Coisotropic rigidity and C^0-symplectic geometry

TL;DR

This paper demonstrates that symplectic homeomorphisms preserve coisotropic submanifolds and their characteristic foliations, extending classical rigidity results and revealing a unified phenomenon in symplectic topology.

Contribution

It generalizes the Gromov-Eliashberg Theorem to include coisotropic submanifolds and introduces a C^0-dynamical property that broadens understanding of Hamiltonian flow analogs.

Findings

Symplectic homeomorphisms preserve coisotropic submanifolds.

Unified rigidity phenomenon for Lagrangians and hypersurfaces.

Establishment of a C^0-dynamical property for coisotropic submanifolds.

Abstract

We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov-Eliashberg Theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C^0-dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C^0-Hamiltonian dynamics: Uniqueness of generators for continuous analogs of Hamiltonian flows.