Asian Journal of Mathematics

Volume 18 (2014)

Number 4

A no breathers theorem for some noncompact Ricci flows

Pages: 727 – 756

DOI: http://dx.doi.org/10.4310/AJM.2014.v18.n4.a8

Author

Qi S. Zhang (Department of Mathematics Nanjing University, Nanjing, China; and Department of Mathematics, University of California at Riverside)

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 [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.

Keywords

Ricci flow, breathers, scaling invariant entropy

2010 Mathematics Subject Classification

35K40, 53C20, 53C44

Full Text (PDF format)