3D convex contact forms and the Ruelle invariant

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 of the Reeb flow on Y to prove that there are constants \(C> c > 0\) independent of Y such that $$\begin{aligned} c \leqslant {\text {ru}}(Y) \cdot {\text {sys}}(Y)^{1/2} \leqslant C \end{aligned}$$Here \({\text {sys}}(Y)\) is the systolic ratio of Y, i.e. the square of the minimal period of a closed Reeb orbit of Y divided by twice the volume of X, and \({\text {ru}}(Y)\) is the volume-normalized Ruelle invariant. We then construct dynamically convex contact forms on \(S^3\) that violate this bound using methods of Abbondandolo–Bramham–Hryniewicz–Salomão. These are the first examples of dynamically convex contact 3-spheres that are not strictly contactomorphic to a convex boundary Y.