Perelman's W-functional and stability of K\"ahler-Ricci flow

TL;DR

This paper investigates the second variation of Perelman's entropy on K"ahler metrics at a K"ahler-Ricci soliton, demonstrating its stability along the K"ahler-Ricci flow, which enhances understanding of geometric flow stability.

Contribution

It proves the stability of Perelman's entropy at K"ahler-Ricci solitons and along the K"ahler-Ricci flow, providing new insights into geometric flow stability analysis.

Findings

Entropy is stable at K"ahler-Ricci solitons

Entropy remains stable along the K"ahler-Ricci flow

Second variation of entropy is non-negative

Abstract

In this expository note, we study the second variation of Perelman's entropy on the space of Kahler metrics at a K\"ahler-Ricci soliton. We prove that the entropy is stable in the sense of variations. In particular, Perelman's entropy is stable along the K\"ahler-Ricci flow. The Chinese version of this note has appeared in a volume in honor of professor K.C.Chang (Scientia Sinica Math., 46 (2016), 685-696).