Fully nonlinear parabolic equations of real forms on Hermitian manifolds

TL;DR

This paper develops PDE techniques for fully nonlinear equations involving real (p,p) forms on Hermitian manifolds, extending complex Monge-Ampère theory to higher cohomology classes.

Contribution

It introduces a parabolic method to establish existence and convergence of solutions for fully nonlinear (p,p) form equations on Hermitian manifolds, addressing a gap in complex geometry.

Findings

Proved long-time existence of solutions

Established convergence to elliptic solutions

Extended PDE methods to higher cohomology classes

Abstract

Over many decades fully nonlinear PDEs, and the complex Monge-Amp\`ere equation in particular played a central role in the study of complex manifolds. Most previous works focused on problems that can be expressed through equations involving real $(1, 1)$ forms. As many important questions, especially those linked to higher cohomology classes in complex geometry involve real $(p, p)$ forms for $p > 1$, there is a strong need to develop PDE techniques to study them. In this paper we consider a fully nonlinear equation for $(p, p)$ forms on compact Hermitian manifolds. We establish the existence of classical solutions for a large class of these equations by a parabolic approach, proving the long-time existence and convergence of solutions to the elliptic case.