2023/10/05 by He, Weiyong
#35K40 #53E50 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2310.03651
Given a symplectic class [ω] on a four torus T4 (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 |∇ log u| stays bounded along the flow, then the flow exists for all time for any initial symplectic form ρ∈ [ω] and it converges to ω smoothly along the flow with uniform control, where u is the volume potential of ρ.