On the weighted forward reduced Entropy of Ricci flow

TL;DR

This paper introduces a new monotone quantity called the weighted forward reduced volume for Ricci flow, which characterizes trivial flows on flat Euclidean space and extends concepts related to Perelman's reduced volume to noncompact manifolds.

Contribution

The paper defines the weighted forward reduced volume for Ricci flow on noncompact manifolds and proves its monotonicity and characterization of trivial flows, extending previous work on Perelman's reduced volume.

Findings

Weighted forward reduced volume is well-defined on noncompact manifolds.

It is monotone non-increasing under Ricci flow.

It characterizes trivial Ricci flows on flat Euclidean space.

Abstract

In this paper, we first introduce the weighted forward reduced volume of Ricci flow. The weighted forward reduced volume, which related to expanders of Ricci flow, is well-defined on noncompact manifolds and monotone non-increasing under Ricci flow. Moreover, we show that, just the same as the Perelman's reduced volume, the weighted reduced volume entropy has the value $(4\pi)^{\frac{n}{2}}$ if and only if the Ricci flow is the trivial flow on flat Euclidean space.