Sharp systolic inequalities for Reeb flows on the three-sphere

• 成果类型: Article
• 署名作者: Abbondandolo, Alberto; Bramham, Barney; Hryniewicz, Umberto L.; Salomao, Pedro A. S.
• 署名单位: Ruhr University Bochum; Universidade Federal do Rio de Janeiro; Universidade de Sao Paulo
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-017-0755-z
• 发表日期: 2018
• 页码: 687-778
• 关键词: isosystolic inequalities; weinstein conjecture; hamiltonian-dynamics; Capacities; MANIFOLDS; geodesics; SURFACES; geometry; equation; CURVES

摘要

The systolic ratio of a contact form on the three-sphere is the quantity rho(sys)(alpha) = T-min(alpha)(2)/vol(S-3, alpha boolean AND d alpha), where is the minimal period of closed Reeb orbits on . A Zoll contact form is a contact form such that all the orbits of the corresponding Reeb flow are closed and have the same period. Our first main result is that in a neighbourhood of the space of Zoll contact forms on , with equality holding precisely at Zoll contact forms. This implies a particular case of a conjecture of Viterbo, a local middle-dimensional non-squeezing theorem, and a sharp systolic inequality for Finsler metrics on the two-sphere which are close to Zoll ones. Our second main result is that is unbounded from above on the space of tight contact forms on .