First variation of the Log Entropy functional along the Ricci flow

TL;DR

This paper derives the first variation formula for the adjusted log entropy functional along the Ricci flow, demonstrating its monotonicity and contributing to the understanding of geometric analysis in Ricci flow.

Contribution

It establishes the first variation formula and monotonicity of the adjusted log entropy functional along the Ricci flow, a novel result in geometric analysis.

Findings

First variation formula of the adjusted log entropy functional derived

Monotonicity of the functional along Ricci flow proven

Enhances understanding of entropy functionals in Ricci flow

Abstract

In this note, we establish the first variation formula of the adjusted log entropy functional $\mathcal Y_a$ introduced by Ye in \cite{Y2}. As a direct consequence, we also obtain the monotonicity of $\mathcal Y_a$ along the Ricci flow.