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A local contact systolic inequality in dimension three · arXivDesk

1902.01249Feb 4, 201942 pages, a revised version of Part I in "A local systolic-diastolic inequality in contact and symplectic geometry" arXiv:1801.00539 (now withdrawn), accepted for publication in JEMS

A local contact systolic inequality in dimension three

Gabriele Benedetti, Jungsoo Kang

Symplectic Geometry Differential Geometry

Abstract

Let $α$ be a contact form on a connected closed three-manifold $Σ$ . The systolic ratio of $α$ is defined as $ρ_{\mathrm{sys}}(α):=\tfrac{1}{\mathrm{Vol}(α)}T_{\min}(α)^2$

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, where

T_{\min}(α)

and

\mathrm{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

S^1

-action on

Σ

. We prove that the set of Zoll contact forms on

Σ

locally maximises the systolic ratio in the

C^3

-topology. More precisely, we show that every Zoll form

α_*

admits a

C^3

-neighbourhood

C^3

in the space of contact forms such that, for every

α\in\mathcal U

, there holds

ρ_{\mathrm{sys}}(α)\leq ρ_{\mathrm{sys}}(α_*)

with equality if and only if

α

is Zoll.

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