Parabolic complex Monge-Amp\`ere equations on compact Hermitian manifolds

TL;DR

This paper establishes the long-term existence and convergence of solutions to a broad class of parabolic complex Monge-Ampère equations on compact Hermitian manifolds, linking them to elliptic solutions.

Contribution

It extends the analysis of parabolic complex Monge-Ampère equations to non-concave cases on Hermitian manifolds, proving existence and convergence results.

Findings

Solutions exist for all time and converge to elliptic Monge-Ampère solutions.

The limiting function solves an elliptic complex Monge-Ampère equation.

The results apply to a general class of parabolic equations, not necessarily concave in the Hessian.

Abstract

We prove the long-time existence and convergence of solutions to a general class of parabolic equations, not necessarily concave in the Hessian of the unknown function, on a compact Hermitian manifold. The limiting function is identified as the solution of an elliptic complex Monge-Amp\`ere equation.