Fully nonlinear parabolic equations of real forms on Hermitian manifolds

Mathew George, Bo Guan

Published 2024-11-17Version 1

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.