Let (M, g) be Riemannian four-manifold. Does there exist a
non-zero function f:M->R such that
(*) f^2 g is flat?
(**) f^2 g satisfies Einstein equations?
Most people know the answer to (*). Nobody (really) knows the full
answer to (**). In this talk I will provide the answer to
(***) f^2 g is Kahler for some Kahler form?

In this talk, using the local Ricci flow, we prove the short-time
existence of the Ricci flow on noncompact manifolds, whose Ricci curvature
has global lower bound and sectional curvature has only local average integral
bound. The short-time existence of the Ricci flow on noncompact manifolds
was studied by Wan-Xiong Shi in 1990s, who required a point-wise bound of
curvature tensors. As a corollary of our main theorem, we get the short-time existence part of Shis theorem in this more general context.

In this talk by using the idea in the proof of Perelman's pseudo locality theorem we will derive a local curvature bound in Ricci flow assuming only local sectional curvature bound and local volume lower bound for the initial metric.

This result is closely related to Theorem 10.3 in Perelman's entropy paper.

Let M be an m-dimensional complete non-compact Reimannian manifold. We prove that any bounded set of p-harmonic k-forms in L^q(M), 0