Mathematische Annalen

, Volume 356, Issue 4, pp 1425–1454

Stability of Kähler-Ricci flow on a Fano manifold

Authors

    • Department of MathematicsPeking University
Article

DOI: 10.1007/s00208-012-0889-7

Cite this article as:
Zhu, X. Math. Ann. (2013) 356: 1425. doi:10.1007/s00208-012-0889-7

Abstract

Let \((M,J)\) be a Fano manifold which admits a Kähler-Einstein metric \(g_{KE}\) (or a Kähler-Ricci soliton \(g_{KS}\)). Then we prove that Kähler-Ricci flow on \((M,J)\) converges to \(g_{KE}\) (or \(g_{KS}\)) in \(C^\infty \) in the sense of Kähler potentials modulo holomorphisms transformation as long as an initial Kähler metric of flow is very closed to \(g_{KE}\) (or \(g_{KS}\)). The result improves Main Theorem in [14] in the sense of stability of Kähler-Ricci flow.

Mathematics Subject Classification (1991)

Primary 53C25; Secondary 53C5558E11

1 Introduction

Ricci flow was first introduced by Hamilton in [5] in 1982. If an underlying manifold \(M\) is Kähler with positive first Chern class \(c_1(M)>0\) (namely, \(M\) is Fano), it is more natural to study the following Kähler-Ricci flow (normalized),
$$\begin{aligned} \frac{\partial g(t,\cdot )}{\partial t} =-\text{ Ric}(g(t,\cdot ))+g(t,\cdot ),\quad g(0,\cdot )=g, \end{aligned}$$
(1.1)
where \(g\) is an initial Kähler metric with its Kähler form \(\omega _g \in 2\pi c_1(M)>0.\) It can be shown that (0.1) preserves the Kähler class and it has a global solution \(g_t=g(t,\cdot )\) for any \(t>0\) [2]. So, the interest and difficulty of (0.1) are to study the limiting behavior of evolved Kähler metrics \(g_t\) as \(t\) tends to \(\infty \) (cf. [3, 7, 14], etc.).

On the other hand, by applying Perelman’s \(W\)-function introduced in [8] to Eq. (1.1), one can show that the limit of \(g_t\) should be a Kähler-Ricci soliton \(g_{KS}\) if \(g_t\) is convergent in \(C^\infty \)- topology of Cheeger-Gromov (cf. [9]). Thus it is a natural problem to study the convergence of \(g_t\) when an initial Kähler metric \(g\) in (1.1) is very closed \(g_{KS}\). In this paper, we will discuss this problem, namely, the stability of Kähler-Ricci flow. We prove

Theorem 1.1

(Main Theorem) Let \((M,J)\) be a Fano manifold which admits a Kähler-Ricci soliton \(g_{KS}\) with its Kähler form \(\omega _{KS}\in 2\pi c_1(M)\). Let \(\psi \) be a Kähler potential of an initial metric \(g\) of (1.1) with \(\omega _g=\omega _{KS}+\sqrt{-1} \partial \bar{\partial }\psi \). Then there exists a small \(\epsilon \) depending on \((M, J,g_{KS})\) such that if \(\Vert \psi \Vert _{C^{2,\alpha }} < \epsilon \)1 there exist a family of holomorphisms \(\sigma _t\) of \(M\) such that Kähler potentials \(((\varphi _t)_{\sigma _t}- \frac{1}{\int _M\omega _{KS}^n}\int _M(\varphi _t)_{\sigma _t} \omega _{KS}^n )\) are \(C^k\)-norm uniformly bounded, where \((\varphi _t)_{\sigma _t} \) are modified Kähler potentials of \(g_t\) defined by
$$\begin{aligned} (\sigma _t)^\star \omega _{g_t}=\omega _{KS} +\sqrt{-1}\partial \overline{\partial } (\varphi _t)_{\sigma _t}. \end{aligned}$$
As a consequence, \(g_t\) converges to \(g_{KS}\) in \(C^\infty \) in the sense of Kähler potentials modulo holomorphisms transformation.

The method to prove Theorem 1.1 is to reduce equation (1.1) to certain parabolic equation of complex Monge-Ampère type for Kähler potentials \(\varphi _t=\varphi (t,\cdot )\) associated to \(g_t\) as in [3, 6, 14], etc. Then the main step in the proof of theorem is to obtain a decay estimate for both \(\dot{\varphi }=\frac{\partial \varphi }{\partial t}\) and \(\varphi \). We will first deal with the case that \((M,J)\) admits a Kähler Einstein metric in Sect. 1 and obtain an exponential decay estimate for both \(\dot{\varphi }\) and \(\varphi \) as long as \(\Vert \psi \Vert _{C^{2,\alpha }}\) is small enough (cf. Theorem 2.1). In general case that \((M,J)\) admits a Kähler-Ricci soliton \((g_{KS}, X_0)\), we need more techniques to prove Theorem 1.1 in Sect. 2 (cf. Theorem 3.1). Basically, we shall use the modified Futaki-invariant and the Gauge Transformation group induced by the reductive subgroup \(\text{ Aut}_r(M)\) of holomorphisms transformation group \(\text{ Aut}(M)\) of \(M\) to control the modified Kähler potentials \(( (\varphi _t)_{\sigma _t}-\underline{(\varphi _t)_{\sigma _t}} )\) along the Kähler-Ricci flow, where \(\underline{(\varphi _t)_{\sigma _t}} = \frac{1}{\int _M\omega _{KS}^n}\int _M(\varphi _t)_{\sigma _t} \omega _{KS}^n\). In particular, in addition if \(\psi \) is \(K_{X_0}\)-invariant, we can further obtain an exponential decay estimate for both \(\dot{\varphi }\) and \(\varphi \). Here \(K_{X_0}\) denotes an one-parameter compact subgroup of \(\text{ Aut}_r(M)\) generated by the imaginary part \(X^{\prime }\) of \(X_0\) [12, 13]. This result was also obtained in [14] where a crucial step is to use the monotonicity and the properness of modified Mabuchi’s K-energy [4]. But at the present paper, we can avoid to use this energy in dealing with our stability problem. This advantage allows us to remove the \(K_{X_0}\)-invariant condition for the initial potential \(\psi \). Unfortunately, we could not improve the exponential convergence of \(((\varphi _t)_{\sigma _t}-\underline{(\varphi _t)_{\sigma _t}} )\) without the assumption of \(K_{X_0}\)-invariant for \(\psi \). But we believe that it is still true if one can extend the Gauge Transformation Group \(\text{ Aut}_r(M)\) to \(\text{ Aut}(M)\) (cf. Proposition 3.9, Proposition 3.10).

In subsequence papers [15, 16], Theorem 1.1 will be used to prove the convergence of Kähler-Ricci flow on a Fano manifold for general initial Kähler metric in Kähler class \(2\pi c_1(M)\).

2 In case of Kähler-Einstein metric

In this section, we assume that \((M,J)\) admits a Kähler-Einstein metric \(g_{KE}\) with its Kähler form \(\omega _{KE}\in 2\pi c_1(M)\). Let \(\Lambda _1(M,\omega _{KE})\) be a finite dimensional linear space of the first non-zero eigenvalue-functions of Lapalace operator \(\triangle _{\omega _{KE}}\) associated to \(\omega _{KE}\). Then by using the Bochner formula, it is well-known that the first non-zero eigenvalue is 1 and \(\Lambda _1(M,\omega _{KE})=\text{ span}\{\theta _X|~X\in \eta (M)\}\), where \(\eta (M)\) is a linear space consisting of holomorphic vector fields on \(M\) which is isomorphic to the Lie algebra of holomorphisms transformation group \(\text{ Aut}(M)\) on \(M\) and \(\theta _X\) is a real-valued potential of \(X\) defined by
$$\begin{aligned} \sqrt{-1}\overline{\partial }\theta _X=i_X (\omega _{KE})~\quad \text{ and} \quad ~\int \limits _M\theta _X\omega _{KE}^n=0. \end{aligned}$$
(2.1)
Thus we have the following conclusion:
$$\begin{aligned} \lambda _1(\omega _{KE})&\ge 1+\delta _0,\quad \text{ if}\,\eta (M)= 0,\\ \lambda _2(\omega _{KE})&\ge 1+\delta _0,\quad \text{ if}\,\eta (M)\ne 0, \end{aligned}$$
where \(\lambda _1( \omega _{KE})\) and \(\lambda _2( \omega _{KE})\) are the first two non-zero eigenvalues of \(\triangle _{\omega _{KE}}\), and \(\delta _0\) is a positive number depending only on \((M,J,\omega _{KE})\).
Recall that the set of Kähler potentials in \(2\pi c_1(M)\) is given by
$$\begin{aligned} \mathcal{M }(\omega _{KE})=\{\phi \in C^\infty (M,\mathbb{R })|\quad \omega _{KE}+\sqrt{-1}\partial \bar{\partial }\phi >0\}. \end{aligned}$$
Then for any Kähler metric \(g\) with its Kähler form \(\omega _g\in 2\pi c_1(M)\), we have \(\omega _g=\omega _{KE}+\sqrt{-1}\partial \bar{\partial }\psi \) for some \(\psi \in \mathcal{M }(\omega _{KE})\) and Kähler-Ricci flow (1.1) is equivalent to the following parabolic equation of complex Monge-Ampère type for Kähler potentials \(\varphi _t=\varphi (t,\cdot )\) with \(\omega _{g_t}=\omega _{KE}+\sqrt{-1}\partial \bar{\partial }\varphi _t\),
$$\begin{aligned} \frac{\partial \varphi }{\partial t}=\log \frac{\omega ^n_\varphi }{\omega _{KE}^n}+\varphi , \quad \varphi (0)=\psi . \end{aligned}$$
(2.2)
Let \(\text{ Aut}_0(M)\) be the connected component of \(\text{ Aut}(M)\) which contains the identity map of \(M\). Then we prove

Theorem 2.1

There exists a small \(\epsilon \) depending only on \((M,J,\omega _{KE})\) such that for any initial data \(\psi \) of (2.2) which satisfies
$$\begin{aligned} \Vert \psi \Vert _{C^{2,\alpha }} < \epsilon , \end{aligned}$$
(2.3)
\(\Vert \varphi _t-\frac{1}{\int _M\omega _{KE}^n}\int _M \varphi _t\omega _{KE}^n\Vert _{C^{2,\alpha }}\) is uniformly bounded. Moreover, there exists a \(\sigma \in \text{ Aut}_0(M)\) such that Kähler potentials \(((\varphi _t)_\sigma - \frac{1}{\int _M\omega _{KE}^n}\int _M(\varphi _t)_{\sigma } \omega _{KE}^n) \) converges exponentially to \(0\) as \(t\rightarrow \infty \), where \((\varphi _t)_\sigma \) are modified Kähler potentials of \(g_t\) defined by \(\sigma ^\star \omega _{g_t}=\omega _{KE}+\sqrt{-1} \partial \overline{\partial }(\varphi _t)_\sigma \). As a consequence, \(\sigma ^*\omega _{g_t}\) converges exponentially to \(\omega _{KE}\).

We need several lemmas to prove Theorem 2.1.

For convenience, we set a Hölder space by
$$\begin{aligned} \mathcal{K }(\epsilon _0)=\{\phi \in \mathcal{M }(\omega _{KE})|\quad \Vert \phi -\underline{\phi }\Vert _{C^{2,\alpha }} <\epsilon _0\}, \end{aligned}$$
where \(\underline{\phi }=\frac{1}{V}\int _M\phi \omega _{KE}^n\) and \(V=\int _M\omega _{KE}^n\).

