Sharp systolic inequalities for Reeb flows on the three-sphere

Author(s)

Abbondandolo, Alberto ^[1] ; Bramham, Barney ^[1] ; Hryniewicz, Umberto L. ^[2] ; Salomao, Pedro A. S. ^[3]

Affiliation

^[1] Ruhr Univ Bochum, Fak Math, Univ Str 150, D-44801 Bochum - Germany

^[2] Univ Fed Rio de Janeiro, Inst Matemat, Dept Matemat Aplicada, Ctr Tecnol, Ilha Fundao, Ave Athos da Silveira Ramos 149, Bloco C, BR-21941909 Rio De Janeiro - Brazil

^[3] Univ Sao Paulo, Inst Matemat & Estat, Dept Matemat, Rua Matao 1010, Cidade Univ, BR-05508090 Sao Paulo - Brazil

• Document type: Journal article
• Source: INVENTIONES MATHEMATICAE; v. 211, n. 2, p. 687-778, FEB 2018.
• Web of Science Citations: 3

Abstract

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 . (AU)

• FAPESP's process: 11/16265-8 - Low dimensional dynamics
• Grantee: Edson Vargas
• Support Opportunities: Research Projects - Thematic Grants