A parabolic Monge-Amp\`ere type equation of Gauduchon metrics

TL;DR

This paper establishes long-term existence and convergence of solutions to a parabolic Monge-Ampère equation on Hermitian manifolds, providing a new proof of the Gauduchon conjecture through parabolic methods.

Contribution

It introduces a parabolic approach to solve the Gauduchon conjecture, proving existence, uniqueness, and convergence of solutions on compact Hermitian manifolds.

Findings

Proves long-time existence and uniqueness of solutions.

Shows convergence of normalized solutions to a Monge-Ampère type equation.

Provides a parabolic proof of the Gauduchon conjecture.

Abstract

We prove the long time existence and uniqueness of solution to a parabolic Monge-Amp\`ere type equation on compact Hermitian manifolds. We also show that the normalization of the solution converges to a smooth function in the smooth topology as $t$ approaches infinity which, up to scaling, is the solution to a Monge-Amp\`ere type equation. This gives a parabolic proof of the Gauduchon conjecture based on the solution of Sz\'ekelyhidi, Tosatti and Weinkove to this conjecture.