[Paper Review] Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group H^n

Manuel Ritoré, César Rosales|ArXiv.org|Apr 21, 2005

Geometric Analysis and Curvature Flows4 references16 citations

TL;DR

This paper classifies rotationally invariant hypersurfaces with constant mean curvature (CMC) in the Heisenberg group ℍⁿ by reducing the CMC partial differential equation to an ordinary differential system via rotational symmetry. The key result establishes a Heisenberg group analog of Delaunay's classification, identifying CMC hypersurfaces as nodoids, spheres, and catenoids, and proving that the only compact, embedded, rotationally symmetric CMC hypersurfaces are the spheres S_H for H > 0.

ABSTRACT

In this paper we study sets in the $n$-dimensional Heisenberg group $\hhn$ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in $\hhn$. We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones. Our main result describes which are the CMC hypersurfaces of revolution in $\hhn$. The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence we classify the rotationally invariant isoperimetric sets in $\hhn$.

Motivation & Objective

• To characterize rotationally invariant hypersurfaces with constant mean curvature (CMC) in the n-dimensional Heisenberg group ℍⁿ.
• To define and justify a notion of mean curvature for hypersurfaces in ℍⁿ that aligns with prior definitions in the literature.
• To determine which CMC hypersurfaces of revolution exist in ℍⁿ, extending the classical Delaunay classification to the sub-Riemannian setting.
• To identify the rotationally invariant isoperimetric sets in ℍⁿ, which are critical for minimizing perimeter under volume constraints.

Proposed method

• Define the sub-Riemannian perimeter in ℍⁿ using horizontal vector fields and the De Giorgi perimeter formulation.
• Introduce a notion of mean curvature as the Riemannian divergence of the horizontal normal vector field ν_H on the hypersurface.
• Leverage rotational symmetry about the t-axis to reduce the CMC PDE to a system of ordinary differential equations in the variables (x, σ, t).
• Analyze the resulting ODE system using geometric and analytic techniques, including the use of holomorphic one-forms on Riemann surfaces.
• Apply symmetry and reflection principles to construct complete, embedded solutions from local solutions of the ODE system.
• Use complex analysis to evaluate integrals related to the t-coordinate and prove that t₂ > 0, implying self-intersections and nodoid-like geometry.

Experimental results

Research questions

• RQ1 What are the complete, embedded, rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍⁿ?
• RQ2 How does the CMC equation in ℍⁿ behave under rotational symmetry, and what types of solutions emerge?
• RQ3 Can the Delaunay classification of constant mean curvature surfaces in Euclidean space be extended to the sub-Riemannian setting of ℍⁿ?
• RQ4 Are the rotationally symmetric isoperimetric sets in ℍⁿ congruent to the spheres S_H described in Example 4.2?
• RQ5 What is the geometric structure of CMC hypersurfaces in ℍⁿ when the mean curvature H ≠ 0 and the energy E > 0?

Key findings

• The only compact, embedded, rotationally symmetric CMC hypersurfaces in ℍⁿ are the spheres S_H for H > 0, as described in Example 4.2.
• For H > 0 and E > 0, the resulting hypersurfaces are periodic, embedded, and resemble Euclidean nodoids, with self-intersections due to t₂ > 0.
• The only minimal (H = 0) rotationally symmetric hypersurfaces in ℍⁿ are Euclidean hyperplanes orthogonal to the t-axis and catenoidal-type surfaces.
• The constructed CMC hypersurfaces are generated by curves that are graphs of x(t) over t, with x(t) strictly decreasing and concave/convex in different intervals.
• The t₂ integral, representing the time to return to symmetry, is strictly positive, confirming the nodoid-like structure and non-compactness of the surface.
• The use of holomorphic one-forms on the Riemann surface associated to w² = x^{4n−2} − (E + Hx^{2n})² allows exact evaluation of the t₂ integral, proving t₂ > 0.