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Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍn

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Abstract

In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍn which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍn.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 ℍn.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 ℍn.

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  1. Departamento de Geometría y Topología, Universidad de Granada, E-18071, Granada, España

    Manuel Ritoré & César Rosales

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  1. Manuel Ritoré
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  2. César Rosales
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Correspondence to Manuel Ritoré.

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Communicated by Fulvio Ricci

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Ritoré, M., Rosales, C. Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍn . J Geom Anal 16, 703–720 (2006). https://doi.org/10.1007/BF02922137

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  • DOI: https://doi.org/10.1007/BF02922137

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