A note on the Almost Schur lemma on smooth metric measure spaces

TL;DR

This paper extends the Almost Schur Lemma to closed smooth metric measure spaces, generalizing previous results and providing new insights into the geometric analysis of these spaces.

Contribution

It proves the Almost Schur Lemma in the context of smooth metric measure spaces, broadening its applicability beyond constant weighted functions.

Findings

Established the Almost Schur Lemma for closed smooth metric measure spaces.

Unified previous results by Cheng and De Lellis-Topping as special cases.

Provided new geometric inequalities related to the lemma.

Abstract

In this paper, we prove almost Schur Lemma on closed smooth metric measure spaces, which implies the results of X. Cheng and De Lellis-Topping whenever the weighted function f is constant.