolimits X} \Omega = 0$ and $[ \mathop {\mathrm {Re}}\nolimits X, \mathop {\mathrm {Im}}\nolimits X ] = 0$, it follows that

$$\begin{aligned} \left( \exp ( s\mathop {\mathrm {Im}}\nolimits X)\right) ^* \left( \exp (-t\mathop {\mathrm {Re}}\nolimits X)\right) ^* \Omega&= \left( \exp (-t\mathop {\mathrm {Re}}\nolimits X)\right) ^* \left( \exp ( s\mathop {\mathrm {Im}}\nolimits X)\right) ^* \Omega \nonumber \\&= \left( \exp (-t\mathop {\mathrm {Re}}\nolimits X)\right) ^* \Omega . \end{aligned}$$

(4.5)

Moreover, since $\mathcal {L}_{\mathop {\mathrm {Im}}\nolimits X} \omega = 0$, we have

$$\begin{aligned} \left( \exp ( s\mathop {\mathrm {Im}}\nolimits X)\right) ^* \tilde{\omega }_0 = \omega . \end{aligned}$$

(4.6)

Therefore, the uniqueness of the solution for (4.4) implies

$$\begin{aligned} \left( \exp ( s\mathop {\mathrm {Im}}\nolimits X)\right) ^* \tilde{\omega }_t = \tilde{\omega }_t, \end{aligned}$$

(4.7)

and hence, $\mathcal {L}_{\mathop {\mathrm {Im}}\nolimits X} \tilde{\omega }_t = 0$.

Thus

$$\begin{aligned} \omega _t = (\exp (t\mathop {\mathrm {Re}}\nolimits X))^* \tilde{\omega }_t \end{aligned}$$

(4.8)

is the unique short-time solution for (4.2). $\square$

5 A Priori Estimates

In this section, let us assume that X has a zero point. Then, from Proposition 2.3, $\alpha _X \equiv 0$. First, we need the following lemma:

Lemma 5.1

(see [2, 9]) Let $(M, \omega )$ be a compact Kähler manifold. Let $\omega _\varphi = \omega + \sqrt{-1}\partial \bar{\partial }\varphi$ be a Kähler form. Suppose that $\mathcal {L}_X \omega$ and $\mathcal {L}_X \omega _\varphi$ are real (1, 1)-forms. Then $\Vert \theta _X(\omega ) \Vert _{C^0} = \Vert \theta _X(\omega _\varphi ) \Vert _{C^0}$.

Proof

First note that $\theta _X(\omega _\varphi )$ and $\theta _X(\omega )$ are real functions. Suppose $\theta _X (\omega _\varphi )$ and $\theta _X (\omega )$ attain their maximum at p and q, respectively. Since $i_X \omega = \sqrt{-1}\bar{\partial }\theta _X(\omega )$ and $i_X \omega _\varphi = \sqrt{-1}\bar{\partial }\theta _X(\omega _\varphi )$, X vanishes at p and q. Thus, from Proposition 2.1, we can see that

$$\begin{aligned}&\theta _X(\omega _\varphi ) (p) = \theta _X (\omega ) (p) \le \theta _X (\omega ) (q), \end{aligned}$$

(5.1)

$$\begin{aligned}&\theta _X (\omega ) (q) = \theta _X (\omega _\varphi ) (q) \le \theta _X (\omega _\varphi ) (p). \end{aligned}$$

(5.2)

Hence $\max \theta _X(\omega ) = \max \theta _X(\omega _\varphi )$. Similarly, we see $\min \theta _X(\omega ) = \min \theta _X(\omega _\varphi )$. $\square$

5.1 Volume Ratio Estimate

Let $\varphi _t$ be the solution for (4.3). Now we shall prove some estimates for $\varphi _t$. Differentiating (4.3), we obtain

$$\begin{aligned} (\partial _t - \triangle _t - X) \dot{\varphi _t} = 0. \end{aligned}$$

(5.3)

Then the maximum principle implies the following:

Proposition 5.2

There exists a positive constant $C_1$ depending only on f and $\theta _X(\omega )$ such that

$$\begin{aligned} |\dot{\varphi _t}| \le C_1 \end{aligned}$$

(5.4)

for all $t \ge 0$.

Moreover, by (4.3), Lemma 5.1 and Proposition 5.2, we obtain the following estimate:

Proposition 5.3

There exists a positive constant $C_2$ depending only on f and $\theta _X(\omega )$ such that

$$\begin{aligned} \Big |\log \frac{\omega _{\varphi _t}^m}{\omega ^m} \Big | \le C_2. \end{aligned}$$

(5.5)

5.2 $C^2$ Estimate

Next let $Y_t := g^{i \bar{j}} g_{t, i \bar{j}} = m + \triangle _{\omega } \varphi _t$. We shall show an estimate for $Y_t$.

Proposition 5.4

$$\begin{aligned} Y_t \le C_3 \end{aligned}$$

(5.6)

for some positive constant $C_3$ independent of t.

Proof

Let

$$\begin{aligned} \psi _t := \varphi _t - \frac{1}{\mathop {\mathrm {Vol}}\nolimits (M)}\int _M \varphi _t \, \frac{\omega ^m}{m!}, \end{aligned}$$

(5.7)

where $\mathop {\mathrm {Vol}}\nolimits (M) = \int _M \omega ^m/m!$. Since $\omega _{\varphi _t} = \omega _{\psi _t}$, we consider $\psi _t$ instead of $\varphi _t$.