Lemma 2.2

Suppose that solution \(\varphi _{t_0-1}\) of (2.2) at \(t=t_0-1\) lies in \(\mathcal{K }(\epsilon _0).\) Then
$$\begin{aligned} \Vert \varphi _{t_0}-\underline{\varphi _{t_0}}\Vert _{C^{k,\alpha }}=O(\epsilon _0), \quad \text{ for} \text{ any} \text{ integer}\,k\ge 0. \end{aligned}$$
(2.4)

Proof

We consider the flow (2.2) with the initial \(\psi \) replaced by \(\tilde{\psi }=\psi -\underline{\varphi _{t_0-1}}e^{-t_0+1}\). Then it is easy to see that \( \tilde{\varphi }-\varphi = -\underline{\varphi _{t_0-1}}e^{t-t_0+1}\), where \( \tilde{\varphi }= \tilde{\varphi }_t\) is the solution with the initial \(\tilde{\psi }\). In particular \(\tilde{\varphi }_{t_0-1}=\varphi _{t_0-1}-\underline{\varphi _{t_0-1}}\). So (2.2) implies
$$\begin{aligned} \Vert \dot{\tilde{\varphi }}_{t_0-1}\Vert _{C^0} \le (n+2)\epsilon _0. \end{aligned}$$
Thus by applying the maximum principle to equation
$$\begin{aligned} \ddot{\tilde{\varphi }}=\triangle \dot{\tilde{\varphi }}+\dot{\tilde{\varphi }}, \end{aligned}$$
(2.5)
we get
$$\begin{aligned} \Vert \dot{\tilde{\varphi }}_{t}\Vert _{C^0} \le (n+2)e^2\epsilon _0, \quad \forall ~t\in [t_0-1,t_0+1]. \end{aligned}$$
(2.6)
It follows
$$\begin{aligned} \Vert \tilde{\varphi }_{t}\Vert _{C^0} \le (2(n+2)e^2+1)\epsilon _0, \quad \forall ~t\in [t_0-1,t_0+1]. \end{aligned}$$
(2.7)
Under the condition of (2.6) and (2.7), one can follow arguments in [17] to obtain any higher order estimates for solution \(\tilde{\varphi }_{t}\) of (2.2) in \([t_0-\frac{1}{2}, t_0+1]\) (see a reference [16], Proposition 9.1). Namely, for any integer \(k\ge 0\), there exists a uniform constant \(C(k)\) such that
$$\begin{aligned} \Vert \tilde{\varphi }_{t}\Vert _{C^{k}}\le C(k), \quad \forall ~t\in \left[t_0-\frac{1}{2}, t_0+1\right]\!. \end{aligned}$$
Again by (2.5), we obtain the Hölder estimate
$$\begin{aligned} \Vert \dot{\tilde{\varphi }}_{t}\Vert _{C^{\alpha }} \le C_1 \epsilon _0, \quad \forall ~t\in \left[t_0-\frac{1}{4},t_0+\frac{1}{4}\right]\!. \end{aligned}$$
Furthermore by the Schauder estimate, for any \(k\ge 4\), we derive
$$\begin{aligned} \Vert \dot{\tilde{\varphi }}_{t}\Vert _{C^{k-2,\alpha } }\le C_k\epsilon _0, \quad \forall ~t\in \left[t_0-\frac{1}{4}, t_0+\frac{1}{4}\right]\!. \end{aligned}$$
(2.8)
Decompose \(\tilde{\varphi }_{t_0}\) into \(\tilde{\varphi }_{t_0}=\phi +\phi ^{\prime }\) with \(\phi \in \Lambda _1(M,\omega _{KE})\) and \(\phi ^{\prime }\in \Lambda _1^\bot (M,\)\(\omega _{KE})\), where \(\Lambda _1^\bot (M,\omega _{KE})\) is the orthogonal complementary space of \(\Lambda _1(M,\)\(\omega _{KE})\cup \mathbb{R }\). Then \(\phi = a+\sum _i a_i\theta _i\) for some constants \(a\) and \(a_i\), where \(\{\theta _i\}\) is a basis of \(\Lambda _1(M,\omega _{KE})\). Clearly, \(a, a_i=O(\epsilon _0)\) and so
$$\begin{aligned} \Vert \phi \Vert _{C^k}\le C_k^{\prime }\epsilon _0. \end{aligned}$$
(2.9)
On the other hand, \(\phi ^{\prime }\) satisfies equation,
$$\begin{aligned} P\left[\log \left(\frac{[\omega _{\phi +\phi ^{\prime }}]^n}{\omega _{KE}^n}\right)\right]-\phi ^{\prime }= P(\dot{\tilde{\varphi }}_{t_0} ), \end{aligned}$$
where \(P\) be a projection from Banach space \(C^{k,\alpha }(M)\) to Banach space \(C^{k-2,\alpha }(M)\cap \Lambda _1^\bot (M,\omega _{KE})\). Thus by using the Implicit Functional Theorem together with (2.8) and (2.9), we get
$$\begin{aligned} \Vert \phi ^{\prime }\Vert _{C^{k,\alpha }}\le C_k^{\prime \prime }\epsilon _0, \end{aligned}$$
It follows
$$\begin{aligned} \Vert \tilde{\varphi }_{t_0}\Vert _{C^{k,\alpha }}\le C\epsilon _0. \end{aligned}$$
This proves the lemma. \(\square \)

Lemma 2.3

Let \(H_0(t)=\frac{1}{V}\int _M(\dot{\varphi }_t-c(t))^2\omega _{\varphi _t}^n\), where \(c(t)=\frac{1}{V}\int _M\dot{\varphi }_t\omega _{\varphi _t}^n.\) Suppose that \(\varphi _t\) lies in \(\mathcal{K }(\epsilon _0)\) for any \(t\in [0,T)\), where \(\epsilon _0\) is a small number. Then there exists a \(\theta >0\) such that
$$\begin{aligned} H_0(t)\le H_0(1)e^{-\theta t}, \quad \forall ~t\in [1,T). \end{aligned}$$
(2.10)

Proof

For simplicity, we let \(\varphi =\varphi _t\). By a direct computation, we have
$$\begin{aligned}&\frac{d}{dt}H_0(t)\\&\quad =2\frac{1}{V}\int _M(\dot{\varphi }-c(t))(\ddot{\varphi }-\dot{c}(t))\omega ^n_\varphi +\frac{1}{V}\int _M(\dot{\varphi }-c(t))^2\triangle _\varphi \dot{\varphi }\omega _\varphi ^n\\&\quad =2\frac{1}{V}\int _M(\dot{\varphi }_t-c(t))(\triangle _\varphi \dot{\varphi }+\dot{\varphi })\omega ^n_\varphi +\frac{1}{V}\int _M(\dot{\varphi }-c(t))^2\triangle _\varphi \dot{\varphi }\omega _\varphi ^n\\&\quad =2H_0(t) -2\frac{1}{V}\int _M(1+\dot{\varphi }-c(t))|\nabla (\dot{\varphi }-c(t))|^2\omega _\varphi ^n. \end{aligned}$$
Note that by the argument to derive the estimate (2.6) it holds
$$\begin{aligned} \Vert \dot{\varphi }_{t} -c(t)\Vert _{C^0}\le (n+2)e\epsilon _0. \end{aligned}$$
(2.11)
It follows
$$\begin{aligned}&\frac{d}{dt}H_0(t)\nonumber \\&\quad \le 2H_0(t) -2(1-9n\epsilon _0)\frac{1}{V}\int _M|\nabla (\dot{\varphi }-c(t))|^2 \omega _\varphi ^n\nonumber \\&\quad \le 2H_0(t) -2(1-9n\epsilon _0)(1-2n\epsilon _0)\frac{1}{V}\int _M|\nabla _{g_{KE}} (\dot{\varphi }-c(t))|_{g_{KE}}^2\omega _{g_{KE}}^n.\nonumber \\ \end{aligned}$$
(2.12)
Case 1, \(\eta (M)=0\). Then by (2.12), we have
$$\begin{aligned} \frac{d}{dt}H_0(t)\le \frac{1}{V} [-2+2(1-9n\epsilon _0)(1+\delta _0)(1-2n\epsilon _0)] \int _M(\dot{\varphi }-c(t))^2 \omega _{g_{KE}}^n. \end{aligned}$$
Note that
$$\begin{aligned} \int _M (\dot{\varphi }-c(t))^2\omega _{g_{KE}}^n\ge (1-2n\epsilon _0)\int _M (\dot{\varphi }-c(t))^2\omega _{\varphi }^n. \end{aligned}$$
(2.13)
Thus by choosing \(\theta =-2+2(1-9n\epsilon _0)(1-2n\epsilon _0)^2(1+\delta _0)\ge \delta _0\) as \(\epsilon _0<<1\), we will get
$$\begin{aligned} H_0(t)\le H_0(0)e^{-\theta t}. \end{aligned}$$
(2.14)
Case 2, \(\eta (M)\ne 0\). Recall that \(\dot{\varphi }-c(t)= \dot{\varphi }_t-c(t)\) is a Ricci potential of metric \(\omega _{\varphi _t}\). Then by [Fu], the Futaki invariant is defined by
$$\begin{aligned} F(X)= \int _M X(\dot{\varphi }-c(t))\omega ^n_\varphi =\int _M \triangle _\varphi (\theta _X+X(\varphi ))(\dot{\varphi }-c(t)) \omega ^n_\varphi , \end{aligned}$$
where \(X\in \eta (M)\), \(\theta _X\) is the potential of \(X\) defined by (2.1) and \(X(\phi )\) is the derivative of \(\phi \) along \(X\). In our case that \((M,J)\) is a Kähler-Einstein manifold,
$$\begin{aligned} F(X)\equiv 0, \quad \forall ~X\in \eta (M). \end{aligned}$$
Thus
$$\begin{aligned} \left|\int _M \theta _X (\dot{\varphi }-c(t))\omega ^n_{KE} \right|&= \left|\int _M \triangle _{\omega _{KE}}\theta _X (\dot{\varphi }-c(t))\omega ^n_{KE}\right|\\&\le (1+O(\epsilon )) \left|\int _M \triangle _{\omega _{KE}} (X(\varphi ))(\dot{\varphi }-c(t))\omega ^n_{KE}\right|. \end{aligned}$$
Thus by Lemma 2.2, we get
$$\begin{aligned} \left|\int _M \theta _X(\dot{\varphi }-c(t))\omega ^n_{KE}\right|\le B\epsilon _0\int _M\left|\dot{\varphi }-c(t)\right|\omega ^n_{KE},\quad \forall ~t\in [1,T), \end{aligned}$$
(2.15)
for any \(X\in \eta (M)\) which satisfies \(\int _M|X|^2_{\omega _{KE}}\omega _{KE}^n=1.\)
Decompose \((\dot{\varphi }-c(t))\) as
$$\begin{aligned} \dot{\varphi }-c(t)=\phi +\phi ^{\prime }, \end{aligned}$$
with \(\phi \in \Lambda _1(M,\omega _{KE})\cup \mathbb{R }\) and \(\phi ^{\prime }\in \Lambda _1^\bot (M,\omega _{KE})\). Then by (2.15), we see that for any \(t\in [1,T)\),
$$\begin{aligned} \int _M|\phi |^2\omega ^n_{KE}\le A_1\epsilon _0^2 \int _M(\dot{\varphi }-c(t))^2\omega ^n_{KE}, \end{aligned}$$
for some uniform constant \(A_1\). It follows
$$\begin{aligned} \int _M|\phi ^{\prime }|^2\omega ^n_{KE}\ge (1-A_1\epsilon _0^2) \int _M(\dot{\varphi }-c(t))^2\omega ^n_{KE}. \end{aligned}$$
(2.16)
Hence
$$\begin{aligned} \int _M |\nabla _{\omega _{KE}}(\dot{\varphi }-c(t))|_{\omega _{KE}}^2\omega ^n_{KE}&\ge \int _M|\nabla _{\omega _{KE}}\phi ^{\prime }|_{\omega _{KE}}^2\omega ^n_{KE}\nonumber \\&\ge (1+\sigma _0)(1-A_1\epsilon _0^2) \int _M(\dot{\varphi }-c(t))^2\omega ^n_{KE}.\nonumber \\&\ge (1+\sigma _0)(1-A_2\epsilon _0) \int _M(\dot{\varphi }-c(t))^2\omega ^n_{\varphi }.\nonumber \\ \end{aligned}$$
(2.17)
By choosing \(\theta = 2(1-9n\epsilon _0)(1-2n\epsilon _0)(1-A_2\epsilon _0)(1+\sigma _0)-2\ge \sigma _0\) as \(\epsilon _0<<1\), we deduce from (2.12),
$$\begin{aligned} \frac{ d H_0(t)}{dt}\le -\theta H_0(t), \quad \forall ~t\in [1,T). \end{aligned}$$
(2.18)
This shows
$$\begin{aligned} H_0(t)\le H_0(1)e^{-\theta t}\!, \quad \forall ~t\in [1,T). \end{aligned}$$
\(\square \)
Next we do \(W^{k,2}\)-estimate for \(\dot{\varphi }\) under the assumption \(\varphi \in \mathcal{K }(\epsilon _0)\). The argument is standard (cf. [3]). Since we will use a generalization of such an estimate to modified Kähler-Ricci flow in next section, we give a short description below. Recall that a \(k\)th-norm of \(\dot{\varphi }\) is defined by
$$\begin{aligned} |\nabla ^k\dot{\varphi }|^2=\sum g^{^{\prime }i_1j_1}\ldots g^{^{\prime }i_kj_k}{\dot{\varphi }}_{i_1\ldots i_k} {\dot{\varphi }}_{j_1\ldots j_k}, \end{aligned}$$
where \({\dot{\varphi }}_{i_1\ldots i_k}\) are components of the \(k\)th-covariant derivative of \(\dot{\varphi }\) with respect to \(g^{\prime }=\omega _\varphi \) as a Riemannian metric.
Since
$$\begin{aligned} {\dot{\varphi }}_{i_1\ldots i_k}=\frac{\partial ^k\dot{\varphi }}{\partial x^{i_1}\ldots \partial x^{i_k}}+\Phi _1(\dot{\varphi },\ldots ,{\dot{\varphi }}_{i_1\ldots i_{k-1}}), \end{aligned}$$
we have
$$\begin{aligned} \frac{ d {\dot{\varphi }}_{i_1\ldots i_k}}{dt}&= \frac{\partial ^k\ddot{\varphi }}{\partial x^{i_1}\ldots \partial x^{i_k}}+\frac{d\Phi _1}{dt}\nonumber \\&= {\ddot{\varphi }}_{i_1\ldots i_k}+\Phi _2(\dot{\varphi }_i,\ldots , {\dot{\varphi }}_{i_1\ldots i_{k-1}},\ddot{\varphi }_i,\ldots , {\ddot{\varphi }}_{i_1\ldots i_{k-1}}), \end{aligned}$$
(2.19)
where \(\Phi _1\) and \(\Phi _2\) are two polynomials with variables \(\dot{\varphi }_i,\ldots ,{\dot{\varphi }}_{i_1\ldots i_{k-1}}\), \(\ddot{\varphi }_i ,\ldots ,\)\({\ddot{\varphi }}_{i_1\ldots i_{k-1}} \), \(g_{ij}^{\prime }, \partial ^l g_{ij}^{\prime }\)\((i,j,i_1,\ldots ,i_{k-1}=1,\ldots ,2n; l=1,\ldots ,k-1)\). Then by Eqs. (1.1) and (2.2), one can compute
$$\begin{aligned}&\frac{d|\nabla ^k \dot{\varphi }|^2}{dt}\nonumber \\&=\sum _{i_1,\ldots ,i_k} \sum _\alpha (R_{ i_\alpha i_\alpha }-g_{i_\alpha i_\alpha }^{\prime }){\dot{\varphi }}_{i_1,\ldots ,i_\alpha ,\ldots ,i_k} {\dot{\varphi }}_{i_1,\ldots ,i_\alpha ,\ldots ,i_k}\nonumber \\&\quad + 2\sum g^{^{\prime }i_1j_1}\ldots g^{^{\prime }i_kj_k}\frac{d{\dot{\varphi }}_{i_1\ldots i_k}}{dt} {\dot{\varphi }}_{j_1\ldots j_k}\nonumber \\&\le C_1 \sum _{i=1}^k|\nabla ^i\dot{\varphi }|^2+ 2\sum g^{^{\prime }i_1j_1}\ldots g^{^{\prime }i_kj_k}(\triangle \dot{\varphi })_{i_1\ldots i_k} {\dot{\varphi }}_{j_1\ldots j_k}. \end{aligned}$$
(2.20)
Denote \(H_k(t)\) by
$$\begin{aligned} H_k(t)=\int _M|\nabla ^k\dot{\varphi }|^2\omega _{\varphi }^n. \end{aligned}$$
Then we prove

