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 K\"ahler 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- \delta )\omega < \omega_{u_0} < (1+\delta )\omega, \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 } \omega_{u_0} > 0, \end{equation*} which should be interpreted as a "continuous K\"ahler metric". A main technical ingredient is Schauder-type estimates for biharmonic heat equation on Riemannian manifolds with time weighted H\"older norms.