Now we compute in normal coordinates with respect to $\omega$ at $p \in M$, i.e., $g_{i \bar{j}}(p) = \delta _{i j}$ and $\partial _k g_{i \bar{j}} (p) = \partial _{\bar{k}} g_{i \bar{j}} (p) = 0$. Furthermore, we may assume that $g_{t, i \bar{j}} (p) = {\uplambda }_i \delta _{i j}$ and then $Y_t(p) = \sum {\uplambda }_\alpha$. First we show the following inequality:

Lemma 5.5

We have

$$\begin{aligned} (\partial _t - \triangle _t - X) \log Y_t (p) \le \frac{1}{Y_t} \left( - R_{i \bar{i} k \bar{k}} (p) \frac{{\uplambda }_i}{{\uplambda }_k} + \Omega _{i \bar{i}} (p)+ {\uplambda }_i \partial _i X^i (p) \right) , \end{aligned}$$

(5.8)

where $R_{i \bar{j} k \bar{l}}$ is the curvature tensor for $\omega$.

Proof of Lemma 5.5

Using (4.3), we have

$$\begin{aligned} Y_t \partial _t \log Y_t = - R_{t, i \bar{i}} + \Omega _{i \bar{i}} + \partial _i \partial _{\bar{i}} (\theta _X(\omega ) + X(\psi _t)). \end{aligned}$$

(5.9)

A straightforward computation gives

$$\begin{aligned} \triangle _t Y_t (p) = R_{i \bar{i} k \bar{k}} (p) \frac{{\uplambda }_i}{{\uplambda }_k} - R_{t, i \bar{i}} (p) + \frac{1}{{\uplambda }_i {\uplambda }_k} \partial _i g_{t, k \bar{j}}(p) \partial _{\bar{i}} g_{t, j \bar{k}}(p). \end{aligned}$$

(5.10)

Furthermore, we have

$$\begin{aligned} |\partial Y_t|^2_{\omega _t} (p)&= \sum \frac{1}{{\uplambda }_i} \partial _i g_{t, j \bar{j}} \partial _{\bar{i}} g_{t, k \bar{k}} \nonumber \\&\le \sum _{j, l} \left( \sum _i \frac{1}{{\uplambda }_i} |\partial _i g_{t, j \bar{j}}|^2\right) ^{\frac{1}{2}} \left( \sum _k \frac{1}{{\uplambda }_k} |\partial _k g_{t, l \bar{l}}|^2 \right) ^{\frac{1}{2}} \nonumber \\&= \left( \sum _j {\uplambda }_j^{\frac{1}{2}} \left( \sum _i \frac{1}{{\uplambda }_j {\uplambda }_i} |\partial _i g_{t, j \bar{j}}|^2\right) ^{\frac{1}{2}} \right) ^2 \nonumber \\&\le \left( \sum _k {\uplambda }_k \right) \left( \sum _{i, j} \frac{1}{{\uplambda }_j {\uplambda }_i} |\partial _i g_{t, j \bar{j}}|^2 \right) \le Y_t \left( \sum _{i, j, l} \frac{1}{{\uplambda }_j {\uplambda }_i} |\partial _i g_{t, j \bar{l}}|^2 \right) . \end{aligned}$$

(5.11)

Here the first and second inequalities follow from the Cauchy–Schwarz inequality. Combining (5.10) and (5.11), we obtain

$$\begin{aligned} - Y_t \triangle _t \log Y_t (p)&= - \triangle _t Y_t (p) + \frac{1}{Y_t} |\partial Y_t|^2_{\omega _t}(p) \nonumber \\&\le - R_{i \bar{i} k \bar{k}} (p) \frac{{\uplambda }_i}{{\uplambda }_k} + R_{t, i \bar{i}} (p). \end{aligned}$$

(5.12)

We also have

$$\begin{aligned} \partial _i \partial _{\bar{i}} (\theta _X(\omega ) + X(\psi _t))&= \partial _i (g_{t, k \bar{i}} X^k) \nonumber \\&= X^k \partial _k g_{i \bar{i}} + g_{t, k \bar{i}} \partial _i X^k \nonumber \\&= X(Y_t) + {\uplambda }_i \partial _i X^i. \end{aligned}$$

(5.13)

Here we used the Kähler identities (2.1). Combining (5.9), (5.12) and (5.13), we complete the proof of Lemma 5.5. $\square$

From Lemma 5.5 and ${\uplambda }_i \le Y_t$, it follows that

$$\begin{aligned} (\partial _t - \triangle _t - X) \log Y_t \le C \sum _i \frac{1}{{\uplambda }_i} + C', \end{aligned}$$

(5.14)

where a positive constant C depends only on $\Omega$ and a lower bound of the bisectional curvature for $\omega$, and $C' = \Vert \nabla ^\omega X \Vert _{C^0(M, \omega )}$. Moreover, from (4.3), Proposition 5.2 and Proposition 5.3, it follows that

$$\begin{aligned} (\partial _t - \triangle _t - X) \psi _t&= \log \frac{\omega _t^m}{\omega ^m} - f + \theta _X(\omega ) - \triangle _t \psi _t + \frac{1}{\mathop {\mathrm {Vol}}\nolimits (M)}\int _M \dot{\varphi _t} \, \frac{\omega ^m}{m!} \nonumber \\&\ge \sum \frac{1}{{\uplambda }_i} - C'', \end{aligned}$$

(5.15)

where $C'' > 0$ depends on f, $\theta _X(\omega )$ and m. Let $w_t := \log Y_t - (C + 1) \psi _t$. Then, by (5.14) and (5.15), we obtain

$$\begin{aligned} (\partial _t - \triangle _t - X) w_t&\le C \sum _i \frac{1}{{\uplambda }_i} + C' - (C + 1) \left( \sum \frac{1}{{\uplambda }_i} - C'' \right) \nonumber \\&= - \sum _i \frac{1}{{\uplambda }_i} + C_4, \end{aligned}$$

(5.16)

where $C_4 = C' + C'' (C + 1)$.

