Stability of Kähler-Ricci flow on a Fano manifold

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.

Appendix

The following proposition is about the existence of almost orthonormality of Kähler potential to $\Lambda _1(M,\omega _{KS})$, the space of first eigenvalue-functions of operator $(P,\omega _{KS})$ defined in Lemma 2.2 in Sect. 2. The proposition is crucial in the proof ofProposition 3.9.

Proposition 4.1

Let $M$ be a compact Kähler manifold $M$ with $c_1(M)>0$ which admits a Kähler-Ricci soliton $(\omega _{KS}, X_0)$. Then for any Kähler potential $\phi \in \mathcal{K }(\epsilon _0)$ there exists a $\sigma \in \text{ Aut}_r(M)$ with $\text{ dist}(\sigma ,\text{ Id})\le A$ ² for some uniform constant $A$ such that for any $Y\in \eta _r(M)$ with $\int _M|Y|^2\omega _{KS}^n=1$, it holds

$$\begin{aligned} \left|\int _M \theta _Y^{\prime } (\sigma ^*\phi +\rho _\sigma )e^{\theta _X}\omega _{KS}^n\right|\le C\Vert X^{\prime }(\phi )\Vert _{C^0}^2=O(\epsilon _0^2), \end{aligned}$$

where $\theta _Y^{\prime }\in \text{ ker}(P,\omega _{KS})$ and $\rho _\sigma$ is a Kähler potential defined by

$$\begin{aligned} \sigma ^*(\omega _{KS})=\omega _{KS}+ \sqrt{-1}\partial \bar{\partial }\rho _\sigma \end{aligned}$$

with $\rho _\sigma \in \Lambda _1^\bot (M,\omega _{KS})$.

Proof

This proposition was proved in [12] if $\phi$ is $K_0$-invariant. The key point in the proof is to use a functional defined on a space of Kähler-Ricci solitons

$$\begin{aligned} \{\omega _{KS}^{\prime }=\sigma ^*(\omega _{KE})=\omega _{KS}+ \sqrt{-1}\partial \bar{\partial }\rho _\sigma |\quad \, \sigma \in \text{ Aut}_r(M)\}, \end{aligned}$$

which was introduced in [18] by

$$\begin{aligned}&(I-J)(\omega _\phi ,\omega _{KS}^{\prime })\\&\quad =\int _0^1 dt\int _M \dot{\phi }_te^{\theta _{X_0}(\phi _t)}\omega _{\phi _t}^n -\int _M (-\phi +\rho )e^{\theta _{X_0}+X(\rho )}(\omega _{KS}^{\prime })^n, \end{aligned}$$

where $\phi _t$ is a $K_{X_0}$-invariant path in $\mathcal{M }(\omega _{KS})$ which connects $0$ and $-\phi +\rho$, and $\theta _{X_0}(\phi _t)$ are potentials of $X_0$ associated to metric $\omega _{\phi _t}$ defined by (2.1). It is proved in [18] that $(I-J)(\omega _\phi ,\omega _{KS}^{\prime })$ is well-defined for a $K_0$-invariant $\phi$, i.e., the functional is independent of choice of a path of $K_0$-invariant Kähler potentials. But for a general Kähler potential $\phi$, $(I-J)(\omega _\phi ,\omega _{KS}^{\prime })$ is not well-defined (e.g., see (4.4) below), so we shall introduce another functional defined on whole space $\mathcal{M }(\omega _{KS})$ to replace it. In fact, we will consider the following functional

$$\begin{aligned} \mathcal{F }(\omega _\phi ,\omega _{KS}^{\prime })&= \text{ Re}\left[\int _0^1 dt\int _M (-\phi +\rho _\sigma )e^{\theta _{X_0}(t(-\phi +\rho _\sigma ))}\omega _{t(-\phi +\rho _\sigma )}^n\right.\nonumber \\ \quad&-\left.\int _M (-\phi +\rho )e^{\theta _{X_0}^{\prime }}(\omega _{KS}^{\prime })^n\right]. \end{aligned}$$

(4.1)

Clearly, the definition of $\mathcal{F }$ just uses a real part of $(I-J)(\omega _\phi ,\omega _{KS}^{\prime })$ while a path of Kähler potentials is chosen by $\phi _t=t(-\phi +\rho _\sigma )$. We now consider a path of Kähler potentials $\rho _t$ induced by an one-parameter subgroup $\sigma _t$ generated by the real part of $Y\in \eta _r(M)$, i.e. $\rho _t$ are defined by $\omega _t=\sigma _t^*\omega _{KS}^{\prime }=\omega _{KS}^{\prime }+\sqrt{-1}\partial \bar{\partial }\rho _t$. Let

$$\begin{aligned} f_Y(t)=\text{ Re}\left[\int _0^{1+t} ds\int _M (\dot{\phi }_s)e^{\theta _{X_0}(\phi _s)}\omega _{\phi _s}^n -\int _M (-\phi +\rho _t)e^{\theta _{X_0}(\omega _t)}\omega _t^n\right], \end{aligned}$$