Lemma 2.4

Suppose that \(\varphi _t\) lies \(\mathcal{K }(\epsilon _0)\) for any \(t\in [0,T)\), where \(\epsilon _0\) is a small number. Let \(k\ge 1\). Then there exist a small number \(\theta ^{\prime }\) and a large number \(A\) depending only on \(k, \epsilon _0\) and the metric \(g_{KE}\) such that
$$\begin{aligned} H_k(t)\le (H_k(1)+AH_0(1)) e^{- \theta ^{\prime }t}, \quad \forall ~t\in [1,T). \end{aligned}$$
(2.21)

Proof

Note that \(\triangle \dot{\varphi } \) is uniformly bounded by Lemma 2.2. Then by (2.20) and the Sobolev interpolatation theorem, we have
$$\begin{aligned} \frac{ dH_k(t)}{dt}&= \int _M \frac{d|\nabla ^k\dot{\varphi }|^2}{dt}\omega _{\varphi }^n+\int _M |\nabla ^k\dot{\varphi }|^2\triangle \dot{\varphi } \omega _{\varphi }^n\nonumber \\&\le -2H_{k+1}(t)+C_2H_k(t)\nonumber +C_3\int _M|\dot{\varphi }-c(t)|^2\omega _{\varphi }^n\\&\le - \theta ^{\prime }H_{k}(t)+C_4H_0(t), \quad \, \forall ~t\in [1,T). \end{aligned}$$
(2.22)
By combining (2.18), we get
$$\begin{aligned} \frac{ d(H_k(t)+AH_0(t))}{dt}\le -\theta ^{\prime } \left[H_{k}(t)+\frac{(A\theta -C_4)}{\theta ^{\prime }}H_0(t)\right], \quad \, \forall ~t\in [1,T), \end{aligned}$$
where \(A\) is a sufficiently large number. It follows
$$\begin{aligned} \frac{ d \ln (H_k(t)+AH_0(t))}{dt}\le -\theta ^{\prime }\frac{ H_{k}(t)+\frac{(A\theta -C_4)}{\theta ^{\prime }}H_0(t)}{H_k(t)+AH_0(t)}\le -\theta ^{\prime } \end{aligned}$$
as long as \(\theta ^{\prime }<\theta \). Therefore
$$\begin{aligned} H_k(t)+AH_0(t)\le (H_k(1)+AH_0(1))e^{-\theta ^{\prime } t},\quad \forall ~t\in [1,T). \end{aligned}$$
This implies (2.21).

By the Sobolev embedding Theorem and Lemma 2.2 , we get the following proposition immediately from Lemma and Lemma 2.4.

Proposition 2.5

Suppose that \(\varphi _t\) lies \(\mathcal{K }(\epsilon _0)\) for any \(t\in [0,T)\), where \(\epsilon _0\) is a small number. Then for any integer \(k\ge 0\) there exists \(B_k, \theta _k>0\) such that
$$\begin{aligned} \Vert \dot{\varphi _t}-c(t)\Vert _{C^k}\le B_ke^{-\theta _kt}, \quad \forall ~t\in [1,T). \end{aligned}$$
Now we begin to prove Theorem 2.1. For fixed a large number \(T\) and \(N\), we can choose a sufficient small \(\epsilon \) depends on \(T\), \(\epsilon _0\) and \(N\) such that for any \(t\le T\), solutions \(\varphi _t\) of (2.2) lies in \(\mathcal{K }(\frac{\epsilon _0}{2})\) and satisfies
$$\begin{aligned} \Vert \dot{\varphi }_t-c(t)\Vert _{C^0}\le \frac{\epsilon _0}{2N}, \quad \text{ and} \quad osc(\varphi _t)\le \frac{\epsilon _0}{4N}, \end{aligned}$$
(2.23)
whenever \(\Vert \psi \Vert _{C^{2,\alpha }}\le \epsilon \). Choose a maximal \(\delta (T)\) such that \(\varphi _t\in \mathcal{K }(\epsilon _0)\) for any \(t<T+\delta (T)\). We want to show that \(\delta (T)\) must be the infinity whenever \(T\) and \(N\) are large enough.

The following lemma is a corollary of Proposition 2.5.

Lemma 2.6

Choose some large \(T\) such that
$$\begin{aligned} \frac{2(n+1)B_0}{ \theta _0}e^{-\theta _0T}\le \frac{\epsilon _0}{4N}, \end{aligned}$$
(2.24)
where \(B_0,\theta _0\) are the two constants determined in Proposition 2.5 in case of \(k=0.\) Then
$$\begin{aligned} |\tilde{\varphi }|\le \frac{3\epsilon _0}{4N}, \quad \, \forall ~t\in [0,T+\delta (T)), \end{aligned}$$
(2.25)
where \(\tilde{\varphi }=\tilde{\varphi }_t=\varphi _t-\frac{1}{V}\int _M\varphi _t \omega _{\varphi _t}^n.\)

Proof

We suffice to prove (2.25) for any \(t\ge T\). Notice that
$$\begin{aligned} \frac{d}{dt}\tilde{\varphi }=h-\frac{1}{V}\int _Mh\triangle _\varphi \varphi \omega _\varphi ^n, \end{aligned}$$
where \(h=h_t=\dot{\varphi }_t-c(t)\). Then by Proposition 2.5, we have
$$\begin{aligned} \tilde{\varphi }&= \tilde{\varphi }_T+\int _T^{T+\delta (T)}h dt -\int _T^{T+\delta (T)}\frac{1}{V}\int _Mh\triangle _\varphi \varphi \omega _\varphi ^ndt\\&\le \frac{\epsilon _0}{2N}+B_0\int _T^{T+\delta (T)}e^{-\theta _0t}dt+ (n+1)B_0\epsilon _0\int _T^{T+\delta (T)}e^{-\theta _0t}dt\\&\le \frac{\epsilon _0}{2N}+\frac{2(n+1)B_0}{\theta }e^{-\theta _0T}\le \frac{3\epsilon _0}{4N}. \end{aligned}$$
\(\square \)

Proposition 2.7

For any small \(\epsilon _0\), there exist a small number \(\epsilon \) depending only on \((M,J,\omega _{KE})\) such that \(\varphi _t\in \mathcal{K }(\epsilon _0)\)\(\forall ~t>0\) if \(\psi \) satisfies (2.3).

Proof

We will use an argument by contradiction. On the contrary, we may assume that there exists a number \(\delta (T)<\infty \) such that \(\varphi _t\in \mathcal{K }(\epsilon _0)\) for any \(t<T+\delta (T)\) and there exists a sequence of \(t_i\rightarrow T+\delta (T)\) such that
$$\begin{aligned} \Vert \overline{\varphi _{t_i}}\Vert _{C^{2,\alpha }}=\Vert \varphi _{t_i} -\underline{\varphi _{t_i}}\Vert _{C^{2,\alpha }}\rightarrow \epsilon _0. \end{aligned}$$
(2.26)
Let \(b_t\) be a constant so that \(\overline{\varphi _t}=\tilde{\varphi _t}+b_t\). Then by Lemma , it is easy to see \(b_t\le \frac{2\epsilon _0}{N}\). As in the proof of Lemma 2.2, we decompose \(\overline{\varphi }\) as \(\overline{\varphi }=\phi +\phi ^{\prime }\), where \(\phi \in \Lambda _1(M,\omega _{KE})\) and \(\phi ^{\prime }\in \Lambda _1^\bot (M,\omega _{KE})\). Thus \(\phi =\sum _i a_i\theta _i\) for some constants \(a_i\). As a consequence, by Lemma , we have \(|a_i|\le \frac{A\epsilon _0}{N}\), so
$$\begin{aligned} \Vert \phi \Vert _{C^{2,\alpha }}\le \frac{A_0\epsilon _0}{N}, \end{aligned}$$
(2.27)
for some uniform constants \(A\) and \(A_0\).
By Eq. (2.2), we have
$$\begin{aligned} \omega _\varphi ^n=\omega _{KE}^ne^{h+\overline{\varphi }+a}, \end{aligned}$$
(2.28)
where \(h=\dot{\varphi }_t-c_t\) and \(a=a_t\) is a constant. By Lemma and Proposition 2.5, it is easy to see that \(|a|\le \frac{4A_0\epsilon _0}{N}\). Let \(P\) be a projection from Banach space \(C^{2,\alpha }(M)\) to Banach space \(C^\alpha (M)\cap \Lambda _1^\bot (M,\omega _{KE})\). Then \(\phi ^{\prime }\) is a solution of equation
$$\begin{aligned} P\left[\log \left(\frac{[\omega _{\phi +\phi ^{\prime }}]^n}{\omega _{KE}^n}\right)\right]-\phi ^{\prime }=P( h+a), \end{aligned}$$
(2.29)
where \(\phi \) and \(h+a\) are regarded as two perturbation functions. On the other hand, by Proposition 2.5, we have
$$\begin{aligned} \Vert P( h+a)\Vert _{C^\alpha }\le |a|+ B_1e^{-\theta _1T}\le \frac{8A_0\epsilon _0}{N}, \end{aligned}$$
when \(T\) is sufficiently large. Thus by using the Implicit Functional Theorem, we get
$$\begin{aligned} \Vert \phi ^{\prime }\Vert _{C^{2,\alpha }}\le c=O\left(\frac{\epsilon _0}{N}\right)\!, \end{aligned}$$
(2.30)
where the constant \(c\) is independent of \(t\) and goes to zero as \(N\rightarrow \infty \). In particular, \(c\le \frac{\epsilon _0}{4}\) by choosing a large number \(N\). Hence by combining (2.27) and (2.30), we obtain
$$\begin{aligned} \Vert \overline{\varphi }\Vert _{C^{2,\alpha }}\le \frac{\epsilon _0}{2}, \quad \forall ~t\in [T,T+\delta (T)). \end{aligned}$$
But this is impossible according to (2.26). Hence the proposition is true. \(\square \)

