Nonlinear Hodge flows in symplectic geometry
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909989085380608 |
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| author | He, Weiyong |
| author_facet | He, Weiyong |
| contents | 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 $ρ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_03651 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Nonlinear Hodge flows in symplectic geometry He, Weiyong Differential Geometry Analysis of PDEs Symplectic Geometry 53E50, 35K40 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 $ρ$. |
| title | Nonlinear Hodge flows in symplectic geometry |
| topic | Differential Geometry Analysis of PDEs Symplectic Geometry 53E50, 35K40 |
| url | https://arxiv.org/abs/2310.03651 |