Evolve nondegenerate two forms

TL;DR

This paper introduces geometric flows on compact almost complex manifolds to evolve nondegenerate two forms into symplectic forms, establishing existence, uniqueness, and exploring long-term behavior.

Contribution

It presents new geometric flow methods for transforming nondegenerate two forms into symplectic forms on almost complex manifolds, including existence and uniqueness results.

Findings

Proved short-time existence and uniqueness of the flows.

Discussed extension and long-term behavior of the flows.

Provided example of long-time flow on warped product.

Abstract

We introduce geometric flows on a compact almost complex manifold, with the aim to flow a nondegenerate two form to a symplectic two form. We discuss mainly two flows, $d^*d$-flow and $d^*d$-Ricci flow. Among others, we prove the uniqueness and short time existence for smooth initial data. We also discuss the extension problem and a long time example on the warped product.