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An instability criterion for volume-preserving area-stationary surfaces with singular curves in sub-Riemannian 3-space forms

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Abstract

We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension 3. We prove a stability inequality and provide sufficient conditions ensuring instability of volume-preserving area-stationary \(C^2\) surfaces with a non-empty singular set of curves. Combined with previous results, this allows to describe any complete, orientable, embedded and stable \(C^2\) surface \(\Sigma \) in the Heisenberg group \({\mathbb {H}}^1\) and the sub-Riemannian sphere \({\mathbb {S}}^3\) of constant curvature 1. In \({\mathbb {H}}^1\) we conclude that \(\Sigma \) is a Euclidean plane, a Pansu sphere or congruent to the hyperbolic paraboloid \(t=xy\). In \({\mathbb {S}}^3\) we deduce that \(\Sigma \) is one of the Pansu spherical surfaces discovered in Hurtado and Rosales (Math Ann 340(3):675–708, 2008). As a consequence, such spheres are the unique \(C^2\) solutions to the sub-Riemannian isoperimetric problem in \({\mathbb {S}}^3\).

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Authors and Affiliations

  1. Departamento de Geometría y Topología and Excellence Research Unit “Modeling Nature” (MNat), Universidad de Granada, 18071, Granada, Spain

    Ana Hurtado & César Rosales

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  1. Ana Hurtado
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  2. César Rosales
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Correspondence to César Rosales.

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Communicated by A. Malchiodi.

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The authors were supported by MINECO Grant No.  MTM2017-84851-C2-1-P and Junta de Andalucía Grant No. FQM-325.

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Hurtado, A., Rosales, C. An instability criterion for volume-preserving area-stationary surfaces with singular curves in sub-Riemannian 3-space forms. Calc. Var. 59, 165 (2020). https://doi.org/10.1007/s00526-020-01834-1

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