Abstract :
Let M be a complete, connected noncompact manifold with bounded geometry. Under a condition near
infinity, we prove that the Log Sobolev functional (1.1) 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 (2002) [22, p. 9, 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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