Asian Journal of Mathematics
Volume 18 (2014)
A no breathers theorem for some noncompact Ricci flows
Pages: 727 – 756
Under suitable conditions near infinity and assuming boundedness of curvature tensor, we prove a no breathers theorem in the spirit of Ivey-Perelman for some noncompact Ricci flows. These include Ricci flows on asymptotically flat (AF) manifolds with positive scalar curvature, which was studied in [DM] and [OW] in connection with general relativity. Since the method for the compact case faces a difficulty, the proof involves solving a new non-local elliptic equation which is the Euler-Lagrange equation of a scaling invariant log Sobolev inequality.
It is also shown that the Ricci flow on AF manifolds with positive scalar curvature is uniformly $\kappa$ noncollapsed for all time. This result, being different from Perelman’s local noncollapsing result which holds in finite time, seems to have implications for the issue of longtime convergence.
Ricci flow, breathers, scaling invariant entropy
2010 Mathematics Subject Classification
35K40, 53C20, 53C44