The Cauchy-Dirichlet problem for parabolic deformed Hermitian-Yang-Mills equation

TL;DR

This paper studies the parabolic deformed Hermitian-Yang-Mills equation with hypercritical phase, proving solution convergence and providing an alternative proof for the Dirichlet problem in complex domains.

Contribution

It introduces a new approach using the J-functional to prove convergence and offers an alternative proof for the Dirichlet problem.

Findings

Solutions to the parabolic deformed Hermitian-Yang-Mills equation converge under certain conditions.

The J-functional is effective in analyzing the equation's behavior.

An alternative proof for the Dirichlet problem is established.

Abstract

The purpose of this paper is to investigate the parabolic deformed Hermitian-Yang-Mills equation with hypercritical phase in a smooth domain $\Omega\subset \mathbb{C}^{n}$. By using $J$-functional, we are able to prove the convergence of solutions. As an application, we give an alternative proof of the Dirichlet problem for deformed Hermitian-Yang-Mills equation.