Fully nonlinear parabolic equations of real forms on Hermitian manifolds

2024/11/17 by George, Mathew, Guan, Bo
#35K10 #35K55 #53C55 #58J35 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2411.10946

Abstract

Over many decades fully nonlinear PDEs, and the complex Monge-Ampère 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.

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