Some quantitative results in mathcal C0

Authors: Lev Buhovsky Emmanuel Opshtein

Publish Date: 2015/10/26

Volume: 205, Issue: 1, Pages: 1-56

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Abstract

This paper proceeds with the study of the mathcal C0symplectic geometry of smooth submanifolds as initiated in Humilière et al Duke Math J 1644 767–799 2015 and Opshtein Ann Sci Éc Norm Supér 425 857–864 2009 with the 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 mathcal C0flexibility while this is impossible for codimension 2 symplectic submanifolds mathcal C0rigidity We also discuss mathcal C0invariants of coistropic and Lagrangian submanifolds proving some rigidity results and formulating some conjectures We finally formulate an EliashbergGromov mathcal C0rigidity type question for submanifolds which we solve in many cases Our main technical tool is a quantitative hprinciple result in symplectic geometryLB was partially supported by the Israel Science Foundation grant 1380/13 and by the Raymond and Beverly Sackler Career Development Chair EO was supported by the grant ANR116JS0101001 We thank Yakov Eliashberg Vincent Humilière Leonid Polterovich Sobhan Seyfaddini and Sasha Sodin for carefully listening to the proofs and for their remarks We thank Yaron Ostrover for providing us a reference which improved our paper We finally thank the referees for valuable suggestions which in particular improved the readability of the paperLet epsilon 0 be a positive real m geqslant 6 be an integer W subset mathbb Rm be an open set u 1 u 2 overlineD rightarrow W be disjoint smoothly embedded discs and assume that there exists a continuous homotopy between u 1 and u 2 in W of size less than epsilon ie a continuous map F overlineD times 01 rightarrow W such that Fz0 = u 1z Fz1 = u 2z for all z in overlineD and that text sizeF epsilon Then there exists a smooth embedded isotopy widetildeF between u 1 and u 2 in W of size less than 2 epsilon ie a smooth embedding widetildeF overlineD times 01 rightarrow W such that widetildeFz0 = u 1z widetildeFz1 = u 2z for all z in overlineD and text sizewidetildeF 2 epsilon Moreover if m 6 then the estimate on the size of the isotopy can be improved to text sizewidetildeF epsilon First we can slightly perturb our homotopy F between u 1 and u 2 to obtain a smooth homotopy between u 1 and u 2 in W of size less than epsilon Hence without loss of generality we may assume that the homotopy F is smoothLet Momega be a connected symplectic manifold let r 0 G = Dr subset mathbb C and let v 1v 2 overlineG rightarrow M be smoothly embedded symplectic discs v 1 omega = v 2 omega = omega text st Then there exists a compactly supported Hamiltonian isotopy of M whose time1 map psi satisfies psi circ v 1 = v 2

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