Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow

TL;DR

This paper investigates the asymptotic entropy of ancient Ricci flow solutions, establishing a gap theorem and linking finite entropy to noncollapsing conditions, supporting conjectures about the structure of such solutions.

Contribution

It proves a new gap theorem for ancient solutions and connects finite entropy with kappa-noncollapsing under certain conditions, advancing understanding of Ricci flow behavior.

Findings

Established a gap theorem for ancient Ricci solutions.

Proved finite entropy implies noncollapsing under specific assumptions.

Supported conjecture relating entropy bounds to geometric noncollapsing.

Abstract

In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota's work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative curvature operator, finite asymptotic entropy implies kappa-noncollapsing on all scales. This provides an evidence for Perelman's more general assertion that on a complete ancient solution with nonnegative curvature operator, bounded entropy is equivalent to kappa-noncollapsing.