3d Convex Contact Forms And The Ruelle Invariant

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 $Y$.