Completion of proof of Theorem 3.1

By Proposition 2.5, 2.7, we see that \(\varphi _t\) and \(\dot{\varphi }_t\) converge exponentially to a potential \(\phi _\infty \) and zero in \(C^\infty \) as \(t\rightarrow \infty \), respectively. Since \(\dot{\varphi }_t\) are Ricci potentials of \(\omega _{\varphi _t}\), the limit \(\phi _\infty \) must be a potential of Kähler-Einstein metric. By the uniqueness of Kähler-Einstein metrics [1], we conclude that there exists a \(\sigma \in \text{ Aut}_0(M)\) such that Kähler potentials \(((\varphi _t)_\sigma -\underline{(\varphi _t)_\sigma })\) with \(\sigma ^\star \omega _{\varphi _t}=\omega _{KE}+\sqrt{-1}\partial \overline{\partial } (\varphi _t)_\sigma \) converge exponentially to \(0\) as \(t\rightarrow \infty \). \(\square \)

3 In case of Kähler-Ricci soliton

In this section, we assume that \((M,J)\) admits a Kähler Ricci soliton \((\omega _{KS}, X_0)\) with respect to holomorphic vector field \(X_0\) on \(M\). Namely, \((\omega _{KS},X_0)\) satisfies equation
$$\begin{aligned} \text{ Ric}(\omega _{KS})-\omega _{KS}=L_{X_0}\omega _{KS}, \end{aligned}$$
where \(L_{X_0}\) is a Lie derivative along \(X_0\).
By the Hodge theorem, one can define a real-valued potential \(\theta _{X_0}\) of \(X_0\) by
$$\begin{aligned} L_{X_0}\omega _{KS}=\sqrt{-1}\partial \overline{\partial }\theta _{X_0}~\text{ and}~ \int _M e^{\theta _{X_0}}\omega _{KS}^n=\int _M\omega _{KS}^n=V. \end{aligned}$$
Since
$$\begin{aligned} L_{X_0}\omega _{\phi }=\sqrt{-1}\partial \overline{\partial }(\theta _X+X_0(\phi )), \end{aligned}$$
(3.1)
for any \(\phi \in \mathcal{M }(\omega _{KS})=\{\phi \in C^\infty (M,\mathbb{R })|\quad \, \omega _{KS}+\sqrt{-1}\partial \bar{\partial }\phi >0\}\), we have
$$\begin{aligned} L_{X}\omega _{\phi }=\sqrt{-1}\partial \overline{\partial }(\theta _{X_0}+X(\phi )) \end{aligned}$$
and
$$\begin{aligned} L_{X^{\prime }}\omega _{\phi }=\sqrt{-1}\partial \overline{\partial }(X^{\prime }(\phi )), \end{aligned}$$
where \(X\) and \(X^{\prime }\) be a real part and imaginary part of \(X_0\), respectively. (3.1) also implies that
$$\begin{aligned} \langle \overline{\partial }( \theta _{X_0}+X_0(\phi )), \overline{\partial }\psi \rangle _{\omega _\phi }=X_0(\psi )=X(\psi )+\sqrt{-1}X^{\prime }(\psi ) \end{aligned}$$
for any \(\psi \in C^\infty (M)\). Thus
$$\begin{aligned} |\langle \nabla ( \theta _{X_0}+X(\phi )),\nabla \psi \rangle _{\omega _\phi }-X(\psi )|\le |X^{\prime }(\phi )||\nabla \psi |_{\omega _\phi }. \end{aligned}$$
(3.2)
We now consider the following modified flow of (1.1) for \(\tilde{g}=\tilde{g}(t,\cdot )=\tilde{g}_t\) as in [14],
$$\begin{aligned} \frac{\partial \tilde{g}}{\partial t}=-\text{ Ric}(\tilde{g})+\tilde{g}+L_X \tilde{g},\quad \, \tilde{g}(0,\cdot )=g. \end{aligned}$$
(3.3)
Then (3.3) is equivalent to a parabolic equation of complex Monge-Ampère type,
$$\begin{aligned} \frac{\partial \varphi }{\partial t}=\log \frac{\omega ^n_\varphi }{\omega _{KS}^n}+\varphi +X(\varphi ), \quad \, \varphi (0)=\psi , \end{aligned}$$
(3.4)
where \(\varphi =\varphi _t\) are potentials of evolved Kähler metrics \(\tilde{g}_t\) of (3.3) and \(\psi \) is a potential of \(g\).

Let \(K_{X_0}\) be an one-parameter compact subgroup of \(\text{ Aut}_0(M)\) generated by the imaginary part \(X^{\prime }\) of \(X_0\). Then one can choose a reductive subgroup \(\text{ Aut}_r(M)\) of \(\text{ Aut}_0(M)\) such that \(\text{ Aut}_r(M)\) contains \(K_{X_0}\). In this section, we shall prove

Theorem 3.1

There exists a small \(\epsilon \) depending only on \((M,J,\omega _{KS})\) such that for any initial data \(\psi \) of (3.4) which satisfies (2.3), there exist a family of \(\sigma _t\in \text{ Aut}_r(M)\) for \(\tilde{g}_t\) of (3.3) such that Kähler potentials \(((\varphi _t)_{\sigma _t}-\frac{1}{V}\int _M (\varphi _t)_{\sigma _t} e^{\theta _{X_0}}\omega _{KS}^n)\) are \(C^k\)-norm uniformly bounded, where \((\varphi _t)_{\sigma _t} \) are modified Kähler potentials of \(\tilde{g}_t\) defined by \((\sigma _t)^\star \omega _{\tilde{g}_t}=\omega _{KS}+\sqrt{-1}\partial \overline{\partial } (\varphi _t)_{\sigma _t}\). As a consequence, \(\omega _{g_t}\) of (1.1) converges to \(\omega _{KS}\) in \(C^\infty \) in the sense of Kähler potentials modulo holomorphisms transformation. Furthermore, if in addition that \(\psi \) is \(K_{X_0}\)-invariant, then there exist a \(\sigma \in \text{ Aut}_r(M)\) such that \(((\varphi _t)_\sigma -\frac{1}{V}\int _M (\varphi _t)_\sigma e^{\theta _{X_0}}\omega _{KS}^n )\) converges exponentially to \(0\) as \(t\rightarrow \infty \), and so Kähler metric \(\sigma ^*\omega _{\tilde{g}_t}\) converges exponentially to \(\omega _{KS}\).

Theorem 3.1 is a generalization of Theorem 2.1 to the case that \(M\) admits a Kähler Ricci soliton and we will generalize the arguments in Section 1 to prove it. As in Section 1, we first estimate \(\dot{\varphi }=\dot{\varphi }_t\) of (3.4).

By introducing a modified functional of \(H_0(t)\),
$$\begin{aligned} \tilde{H}_0(t)=\frac{1}{V}\int _M(\dot{\varphi }-c(t))^2 e^{\theta _{X_0}+X(\varphi ) }\omega ^n_\varphi , \end{aligned}$$
where \( c(t)=\frac{1}{V}\int _M \dot{\varphi }e^{\theta _{X_0}+X(\varphi ) }\omega _{\varphi }^n\), we compute
$$\begin{aligned} \frac{d}{dt}\tilde{H}_0(t)&= 2\frac{1}{V}\int _M(\dot{\varphi }-c(t))(\ddot{\varphi }-\dot{c}(t)) e^{\theta _{X_0}+X(\varphi ) }\omega ^n_\varphi \nonumber \\ \quad&+\frac{1}{V}\int _M(\dot{\varphi }-c(t))^2(\triangle _\varphi \dot{\varphi }+X(\dot{\varphi }))e^{\theta _{X_0}+X(\varphi ) }\omega _\varphi ^n\nonumber \\ v&= 2\frac{1}{V}\int _M [\dot{\varphi }-c(t)](\triangle _\varphi \dot{\varphi }+X(\dot{\varphi })) e^{\theta _{X_0}+X(\varphi ) }\omega ^n_\varphi \nonumber \\ \quad&+2\frac{1}{V}\int _M(\dot{\varphi }-c(t))^2(1+\triangle _\varphi \dot{\varphi }+X(\dot{\varphi })) e^{\theta _{X_0}+X(\varphi ) }\omega ^n_\varphi . \end{aligned}$$
(3.5)
On the other hand, one sees
$$\begin{aligned}&\int _M [\dot{\varphi }-c(t)](\triangle _\varphi \dot{\varphi }+X(\dot{\varphi })) e^{\theta _{X_0}+X(\varphi ) } \omega ^n_\varphi \\ \quad&=-\int _M |\nabla (\dot{\varphi }-c(t))|^2 e^{\theta _{X_0}+X(\varphi ) } \omega ^n_\varphi \nonumber \\ \qquad&+ \int _M (\dot{\varphi }-c(t))[X(\dot{\varphi })-\langle \nabla (\theta _X+X(\phi )-\dot{\varphi }),\nabla \dot{\varphi }\rangle ] e^{\theta _{X_0}+X(\varphi ) }\omega ^n_\varphi . \end{aligned}$$
By (3.2), it follows
$$\begin{aligned}&\int _M [\dot{\varphi }-c(t)](\triangle _\varphi \dot{\varphi }+X(\dot{\varphi })) e^{\theta _{X_0}+X(\varphi ) } \omega ^n_\varphi \\&\le -\int _M |\nabla (\dot{\varphi }-c(t))|^2 e^{\theta _{X_0}+X(\varphi ) } \omega ^n_\varphi \nonumber + V\Vert \dot{\varphi }-c(t)\Vert _{C^0} \Vert X^{\prime }(\dot{\varphi })\Vert _{C^0}\Vert \nabla {\dot{\varphi }}\Vert _{C^0}. \end{aligned}$$
Thus inserting the above inequality into (3.5), we get
$$\begin{aligned} \frac{d}{dt}\tilde{H}_0(t)&\le 2\frac{1}{V}\int _M(\dot{\varphi }-c(t))^2(1+\triangle _\varphi \dot{\varphi }+X (\dot{\varphi }) ) e^{\theta _{X_0}+X(\varphi ) } \omega ^n_\varphi \nonumber \\&\quad -2\frac{1}{V}\int _M |\nabla (\dot{\varphi }-c(t))|^2 e^{\theta _{X_0}+X(\varphi ) } \omega _\varphi ^n\nonumber \\&\quad + 2\Vert \dot{\varphi }-c(t)\Vert _{C^0} \Vert X^{\prime }(\dot{\varphi })\Vert _{C^0}\Vert \nabla {\dot{\varphi }}\Vert _{C^0}. \end{aligned}$$
(3.6)
To estimate the \(L^2\)-integral of \(\nabla \dot{\varphi }\), we need the following lemma. Let \( \text{ ker}(P,\omega _{KS})\) be the kernel of linear operator
$$\begin{aligned} P\psi =\triangle _{\omega _{KS}} \psi +\psi +\langle \overline{\partial }\theta _X, \overline{\partial }\psi \rangle _{\omega _{KS}}. \end{aligned}$$
Then \(\text{ ker}(P,\omega _{KS}) \cong \eta _r(M)\) [14]. By setting a Hölder space by
$$\begin{aligned} \overline{\mathcal{K }}(\epsilon _0)=\{\phi \in \mathcal{M } (\omega _{KS})|\quad \, \Vert \phi -\underline{\phi }\Vert _{C^{2,\alpha }} <\epsilon _0\}, \end{aligned}$$
where \(\underline{\phi }=\frac{1}{V}\int _M \phi \omega _{KS}^n,\) we have

