Some quantitative results in C⁰ symplectic geometry

2014/04/03 by Lev Buhovsky, Buhovsky, Lev, Emmanuel Opshtein +1 · 2 citations
Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1404.0875

openalex publication_date 2014/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper studies the action of symplectic homeomorphisms on smooth submanifolds, with a main focus on the behaviour of symplectic homeomorphisms with respect to numerical invariants like capacities. Our main result is that a symplectic homeomorphism may preserve and squeeze codimension 4 symplectic submanifolds (C⁰-flexibility), while this is impossible for codimension 2 symplectic submanifolds (C⁰-rigidity). We also discuss C⁰-invariants of coistropic and Lagrangian submanifolds, proving some rigidity results and formulating some conjectures. We finally formulate an Eliashberg-Gromov C⁰-rigidity type question for submanifolds, which we solve in many cases. Our main technical tool is a quantitative h-principle result in symplectic geometry.

Cited by

• C⁰-rigidity of Legendrians and coisotropics via sheaf quantization
• C⁰-rigidity of the Hamiltonian diffeomorphism group of symplectic rational surfaces

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