Convergence of the Hesse–Koszul flow on compact Hessian manifolds

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, Delanoë and Caffarelli–Viaclovsky.