New steps in C0 symplectic and contact geometry of smooth submanifolds

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 that takes a subcritical isotropic curve to a transverse one. On the rigid side, we give another proof of the Dimitroglou Rizell and Sullivan theorem 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.