An equation linking $\mathscr{W}$-entropy with reduced volume

TL;DR

This paper provides an alternative proof linking $ abla$-entropy and reduced volume in Ricci flow, and confirms the equation's validity for Type I $ ext{ extkappa}$-solutions, enhancing understanding of Ricci flow behavior.

Contribution

It offers a new proof of Ni's equation relating $ abla$-entropy and reduced volume, and extends its validity to Type I $ ext{ extkappa}$-solutions of Ricci flow.

Findings

Established an alternative proof of Ni's equation for Ricci flow.

Confirmed the equation holds for Type I $ ext{ extkappa}$-solutions.

Enhanced understanding of the relationship between entropy and volume in Ricci flow.

Abstract

$\mathscr{W}$-entropy and reduced volume for the Ricci flow were introduced by Perelman, which had proved their importance in the study of the Ricci flow. L. Ni studied the analogous concepts for the linear heat equation on the static manifolds, and established an equation which links the large time behavior of these two. Due to the surprising similarity between those concepts in the Ricci flow and the linear heat equation, a natural question whether such equation holds for the Ricci flow ancient solution was asked by L. Ni. In this paper, we gave an alternative proof to L. Ni's original equation based on a new method. And following the same philosophy of this method, we answer L. Ni's question positively for Type I $\kappa$-solutions of the Ricci flow.