Area-stationary surfaces in the Heisenberg group H1

AbstractWe consider area-stationary surfaces, perhaps with a volume constraint, in the Heisenberg group H1 endowed with its Carnot–Carathéodory distance. By analyzing the first variation of area, we characterize C2 area-stationary surfaces 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 the Bernstein type theorem given in [Jih-Hsin Cheng, Jenn-Fang Hwang, Andrea Malchiodi, Paul Yang, Minimal surfaces in pseudohermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (1) (2005), 129–177. MR 2165405] and [Nicola Garofalo, Scott Pauls, The Bernstein problem in the Heisenberg group, arXiv: math.DG/0209065] to describe all C2 entire area-stationary graphs over the xy-plane in H1. A calibration argument shows that these graphs are globally area-minimizing.Finally, by using the description of the singular set in [Jih-Hsin Cheng, Jenn-Fang Hwang, Andrea Malchiodi, Paul Yang, Minimal surfaces in pseudohermitian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (1) (2005), 129–177. MR 2165405], the characterization of area-stationary surfaces, and the ruling property of constant mean curvature surfaces, we prove our main results where we classify area-stationary surfaces in H1, with or without a volume constraint, and non-empty singular set. In particular, we deduce the following counterpart to Alexandrov uniqueness theorem in Euclidean space: any compact, connected, C2 surface in H1, area-stationary under a volume constraint, must be congruent to 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 H1 assuming C2 smoothness of the solutions