Lemma 3.2

Let \(\varphi =\varphi _{t}\) be solution of (3.4) at \(t\) and \(\theta _Y^{\prime }\in \text{ ker}(P,\omega _{KS})\) be a potential of \(Y\in \eta _r(M)\) with \(\int _M |Y|^2\omega _{KS}^n=1\). If \(\varphi _{t-1}\in \overline{\mathcal{K }}(\epsilon _0)\), then there exist two uniform constants \(C_1\) and \(C_2\) such that
$$\begin{aligned}&\left|\int _M \theta _Y^{\prime } (\dot{\varphi }-c(t))e^{\theta _{X_0}+X(\varphi )}\omega _\varphi ^n\right|\nonumber \\&\quad \le C_1\epsilon _0\int _M|\dot{\varphi }-c(t)|e^{\theta _{X_0}+X(\varphi )} \omega _\varphi ^n+C_2\epsilon _0^2. \end{aligned}$$
(3.7)

Proof

Recall a generalized Futaki-invariant [13]
$$\begin{aligned} F_{X_0}(Y)=\int _M Y[h_{\omega _\phi }-(\theta _{X_0}+X_0(\phi ))]e^{\theta _{X_0}+X_0(\phi ))}\omega _\phi ^n,\quad \forall ~ Y\in \eta (M). \end{aligned}$$
\(F_{X_0}(\cdot )\) is independent of the choice of Kähler metric \(\omega _\phi \) on \(M\) and it vanishes if \(M\) admits the Kähler-Ricci soliton \((\omega _{KS}, X_0)\). So in our case, we have
$$\begin{aligned} F_{X_0}(Y)\equiv 0,\quad \forall ~ Y\in \eta (M). \end{aligned}$$
(3.8)
By applying the metrics \(\omega _\varphi \) to (3.8), one sees
$$\begin{aligned} \int _M Y[\dot{\varphi }-c(t)-\sqrt{-1}X^{\prime }(\varphi )]e^{\theta _{X_0}+X_0(\varphi )} \omega _\varphi ^n=0,\quad \forall ~Y\in \eta _r(M). \end{aligned}$$
It follows
$$\begin{aligned}&\left|\text{ Re}\left(\int _M Y[\dot{\varphi }-c(t)-\sqrt{-1}X^{\prime }(\varphi )]e^{\theta _{X_0}+X(\varphi )+ \ln \cos (X^{\prime }(\varphi ))}\omega _\varphi ^n\right)\right|\nonumber \\&\le C\Vert \varphi \Vert _{C^2}\Vert X^{\prime }(\varphi )\Vert _{C^0}. \end{aligned}$$
(3.9)
On the other hand, under the condition in Lemma 3.2, by using the argument in the proof of Lemma 2.2 to Eq. (3.4), one can prove
$$\begin{aligned} \Vert \varphi -\underline{\varphi }\Vert _{C^{k}}\le A\epsilon _0, \quad \forall ~k\ge 2, \end{aligned}$$
(3.10)
where \(A\) is a uniform constant depending only on the metric \(\omega _{KS}\) and \(k\). Thus by using the Stoke’s formula, we obtain
$$\begin{aligned}&\int _M Y[\dot{\varphi }-c(t)-\sqrt{-1}X^{\prime }(\varphi )]e^{\theta _{X_0}+X(\varphi )+ \ln \cos (X^{\prime }(\varphi ))}\omega _\varphi ^n\\&=-\int _M(\dot{\varphi }-c(t))-\sqrt{-1}X^{\prime }(\varphi ))\\&\quad \times [\triangle (\theta _Y^{\prime }+Y(\varphi ))+\langle \overline{\partial }(\theta _Y^{\prime }+Y(\varphi )),\overline{\partial }(\theta _{X_0}+X(\varphi )+ \ln \cos (X^{\prime }(\varphi )))\rangle ]\\&\quad \times e^{\theta _{X_0}+X(\varphi )+ \ln \cos (X^{\prime }(\varphi ))}\omega _\varphi ^n\\&=-\int _M(\dot{\varphi }-c(t)-\sqrt{-1}X^{\prime }(\varphi ))(\triangle \theta _Y^{\prime }+\langle \overline{\partial }\theta _Y^{\prime },\overline{\partial }\theta _{X_0}\rangle )\\&\quad \times e^{\theta _{X_0}+X(\varphi )+ \ln \cos (X^{\prime }(\varphi ))}\omega _\varphi ^n+O(\epsilon _0^2)\\&= -\int _M(\dot{\varphi }-c(t)-\sqrt{-1}X^{\prime }(\varphi ))(\triangle _{\omega _{KS}}\theta _Y^{\prime }+\langle \overline{\partial }\theta _Y^{\prime }, \overline{\partial }\theta _{X_0}\rangle _{\omega _{KS}})\\&\quad \times e^{\theta _{X_0}+X(\varphi )+ \ln \cos (X^{\prime }(\varphi )) }\omega _\varphi ^n+O(\epsilon _0^2). \end{aligned}$$
Note that
$$\begin{aligned} \triangle _{\omega _{KS}}\theta _Y^{\prime }+\langle \overline{\partial }\theta _Y^{\prime }, \overline{\partial }\theta _{X_0}\rangle _{\omega _{KS}}=\triangle _{\omega _{KS}}\theta _Y^{\prime }+\langle \overline{\partial }\theta _{X_0}, \overline{\partial }\theta _Y^{\prime }\rangle _{\omega _{KS}}=-\theta _Y^{\prime }. \end{aligned}$$
Therefore, combining the above two relations with (3.9), one will get (3.7). \(\square \)
By (3.7), one sees that there exist two constants \(C\) and \(A_0\) such that for any \(t\ge 1\) and any \(\theta _Y^{\prime }\in \text{ ker}(P,\omega _{KS})\) in Lemma 3.2
$$\begin{aligned}&\left|\int _M \theta _Y^{\prime } (\dot{\varphi }-c(t))e^{\theta _{X_0}}\omega _{KS}^n\right|\nonumber \\&\quad \le C\epsilon _0\int _M|\dot{\varphi } -c(t)|e^{\theta _{X_0}}\omega _{KS}^n+A_0\epsilon _0^2. \end{aligned}$$
(3.11)
Now as same as in the proof of Lemma , we decompose \(\dot{\varphi }_t-c(t)\) as \(\phi +\phi ^{\prime }\) with \(\phi \in \Lambda _1(M,\omega _{KS})\cup \mathbb{R }\) and \(\phi ^{\prime }\in \Lambda _1^\bot (M,\omega _{KS})\), where \(\Lambda _1(M,\omega _{KS})\) is an orthogonal complementary space of \(\Lambda _1(M,\omega _{KS})\cup \mathbb{R }\) with respect to \(L^2\)-weighted product in the following sense,
$$\begin{aligned} \int _M \psi \psi ^{\prime } e^{\theta _{X_0}} \omega _{KS}^n=0,\quad \forall ~\psi \in \Lambda _1(M,\omega _{KS}), \psi ^{\prime }\in \Lambda _1^\bot (M,\omega _{KS}). \end{aligned}$$
Then from (3.11), we get
$$\begin{aligned} \int _M|\phi |^2 e^{\theta _{X_0}}\omega _{KS}^n \le C^{\prime }\epsilon _0^2 \int _M(\dot{\varphi }-c(t))^2e^{\theta _{X_0}}\omega _{KS}^n + 2nA_0^2\epsilon _0^4, \end{aligned}$$
and so
$$\begin{aligned}&\int _M|\phi ^{\prime }|^2e^{\theta _{X_0}}\omega _{KS}^n\nonumber \\&\quad \ge (1-C^{\prime }\epsilon _0^2) \int _M(\dot{\varphi }-c(t))^2e^{\theta _{X_0}} \omega _{KS}^n- 2nA_0^2\epsilon _0^4. \end{aligned}$$
(3.12)
On the other hand, since \(\phi ^{\prime }\in \Lambda _1(M,\omega _{KS})\), we have
$$\begin{aligned} \int _M|\nabla \phi ^{\prime }|^2 e^{\theta _{X_0}}\omega _{KS}^n \ge (1+\delta _0) \int _M (\phi ^{\prime })^2 e^{\theta _{X_0}}\omega _{KS}^n, \end{aligned}$$
where \(\delta _0\) is the gap of the first two nonzero eigenvalues of operator \(P\). Thus by (3.12), we obtain
$$\begin{aligned}&\int _M |\nabla (\dot{\varphi } -c(t))|^2 e^{\theta _{X_0}}\omega _{KS}^n \\&\quad \ge (1+\delta _0)(1-C^{\prime }\epsilon _0^2) \int _M(\dot{\varphi } -c(t))^2 e^{\theta _{X_0}}\omega _{KS}^n-2 nA_0^2\epsilon _0^4. \end{aligned}$$
It follows
$$\begin{aligned}&\int _M |\nabla (\dot{\varphi } -c(t))|^2 e^{\theta _{X_0}+X(\varphi )}\omega _{\varphi }^n \nonumber \\&\quad \ge (1+\delta _0)(1-C^{\prime \prime }\epsilon _0) \int _M(\dot{\varphi } -c(t))^2 e^{\theta _{X_0}+X(\varphi )}\omega _{\varphi }^n- 2nA_0^{\prime 2}\epsilon _0^4 \end{aligned}$$
(3.13)
for some uniform constants \(C^{\prime \prime }\) and \(A_0^{\prime }\).

By (3.13), we get the following \(L^2\)-estimate of \(\dot{\varphi }\).

Lemma 3.3

Let \(\epsilon _0<<1\). Then
$$\begin{aligned} \tilde{H}_0(t)\le \tilde{H}_0(1) e^{-\theta t}+ \frac{B_0}{\theta }\epsilon _0^3, \quad \forall t\in [1,T), \end{aligned}$$
(3.14)
if \(\tilde{H}_0(t)\ge \frac{B_0}{\theta }\epsilon _0^3\) and \( \varphi _t\in \overline{\mathcal{K }}(\epsilon _0)\). Here the constant \(B_0=B_0(\Vert X^{\prime }\Vert _{C^0})\) depends only on \(\Vert X^{\prime }\Vert _{C^0}\) and the constant \(\theta >0\) depends only on the gap of the first two nonzero eigenvalues of operator \(P\).

