A no breathers theorem for some noncompact Ricci flows. (English) Zbl 1305.53071

Zhang, Qi S.

Asian J. Math. 18, No. 4, 727-756 (2014).

Summary: 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 [X. Dai and J. Zhou, J. Lanzhou Univ., Nat. Sci. 44, No. 1, 65–70 (2008; Zbl 1199.94057)] and [T. A. Oliynyk and E. Woolgar, Commun. Anal. Geom. 15, No. 3, 535–568 (2007; Zbl 1138.53057)] 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.

MSC:

53C44	Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
35K40	Second-order parabolic systems
53C20	Global Riemannian geometry, including pinching

Keywords:

Ricci flow; breathers; scaling invariant entropy

Citations:

Zbl 1199.94057; Zbl 1138.53057