20,449 research outputs found
Extremal of Log Sobolev inequality and 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
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
- …