Quantitative $h$-principle for isotropic embeddings and applications to $C^0$-symplectic geometry

TL;DR

This paper establishes a quantitative $h$-principle for isotropic embeddings, leading to new flexibility and rigidity results in $C^0$-symplectic geometry, including the behavior of symplectic homeomorphisms on discs and coisotropic submanifolds.

Contribution

It introduces a quantitative $h$-principle for isotropic embeddings and applies it to derive novel $C^0$-flexibility and rigidity results in symplectic geometry.

Findings

Symplectic homeomorphisms can map symplectic discs to smooth isotropic ones.

A $C^0$-rigidity result for the action on coisotropic submanifold reductions.

Established a quantitative $h$-principle applicable to isotropic embeddings.

Abstract

We prove here a quantitative $h$-principle statement that applies to isotropic embeddings of discs. We then apply it to get $C^0$-flexibility and rigidity results in symplectic geometry. On the flexible side, we prove that a symplectic homeomorphism might take a symplectic disc to a smooth isotropic one. We also get a $C^0$-rigidity result for the action of a symplectic homeomorphism on the reduction of a coisotropic submanifold.