Now we compute $\sum _i {\uplambda }_i^{- 1}$. We have

$$\begin{aligned} Y_t e^{- \log \frac{\omega _t^m}{\omega ^m}}&= \sum _i {\uplambda }_i \prod _k \frac{1}{{\uplambda }_k} \nonumber \\&= \sum _i \prod _{k \ne i} \frac{1}{{\uplambda }_k} \nonumber \\&\le \left( \sum \frac{1}{{\uplambda }_i} \right) ^{m - 1}. \end{aligned}$$

(5.17)

By using Proposition 5.3, we obtain

$$\begin{aligned} Y_t \le e^{C_2} \Big ( \sum \frac{1}{{\uplambda }_i} \Big )^{m - 1}. \end{aligned}$$

(5.18)

We now need the following lemma (see [6]).

Lemma 5.6

Let f be a positive function on $(M, \omega )$. Suppose $\varphi$ satisfies

$$\begin{aligned} \frac{\omega _\varphi ^m}{\omega ^m} = f. \end{aligned}$$

(5.19)

Then we have

$$\begin{aligned} \mathop {\mathrm {osc}}_M \varphi := \sup _M \varphi - \inf _M \varphi \le C \end{aligned}$$

(5.20)

for some positive constant depending only on $(M, \omega )$ and $\Vert f \Vert _{C^0}$.

From Proposition 5.3, Lemma 5.6 and (5.18), we see that

$$\begin{aligned} e^{w_t} = e^{-(C + 1) \psi _t} Y_t < C_5 \left( \sum \frac{1}{{\uplambda }_i} \right) ^{m - 1} \end{aligned}$$

(5.21)

for some positive constant $C_5$ independent of t. Thus, from (5.16) and (5.21), it follows that

$$\begin{aligned} (\partial _t - \triangle _t - X) w_t < - \frac{1}{C_5} e^{\frac{w_t}{m - 1}} + C_4. \end{aligned}$$

(5.22)

Note that we can choose positive constants $C_4$ and $C_5$ such that

$$\begin{aligned} w_0 \equiv \log m < (m - 1) \log C_4 C_5. \end{aligned}$$

(5.23)

Now we prove

$$\begin{aligned} w_t < (m - 1) \log C_4 C_5 \end{aligned}$$

(5.24)

for all $t > 0$ by contradiction. We assume that

$$\begin{aligned} \max w_{t_0} = w_{t_0} (p_0) = (m - 1) \log C_4 C_5, \end{aligned}$$

(5.25)

$$\begin{aligned} w_t < (m - 1) \log C_4 C_5 \qquad (t < t_0 ) \end{aligned}$$

(5.26)

for some $t_0 > 0$. From (5.22), (5.25) and (5.26), we have

$$\begin{aligned} 0 \le \frac{d}{dt} w_{t_0}(p_0) < 0, \end{aligned}$$

(5.27)

a contradiction. Thus we obtain

$$\begin{aligned} w_t < (m - 1) \log C_4 C_5 \end{aligned}$$

(5.28)

and hence we complete the proof. $\square$

Propositions 5.3 and 5.4 immediately imply the following proposition:

Proposition 5.7

$$\begin{aligned} C_6^{-1} \omega \le \omega _t \le C_6 \omega \end{aligned}$$

(5.29)

for some positive constant $C_6$ independent of t.

Proof

We have ${\uplambda }_i < Y_t \le C_3$ from Proposition 5.4. On the other hand, from Propositions 5.3 and 5.4, it follows immediately that

$$\begin{aligned} \frac{1}{{\uplambda }_i} < \sum _i \frac{1}{{\uplambda }_i} \le \prod _k \frac{1}{{\uplambda }_k} \left( \sum {\uplambda }_i \right) ^{m - 1} \le C \end{aligned}$$

(5.30)

for some positive constant C independent of t. $\square$

5.3 $C^3$ Estimate

In this subsection, we shall show the following proposition:

Proposition 5.8

There exists a positive constant $C_7$ independent of t such that

$$\begin{aligned} \big |\nabla ^0 g_t \big |^2_{\omega _t} \le C_7, \end{aligned}$$

(5.31)

where $\nabla ^0$ is the Levi-Civita connection for $\omega _0$.

