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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
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