Proof

First we note that under the condition that \(\varphi = \varphi _t\in \overline{\mathcal{K }}(\epsilon _0)\), one can prove that (2.8) in Lemma 2.2 is still true when \(t\ge 1\). In particular,
$$\begin{aligned} |\triangle _\varphi \dot{\varphi }+X(\dot{\varphi })|\le A \epsilon _0, \quad \text{ and}\quad |\dot{\varphi }-c(t)|, |X^{\prime }(\dot{\varphi })|, |\nabla {\dot{\varphi }}|=O(\epsilon _0). \end{aligned}$$
Then by inserting (3.13) into (3.6), we obtain
$$\begin{aligned}&\frac{ d \tilde{H}_0(t)}{dt}\nonumber \\&\quad \le - 2[(1-2\epsilon _0)(1+\delta _0)(1-C^{\prime \prime }\epsilon _0)-(1+A\epsilon _0)] \tilde{H}_0(t)+ B_0\epsilon _0^3\nonumber \\&\quad \le -\theta \tilde{H}_0(t)+ B_0\epsilon _0^3, \quad \forall ~t\in [1,T), \end{aligned}$$
(3.15)
where the constant \(B_0=B_0(\Vert X^{\prime }\Vert _{C^0})\) depends only on \(\Vert X^{\prime }\Vert _{C^0}\) and \(\theta = 2[(1-2\epsilon _0)(1+\delta _0)(1-C^{\prime \prime }\epsilon _0)-(1+A\epsilon _0)]\ge \delta _0\). It follows
$$\begin{aligned} \frac{ d (\tilde{H}_0(t)-\frac{B_0\epsilon _0^3}{\theta })}{dt}\le -\theta \left(\tilde{H}_0(t)-\frac{B_0\epsilon _0^3}{\theta }\right), \quad \forall ~t\in [1,T). \end{aligned}$$
Since \(\tilde{H}_0(t)\ge \frac{B_0\epsilon _0^3}{\theta }\), we get
$$\begin{aligned} \tilde{H}_0(t)&\le e^{-\theta t}\left( \tilde{H}_0(1)-\frac{B_0\epsilon _0^3}{\theta }\right)+ \frac{B_0\epsilon _0^3}{\theta }\\&\le e^{-\theta t} \tilde{H}_0(1)+ \frac{B_0\epsilon _0^3}{\theta }, \quad \forall ~t\in [1,T). \end{aligned}$$
\(\square \)

Remark 3.4

From (3.6) and (3.9), we see that if in addition that the initial Kähler potential \(\psi \) is \(K_{X_0}\)-invariant, then (3.14) can be improved to
$$\begin{aligned} \tilde{H}_0(t)\le \tilde{H}_0(1) e^{-\theta t}\!, \quad \forall t\in [1,T), \end{aligned}$$
whenever \(\varphi _t\) lies in \(\overline{\mathcal{K }}(\epsilon _0)\).
To get \(W^{k,2}\)-estimates (\(k\ge 1\)) for \(\dot{\varphi }\), we introduce
$$\begin{aligned} \tilde{H}_k(t)=\int _M|\nabla ^k\dot{\varphi }|^2e^{\theta _{X_0}+X(\varphi )}\omega _\varphi ^n. \end{aligned}$$
Then

Lemma 3.5

Let \(\epsilon _0<<1\) and \(k\ge 1\). Then there exist two uniform constants \(\theta ^{\prime }\) and \( B>0\) which depend only on the metric \(\omega _{KS}\) and \(k\) such that for any \(t\in [1,T)\),
$$\begin{aligned} \tilde{H}_k(t)+B\tilde{H}_0(t))\le e^{-\theta ^{\prime } t} (\tilde{H}_k(1)+B\tilde{H}_0(1))+\frac{B_0B}{\theta ^{\prime }}\epsilon _0^3, \end{aligned}$$
(3.16)
if \(\tilde{H}_k(t)+B\tilde{H}_0(t))\ge \frac{B_0B}{\theta ^{\prime }}\epsilon _0^3\) and \(\varphi _t\in \overline{\mathcal{K }}(\epsilon _0)\) for any \(t\le T\), where \(B_0\) is the constant determined in Lemma 3.3.

Proof

First we notice that similarly to (2.20) in Sect. 1 it holds,
$$\begin{aligned} \frac{d|\nabla ^k \dot{\varphi }|^2}{dt} \le 2\sum g^{^{\prime }i_1j_1}\ldots g^{^{\prime }i_kj_k}\frac{d{\dot{\varphi }}_{i_1\ldots i_k}}{dt}{\dot{\varphi }}_{j_1\ldots j_k} +C_1\sum _{i=1}^k|\nabla ^i\dot{\varphi }|^2. \end{aligned}$$
Then by the Sobolev interpolatation theorem we have,
$$\begin{aligned}&\frac{ d \tilde{H}_k(t)}{dt}\\&\quad =\int _M \frac{d|\nabla ^k\dot{\varphi }|^2}{dt}e^{\theta _{X_0}+X(\varphi ) }\omega _\varphi ^n+\int _M |\nabla ^k\dot{\varphi }|^2(\triangle \dot{\varphi }+X(\dot{\varphi })) e^{\theta _{X_0}+X(\varphi ) }\omega _\varphi ^n\\&\quad \le -2\tilde{H}_{k+1}(t)+C_1^{\prime }\tilde{H}_k(t)\nonumber +C_2\tilde{H}_0(t)\\&\quad \le -\theta ^{\prime } \tilde{H}_{k}(t)+C_3\tilde{H}_0(t) , \quad \forall ~t\in [1,T). \end{aligned}$$
On the other hand, by (3.15), one sees
$$\begin{aligned} \frac{d \tilde{H}_0(t)}{dt}\le -\theta \tilde{H}_0(t)+ B_0\epsilon _0^3, \quad \forall ~t\in [1,T). \end{aligned}$$
Thus combining the above two inequalities, we derive
$$\begin{aligned} \frac{d( \tilde{H}_k(t)+B\tilde{H}_0(t))}{dt}&\le -\theta ^{\prime } [\tilde{H}_{k}(t)+\frac{(B\theta -C_3)}{\theta ^{\prime }}\tilde{H}_0(t)]+B_0B\epsilon _0^3\\&\le -\theta ^{\prime } (\tilde{H}_k(t)+B\tilde{H}_0(t))+B_0B\epsilon _0^3 \end{aligned}$$
as long as \(\theta ^{\prime }<\theta \). Here \(B\) is a sufficiently large constant independent of \(\epsilon _0\). (3.16) follows from the above differential inequality immediately (see an argument in the proof of Lemma 3.3). \(\square \)

By the Sobolev embedding theorem, we get

Corollary 3.6

Let \(\epsilon _0<<1\). Then for any positive \(\alpha <1\), there exist an integer \(k_0>n\) and three uniform constants \(\theta _0\), \(B\) and \(C_0\) which depend only on \(\alpha \) and the metric \(\omega _{KS}\) such that
$$\begin{aligned} \Vert \dot{\varphi _t}-c(t) \Vert _{C^{\alpha }}\le C_0 [\tilde{H}_{k_0}(1) +B\tilde{H}_0(1))^{\frac{1}{2}}e^{-\theta _0 t}+ \epsilon _0^{\frac{3}{2}}], \quad \forall ~t\in [1,T), \end{aligned}$$
(3.17)
if \(\tilde{H}_{k_0}(t)+B\tilde{H}_0(t)) \ge C_0\epsilon _0^3\) and \(\varphi _t\in \overline{\mathcal{K }}(\epsilon _0)\) for any \(t\le T\). Consequently, under the same condition above,
$$\begin{aligned} \Vert \dot{\varphi _t}-c(t) \Vert _{C^{\alpha }}\le C_0^{\prime }( \epsilon _0 e^{-\theta _0 t}+ \epsilon _0^{\frac{3}{2}}), \quad \forall ~t\in [1,T). \end{aligned}$$
(3.18)

Proof

We suffice to prove the second relation (3.18) in Corollary 3.6. Notice that by (3.10) we have
$$\begin{aligned} \Vert \varphi _t-\underline{\varphi _t}\Vert _{C^{k_0+2}}=O(\epsilon _0), \quad \forall ~t\in [1,T). \end{aligned}$$
It follows by (3.4),
$$\begin{aligned} \Vert \dot{\varphi _t}-c(t) \Vert _{C^{k_0}}=O(\epsilon _0), \quad \forall ~t\in [1,T). \end{aligned}$$
Also similarly to (2.11), we have
$$\begin{aligned} \Vert \dot{\varphi _t}-c(t) \Vert _{C^{0}}=O(\epsilon _0), \quad \forall ~t\in [1,T). \end{aligned}$$
Thus in particular,
$$\begin{aligned} \tilde{H}_{k_0}(1)+B\tilde{H}_0(1))=O(\epsilon _0^2). \end{aligned}$$
(3.19)
(3.18) follows from (3.17) immediately. \(\square \)

Remark 3.7

By Remark 3.4, according to the proof of Lemma 3.5, we see that if in addition that the initial Kähler potential \(\psi \) is \(K_{X_0}\)-invariant, then (3.16) can be improved to
$$\begin{aligned} \tilde{H}_k(t)\le (\tilde{H}_0(1)+B\tilde{H}_k(1)) e^{-\theta ^{\prime } t}, \quad \forall ~t\in [1,T), \end{aligned}$$
whenever \(\varphi _t\) lies in \(\overline{\mathcal{K }}(\epsilon _0)\). Thus we get
$$\begin{aligned} \Vert \dot{\varphi _t}-c(t)\Vert _{C^k}\le B_ke^{-\theta _k t}, \quad \forall ~t\in [1,T), \end{aligned}$$
(3.20)
where \(k\ge 0\) is any integer and \(B_k, \theta _k\) are two uniform constants.

Next we estimate \(\varphi =\varphi _t\). We begin with

3.8

Suppose that
$$\begin{aligned} \Vert \tilde{\dot{\varphi }}-c(t)\Vert _{C^\alpha }\le C_0\epsilon _0^{\frac{3}{2}}, \end{aligned}$$
where \(C_0\) is a fixed constant. Then there exist a \(\sigma \in \text{ Aut}_r(M)\) with \(\text{ dist}(\sigma ,\text{ Id})\)\(\le C\) for some uniform constant \(C\) such that
$$\begin{aligned} \Vert \varphi _{\sigma }-\underline{\varphi _{\sigma }}\Vert _{C^{2,\alpha }}\le AC_0\epsilon _0^{\frac{3}{2}}, \end{aligned}$$
(3.21)
if \(\varphi \in \overline{\mathcal{K }}(\epsilon _0)\), where \(\varphi _{\sigma }=(\varphi _t)_\sigma \) is a modified Kähler potential of \(\varphi _t\) defined by \(\sigma ^\star \omega _{g_t}=\omega _{KS}+\sqrt{-1}\partial \overline{\partial } (\varphi _t)_\sigma \) and \(A\) is a uniform constant depending only on the metric \(g_{KS}\).

Proof

The proof is a modification of one of Lemma 2.2 and we need to use the gauge transformation group \(\text{ Aut}_r(M)\). In fact by Proposition 3.10 in Appendix, we see that there exists \(\sigma \) with bounded \(\text{ dist}(\sigma ,\text{ Id})\) such that for any \(Y\in \eta _r(M)\) with \(\int _M|Y|^2\omega _{KS}^n=1\), it holds
$$\begin{aligned} \left|\int _M \theta _Y^{\prime } \tilde{\varphi }_\sigma e^{\theta _{X_0}}\omega _{KS}^n\right|\le O(\epsilon _0^2), \end{aligned}$$
(3.22)
where \(\tilde{\varphi }_\sigma =\varphi _{\sigma }-\frac{1}{V}\int _M\varphi _{\sigma }e^{\theta _{X_0}}\omega _{KS}^n \) and \(\varphi _{\sigma }=\sigma ^*\varphi +\rho _\sigma \). We can decompose \(\tilde{\varphi _\sigma }\) as \(\tilde{\varphi _\sigma }=\phi +\phi ^{\prime }\) with \(\phi \in \Lambda _1(M,\omega _{KS})\) and \(\phi ^{\prime }\in \Lambda _1^\bot (M,\omega _{KS})\). Then \(\phi =\sum _i a_i\theta _i\) for some constants \(a_i\), where \(\{\theta _i\}\) is a basis of the space \(\Lambda _1(M,\omega _{KS})\). It is clear by (3.22) that \(a_i=O(\epsilon _0^2)\) and so
$$\begin{aligned} \Vert \phi \Vert _{C^{2,\alpha }}\le O(\epsilon _0^2). \end{aligned}$$
(3.23)
Since \(\rho =\rho _\sigma \) satisfies equation,
$$\begin{aligned} \omega _\rho ^n=\omega _{KS}^n e^{-\rho -X(\rho )+c}, \end{aligned}$$
for some constant \(c\), by Eq. (3.4), we have
$$\begin{aligned} \omega _{\tilde{\varphi }_\sigma }^n=\omega _{KS}^ne^{\sigma ^*(\tilde{\dot{\varphi }})-\tilde{\varphi }_\sigma -X(\tilde{\varphi }_\sigma )+b}, \end{aligned}$$
for some constant \(b\), where \(\tilde{\dot{\varphi }}=\dot{\varphi }_t-c(t)\). Then \(\phi ^{\prime }\) is a solutions of equation
$$\begin{aligned} P\left[\log \left(\frac{[\omega _{\phi +\phi ^{\prime }}]^n}{\omega _{KS}^n}\right)\right] +\varphi ^\bot +X(\phi ^{\prime })=P[\sigma ^*(\tilde{\dot{\varphi }})+b-X(\phi )], \end{aligned}$$
where \(P\) is a projection from Banach space \(C^{2,\alpha }(M)\) to Banach space \(C^\alpha (M)\)\(\cap \Lambda _1^\bot (M,\omega _{KS})\), and \(\phi \) and \(\sigma ^*(\tilde{\dot{\varphi }})\) are regarded as two perturbation functions. Note that
$$\begin{aligned} \Vert P[\sigma ^*(\tilde{\dot{\varphi }})+b-X(\phi )] \Vert _{C^\alpha }=\Vert P[\sigma ^*(\tilde{\dot{\varphi }})-X(\phi )] \Vert _{C^\alpha }\le aC_0\epsilon _0^{\frac{3}{2}} \end{aligned}$$
for some constant \(a\). Thus by the Implicit Functional Theorem, we get
$$\begin{aligned} \Vert \phi ^{\prime }\Vert _{C^{2,\alpha }}\le a^{\prime }(\Vert P[\sigma ^*(\tilde{\dot{\varphi }})+b-X(\phi )] \Vert _{C^\alpha } + \Vert \phi \Vert _{C^{2,\alpha }})\le \frac{A}{2}C_0\epsilon _0^{\frac{3}{2}}. \end{aligned}$$
(3.24)
Combining this estimate with (3.23), we obtain (3.21). \(\square \)

