On the noncollapsedness of positively curved Type I ancient Ricci flows

TL;DR

This paper proves that certain positively curved ancient Ricci flows are noncollapsed on all scales, leading to a classification of three-dimensional solutions without assuming noncollapsing.

Contribution

It establishes noncollapsing for complete and noncompact, as well as even-dimensional closed, Type I ancient Ricci flows with positive curvature, and classifies three-dimensional solutions.

Findings

All complete, noncompact ancient solutions are noncollapsed.

Even-dimensional closed solutions are noncollapsed.

Provides classification for three-dimensional noncompact solutions.

Abstract

In this article, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such ancient solutions must be noncollapsed on all scales. This furthermore gives a complete classification for three-dimensional noncompact Type I ancient solutions without assuming the noncollapsing condition.