Some quantitative results in $C^0$ symplectic geometry

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 ($C^0$-flexibility), while this is impossible for codimension $2$ symplectic submanifolds ($C^0$-rigidity). We also discuss $C^0$-invariants of coistropic and Lagrangian submanifolds, proving some rigidity results and formulating some conjectures. We finally formulate an Eliashberg-Gromov $C^0$-rigidity type question for submanifolds, which we solve in many cases. Our main technical tool is a quantitative $h$-principle result in symplectic geometry.

References in corpus (2)

• Higher dimensional contact topology via holomorphic disks
• $C^0$-rigidity of characteristics in symplectic geometry

Cited by in corpus (7)

• The action spectrum and C^0 symplectic topology
• Reduction of symplectic homeomorphisms
• A C^0 counterexample to the Arnold conjecture
• Proof of the simplicity conjecture
• ${\mathcal{C}}^0$-rigidity of Lagrangian submanifolds and punctured holomorphic discs in the cotangent bundle
• Fragility and Persistence of Leafwise Intersections
• Dehn-Seidel twist, $C^0$ symplectic topology and barcodes