Area-stationary surfaces in the Heisenberg group H¹

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We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H¹ endowed with its Carnot-Carathéodory distance. By analyzing the first variation of area, we characterize C² 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 H¹. 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 H¹ with 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 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 H¹ assuming C² smoothness of the solutions.