The Calabi flow with rough initial data

Abstract

In this paper, we prove that there exists a dimensional constant $\delta >0$ such that given any background Kähler metric $\omega$, the Calabi flow with initial data $u_0$ satisfying \begin{equation*} \partial \bar \partial u_0 \in L^\infty (M) \text{ and } (1- δ)ω< ω_{u_0} < (1+δ)ω, \end{equation*} admits a unique short time solution and it becomes smooth immediately, where ${\omega }_{u_0}:=\omega +\sqrt{-1}\partial \bar{\partial }{u}_0$. The existence time depends on initial data $u_0$ and the metric $\omega$. As a corollary, we get that Calabi flow has short time existence for any initial data satisfying \begin{equation*} \partial \bar \partial u_0 \in C^0(M) \text{ and } ω_{u_0} > 0, \end{equation*} which should be interpreted as a "continuous Kähler metric". A main technical ingredient is Schauder-type estimates for biharmonic heat equation on Riemannian manifolds with time weighted Hölder norms.

We improved our previous result to initial L^\infty Kähler metrics with small L^\infty oscillation. We also added the application, see Theorem 1.7

References in corpus (3)

• On the existence of constant scalar curvature Kähler metric: a new perspective
• Approximation of weak geodesics and subharmonicity of Mabuchi energy
• Convexity of the K-energy on the space of Kahler metrics and uniqueness of extremal metrics