Nonlinear Hodge flows in symplectic geometry

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 $\rho$.