Concavity of Perelman's $\mathcal{W}$-functional over the space of K\"ahler potentials

TL;DR

This paper demonstrates that Perelman's $ ext{W}$-functional is concave near K"ahler-Ricci solitons within the space of K"ahler potentials, linking it to the author's prior work on stability and providing a simplified proof.

Contribution

It establishes the concavity of Perelman's $ ext{W}$-functional around K"ahler-Ricci solitons as a consequence of the author's stability results, with a new elementary proof.

Findings

Concavity of $ ext{W}$-functional near K"ahler-Ricci solitons.

Connection between stability results and functional concavity.

Simplified proof based on elementary formulas.

Abstract

In this short note we observe that the concavity of Perelman's $\mathcal{W}$-functional over a neighborhood of a K\"ahler-Ricci soliton inside the space of K\"ahler potentials is a direct consequence of author's solution of the variational stability problem for K\"ahler-Ricci solitons. Independently, we provide a rather simple proof of this fact based on some elementary formulas obtained in our previous work.