Our proof is a slight modification of the argument in [5] (see also [7]). Let $\sigma _t := \exp (-t\mathop {\mathrm {Re}}\nolimits X)$. We prove Proposition 5.8 by computing pullbacks under $\sigma _t$. More precisely, we put

$$\begin{aligned} \tilde{\omega }_t := \sigma _t^* \omega _t, \end{aligned}$$

(5.32)

$$\begin{aligned} \hat{\omega }_t := \sigma _t^* \omega _0, \end{aligned}$$

(5.33)

and let $\tilde{\nabla }$, $\hat{\nabla }$ be the Levi-Civita connections for $\tilde{\omega }_t$, $\hat{\omega }_t$, respectively. Then

$$\begin{aligned} S := \big |\hat{\nabla } \tilde{g}_t \big |^2_{\tilde{\omega }_t} = \sigma _t^*\big | \nabla ^0 g_t \big |^2_{\omega _t}. \end{aligned}$$

(5.34)

Therefore we show the uniform boundedness of S instead of $| \nabla ^0 g_t |^2_{\omega _t}$. We define a tensor $\Psi ^k_{i, p}$ by

$$\begin{aligned} \Psi := \tilde{\nabla } - \hat{\nabla }. \end{aligned}$$

(5.35)

Then we can express $\Psi$ as

$$\begin{aligned} \Psi ^k_{i p} = \tilde{g}^{k \bar{l}} \partial _i \tilde{g}_{p \bar{l}} - \hat{g}^{k \bar{l}} \partial _i \hat{g}_{p \bar{l}} \end{aligned}$$

(5.36)

and S as

$$\begin{aligned} S = | \Psi |^2_{\tilde{\omega }_t} = \tilde{g}^{i \bar{j}} \tilde{g}^{k \bar{l}} \tilde{g}_{p \bar{q}} \Psi ^k_{i p} \overline{\Psi ^l_{j q}}. \end{aligned}$$

(5.37)

Now let us prove Proposition 5.8.

Proof of Proposition 5.8

We compute in normal coordinates with respect to $\tilde{\omega }_t$ at $p \in M$. First, a straightforward computation gives

$$\begin{aligned} \tilde{\triangle } S&= \tilde{R}_{j \bar{i}} \Psi ^k_{i p} \overline{\Psi ^k_{j p}} + \tilde{R}_{q \bar{p}} \Psi ^k_{i p} \overline{\Psi ^k_{i q}} - \tilde{R}_{k \bar{l}} \Psi ^k_{i p} \overline{\Psi ^l_{i p}} \nonumber \\&\quad + 2 \mathop {\mathrm {Re}}\nolimits \left( \tilde{\nabla }_\alpha \tilde{\nabla }_{\bar{\alpha }} \Psi ^k_{i p} \overline{\Psi ^k_{i p}}\right) + \big |\overline{\tilde{\nabla }} \Psi \big |^2_{\tilde{\omega }_t} + \big |\tilde{\nabla } \Psi \big |^2_{\tilde{\omega }_t}, \end{aligned}$$

(5.38)

where $\tilde{\triangle }$ is the Laplacian with respect to $\tilde{\omega }_t$ and $\tilde{R}_{i \bar{j}}$ is the Ricci tensor of $\tilde{\omega }_t$.

Recall that $\tilde{\omega }_t$ satisfies (4.4). Moreover, we have

$$\begin{aligned} \frac{d}{dt} \hat{\omega }_t = \mathcal {L}_X \hat{\omega }_t = \sqrt{-1}\partial \bar{\partial }\sigma _t^* \theta _X(\omega _0) =: \sqrt{-1}\partial \bar{\partial }\hat{\theta }_t. \end{aligned}$$

(5.39)

By using (4.4), (5.36) and (5.39), we obtain

$$\begin{aligned} \partial _t S =&\left( \tilde{R}_{j \bar{i}} - \Omega _{t, j \bar{i}}\right) \Psi ^k_{i p} \overline{\Psi ^k_{j p}} + \left( \tilde{R}_{q \bar{p}} - \Omega _{t, q \bar{p}}\right) \Psi ^k_{i p} \overline{\Psi ^k_{i q}} - \left( \tilde{R}_{k \bar{l}} - \Omega _{t, k \bar{l}}\right) \Psi ^k_{i p} \overline{\Psi ^l_{i p}} \nonumber \\&+ \partial _t {\Psi ^k_{i p} \overline{\Psi ^k_{i p}}} + \Psi ^k_{i p} \overline{\partial _t{\Psi ^k_{i p}}} \nonumber \\ =&\left( \tilde{R}_{j \bar{i}} - \Omega _{t, j \bar{i}}\right) \Psi ^k_{i p} \overline{\Psi ^k_{j p}} + \left( \tilde{R}_{q \bar{p}} - \Omega _{t, q \bar{p}}\right) \Psi ^k_{i p} \overline{\Psi ^k_{i q}} - \left( \tilde{R}_{k \bar{l}} - \Omega _{t, k \bar{l}}\right) \Psi ^k_{i p} \overline{\Psi ^l_{i p}} \nonumber \\&+ 2 \mathop {\mathrm {Re}}\nolimits \left( \left( - \tilde{\nabla }_i R_{p \bar{k}} + \tilde{\nabla }_i \Omega _{t, p \bar{k}} + \hat{g}^{k \bar{\delta }} \hat{g}^{\gamma \bar{l}} \tilde{\nabla }_i \hat{g}_{p \bar{l}} \partial _p \partial _{\bar{q}} \hat{\theta }_t + \sqrt{-1}\hat{g}^{k \bar{l}} \tilde{\nabla }_i (\mathcal {L}_X \hat{\omega }_t)_{p \bar{l}}\right) \overline{\Psi ^k_{i p}} \right) . \end{aligned}$$

(5.40)

