The Calabi flow with rough initial data
In this paper, we prove that there exists a dimensional constant such that given any background Kähler metric , the Calabi flow with initial data satisfying \beginequation* \partial \bar \partial u_0 \in L^\infty (M) and (1- δ)ω< ω_u_0 < (1+δ)ω, \endequation* admits a unique short time solution and it becomes smooth immediately, where . The existence time depends on initial data and the metric . As a corollary, we get that Calabi flow has short time existence for any initial data satisfying \beginequation* \partial \bar \partial u_0 \in C^0(M) and ω_u_0 > 0, \endequation* 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.