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

2020/02/27 by Hurtado, Ana, Rosales, Césa
#49Q20 #53C17 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2002.12057

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 C² surfaces with a non-empty singular set of curves. Combined with previous results, this allows to describe any complete, orientable, embedded and stable C² surface Σ in the Heisenberg group ℍ¹ and the sub-Riemannian sphere \mathbbS³ of constant curvature 1. In ℍ¹ we conclude that Σ is a Euclidean plane, a Pansu sphere or congruent to the hyperbolic paraboloid t=xy. In \mathbbS³ we deduce that Σ is one of the Pansu spherical surfaces discovered in [28]. As a consequence, such spheres are the unique C² solutions to the sub-Riemannian isoperimetric problem in \mathbbS³.

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