Parabolic Complex Monge-Amp\`ere Type Equations on Closed Hermitian Manifolds

TL;DR

This paper establishes uniform estimates and convergence results for parabolic complex Monge-Ampère equations on closed Hermitian manifolds, providing a new approach for solving elliptic Monge-Ampère equations via continuity methods.

Contribution

It introduces uniform a priori estimates and convergence analysis for parabolic Monge-Ampère equations on Hermitian manifolds, facilitating solutions to elliptic equations.

Findings

Established uniform $C^ abla$ estimates for solutions.

Proved $C^ abla$ convergence of solutions.

Provided a method for continuity approach in elliptic equations.

Abstract

We study the parabolic complex Monge-Amp\`ere type equations on closed Hermitian manfolds. We derive uniform $C^\infty$ {\em a priori} estimates for normalized solutions, and then prove the $C^\infty$ convergence. The result also yields a way to carry out method of continuity for elliptic Monge-Amp\'ere type equations.