$C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916922775306240 |
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| author | Atallah, Marcelo Mak, Cheuk Yu Wu, Weiwei |
| author_facet | Atallah, Marcelo Mak, Cheuk Yu Wu, Weiwei |
| contents | We investigate the $C^0$-topology of the group of symplectic diffeomorphisms of positive symplectic rational surfaces. For all but a few exceptions, we prove that the group of Hamiltonian diffeomorphisms forms a connected component in the $C^0$-topology. This provides the first nontrivial case in which the group of Hamiltonian diffeomorphisms is known to be $C^0$-closed inside the group of symplectic diffeomorphisms.
The key to our approach is to build a bridge between techniques from symplectic mapping class groups and problems in $C^0$-symplectic topology. Via a careful adaptation of tools from $J$-holomorphic foliation and inflation, we establish the necessary $C^0$-distance estimates. We hope that this serves as an example of how these two subfields can interact fruitfully, and also propose several questions arising from this interplay. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20285 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces Atallah, Marcelo Mak, Cheuk Yu Wu, Weiwei Symplectic Geometry Geometric Topology We investigate the $C^0$-topology of the group of symplectic diffeomorphisms of positive symplectic rational surfaces. For all but a few exceptions, we prove that the group of Hamiltonian diffeomorphisms forms a connected component in the $C^0$-topology. This provides the first nontrivial case in which the group of Hamiltonian diffeomorphisms is known to be $C^0$-closed inside the group of symplectic diffeomorphisms. The key to our approach is to build a bridge between techniques from symplectic mapping class groups and problems in $C^0$-symplectic topology. Via a careful adaptation of tools from $J$-holomorphic foliation and inflation, we establish the necessary $C^0$-distance estimates. We hope that this serves as an example of how these two subfields can interact fruitfully, and also propose several questions arising from this interplay. |
| title | $C^0$-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces |
| topic | Symplectic Geometry Geometric Topology |
| url | https://arxiv.org/abs/2508.20285 |