From (5.38) and (5.40), we obtain

$$\begin{aligned}&\left( \partial _t - \tilde{\triangle }\right) S \le CS - 2 \mathop {\mathrm {Re}}\nolimits \left( \tilde{\nabla }_\alpha \tilde{\nabla }_{\bar{\alpha }} \Psi ^k_{i p} \overline{\Psi ^k_{i p}}\right) \nonumber \\&\quad + 2 \mathop {\mathrm {Re}}\nolimits \left( \left( - \tilde{\nabla }_i R_{p \bar{k}} + \tilde{\nabla }_i \Omega _{t, p \bar{k}} + \hat{g}^{k \bar{\delta }} \hat{g}^{\gamma \bar{l}} \tilde{\nabla }_i \hat{g}_{p \bar{l}} \partial _p \partial _{\bar{q}} \hat{\theta }_t + \sqrt{-1}\hat{g}^{k \bar{l}} \tilde{\nabla }_i (\mathcal {L}_X \hat{\omega }_t)_{p \bar{l}}\right) \overline{\Psi ^k_{i p}} \right) \end{aligned}$$

(5.41)

for some positive constant C independent of t.

From (5.36), we have

$$\begin{aligned} \partial _{\bar{\alpha }} \Psi ^k_{i p}&= - \tilde{R}^k_{j i \bar{\alpha }} + \hat{R}^k_{j i \bar{\alpha }}, \end{aligned}$$

(5.42)

$$\begin{aligned} \tilde{\nabla }_{\alpha } \tilde{\nabla }_{\bar{\alpha }} \Psi ^k_{i p}&= - \tilde{\nabla }_{\alpha } \tilde{R}^k_{j i \bar{\alpha }} + \tilde{\nabla }_{\alpha } \hat{R}^k_{j i \bar{\alpha }} \nonumber \\&= - \tilde{\nabla }_i \tilde{R}_{p \bar{k}} + \tilde{\nabla }_{\alpha } \hat{R}^k_{j i \bar{\alpha }}. \end{aligned}$$

(5.43)

Here we used the Bianchi identities $\tilde{\nabla }_{\alpha } \tilde{R}^k_{j i \bar{\beta }} = \tilde{\nabla }_i \tilde{R}^k_{j \alpha \bar{\beta }}$. Combining Proposition 5.7, (5.41) and (5.42), we obtain

$$\begin{aligned} (\partial _t - \tilde{\triangle } ) S&\le CS + 2 \mathop {\mathrm {Re}}\nolimits \left( \left( - \tilde{\nabla }_{\alpha } \hat{R}^k_{j i \bar{\alpha }} + \tilde{\nabla }_i \Omega _{t, p \bar{k}} \right. \right. \nonumber \\&\quad \left. \left. +\, \hat{g}^{k \bar{\delta }} \hat{g}^{\gamma \bar{l}} \tilde{\nabla }_i \hat{g}_{p \bar{l}} \partial _p \partial _{\bar{q}} \hat{\theta }_t + \sqrt{-1}\hat{g}^{k \bar{l}} \tilde{\nabla }_i (\mathcal {L}_X \hat{\omega }_t)_{p \bar{l}}\right) \overline{\Psi ^k_{i p}}\right) \nonumber \\&\le C'\left( S + \sqrt{S}\right) \le 2C' (S + 1) \end{aligned}$$

(5.44)

for some positive constant $C'$ independent of t.

Furthermore, put $\tilde{Y}_t := \hat{g}^{i \bar{j}} \tilde{g}_{i \bar{j}} = \sigma ^*_t Y_t$. Then, from Proposition 5.7, (5.9), (5.10) and (5.13), we have

$$\begin{aligned} \left( \partial _t - \tilde{\triangle }\right) \tilde{Y}_t \le C'' - \frac{1}{C''}S \end{aligned}$$

(5.45)

for some positive constants $C''$ independent of t.

Let $Q := S +C''(2C' + 1) \tilde{Y}_t$. Then, by (5.44) and (5.45), we obtain

$$\begin{aligned} \left( \partial _t - \tilde{\triangle }\right) Q \le - S + C_7 \end{aligned}$$

(5.46)

for some positive constant $C_7$ independent of t. Note that we can choose the positive constant $C_7$ such that

$$\begin{aligned} m C''(2 C' + 1) < C_7. \end{aligned}$$

(5.47)

Then the same argument in Sect. 5.2 implies

$$\begin{aligned} S \le C_7, \end{aligned}$$

(5.48)

and hence, we complete the proof. $\square$

Combining the above estimates and standard Schauder theory, we conclude that a solution for (4.3) exists for a long time.

Theorem 5.9

A solution $\varphi _t$ for (4.3) exists for all time $t \in [ 0, \infty )$.

Proof

Let T be the maximal time. Assume $T < \infty$. From the estimates which we show in this section and standard Schauder theory (see [3]), there exists a positive constant C independent of $t \in [0, T)$ such that $\Vert \varphi _t \Vert _{C^{2, \varepsilon }} \le C$. Now let $(z^1, \dots , z^m)$ be local coordinates of M and $z^i = x^i + \sqrt{-1}x^{m + i}$. Differentiating (4.3) with respect to $x^l$, we obtain

$$\begin{aligned} (\partial _t - \triangle _t - v ) \frac{\partial \varphi _t}{\partial x^l} = g_t^{i \bar{j}} \frac{\partial g_{i \bar{j}}}{\partial x^l} - g^{i \bar{j}} \frac{\partial g_{i \bar{j}}}{\partial x^l} - \frac{\partial f}{\partial x^l} + \frac{\partial \theta (\omega ) }{\partial x^l} + \frac{\partial v^i}{\partial x^l} \frac{\partial \varphi _t}{\partial x^i}, \end{aligned}$$

