Area-stationary surfaces in the Heisenberg group H1

By Manuel Ritor'e, César Rosales. 2005

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We consider area-stationary surfaces, perhaps with a volume constraint, in the Heisenberg group H1 endowed with its Carnot�Caratheodory 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 . 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 , 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 , 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 assuming C2 smoothness of the solutions. 13. Completely reducible hypersurfaces in a pencil Pages 672-688 J.V. Pereira, S. Yuzvinsky Preview PDF (324 K) | Related Articles 14. Modules over motivic cohomology Pages 689-727 Oliver Rondigs, Paul Arne Ostvaer Preview PDF (303 K) | Related Articles 15. The fundamental group of period domains over finite fields Pages 728-741 Sascha Orlik Preview PDF (154 K) | Related Articles articles 1 - 15

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Citations in the corpus

Smoothness of Lipschitz minimal intrinsic graphs in Heisenberg groups , n > 1 (2010, background): “Several first variation formula have been independently established in receent years, see for instance [18], [7], [8], [4], [32], [33], [27], [28], [35], [36].”
Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$ (2008, background): “To quote a few: H0 can be defined in terms of the first variation of the area functional [20, 29, 9, 36, 40, 31, 7], as horizontal divergence of the horizontal unit normal or as limit of the mean curvatures Hε in the Riemannian metrics gε as ε → 0.”
Examples of area-minimizing surfaces in the subriemannian Heisenberg group H^1 with low regularity (2008, methodology): “Characterization in H1 of solutions of the Bernstein problem for C2 surfaces has been obtained by Cheng, Hwang, Malchiodi and Yang [8], and Ritoré and Rosales [29] for tgraphs, and by Barone Adessi, Serra Cassano and Vittone [4] and Garofalo and Pauls [23] for vertical graphs.”
Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, background): “When X = aν + kT , then a proof based on CR-geometry can be found in [15], and [84,85].”
Convex isoperimetric sets in the Heisenberg group (2006, background): “On the other hand, isoperimetric sets can be also determined assuming only the C2 regularity of the boundary and no symmetry (see [23]).”
Regularity of C1 smooth surfaces with prescribed p-mean curvature in the Heisenberg group (2007, methodology): “In [ 20 ], Ritor´e and Rosales proved Pansu’s conjecture for isoperimetric sets of class C2 without any symmetry assumption.”
The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations (2006, methodology): “Very recently the more delicate case when C(S) 6= ∅ has been studied in [45] and [13].”