Nonlinear Hodge flows in symplectic geometry

TL;DR

This paper introduces nonlinear Hodge heat flows on symplectic four-manifolds to address the isotopy problem of symplectic forms within a fixed class, demonstrating stability and convergence under certain conditions.

Contribution

It develops a new family of nonlinear Hodge flows tailored for symplectic geometry and proves their stability and convergence properties near almost Kähler structures.

Findings

Flow remains stable near almost Kähler structures.

Flow converges smoothly to the original symplectic form under bounded gradient conditions.

Existence of the flow for all time under specific boundedness assumptions.

Abstract

Given a symplectic class $[\omega]$ on a four torus $T^4$ (or a $K3$ surface), a folklore problem in symplectic geometry is whether symplectic forms in $[\omega]$ are isotropic to each other. We introduce a family of nonlinear Hodge heat flows on compact symplectic four manifolds to approach this problem, which is an adaption of nonlinear Hodge theory in symplectic geometry. As a particular example, we study a conformal Hodge heat flow in detail. We prove a stability result of the flow near an almost Kahler structure $(M, \omega, g)$. We also prove that, if $|\nabla \log u|$ stays bounded along the flow, then the flow exists for all time for any initial symplectic form $\rho\in [\omega]$ and it converges to $\omega$ smoothly along the flow with uniform control, where $u$ is the volume potential of $\rho$.