(5.49)

where $v = v^i \partial _{x^i} = \mathop {\mathrm {Re}}\nolimits X$. We have a uniform $C^{0, \varepsilon }$ estimate for the right-hand side. Therefore, standard Schauder theory implies that for arbitrary k, there exists a positive constant $C_k$ independent of $t \in [0, T)$ such that $\Vert \varphi _t \Vert _{C^k} \le C_k$. Then there exists a smooth function $\varphi _T$ and a time sequence $\{ t_n \}$ which converges to T such that $\varphi _{t_n} \rightarrow \varphi _T$ in $C^k$. This is a contradiction, and hence $\varphi _t$ exists for all time $t \in [0, \infty )$. $\square$

6 Convergence of the Flow

In this section, we show the convergence of the flow (4.2). Let $\psi _t$ be the function defined as in (5.7). First note that from the estimates in the previous section, we have uniform $C^k$ estimates for $\psi _t$ ($k = 0, 1, \ldots$). Hence there is a sequence $\{ t_n \}$ such that

$$\begin{aligned} \psi _{t_n} \rightarrow \psi _\infty \end{aligned}$$

(6.1)

in $C^\infty$ for some smooth function $\psi _\infty$ on M.

Now, we prove the following lemma in order to show the convergence of the flow:

Lemma 6.1

Let M be a compact Riemannian manifold and $g_t$ be Riemannian metrics such that for any nonnegative integers k, l $| \partial _t^k \nabla ^l g_t |$ is uniformly bounded. Here $\nabla$ is the Levi-Civita connection for $g_0$ and $| \cdot |$ is the norm with respect to $g_0$. For a smooth vector field v on M, we consider the following equation:

$$\begin{aligned} (\partial _t - \triangle _t - v) f_t = 0. \end{aligned}$$

(6.2)

Then there exists a positive constant $0 < \gamma < 1$ independent of t such that

$$\begin{aligned} \mathop {\mathrm {osc}}f_{t_2} \le \gamma \mathop {\mathrm {osc}}f_{t_1} \qquad (t_1 + 1 \le t_2) \end{aligned}$$

(6.3)

for arbitrary solution $f_t$ for (6.2).

Proof

First note that the maximum principle implies that, for arbitrary solution $f_t$ for (6.2), $\sup _M f_t$ is monotonically decreasing and $\inf _M f_t$ is monotonically increasing, and hence, $\mathop {\mathrm {osc}}f_t$ is monotonically decreasing. Therefore, we have only to consider the case $t_2 = t_1 + 1$. We prove this lemma by contradiction. We assume that the statement is not true. Then for arbitrary positive integer n, there exists a solution $f^{(n)}_t$ for (6.2) and $t^{(n)}_1$ such that

$$\begin{aligned} \mathop {\mathrm {osc}}_M f^{(n)}_{t^{(n)}_1 + 1} > \left( 1 - \frac{1}{n}\right) \mathop {\mathrm {osc}}_M f^{(n)}_{t^{(n)}_1}. \end{aligned}$$

(6.4)

We may assume that

$$\begin{aligned} \sup _M f^{(n)}_{t^{(n)}_1} = 1, \qquad \inf _M f^{(n)}_{t^{(n)}_1} = 0. \end{aligned}$$

(6.5)

(6.4) implies

$$\begin{aligned} \inf _M f^{(n)}_{t^{(n)}_1 + 1} < \frac{1}{n} \quad \mathrm {or} \quad \sup _M f^{(n)}_{t^{(n)}_1 + 1} > 1 - \frac{1}{n}. \end{aligned}$$

(6.6)

Replacing $f^{(n)}$ by $1 - f^{(n)}$, if necessary, we may assume that

$$\begin{aligned} \inf _M f^{(n)}_{t^{(n)}_1 + 1} < \frac{1}{n}. \end{aligned}$$

(6.7)

Put $h^{(n)}_s := f^{(n)}_{t^{(n)}_1 + \frac{1}{2} + s}$. Then we have

$$\begin{aligned} 0 \le h^{(n)}_s \le 1, \quad \mathop {\mathrm {osc}}_M h^{(n)}_{\frac{1}{2}} > 1 - \frac{1}{n}, \quad \inf _M h^{(n)}_{\frac{1}{2}} < \frac{1}{n}. \end{aligned}$$

(6.8)

Moreover, $h^{(n)}_s$ satisfies

$$\begin{aligned} \left( \partial _s - \triangle _{t^{(n)}_1 + \frac{1}{2} + s} - v\right) h^{(n)}_s = 0. \end{aligned}$$

(6.9)

Since we have uniform $C^k$ estimates ($k = 0, 1, \dots$) for $g_t$, sequences $\{ g^{(n)}_s := g_{t^{(n)}_1 + \frac{1}{2} + s} \}$ and $\{ h^{(n)}_s \}$ have convergent subsequences with limits $\hat{g}_s$ and $\hat{h}_s$, respectively. Note that $\hat{g}_s$ are smooth Riemannian metrics and $\hat{h}_s$ are functions that are smooth as functions on M and of class $C^1$ as functions of $s \in [ 0, \infty )$ such that

$$\begin{aligned}&(\partial _s - \triangle _{\hat{g}_s} - v) \hat{h}_s = 0, \end{aligned}$$

