Some quantitative results in symplectic geometry

• 成果类型: Article
• 署名作者: Buhovsky, Lev; Opshtein, Emmanuel
• 署名单位: Tel Aviv University; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universites de Strasbourg Etablissements Associes; Universite de Strasbourg; Universites de Strasbourg Etablissements Associes; Universite de Strasbourg; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0626-4
• 发表日期: 2016
• 页码: 1-56
• 关键词: hofer-zehnder capacity; holomorphic-curves; MANIFOLDS; energy

摘要

This paper proceeds with the study of the -symplectic geometry of smooth submanifolds, as initiated in HumiliSre et al. (Duke Math J 164(4), 767-799, 2015) and Opshtein (Ann Sci A parts per thousand c Norm Sup,r 42(5), 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 (-flexibility), while this is impossible for codimension 2 symplectic submanifolds (-rigidity). We also discuss -invariants of coistropic and Lagrangian submanifolds, proving some rigidity results and formulating some conjectures. We finally formulate an Eliashberg-Gromov -rigidity type question for submanifolds, which we solve in many cases. Our main technical tool is a quantitative h-principle result in symplectic geometry.