Systolic ratio, index of closed orbits and convexity for tight contact forms on the three-sphere

TL;DR

This paper constructs examples of tight contact forms on the three-sphere with arbitrarily high systolic ratios, challenging existing conjectures and exploring the relationship between convexity, closed orbits, and systolic geometry.

Contribution

It provides explicit constructions of tight contact forms with high systolic ratios, addressing a conjecture of Viterbo and analyzing the properties of closed Reeb orbits.

Findings

Constructed a contact form with systolic ratio close to 2

Created a sequence of forms with systolic ratio approaching n

Established bounds on the mean rotation number of closed orbits

Abstract

We construct a dynamically convex contact form on the three-sphere whose systolic ratio is arbitrarily close to 2. This example is related to a conjecture of Viterbo, whose validity would imply that the systolic ratio of a convex contact form does not exceed 1. We also construct a sequence of tight contact forms $\alpha_n$, $n\geq 2$, with systolic ratio arbitrarily close to $n$ and suitable bounds on the mean rotation number of all the closed orbits of the induced Reeb flow.