3d Convex Contact Forms And The Ruelle Invariant

TL;DR

This paper establishes bounds relating the Ruelle invariant, volume, and systolic ratio for convex domains in four-dimensional space, and constructs examples of dynamically convex contact forms on S^3 that defy these bounds.

Contribution

It proves universal bounds connecting curvature, Ruelle invariant, and systolic ratio for convex domains, and constructs the first dynamically convex contact 3-spheres not contactomorphic to convex boundaries.

Findings

Established bounds involving Ruelle invariant, volume, and systolic ratio.

Constructed examples of dynamically convex contact forms on S^3 violating these bounds.

First examples of such contact forms not contactomorphic to convex boundaries.

Abstract

Let $X \subset \mathbb{R}^4$ be a convex domain with smooth boundary $Y$. We use a relation between the extrinsic curvature of $Y$ and the Ruelle invariant $\text{Ru}(Y)$ of the natural Reeb flow on $Y$ to prove that there exist constants $C > c > 0$ independent of $Y$ such that \[c < \frac{\text{Ru}(Y)^2}{\text{vol}(X)} \cdot \text{sys}(Y) < C\] Here $\text{sys}(Y)$ is the systolic ratio, i.e. the square of the minimal period of a closed Reeb orbit of $Y$ divided by twice the volume of $X$. We then construct dynamically convex contact forms on $S^3$ that violate this bound using methods of Abbondandolo-Bramham-Hryniewicz-Salom\~{a}o. These are the first examples of dynamically convex contact $3$-spheres that are not strictly contactomorphic to a convex boundary $Y$.