Extremal of Log Sobolev inequality and WW entropy on noncompact manifolds

Let \M be a complete, connected noncompact manifold with bounded geometry. Under a condition near infinity, we prove that the Log Sobolev functional (\ref{logfanhan}) has an extremal function decaying exponentially near infinity. We also prove that an extremal function may not exist if the condition is violated. This result has the following consequences. 1. It seems to give the first example of connected, complete manifolds with bounded geometry where a standard Log Sobolev inequality does not have an extremal. 2. It gives a negative answer to the open question on the existence of extremal of Perelman's $W$ entropy in the noncompact case, which was stipulated by Perelman \cite{P:1} p9, 3.2 Remark. 3. It helps to prove, in some cases, that noncompact shrinking breathers of Ricci flow are gradient shrinking solitons

A compactness result for Fano manifolds and K\"ahler Ricci flows

We obtain a compactness result for Fano manifolds and K\"ahler Ricci flows. Comparing to the more general Riemannian versions by Anderson and Hamilton, in this Fano case, the curvature assumption is much weaker and is preserved by the K\"ahler Ricci flows. One assumption is the boundedness of the Ricci potential and the other is the smallness of Perelman's entropy. As one application, we obtain a new local regularity criteria and structure result for K\"ahler Ricci flows. The proof is based on a H\"older estimate for the gradient of harmonic functions, which may be of independent interest