Pseudolocality theorems of Ricci flows on incomplete manifolds

In this paper we study the pseudolocality theorems of Ricci flows on incomplete manifolds. We prove that if a relatively compact ball in an incomplete manifold has the small scalar curvature lower bound and almost Euclidean isoperimetric constant, or almost Euclidean local ν constant, then we can construct a solution of Ricci flow in a smaller ball for which the pseudolocality theorems hold on a uniform time interval. We also give two applications. First, we prove the short-time existence of Ricci flows on complete manifolds with scalar curvature bounded below uniformly and almost Euclidean isoperimetric inequality holds locally. Second, we obtain a rigidity theorem that any complete manifold with nonnegative scalar curvature and Euclidean isoperimetric inequality must be isometric to the Euclidean space.