Perelman-type no breather theorem for noncompact Ricci flows

Abstract

In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete 3-dimensional shrinking breathers. Second, we can show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient-a generalization of Naber's result in [11]. Furthermore, we develop a general condition for the existence of the asymptotic shrinking gradient Ricci soliton, which hopefully will contribute to the study of ancient solutions.

On the noncollapsedness of positively curved Type I ancient Ricci flows

“…These classical theorems imply that an ancient solution as described in Theorem 1.2 or Theorem 1.3 has a Type I injectivity radius lower bound. This is sufficient to conclude the existence of an asymptotic shrinker, which in turn implies noncollapsedness; the authors used a similar idea in [7] to prove the noncollapsedness of a more general type of ancient Ricci flows-the locally uniformly Type I ancient solutions. Here we emphasize that the Type I injectivity radius lower bound itself does not directly imply the noncollapsedness; see section 2 for more details concerning this point.…”