  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.

Differential Geometry;Analysis of PDEs