Perelman's entropy on ancient Ricci flows

TL;DR

This paper extends Perelman's entropy results for ancient Ricci flows by relaxing curvature conditions, showing bounded entropy implies noncollapsing under weaker assumptions, and applies to steady Ricci solitons with nonnegative Ricci curvature.

Contribution

It demonstrates that nonnegative curvature operator is not necessary, requiring only a consequence of Hamilton's trace Harnack inequality, broadening the applicability of entropy and noncollapsing equivalence.

Findings

Bounded entropy implies noncollapsing under weaker conditions.

Hamilton's trace Harnack inequality suffices for the main results.

The condition holds for steady Ricci solitons with nonnegative Ricci curvature.

Abstract

In [ZY2], the second author proved Perelman's assertion, namely, for an ancient Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. In this paper, we continue this discussion. It turns out that the curvature operator nonnegativity is not a necessary condition, and we need only to assume a consequence of Hamilton's trace Harnack. Furthermore, we show that this condition holds for steady Ricci solitons with nonnegative Ricci curvature.