New steps in $C^0$ symplectic and contact geometry of smooth submanifolds

TL;DR

This paper advances $C^0$ symplectic and contact geometry by providing counterexamples, establishing a quantitative $h$-principle, and demonstrating rigidity phenomena for Legendrian knots under contact homeomorphisms.

Contribution

It introduces a $C^0$ counterexample to the Lagrangian Arnold conjecture, proves a quantitative $h$-principle for subcritical isotropic embeddings, and offers new proofs of Legendrian knot rigidity.

Findings

Counterexample to Lagrangian Arnold conjecture in cotangent bundles

Quantitative $h$-principle for subcritical isotropic embeddings

Legendrian knots are preserved by contact homeomorphisms if smooth

Abstract

We provide a $C^0$ counterexample to the Lagrangian Arnold conjecture in the cotangent bundle of a closed manifold. Additionally, we prove a quantitative $h$-principle for subcritical isotropic embeddings in contact manifolds, and provide an explicit construction of a contact homeomorphism which takes a subcritical isotropic curve to a transverse one. On the rigid side, we give another proof of the Dimitroglou Rizell and Sullivan theorem \cite{RS22} which states that Legendrian knots are preserved by contact homeomorphisms, provided their image is smooth. Moreover, our method gives related examples of rigidity in higher dimensions as well.