3d Convex Contact Forms And The Ruelle Invariant

2020/12/23 by Chaidez, Julian, Edtmair, Oliver · 3 citations
#37J55 #53C99 #53D35 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2012.12869

Abstract

Let X ⊂ ℝ⁴ be a convex domain with smooth boundary Y. We use a relation between the extrinsic curvature of Y and the Ruelle invariant Ru(Y) of the natural Reeb flow on Y to prove that there exist constants C > c > 0 independent of Y such that c lt; (Ru(Y)²)/(vol(X)) ⋅ sys(Y) lt; C Here 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 S³ 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.

Cited by

• Topological Obstructions to Dynamical Convexity
• Dynamical extensions of Zoll to nonsmooth convex bodies
• On dynamical invariants of coadjoint orbits of 3D volume-preserving diffeomorphisms

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