A no breathers theorem for some noncompact Ricci flows

Abstract

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 "Mass under the Ricci flow," [X. Dai and L. Ma, Comm. Math. Phys. 274:1 (2007), pp. 65–80] and "Rotationally symmetric Ricci flow on asymptotically flat manifolds," [T. A. Oliynyk, and E. Woolgar, Comm. Anal. Geom., 15:3 (2007), pp. 535–568] 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.