[Paper Review] A symmetry result on Reinhardt domains

Vittorio Martino|arXiv (Cornell University)|Nov 18, 2010

Quantum chaos and dynamical systems14 references4 citations

TL;DR

This paper establishes a symmetry result for bounded Reinhardt domains in ℂⁿ⁺¹: if the characteristic curvature h(T,T) — the normal curvature of the Hamiltonian flow along the characteristic direction T — is constant across the boundary hypersurface M, then M must be a sphere. The proof leverages CR geometry and Hamiltonian dynamics, showing that constant h(T,T) implies spherical symmetry via conservation laws on torus-structured integral curves of T.

ABSTRACT

We show the following symmetry property of a bounded Reinhardt domain $Ω$ in $\mathbb{C}^{n+1}$: let $M=\partialΩ$ be the smooth boundary of $Ω$ and let $h$ be the Second Fundamental Form of $M$; if the coefficient $h(T,T)$ related to the characteristic direction $T$ is constant then $M$ is a sphere. In Appendix we state the result from an hamiltonian point of view.

Motivation & Objective

• To characterize bounded Reinhardt domains in ℂⁿ⁺¹ whose boundary has constant characteristic curvature h(T,T).
• To establish a symmetry result analogous to Alexandrov's theorem, but based on the characteristic curvature rather than Levi curvatures.
• To explore the geometric implications of h(T,T) being constant across the boundary, using Hamiltonian and CR-geometric tools.
• To show that under this condition, the boundary must be a sphere, extending known results on pseudoconvex hypersurfaces with constant curvature.
• To connect the geometric condition h(T,T) = const to conserved quantities in Hamiltonian systems on tori, leveraging action-angle variables.

Proposed method

• Uses the second fundamental form h and the characteristic direction T = J·N to define the characteristic curvature h(T,T).
• Applies CR geometry: decomposes the tangent space of the boundary M into horizontal (HₚM) and vertical (ℝTₚ) subspaces using the complex structure J.
• Relates the Levi form l to the second fundamental form h via the identity l(Z,Z) = h(X,X) + h(JX,JX), linking CR and Riemannian geometry.
• Employs Hamiltonian dynamics on ℂⁿ⁺¹ ≈ ℝ²ⁿ⁺², identifying the characteristic direction T as the Hamiltonian vector field Xᴴ = J·∇f for a defining function f.
• Exploits the fact that for Reinhardt domains, f(z) = g(r₁,…,rₙ₊₁) depends only on |zₖ|² = rₖ, leading to conserved actions rₖ along integral curves.
• Uses the Liouville-Arnold theorem to show that integral curves of T lie on (n+1)-dimensional tori 𝕋ⁿ⁺¹, where h(T,T) and all Levi curvatures are constant.

Experimental results

Research questions

• RQ1 Under what conditions on the characteristic curvature h(T,T) does the boundary of a bounded Reinhardt domain in ℂⁿ⁺¹ become a sphere?
• RQ2 How does the constancy of h(T,T) across all integral curves of the characteristic direction T imply global spherical symmetry?
• RQ3 Can the symmetry of the boundary be deduced from a single curvature quantity (h(T,T)) rather than from Levi curvatures or mean curvature?
• RQ4 What is the role of Hamiltonian dynamics and action-angle variables in characterizing the geometry of Reinhardt domains?
• RQ5 How do the conserved quantities (rₖ, h(T,T), and Levi curvatures) on tori relate to the global shape of the boundary?

Key findings

• If the characteristic curvature h(T,T) is constant on the entire boundary M of a bounded Reinhardt domain Ω ⊂ ℂⁿ⁺¹, then M is a sphere.
• The value of h(T,T) is constant along each integral curve of the characteristic direction T, and these curves lie on (n+1)-dimensional tori 𝕋ⁿ⁺¹.
• All j-th Levi curvatures Lʲ are conserved quantities on each torus 𝕋ⁿ⁺¹, and so is h(T,T), due to the radial symmetry of the defining function f(z) = g(r₁,…,rₙ₊₁).
• The Hamiltonian vector field Xᴴ = J·∇f coincides with the characteristic direction T, and its integral curves are periodic orbits on the tori.
• The constancy of h(T,T) across all such tori forces the domain to be spherically symmetric, leading to the conclusion that Ω is a ball.
• The result extends previous Alexandrov-type theorems by replacing assumptions on Levi curvatures with a single condition on the characteristic curvature h(T,T).