3D convex contact forms and the Ruelle invariant

• 成果类型: Article
• 署名作者: Chaidez, J.; Edtmair, O.
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-022-01107-y
• 发表日期: 2022
• 页码: 243-301
• 关键词: periodic-solutions; reeb flows; index; systems; Orbits

摘要

Let X subset of R-4 be a convex domain with smooth boundary Y. We use a relation between the extrinsic curvature of Y and the Ruelle invariant of the Reeb flow on Y to prove that there are constants C > c > 0 independent of Y such that c <= ru(Y).sys(Y)(1/2) <= C Here sys(Y) is the systolic ratio of Y, i.e. the square of the minimal period of a closed Reeb orbit of Y divided by twice the volume of X, and ru(Y) is the volume-normalized Ruelle invariant. We then construct dynamically convex contact forms on S-3 that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salomdo. These are the first examples of dynamically convex contact 3-spheres that are not strictly contactomorphic to a convex boundary Y.