Since the solution \(\varphi _t\) of (3.4) grows at most exponentially, it is easy to see that for any large number \(T\) one can choose a big enough number \(N\) so that the potential \(\varphi _t\) lies in \(\mathcal{K }(\epsilon _0)\) for any \(t<T\) as long as the initial potential \(\psi \) is belonged to \(\mathcal{K }(\frac{\epsilon _0}{N})\). Moreover we prove

Proposition 3.9

There exist a small \(\epsilon _0\) and a large number \(N\) such that if the initial data \(\psi \) in (3.4) lies in \(\mathcal{K }(\frac{\epsilon _0}{N})\) then there exist a family of \(\sigma _t\in \text{ Aut}_r(M)\) such that
$$\begin{aligned} \Vert \varphi _{\sigma _t}-\underline{\varphi _{\sigma _t}}\Vert _{C^{2,\alpha }}\le \epsilon _0,\quad \, \forall ~t\ge 0. \end{aligned}$$
(3.25)

Proof

We first notice that under the assumption that \(\psi \) in (3.4) lies in \(\mathcal{K }(\frac{\epsilon _0}{N})\), by using the argument in the proof of Corollary 3.6, one can prove
$$\begin{aligned} \tilde{H}_{k_0}(1)+B\tilde{H}_0(1)=O\left(\left(\frac{\epsilon _0}{N}\right)^2\right)\!, \end{aligned}$$
where the integer \(k_0>n\) and the constant \(B\) are determined in Corollary 3.6. In fact, the above estimate is similar to (3.19) with small \(\epsilon _0\) order replaced by \(\frac{\epsilon _0}{N}\). Let \(C_0\) and \(\theta _0\) be two another constants determined in Corollary 3.6 and \(N_0\) a big enough number with \(N_0\le N\le \frac{1}{\epsilon _0 ^{1/4}}\) and \(C_0e^{-\theta _0 T}\le \frac{1}{N_0}\). In the following, we deal with the proof of Proposition 3.9 in three cases.
Case 1. There exists some time \(t_0\in [0,T]\) such that
$$\begin{aligned} \tilde{H}_{k_0}(t_0)+B\tilde{H}_0(t_0)= C_0\epsilon _0^3. \end{aligned}$$
(3.26)
Then by the Sobolev embedding theorem, we have at \(t=t_0\),
$$\begin{aligned} \Vert \dot{\varphi _t}-c(t) \Vert _{C^{\alpha }}\le C_0^{\prime }\epsilon _0^{\frac{3}{2}}. \end{aligned}$$
Thus by Lemma 3.8, there exists \(\sigma _{t_0}\in \text{ Aut}_r(M)\) such that
$$\begin{aligned} \Vert \psi ^{\prime }\Vert _{C^{2,\alpha }}\le AC_0^{\prime }\epsilon _0^{\frac{3}{2}} < \frac{\epsilon _0}{2N}, \end{aligned}$$
(3.27)
where \(\psi ^{\prime }=\varphi _{\sigma _{t_0}}-\underline{\varphi _{\sigma _{t_0}}}\) and \( \varphi _{\sigma _{t_0}}=\sigma _{t_0}^\star \varphi _{t_0}+\rho _{\sigma _{t_0}}\) is a modified Kähler potential of \(\varphi _{t_0}\) by \(\sigma _{t_0}\). Now we consider a new flow \((\varphi _t^{\prime }; \psi ^{\prime })\) of (3.4) with a replaced initial Kähler potential \(\psi ^{\prime }\). Since \(N=O(\frac{1}{\epsilon _0 ^{1/4}})\) and \(\Vert \psi ^{\prime }\Vert _{C^{2,\alpha }}= O(\epsilon _0^{\frac{3}{2}})\), by the choice of \(T\) we see that either solution \(\varphi ^{\prime }_t\) lies in \(\mathcal K (\frac{\epsilon _0}{N+1})\) for any \(t\) or there exists some time \(T_0>T\) such that \(\Vert \varphi ^{\prime }_{T_0}\Vert _{C^{2,\alpha }}=\frac{\epsilon _0}{N+1}\) for solution \(\varphi ^{\prime }_{T_0}\) of \((\varphi _t^{\prime }; \psi ^{\prime })\) at time \(T_0\). If the first case happens, then we will finish the proof of Proposition 3.9. If the second case happens, then we can continue to consider a flow \((\varphi _t^{(2)}; \psi ^{(2)})\) of (3.4) with a replaced initial Kähler potential \(\psi ^{(2)}=\varphi ^{\prime }_{T_0}\) as in the beginning of discussion for the original flow \((\varphi _t^{(1)}; \psi ^{(1)})=(\varphi _t; \psi )\).
Case 2.
$$\begin{aligned} \tilde{H}_{k_0}(t)+B\tilde{H}_0(t)) < C_0\epsilon _0^3,\quad \forall ~t\le T. \end{aligned}$$
Then
$$\begin{aligned} \tilde{H}_{k_0}(t)+B\tilde{H}_0(t) < C_0\epsilon _0^3,\quad \forall ~t>0, \end{aligned}$$
(3.28)
or there exists some time \(t_0>T\) such that
$$\begin{aligned} \sup _{t\le t_0} (\tilde{H}_{k_0}(t)+B\tilde{H}_0(t)) =\tilde{H}_{k_0}(t_0)+B\tilde{H}_0(t_0)= C_0\epsilon _0^3. \end{aligned}$$
(3.29)
If (3.28) is satisfied, then as in Case 1, we see that there exists a family of \(\sigma _t\in \text{ Aut}_r(M)\) such that (3.27) is satisfied for modified Kähler potential \(\varphi _{\sigma _{t}}\) of \(\varphi _t\) by \(\sigma _{t}\) for any \(t>0\). This will finish the proof of Proposition 3.9. If (3.29) is satisfied, then Case 2 becomes Case 1 while we need to consider the flow (3.4) starting at \(t_0\) as in Case 1.
Case 3.
$$\begin{aligned} \tilde{H}_{k_0}(t)+B\tilde{H}_0(t)) \ge C_0\epsilon _0^3,\quad \forall ~t\le T. \end{aligned}$$
(3.30)
Recall that in this case there exists a uniform constant \(C_1\) by (3.17) in Corollary 3.6 such that
$$\begin{aligned} \Vert \tilde{\varphi }_{T} -c(T)\Vert _{C^\alpha } \le \frac{C_1\epsilon _0}{NN_0}. \end{aligned}$$
(3.31)
As in the proof of Lemma 3.8, we see that there exists \(\sigma =\sigma _{T}\) with bounded \(\text{ dist}(\sigma ,\text{ Id})\) such that \(\tilde{\varphi }_{\sigma }=\sigma ^*\varphi +\rho _\sigma - \frac{1}{V}\int _M ( \sigma ^*\varphi +\rho _\sigma )e^{\theta _X}\omega _{KS}^n\) can be decomposd as \(\tilde{\varphi }_\sigma =\phi +\phi ^{\prime }\) with properties: 1) \(\phi ^{\prime }\in \Lambda _1^\bot (M,\omega _{KS})\) and \(\phi \in \Lambda _1(M,\omega _{KS})\) with
$$\begin{aligned} \Vert \phi \Vert _{C^{2,\alpha }}\le O(\epsilon _0^2); \end{aligned}$$
(3.32)
2) \(\phi ^{\prime }\) satisfies an equation,
$$\begin{aligned} P\left[\log \left(\frac{[\omega _{\phi +\phi ^{\prime }}]^n}{\omega _{KS}^n}\right)\right] +\phi ^{\prime }+X(\psi )=P[\sigma ^*(\tilde{\dot{\varphi }})+b-X(\phi )], \end{aligned}$$
where \(b\) is a constant, \(\tilde{\dot{\varphi }}=\dot{\varphi }_t-c(t)\) and \(P\) is a projection from Banach space \(C^{2,\alpha }(M)\) to Banach space \(C^\alpha (M)\cap \Lambda _1^\bot (M,\omega _{KS}).\) On the other hand, by (3.31) and (3.32), we have
$$\begin{aligned} \Vert P[\sigma ^*(\tilde{\dot{\varphi }})+b-X(\phi )] \Vert _{C^\alpha }=\Vert P[\sigma ^*(\tilde{\dot{\varphi }})-X(\phi )]\Vert _{C^\alpha } \le \frac{C_1^{\prime }\epsilon _0}{NN_0}. \end{aligned}$$
Thus by the implicit functional theorem, we get
$$\begin{aligned} \Vert \phi ^{\prime }\Vert _{C^{2,\alpha }}\le A^{\prime }(\Vert P[\sigma ^*(\tilde{\dot{\varphi }})+b-X(\phi )] \Vert _{C^\alpha } + \Vert \phi \Vert _{C^{2,\alpha }})\le \frac{2A^{\prime }C_1^{\prime }\epsilon _0}{NN_0}, \end{aligned}$$
(3.33)
where \(A^{\prime }\) is a uniform constant depending only on \(\omega _{KS}\). Combining (3.32) and (3.33), we get
$$\begin{aligned} \Vert \tilde{\varphi }_\sigma \Vert _{C^{2,\alpha }}\le \frac{4A^{\prime }C_1^{\prime }\epsilon _0}{NN_0}. \end{aligned}$$
Consequently,
$$\begin{aligned} \Vert \varphi _\sigma -\underline{\varphi _\sigma } \Vert _{C^{2,\alpha }}\le \frac{8A^{\prime }C_1^{\prime }\epsilon _0}{NN_0}<\frac{\epsilon _0}{N}. \end{aligned}$$
(3.34)
This shows \(\varphi _\sigma \in \mathcal{K }(\frac{\epsilon _0}{N})\) when \(N_0\) is large enough.
Next we consider a flow \((\varphi _t^{(2)}; \psi ^{(2)})\) of Eq. (3.4) with \(\psi ^{(2)}=\varphi _{\rho }-\underline{\varphi _\sigma } \) as an initial potential to replace \(\psi \). By the discussion in Case 1, we may assume that the condition (3.30) still holds for the flow \((\varphi _t^{(2)}; \psi ^{(2)})\). Then similarly to derive the estimate (3.34), one sees that there exists \(\sigma _{(2)}\in \text{ Aut}_r(M)\) such that
$$\begin{aligned} \Vert \varphi _{\sigma _{(2)}}^{(2)}-\underline{\varphi _{\sigma _{(2)}}^{(2)}}\Vert _{C^{2,\alpha }} \le \left(\frac{8A^{\prime }C_1^{\prime }}{N_0}\right)^2 \frac{\epsilon _0}{N}<\frac{\epsilon _0}{N}. \end{aligned}$$
(3.35)
Repeating to use the above step for finite times, we conclude that there exist some integer \(l\ge 1\) and some time \(t_0^{\prime }\in (0, T]\) such that
$$\begin{aligned} \Vert \varphi _{\sigma _{t_0^{\prime }}}^{(l)}-\underline{\varphi _{\sigma _{t_0^{\prime }}}^{(l)}}\Vert _{C^{2,\alpha }} =O\left(\epsilon _0^{\frac{3}{2}}\right)<\frac{\epsilon _0}{2N}, \end{aligned}$$
and \(\varphi _t^{(l)}\in \mathcal K (\epsilon _0)\) for any \(t\le t_0^{\prime }\). Here \(\varphi _{t}^{(l)}\) is a solution of Eq. (3.4) with \(\psi ^{(l-1)}=\varphi _{\sigma _{(l-1)}}^{(l-1)}-\underline{\varphi _{\sigma _{(l-1)}}^{(l-1)}}\) as an initial potential to replace \(\psi \), and \(\varphi _{\sigma _{t_0^{\prime }}}^{(l)}\) is a modified Kähler potential of \(\varphi _t^{(l)}\) at \(t=t_0^{\prime }\) by a holomorphism \(\sigma _{t_0^{\prime }}\in \text{ Aut}_r(M)\). Now we consider a new flow \((\varphi _t^{\prime }; \psi ^{\prime })\) of (3.4) with a replaced initial Kähler potential \(\psi ^{\prime }=\varphi _{\sigma _{t_0^{\prime }}}^{(l)}-\underline{\varphi _{\sigma _{t_0^{\prime }}}^{(l)}}\). By the argument in Case 1, we see that either the Kähler potentials \(\varphi ^{\prime }_t\) lies in \(\mathcal K (\frac{\epsilon _0}{N+1})\) for any \(t\) or there exists a flow \((\varphi _t^{(l+1)}; \psi ^{(l+1)})\) of (3.4) with a replaced initial Kähler potential \(\psi ^{(l+1)}=\varphi ^{\prime }_{T_0}\), where \(\varphi ^{\prime }_{T_0}\) is a solution of the flow \((\varphi _t^{\prime (1)}; \psi ^{\prime (1)})=(\varphi _t^{\prime }; \psi ^{\prime })\) at \(t=T_0>T\), which satisfies \(\Vert \varphi ^{\prime }_{T_0}\Vert _{C^{2,\alpha }}=\frac{\epsilon _0}{N+1}\).
In Case 1 and Case 3, we have constructed the flows \((\varphi _t^{(2)}; \psi ^{(2)})\) and \((\varphi _t^{(l+1)}; \psi ^{(l+1)}) (l\ge 1)\), respectively. By the choice of initial Kähler potentials \(\psi ^{(2)}\) and \(\psi ^{(l+1)}\), one can check that (3.30) is satisfied for both flow \((\varphi _t^{(2)}; \psi ^{(2)})\) and \((\varphi _t^{(l+1)}; \psi ^{(l+1)})\) at least for small \(t\) compared to the estimate (3.27 ). This implies that the steps in Case 1 and Case 3 can be repeated. Combining the step in Case 2, we can finally construct a sequence of flows
$$\begin{aligned}&(\varphi _t^{(1)}; \psi ^{(1)}), (\varphi _t^{(2)}; \psi ^{(2)}), \ldots \ldots , (\varphi _t^{(L)};\psi ^{(L)}),\\&(\varphi _t^{\prime (1)}; \psi ^{\prime (1)}), (\varphi _t^{\prime (2)}; \psi ^{\prime (2)}), \ldots \ldots , (\varphi _t^{\prime (L^{\prime })};\psi ^{\prime (L^{\prime })}), \end{aligned}$$
which satisfy the following properties: (1) \(L\ge L^{\prime }\) and the integer \(L^{\prime }\) may be taken to be infinite; (2) there exists a family of \(\sigma _t\) such that (3.27) is satisfied for modified Kähler potentials \(\varphi _{\sigma _{t}}\) of \(\varphi _t\) by \(\sigma _{t}\) for any \(t>0\), if \(L=1\) and \(L^{\prime }=0\); (3) \(\varphi _t^{(l)}\in \mathcal{K }(\epsilon _0)\), for any \( t\le T\), \(l=1,\ldots ..,L\); (4) \(\varphi _t^{\prime (l)}\in \mathcal{K }(\frac{\epsilon _0}{N+1})\), for any \( t\le T_0\)\((>T)\), \(l=1,\ldots ..,L^{\prime }-1\); (5) \(\varphi _t^{\prime (L^{\prime })}\in \mathcal{K }(\frac{\epsilon _0}{N+1})\), for any \( t\ge 0\), if \(L^{\prime }\) is finite. Therefore by connecting the above flows we conclude that there exist a family of \(\sigma _t\in \text{ Aut}_r(M)\) such that (3.25) is satisfied for the original flow \((\varphi _t; \psi )\). \(\square \)

