Evolution Equations for Non-Degenerate 2-Forms

Abstract

Abstract In this paper we introduce several geometric flows that evolve primarily non-degenerate 2-forms, with the motivation to develop a geometric flow to approach the existence of the symplectic forms on a compact manifold that supports a non-degenerate 2-form. In particular, we introduce $\mathrm{d}^{\ast }\mathrm{d}$-flow and $\mathrm{d}^{\ast }\mathrm{d}$-Ricci flow for a compatible pair $(\omega , J)$ of an almost Hermitian structure. We prove the short time existence and uniqueness of these flows with smooth initial data, and give some examples of long time existence and convergence.