Area-stationary surfaces in the Heisenberg group H^1

Abstract

We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carathéodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero (or constant if a volume-preserving condition is assumed) and such that the characteristic curves meet orthogonally the singular curves. Moreover, a Minkowski type formula relating the area, the mean curvature, and the volume is obtained for volume-preserving area-stationary surfaces enclosing a given region. As a consequence of the characterization of area-stationary surfaces, we refine previous Bernstein type theorems in order to describe entire area-stationary graphs over the xy-plane in H^1. A calibration argument shows that these graphs are globally area-minimizing. Finally, by using the known description of the singular set, the characterization of area-stationary surfaces, and the ruling property of constant mean curvature surfaces, we prove our main results where we classify volume-preserving area-stationary surfaces in H^1 with non-empty singular set. In particular, we deduce the following counterpart to Alexandrov uniqueness theorem in Euclidean space: any compact, connected, C^2 surface in H^1 area-stationary under a volume constraint must be congruent with a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points. As a consequence, we solve the isoperimetric problem in H^1 assuming C^2 smoothness of the solutions.

37 pages, 3 figures; corrected typos, very recent references added

References in corpus (2)

• The Bernstein Problem in the Heisenberg Group
• H-minimal graphs of low regularity in the Heisenberg group

Cited by in corpus (9)

• First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds
• Variation of Perimeter Measure in sub-Riemannian geometry
• Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$
• Tubular neighborhoods in the sub-Riemannian Heisenberg groups
• Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups
• The kinematic formula in the 3D-Heisenberg group
• Convex isoperimetric sets in the Heisenberg group
• Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space
• The existence of horizontal envelopes in the 3D-Heisenberg group