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Contactomorphic vertically convex domains

Jan Eyll, Jonas Fritsch, Kai Zehmisch

TL;DR

The paper classifies contactomorphism types of strictly vertically convex domains in a Darboux setup with radial symmetry, showing that two radially symmetric shapes are contactomorphic up to the boundary precisely when their second-derivative data at the umbilic point agree, equivalently their mean curvature matches in sign. It introduces robust invariants based on characteristic length and total characteristic action, and proves a Moser-based integrating method that realizes contactomorphisms via vertical isotopies while preserving the equator. The framework extends to cotangent bundles and Stein-type symplectisations, establishing analogous invariants and demonstrating how equator-preserving twists and Reeb monodromy govern the equivalence of shadow domains. The result provides explicit, computable criteria for contactomorphic equivalence of vertically convex domains, and furnishes a versatile toolbox for constructing shape perturbations that preserve contact type across a range of ambient geometries. Overall, the work unifies local curvature data, Floer-theoretic-type foliations, and global shadow invariants to classify boundary-contact domains in both classical and symplectised contexts.

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.

Contactomorphic vertically convex domains

TL;DR

The paper classifies contactomorphism types of strictly vertically convex domains in a Darboux setup with radial symmetry, showing that two radially symmetric shapes are contactomorphic up to the boundary precisely when their second-derivative data at the umbilic point agree, equivalently their mean curvature matches in sign. It introduces robust invariants based on characteristic length and total characteristic action, and proves a Moser-based integrating method that realizes contactomorphisms via vertical isotopies while preserving the equator. The framework extends to cotangent bundles and Stein-type symplectisations, establishing analogous invariants and demonstrating how equator-preserving twists and Reeb monodromy govern the equivalence of shadow domains. The result provides explicit, computable criteria for contactomorphic equivalence of vertically convex domains, and furnishes a versatile toolbox for constructing shape perturbations that preserve contact type across a range of ambient geometries. Overall, the work unifies local curvature data, Floer-theoretic-type foliations, and global shadow invariants to classify boundary-contact domains in both classical and symplectised contexts.

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.
Paper Structure (32 sections, 12 theorems, 137 equations)

This paper contains 32 sections, 12 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Radial symmetric functions
  3. Morse lemma and consequences
  4. About the Hessian
  5. Invariants of contactomorphisms
  6. Characteristic foliation
  7. Hamiltonian representation
  8. Contactisation
  9. Radial symmetry
  10. Preserving the equator
  11. Integrating a contactomorphism
  12. Vertically diffeomorphic
  13. Interpolation
  14. Moser problem
  15. Decomposition
  16. ...and 17 more sections

Key Result

Theorem 1

Let $(S,V_S,f_{\pm})$ and $(T,V_T,g_{\pm})$ be equivalent shapes in $\mathbb R\times\mathbb R^{2n}$ with shadow set $V_S=V_T$ equal to a round ball about $0$ in $\mathbb R^{2n}$. Assume that $f_{\pm}$ and $g_{\pm}$ are all radially symmetric. Then the domains $(D_S,\xi_{\mathop{\mathrm{\mathrm{st}}}

Theorems & Definitions (33)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2.1.1: Root lemma
  • proof
  • Proposition 3.1.1
  • proof
  • Proposition 3.2.1
  • proof
  • ...and 23 more