(6.10)

$$\begin{aligned}&0 \le \hat{h}_s \le 1, \quad \mathop {\mathrm {osc}}_M \hat{h}_{\frac{1}{2}} \ge 1, \quad \inf _M \hat{h}_{\frac{1}{2}} = 0. \end{aligned}$$

(6.11)

By the maximum principle, we can see that $\hat{h}_s \equiv 0$. This is a contradiction. $\square$

Since $\dot{\varphi _t}$ satisfies (5.3), Lemma 6.1 implies

$$\begin{aligned} \mathop {\mathrm {osc}}\dot{\varphi }_{t_2} \le e^{- a} \mathop {\mathrm {osc}}\dot{\varphi }_{t_1} \qquad (t_1 + 1 \le t_2) \end{aligned}$$

(6.12)

for some positive constant a independent of t. From the definition of $\psi _t$, we see that $\mathop {\mathrm {osc}}\dot{\varphi }_t = \mathop {\mathrm {osc}}\dot{\psi }_t$. Thus we have

$$\begin{aligned} \sup \dot{\psi }_t \le \mathop {\mathrm {osc}}\dot{\psi }_t \le C e^{- a t} \end{aligned}$$

(6.13)

for some positive constant C independent of t. Hence, we obtain the following theorem:

Theorem 6.2

$\psi _t$ converges in $C^\infty$ to $\psi _\infty$ and $\dot{\varphi _t}$ converges in $C^\infty$ to a constant.

Next, we prove the uniqueness of the solution for (3.1). Let $\omega ^0$, $\omega ^1$ be solutions for (3.1) and $\omega ^s := (1 - s) \omega ^0 + s \omega ^1$. Then, we have a solution $\omega ^s_t$ for

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d}{dt} \omega ^s_t = - \mathop {\mathrm {Ric}}\nolimits (\omega ^s_t) + \Omega + \mathcal {L}_X \omega ^s_t, \\ \omega ^s_0 = \omega ^s. \end{array}\right. } \end{aligned}$$

(6.14)

Note that $\omega ^0_t \equiv \omega ^0$ and $\omega ^1_t \equiv \omega ^1$. From Theorem 6.2, $\omega ^s_t$ converges to some Kähler metric $\omega ^s_\infty = \omega + \sqrt{-1}\partial \bar{\partial }\psi ^s_\infty$ and $\omega ^s_\infty$ satisfies

$$\begin{aligned} 0 = - \mathop {\mathrm {Ric}}\nolimits (\omega ^s_\infty ) + \Omega = \mathcal {L}_X \omega ^s_\infty . \end{aligned}$$

(6.15)

Differentiating (6.15) with respect to s, we obtain

$$\begin{aligned} \sqrt{-1}\partial \bar{\partial }\left( (\triangle _{\omega ^s_\infty } + X) \frac{d}{ds} \psi ^s_\infty \right) = 0. \end{aligned}$$

(6.16)

Therefore, the maximum principle implies $\omega ^s_\infty = \omega ^0 = \omega ^1$.

Consequently, we complete the proof of the sufficiency part of Theorem 1.5.

7 More General Cases

In this section, we consider the case of a nowhere vanishing holomorphic vector field X. First, note that in Sect. 5 we use the assumption that X has a zero point only in the proof of Lemma 5.1. Hence, if we show Lemma 5.1 under the condition that X has a zero point, then, by using the argument in Sect. 5, we can show the existence of the solution for (1.3).

The goal of this section is to show the following lemma:

Lemma 7.1

Let $(M, \omega )$ be a compact Kähler manifold and $\omega _\varphi = \omega + \sqrt{-1}\partial \bar{\partial }\varphi$ be a Kähler form. Let X be a nowhere vanishing holomorphic vector field. Suppose that $\mathcal {L}_X \omega$ and $\mathcal {L}_X \omega _\varphi$ are real (1, 1)-forms. Assume both $\{ \exp ( t \mathop {\mathrm {Re}}\nolimits X) \}_{t \in \mathbb {R}}$ and $\{ \exp ( t \mathop {\mathrm {Im}}\nolimits X) \}_{t \in \mathbb {R}}$ are periodic. Then there exists a constant C independent of $\varphi$ such that

$$\begin{aligned} |X(\varphi )| \le C. \end{aligned}$$

(7.1)

7.1 The Case $m = 1$

Let us prove Lemma 7.1 when $m = 1$. Hence, we consider the case where M is a 1-complex torus. When M is a 1-complex torus, we can remove the assumption that $\{ \exp ( t \mathop {\mathrm {Re}}\nolimits X) \}_{t \in \mathbb {R}}$ and $\{ \exp ( t \mathop {\mathrm {Im}}\nolimits X) \}_{t \in \mathbb {R}}$ are periodic. Now we shall show the following:

Lemma 7.2

Let M be a 1-complex torus and $\omega _\varphi = \omega + \sqrt{-1}\partial \bar{\partial }\varphi$ be a Kähler form. Let X be a holomorphic vector field. Suppose that $\mathcal {L}_X \omega$ and $\mathcal {L}_X \omega _\varphi$ are real (1, 1)-forms. Then

$$\begin{aligned} |X(\varphi )| \le C \end{aligned}$$

(7.2)

for some constant C depending only on M, X and $\omega$.

