Area-stationary and stable surfaces of class C¹ in the sub-Riemannian\n Heisenberg group mathbb H¹

2014/10/14 by Matteo Galli, Galli, Matteo, Manuel Ritoré +1 · 2 citations
Computer Science · Mathematics · #49Q20 #53C17 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1410.3619

openalex publication_date 2014/10/14 · openalex created_date 2022/09/11 · openalex updated_date 2026/07/28

Abstract

We consider surfaces of class C¹ in the 3-dimensional sub-Riemannian\nHeisenberg group mathbb H¹. Assuming the surface is area-stationary,\ni.e., a critical point of the sub-Riemannian perimeter under compactly\nsupported variations, we show that its regular part is foliated by horizontal\nstraight lines. In case the surface is complete and oriented, without singular\npoints, and stable, i.e., a second order minimum of perimeter, we prove that\nthe surface must be a vertical plane. This implies the following Bernstein type\nresult: a complete locally area-minimizing intrinsic graph of a C¹ function\nin mathbb H¹ is a vertical plane.\n

Cited by

• The Bernstein problem for Sobolev intrinsic graphs in the Heisenberg group
• Plateau's Problem for intrinsic graphs in the Heisenberg Group

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