Sharp systolic inequalities for invariant tight contact forms on principal S1-bundles over S2

2024/03/04 by Simon Vialaret, Vialaret, Simon · 2 citations
Mathematics · #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2403.02228

openalex publication_date 2024/03/04 · openalex created_date 2024/03/06 · openalex updated_date 2026/07/28

Abstract

The systole of a contact form α is defined as the shortest period of closed Reeb orbits of α. Given a non-trivial \mathbb S¹-principal bundle over \mathbb S² with total space M, we prove a sharp systolic inequality for the class of tight contact form on M invariant under the \mathbb S¹-action. This inequality exhibits a behavior which depends on the Euler class of the bundle in a subtle way. As applications, we prove a sharp systolic inequality for rotationally symmetric Finsler metrics on \mathbb S², a systolic inequality for the shortest contractible closed Reeb orbit, and a particular case of a conjecture by Viterbo.

Cited by

• Symplectic capacities of S¹-invariant dynamically convex domains in ℝ⁴
• Bott-integrable contact forms with large systolic ratio

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