JournalsjemsVol. 23 / No. 3DOI 10.4171/jems/1022

A local contact systolic inequality in dimension three

  • Gabriele Benedetti

    Universität Heidelberg, Germany
  • Jungsoo Kang

    Seoul National University, South Korea
A local contact systolic inequality in dimension three cover
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Let α\alpha be a contact form on a connected closed three-manifold Σ\Sigma. The systolic ratio of α\alpha is defined as ρsys(α):=1Vol(α)Tmin(α)2\rho_{\mathrm {sys}}(\alpha) := \frac {1}{\mathrm {Vol}(\alpha)} T_{\mathrm {min}} (\alpha)^2, where Tmin(α)T_{\mathrm {min}} (\alpha) and Vol(α){\mathrm {Vol}} (\alpha) denote the minimal period of periodic Reeb orbits and the contact volume. The form α\alpha is said to be Zoll if its Reeb flow generates a free S1S^1-action on Σ\Sigma. We prove that the set of Zoll contact forms on Σ\Sigma locally maximises the systolic ratio in the C3C^3-topology. More precisely, we show that every Zoll form α\alpha_{\ast} admits a C3C^3-neighbourhood U\mathcal U in the space of contact forms such that ρsys(α)ρsys(α)\rho_{\mathrm {sys}}(\alpha) \leq \rho_{\mathrm {sys}}(\alpha_{\ast}) for every αU\alpha \in \mathcal U, with equality if and only if α\alpha is Zoll.