Convergence of the Hesse-Koszul flow on compact Hessian manifolds

TL;DR

This paper investigates the long-term behavior of the Hesse-Koszul flow on compact Hessian manifolds, establishing convergence to Hesse-Einstein metrics under certain conditions and providing alternative proofs for key existence theorems.

Contribution

It proves convergence of the Hesse-Koszul flow to Hesse-Einstein metrics on compact Hessian manifolds with negative first affine Chern class, offering new proofs for classical results.

Findings

Flow converges to Hesse-Einstein metric when the first affine Chern class is negative.

A convergence result for twisted Hesse-Koszul flow on any compact Hessian manifold.

Provides alternative proofs for the existence of Hesse-Einstein metrics and the real Calabi theorem.

Abstract

We study the long time behavior of the Hesse-Koszul flow on compact Hessian manifolds. When the first affine Chern class is negative, we prove that the flow converges to the unique Hesse-Einstein metric. We also derive a convergence result for a twisted Hesse-Koszul flow on any compact Hessian manifold. These results give alternative proofs for the existence of the unique Hesse-Einstein metric by Cheng-Yau and Caffarelli-Viaclovsky as well as the real Calabi theorem by Cheng-Yau, Delano\"e and Caffarelli-Viaclovsky.