Now we can finish the proof of Theorem 2.1.

Completion of Proof of Theorem 2.1

By proposition 3.9, we suffice to do higher-order estimates for the modified Kähler potentials \(\varphi _\sigma =\varphi _{\sigma _t}\) in Proposition 3.9. We will choose a family of modified holomorphic transformations \(\overline{\sigma }_t\in \text{ Aut}_r(M)\) of \(\sigma _t\) (\(0< t<\infty \), \(\overline{\sigma }_0=\text{ Id}\)) as in [3] and [14] such that for any \(t\in (0,\infty )\)
$$\begin{aligned} \text{ dist} (\sigma _t^{-1}\overline{\sigma }_t, \text{ Id})\le C, \end{aligned}$$
and
$$\begin{aligned} \left\Vert(\overline{\sigma }_t^{-1})_*\frac{\partial \overline{\sigma }_t}{\partial t}\right\Vert_{g_{KS}}\le C, \end{aligned}$$
where \((\overline{\sigma }_t^{-1})_*\frac{\partial \overline{\sigma }_t}{\partial t}=\overline{X}_t\in \eta _r(M)\) is a family of holomorphic vector fields on \(M\). Furthermore, for any \(k \ge 0\), we may assume that there is a constant \(C_k\) such that
$$\begin{aligned} \left\Vert\frac{\partial ^k \overline{X}_t}{\partial t^k}\right\Vert_{g_{KS}}\le C_k. \end{aligned}$$
Note that the choice of such \(\overline{\sigma }_t\) just depends on the \(C^0\)-estimate of \(\tilde{\varphi }=\widetilde{\varphi _\sigma }=((\sigma _t)^*\varphi _t+\rho _t)-\frac{1}{V}\int _M ((\sigma _t)^*\varphi _t+\rho _t)e^{\theta _{X_0}}\omega _{KS}^n\).
Let \(\overline{\varphi }=\varphi _{\overline{\sigma }_t}=(\overline{\sigma }_t)^*(\varphi _t)+\overline{\rho }_t\) be a family of new modified potentials corresponding to holomorphic transformations \(\overline{\sigma }_t\). Then \(\overline{\varphi }\) satisfies an equation,
$$\begin{aligned} \frac{\partial \overline{\varphi }}{\partial t}=\log \frac{\omega ^n_{\overline{\varphi }}}{\omega _{KS}^n}+\overline{\varphi }+[\text{ re}(\overline{X})](\overline{\varphi }), \quad \, \overline{\varphi }(0)=\psi -\underline{\psi }. \end{aligned}$$
(3.36)
Since
$$\begin{aligned} \sqrt{-1}\partial \overline{\partial }\overline{\phi } =(\rho _t^{-1}\overline{\rho }_t-I)^*\omega _{KS}+ \sqrt{-1}\partial \overline{\partial }((\rho _t^{-1}\overline{\rho }_t)^*\tilde{\phi }), \end{aligned}$$
(3.37)
by Proposition (3.9), we see that there exists a uniform constant \(C\) such that
$$\begin{aligned} \left\Vert\overline{\varphi }-\frac{1}{V}\int _M\overline{\varphi } \omega _{KS}^n\right\Vert_{C^{2,\alpha }}\le C. \end{aligned}$$
This means that (3.36) is a strictly parabolic equation. Differentiating this equation on \(z_k\) in a coordinate chart of \(M\) with local holomorphic coordinates \((z_1,\ldots ,z_n)\), we get
$$\begin{aligned} \left(\Delta -\frac{\partial }{\partial t}\right)\left(\frac{\partial \overline{\phi }}{\partial z_k}\right)= g_{KS}^{i\overline{j}} \frac{\partial [(g_{KS})_{ i\overline{j}}]}{\partial z^k} - g^{i\overline{j}} \frac{\partial [g_{KS})_{i\overline{j}}]}{\partial z^k} -\overline{\phi }_k-([\text{ re}(\overline{X})](\overline{\phi })))_k. \end{aligned}$$
Thus by the regularity theory of parabolic equation, we will derive all \(C^k\)-norms of \((\overline{\varphi }_t-\frac{1}{V}\int _M\overline{\varphi }_t\omega _{KS}^n)\), and so all \(C^k\)-norms of \(\widetilde{\varphi _{\sigma _t}}\).

From the above estimates, we see that for any sequence of Kähler metrics \(\omega _{\tilde{\varphi }_{\sigma _i}}\), there exists a \(C^k\)- limit Kähler metric \(\omega _{\infty }\) of subsequence of \(\omega _{\tilde{\varphi }_{\sigma _i}}\). Applying the Perelman’s \(W\)-function in [8] to the family of \(\omega _{\tilde{\varphi }_{\sigma _t}}\), one concludes that \(\omega _{\infty }\) must be a Kähler-Ricci soliton of \((M,J)\) (cf. [9]). Since the Kähler-Ricci solition is unique, we see that there exists an element \(\tau _\infty \in \text{ Aut}_0(M)\) such that \(\omega _{\infty }=\tau _\infty ^*\omega _{KS}\). By using the fact that the convergent sequence is arbitary, the above implies that there exists a family of \(\tau _t\in \text{ Aut}_0(M)\) such that \(\tau _t^*g\) converge to \(g_{KS}\) smoothly.

If in addition that the initial Kähler potential \(\psi \) is \(K_{X_0}\)-invariant, by Remark 3.7, one can follow the argument in the proof of Proposition 2.7 to see that for any small \(\epsilon _0\), there exist a small number \(\epsilon \) such that \(\varphi _t\in \overline{\mathcal{K }}(\epsilon _0)\) for any \(t>0\) if \(\psi \) satisfies \(\Vert \psi \Vert _{C^{2,\alpha }}\le \epsilon .\) Moreover, \(\varphi _t\) and \(\dot{\varphi }_t -\frac{1}{V}\int _M \dot{\varphi }_t e^{\theta _{X_0}+X(\varphi _t) }\omega _{\varphi _t}^n\) converge exponentially to a potential \(\phi _\infty \) and zero in \(C^\infty \) as \(t\rightarrow \infty \), respectively. Since \(\dot{\varphi }_t\) are modified Ricci potentials of \(\omega _{\varphi _t}\), the limit \(\phi _\infty \) must be a potential of Kähler-Ricci soliton. By the uniqueness of Kähler-Ricci solitons [12], there exists a \(\sigma \in \text{ Aut}_0(M)\) such that Kähler potentials \(((\varphi _t)_\sigma -\underline{(\varphi _t)_\sigma })\) with \(\sigma ^\star \omega _{\varphi _t}=\omega _{KS}+\sqrt{-1}\partial \overline{\partial } (\varphi _t)_\sigma \) converge exponentially to \(0\) as \(t\rightarrow \infty \). \(\square \)

Footnotes
1

Throughout the paper, \(\Vert \psi \Vert _{C^{2,\alpha }}= \sum _\delta \sum _{|l|\le 2} \Vert \frac{\partial ^{l}\psi }{\partial ^{l} x^\delta }\Vert _{C^0(U^\delta )}+\sum _\delta \sum _{|l|=2} \Vert \frac{\partial ^{l}\psi }{\partial ^{l} x^\delta }\Vert _{C^\alpha (U^\delta )}\) denotes the usual Hölder \(C^{2,\alpha }\)-norm for a smooth function \(\psi \) with a fixed local complex coordinates system \(\{(U^\delta ; x^\delta )\}\) on \(M\).

 
2

\(\text{ dist}(\sigma ,\sigma ^{\prime })\) denotes the distance between two pints \(\sigma \) and \(\sigma ^{\prime }\) in the Lie group \(\text{ Aut}_r(M)\) with a non-compact complete Riemannian metric.

 

Acknowledgments

The paper is a revised version of preprint [19] (In case that the underlying manifold is a Fano Käher-Einstein one, Sun and Wang recently proved a more general stability theorem of Kähler-Ricci flow in the sense of Cheeger-Gromov topology [11].) posted in arXiv in 2009 which was partially finished when the author was visiting The Hausdorff Institute of Mathematics, University of Bonn in the Autumn of 2008. The author would like to thank her hospitality and financial support. The author would also like to thank professor Gang Tian and professor Xiuxiong Chen for their valuable discussions. Finally, the author is appreciated to the referee for suggestions to improve presentation of the paper, particularly, valuable discussions on Lemma 1.2.

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