The Calabi flow with rough initial data
In this paper, we prove that there exists a dimensional constant $δ> 0$ such that given any background Kähler metric $ω$, 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 $ω_{u_0} : = ω+\sqrt{-1}\partial \bar\partial u_0$. The existence time depends on initial data $u_0$ and the metric $ω$. 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.