Proof

We may assume that

$$\begin{aligned} M = \mathbb {C} / (\xi _1 \mathbb {Z} + \xi _2 \mathbb {Z}) \qquad (\xi _1, \xi _2 \in \mathbb {C}, \, \mathop {\mathrm {Re}}\nolimits \xi _1 \ne 0) \end{aligned}$$

(7.3)

and $X = \partial / \partial z$. Note that for any Kähler form $\omega$ on M,

$$\begin{aligned} k \sqrt{-1}dz \wedge d \bar{z} \in [ \omega ], \end{aligned}$$

(7.4)

where $k = \int _M [\omega ] / \int _M \sqrt{-1}dz \wedge d \bar{z}$. Therefore, we may assume $\omega = k \sqrt{-1}dz \wedge d \bar{z}$.

Put $z = x + \sqrt{-1}y$. Suppose $\omega _\varphi = \omega + \sqrt{-1}\partial \bar{\partial }\varphi$ is a Kähler form and $X(\varphi )$ is a real function. Using the natural projection $\pi : \mathbb {C} \longrightarrow \mathbb {C} / (\xi _1 \mathbb {Z} + \xi _2 \mathbb {Z})$, we identify $\varphi$ and $\omega$ with their pullbacks. Since $\omega _\varphi$ is a Kähler form, $\varphi$ satisfies

$$\begin{aligned} k + \frac{1}{4} \left( \frac{\partial ^2}{\partial x^2} + \frac{\partial ^2}{\partial y^2} \right) \varphi > 0. \end{aligned}$$

(7.5)

From $0 = 2\mathop {\mathrm {Im}}\nolimits X (\varphi ) = \partial _y \varphi$, we see that

$$\begin{aligned} \varphi (x, y)= \varphi (x). \end{aligned}$$

(7.6)

Hence, from (7.5), we have

$$\begin{aligned} k + \frac{1}{4} \frac{\partial ^2}{\partial x^2} \varphi > 0. \end{aligned}$$

(7.7)

Moreover,

$$\begin{aligned} \varphi (x + \mathop {\mathrm {Re}}\nolimits \xi _1 ) = \varphi (x + \mathop {\mathrm {Re}}\nolimits \xi _1, \mathop {\mathrm {Im}}\nolimits \xi _1) = \varphi (x), \end{aligned}$$

(7.8)

i.e., $\varphi$ is $|\mathop {\mathrm {Re}}\nolimits \xi _1|$-periodic. Note that $\partial _x \varphi$ and $\partial _x^2 \varphi$ are also $|\mathop {\mathrm {Re}}\nolimits \xi _1|$-periodic.

Let $x_0 \in \mathbb {R}$ be a minimizer of $\varphi$. For arbitrary $\zeta \in [0, |\mathop {\mathrm {Re}}\nolimits \xi _1| ]$, we have

$$\begin{aligned} \frac{\partial \varphi }{\partial x}(\zeta + x_0)&= \int ^{\zeta + x_0}_{x_0} \frac{\partial ^2 \varphi }{\partial x^2} \, dx \nonumber \\&> -4 k \int ^{\zeta + x_0}_{x_0} dx > -4 k |\mathop {\mathrm {Re}}\nolimits \xi _1|. \end{aligned}$$

(7.9)

Hence we obtain $2 \mathop {\mathrm {Re}}\nolimits X (\varphi ) > -4 k |\mathop {\mathrm {Re}}\nolimits \xi _1|$.

On the other hand, let $x_1 \in \mathbb {R}$ be a maximizer of $\partial _x \varphi$. We may assume $\partial _x \varphi (x_1) > 0$. Then there exists $\zeta ' \in (0, |\mathop {\mathrm {Re}}\nolimits \xi _1| )$ such that $\zeta ' + x_1$ is a minimizer of $\varphi$. Thus we have

$$\begin{aligned} - \frac{\partial \varphi }{\partial x}(x_1)&= \int ^{\zeta ' + x_1}_{x_1} \frac{\partial ^2 \varphi }{\partial x^2} \, dx \nonumber \\&> -4 k \int ^{\zeta ' + x_1}_{x_1} du > -4 k |\mathop {\mathrm {Re}}\nolimits \xi _1|. \end{aligned}$$

(7.10)

Hence we conclude

$$\begin{aligned} |X(\varphi )| = |\mathop {\mathrm {Re}}\nolimits X (\varphi )| \le 2 k |\mathop {\mathrm {Re}}\nolimits \xi _1|. \end{aligned}$$

(7.11)

$\square$

7.2 The Case $m \ge 2$

Now we prove Lemma 7.1 when $m \ge 2$.

Proof of Lemma 7.1

The proof is similar to the proof of Corollary 5.3 in [9]. Since $[ \mathop {\mathrm {Re}}\nolimits X, \mathop {\mathrm {Im}}\nolimits X ] = 0$, $\{\mathop {\mathrm {Re}}\nolimits X, \mathop {\mathrm {Im}}\nolimits X \}$ defines a holomorphic foliation $\mathcal {F}_X$ on M. From the assumption that both $\{ \exp ( t \mathop {\mathrm {Re}}\nolimits X) \}_{t \in \mathbb {R}}$ and $\{ \exp ( t \mathop {\mathrm {Im}}\nolimits X) \}_{t \in \mathbb {R}}$ are periodic, we see that every leaf of $\mathcal {F}_X$ is a compact Riemann surface and the leaf space $M/\mathcal {F}_X$ is compact. The condition $X_p \ne 0$ for arbitrary $p \in M$ implies that every leaf of $\mathcal {F}_X$ is a 1-complex torus. Therefore, applying Lemma 7.2 to each leaf of $\mathcal {F}_X$, we obtain the desired uniform estimate. $\square$