Hessian of the Ricci Calabi functional

TL;DR

This paper investigates the Hessian of the Ricci Calabi functional on Fano manifolds, proving its non-negativity at generalized Kähler Einstein metrics and exploring related geometric decompositions and flows.

Contribution

It establishes the non-negativity of the Hessian at critical points and provides a new proof of a decomposition theorem, connecting to recent inverse Monge-Ampère flow research.

Findings

Hessian is non-negative at generalized Kähler Einstein metrics

Provides a new proof of Matsushima's type decomposition theorem

Discusses relations to inverse Monge-Ampère flow

Abstract

The Ricci Calabi functional is a functional on the space of K\"ahler metrics of Fano manifolds. Its critical points are called generalized K\"ahler Einstein metrics. In this article, we show that the Hessian of the Ricci Calabi functional is non-negative at generalized K\"ahler Einstein metrics. As its application, we give another proof of a Matsushima's type decomposition theorem for holomorphic vector fields, which was originally proved by Mabuchi. We also discuss a relation to the inverse Monge-Amp\`ere flow developed recently by Collins-Hisamoto-Takahashi.