Fully nonlinear parabolic equations of real forms on Hermitian manifolds

Mathew George; Bo Guan

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arXiv · 2411.10946

Fully nonlinear parabolic equations of real forms on Hermitian manifolds

Mathew George·Bo Guan

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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.

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Mathew George, Bo Guan. 2024-11-17. Fully nonlinear parabolic equations of real forms on Hermitian manifolds. https://arxiv.org/abs/2411.10946

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