3D convex contact forms and the Ruelle invariant

Let $$X \subset {\mathbb {R}}^4$$ be a convex domain with smooth boundary Y. We use relation between the extrinsic curvature of Y and Ruelle invariant Reeb flow on to prove that there are constants $$C> c > 0$$ independent such $$\begin{aligned} \leqslant {\text {ru}}(Y) \cdot {sys}}(Y)^{1/2} C \end{aligned}$$ Here $${\text {sys}}(Y)$$ is systolic ratio Y, i.e. square minimal period closed orbit divided by twice volume X, {ru}}(Y)$$ volume-normalized invariant. then construct dynamically contact forms $$S^3$$ violate this bound using methods Abbondandolo–Bramham–Hryniewicz–Salomão. These first examples 3-spheres not strictly contactomorphic