(4.2)

where $\phi _s$ is a path in $\mathcal{M }(\omega _{KS})$ defined by $\phi _s=s(-\phi +\rho _\sigma ), ~\forall ~0\le s\le 1$ and $\phi _s=-\phi +\rho _\sigma +\rho _t$, $1\le s\le 1+t$. It is easy to see

$$\begin{aligned} \frac{d}{dt} f_Y(t)|_{t=0}=\int _M \theta _Y^{\prime } (-\varphi +\rho _\sigma )e^{\theta _{X_0}^{\prime }}(\omega _{KS}^{\prime })^n. \end{aligned}$$

This implies

$$\begin{aligned} \frac{d}{dt} f_Y(t)|_{t=0}=-\int _M \theta _Y^{\prime } ((\sigma ^{-1})^*\varphi +\rho _{\sigma ^{-1}})e^{\theta _X}\omega _{KS}^n \end{aligned}$$

(4.3)

The gap between $f_Y(t)$ and $\mathcal{F }(\omega _\phi ,\omega _t)$ can be computed as follows. Let $\Delta =\{(\tau ,s)|~0\le \tau \le 1,~0\le s\le \tau +(1-\tau )(1+t)$} be a domain in $\mathbb{R }^2$. Let $\Phi =\Phi (\tau ,s;\cdot )$ be a family of Kähler potentials with two parameters $(\tau , s)\in \Delta$ which satisfy:

$$\begin{aligned} \Phi&= s(-\phi +\rho _\sigma +\rho _t),\quad \, 0\le s\le 1,\quad \, ~\text{ as}~ \tau =1;\\ \Phi&= \phi _s, ~0\le s\le 1+t,~\text{ as}~ \tau =0;\\ \Phi&= 0,~\text{ as}~s=0;\\ \Phi&= -\phi +\rho _\sigma +\rho _t,~s=\tau +(1-\tau )(1+t). \end{aligned}$$

Then by using the Stoke’s formula, we have

$$\begin{aligned}&|f_Y(t)-\mathcal{F }(\omega _\phi ,\omega _t)|\nonumber \\ \quad&=\left|\text{ Re}\left\{ \int _{\partial \Delta } \int _M d_{\tau ,s}\Phi (\tau ,s;\cdot ) e^{\theta _{X_0}(\phi _s)}\omega _{\phi _s}^n \right\} \right|\nonumber \\ \quad&=\left|\text{ Re}\left\{ \int _\Delta d\tau ds\int _M \dot{\Phi }_\tau (\langle \overline{\partial }\dot{\Phi }_s,\overline{\partial }\theta _{X_0}(\Phi )\rangle \nonumber \right.\right.\\ \qquad&\left.\left.-\langle \overline{\partial }\theta _{X_0}(\Phi ),\overline{\partial }\dot{\Phi }_s\rangle )e^{\theta _{X_0}(\Phi )}\omega _{\Phi }^n\right\} \right|\nonumber \\ \quad&=2\left|\text{ Re}\left\{ \int _\Delta d\tau ds\int _M \dot{\Phi }_\tau \text{ Im}(X_0(\Phi _s))e^{\theta _{X_0}(\Phi )}\omega _{\Phi }^n\right\} \right|\nonumber \\ \quad&\le C\Vert X^{\prime }(\phi )\Vert _{C^0}^2. \end{aligned}$$

(4.4)

At the last inequality, we used a fact that $X_0(\rho _\sigma )$ and $X_0(\rho _t)$ are both real-valued. Similarly, we can get

$$\begin{aligned} \left|\frac{d}{dt}(f_Y(t)-\mathcal{F }(\omega _\phi ,\omega _t))\right|_{t=0} \le C\Vert X^{\prime }(\phi )\Vert _{C^0}^2. \end{aligned}$$

(4.5)

Next we claim

$$\begin{aligned} \mathcal{F }(\sigma )=\mathcal{F }(\omega _\phi ,\omega _{KS}^{\prime })= O(\text{ osc}_M\rho _\sigma )\rightarrow +\infty ,\quad \, \text{ as} ~\text{ dist}(\sigma , \text{ Id})\rightarrow \infty . \end{aligned}$$

(4.6)

In fact, as in the proof of Lemma 3.3 in [12] together with the assumption of $\phi \in \mathcal{K }(\epsilon _0)$, one can estimate

$$\begin{aligned} \mathcal{F }(\sigma )&= O\left(\int _M (-\phi +\rho _\sigma )(\omega _\phi ^n-(\omega _{KS}^{\prime })^n)\right).\\&= O\left(\int _M \rho _\sigma (\omega _{KS}^n-(\omega _{KS}^{\prime })^n)\right). \end{aligned}$$

By using the Green formula, it follows

$$\begin{aligned} \mathcal{F }(\sigma ) = O(\text{ osc}_M\rho _\sigma ). \end{aligned}$$

Thus the claim is true.

By the above claim, we can take a minimizing sequence of $\mathcal{F }(\sigma )$ for $\sigma$ in $\text{ Aut}_r(M)$ so that for any small $\epsilon \le \epsilon _0^2$, there exists a $\sigma \in \text{ Aut}_r(M)$ with bounded $\text{ dist}(\sigma , \text{ Id})$ such that

$$\begin{aligned} |D\mathcal{F }(\sigma )(Y)|\le \epsilon , \end{aligned}$$

(4.7)

for any $Y\in \eta _r(M)$ with $\int _M|Y|^2\omega _{KS}^n=1$. Therefore combining (4.3)–(4.5) with (4.7), we prove Proposition 3.10 while $\sigma$ is replaced by $\sigma ^{-1}$. $\square$