Contactomorphic vertically convex domains

TL;DR

This paper classifies contactomorphisms between certain symmetric, convex domains in a Darboux space, revealing invariants like mean curvature and characteristic length that distinguish these domains up to boundary-preserving contactomorphisms.

Contribution

It provides a classification of contactomorphism classes among vertically convex domains with symmetry, introducing invariants such as mean curvature and characteristic length for these domains.

Findings

Contactomorphism classes are characterized by mean curvature at umbilic points.

Invariants like total characteristic action distinguish domains in generalized settings.

The results extend to codisc bundles and symplectisations of non-Besse contact manifolds.

Abstract

We consider the standard Darboux space equipped with the radial symmetric contact form. We study co-orientation preserving contactomorphisms between relatively compact domains up to the boundary. We determine the contactomorphism classes among all strict vertically convex domains over a round ball in the Liouville hyperplane that are radially symmetric about the Reeb axis and whose boundary coincide along a neighbourhood of the common equator. The total invariant is the mean curvature of the bounding sphere at the umbilic points with the same sign. Replacing the Liouville hyperplane by codisc bundles of closed non-Besse Riemannian manifolds or finite symplectisations of closed non-Besse strict contact manifolds analogous results are formulated in terms of characteristic length and total characteristic action, resp.