Nonlinear Hodge flows in symplectic geometry

Given a symplectic class $[ω]$ on a four torus $T^4$ (or a $K3$ surface), a folklore problem in symplectic geometry is whether symplectic forms in $[ω]$ 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 $(M, ω, g)$. 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 $ρ\in [ω]$ and it converges to $ω$ smoothly along the flow with uniform control, where $u$ is the volume potential of $ρ$.

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