C⁰-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces

2025/08/27 by Atallah, Marcelo, Mak, Cheuk Yu, Wu, Weiwei
#FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2508.20285

Abstract

We investigate the C⁰-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 C⁰-topology. This provides the first nontrivial case in which the group of Hamiltonian diffeomorphisms is known to be C⁰-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 C⁰-symplectic topology. Via a careful adaptation of tools from J-holomorphic foliation and inflation, we establish the necessary C⁰-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.

Citations

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• Symplectic embeddings of 4-dimensional ellipsoids
• Lectures on four-dimensional Dehn twists
• Topology of symplectomorphism groups of rational ruled surfaces
• Periodic Hamiltonian flows on four dimensional manifolds

Related

• The sharp C⁰-fragmentation property for Hamiltonian diffeomorphisms and homeomorphisms on surfaces
• C⁰-rigidity of characteristics in symplectic geometry
• Some properties of Hamiltonian homeomorphisms on closed aspherical surfaces
• On the Flux Conjectures
• Compatible complex structures on symplectic rational ruled surfaces