No breathers theorem for some noncompact Ricci flows

TL;DR

This paper proves a no breathers theorem for certain noncompact Ricci flows, including those on asymptotically flat manifolds with positive scalar curvature, by solving a new non-local elliptic equation related to a log Sobolev inequality.

Contribution

It extends the no breathers theorem to noncompact Ricci flows under specific conditions, introducing a novel method involving a non-local elliptic equation.

Findings

No breathers exist for the considered noncompact Ricci flows.

The proof involves solving a new non-local elliptic equation.

The method adapts techniques from the compact case to noncompact settings.

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. 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.