A local contact systolic inequality in dimension three

2019/02/04 by Gabriele Benedetti, Jungsoo Kang, Benedetti, Gabriele +1 · 2 citations
Mathematics · #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1902.01249

Abstract

Let α be a contact form on a connected closed three-manifold Σ. The systolic ratio of α is defined as ρ_sys(α):=\tfrac1Vol(α)T_min(α)², where T_min(α) and Vol(α) denote the minimal period of periodic Reeb orbits and the contact volume. The form α is said to be Zoll if its Reeb flow generates a free S¹-action on Σ. We prove that the set of Zoll contact forms on Σ locally maximises the systolic ratio in the C³-topology. More precisely, we show that every Zoll form α_* admits a C³-neighbourhood \mathcal U in the space of contact forms such that, for every α∈\mathcal U, there holds ρ_sys(α)≤ ρ_sys(α_*) with equality if and only if α is Zoll.

Citations

• On a local systolic inequality for odd-symplectic forms

Cited by

• Max-min energy of pseudoholomorphic curves and periodic Reeb flows in dimension 3
• On Zoll Contact 5-Spheres

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