A uniform Sobolev inequality for ancient Ricci flows with bounded Nash entropy

TL;DR

This paper proves that ancient Ricci flows with bounded Nash entropy also have bounded $ u$-functional, leading to uniform logarithmic Sobolev and Sobolev inequalities, extending the understanding of geometric analysis under these conditions.

Contribution

It establishes a link between bounded Nash entropy and bounded $ u$-functional for ancient Ricci flows, enabling uniform Sobolev inequalities.

Findings

Bounded Nash entropy implies bounded $ u$-functional.

Uniform logarithmic Sobolev inequalities hold for such flows.

Results depend on the validity of the theory in [Bam20c].

Abstract

This note is a continuation of [CMZ21]. We shall show that an ancient Ricci flow with uniformly bounded Nash entropy must also have uniformly bounded $\nu$-functional. Consequently, on such an ancient solution there are uniform logarithmic Sobolev and Sobolev inequalities. We emphasize that the main theorem in this paper is true so long as the theory in [Bam20c] is valid, and in particular, when the underlying manifold is closed.