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60
Constant mean curvature surfaces in SubRiemannian spaces
, 2005
"... Abstract. We investigate the minimal and isoperimetric surface problems in a large class of subRiemannian manifolds, the socalled Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of ..."
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Cited by 34 (8 self)
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Abstract. We investigate the minimal and isoperimetric surface problems in a large class of subRiemannian manifolds, the socalled Vertically Rigid spaces. We construct an adapted connection for such spaces and, using the variational tools of Bryant, Griffiths and Grossman, derive succinct forms of the EulerLagrange equations for critical points for the associated variational problems. Using the EulerLagrange equations, we show that minimal and isoperimetric surfaces satisfy a constant horizontal mean curvature conditions away from characteristic points. Moreover, we use the formalism to construct a horizontal second fundamental form, II0, for vertically rigid spaces and, as a first application, use II0 to show that minimal surfaces cannot have points of horizontal positive curvature and, that minimal surfaces in Carnot groups cannot be locally horizontally geometrically convex. We note that the convexity condition is distinct from others currently in the literature. 1.
Properly embedded and immersed minimal surfaces in the Heisenberg group
 Bull. Austral. Math. Soc
"... We study properly embedded and immersed p(pseudohermitian)minimal surfaces in the 3dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according ..."
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Cited by 29 (7 self)
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We study properly embedded and immersed p(pseudohermitian)minimal surfaces in the 3dimensional Heisenberg group. From the recent work of Cheng, Hwang, Malchiodi, and Yang, we learn that such surfaces must be ruled surfaces. There are two types of such surfaces: band type and annulus type according to their topology. We give an explicit expression for these surfaces. Among band types there is a class of properly embedded pminimal surfaces of so called helicoid type. We classify all the helicoid type pminimal surfaces. This class of pminimal surfaces includes all the entire pminimal graphs (except contact planes) over any plane. Moreover, we give a necessary and sufficient condition for such a pminimal surface to have no singular points. For general complete immersed pminimal surfaces, we prove a half space theorem and give a criterion for the properness.
Existence and uniqueness for parea minimizers in the Heisenberg group, preprint
, 2006
"... Abstract. In [3], we study pmean curvature and associated pminimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized parea and associated (p) minimizers in gene ..."
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Cited by 27 (6 self)
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Abstract. In [3], we study pmean curvature and associated pminimal surfaces in the Heisenberg group from the viewpoint of PDE and differential geometry. In this paper, we look into the problem through the variational formulation. We study a generalized parea and associated (p) minimizers in general dimensions. We prove the existence and elaborate on the uniqueness of minimizers. Since this is reduced to solving a degenerate elliptic equation, we need to consider the effect of the singular set and this requires a careful study. We define the notion of weak solution and prove that in a certain Sobolev space, a weak solution is a minimizer and vice versa. We also give many interesting examples in dimension 2. An intriguing point is that, in dimension 2, a C 2 smooth solution from the PDE viewpoint may not be a minimizer. However, this statement is true for higher dimensions due to the relative smallness of the size of the singular set.
SubRiemannian calculus on hypersurfaces in Carnot groups
 Advances in Math. 215
, 2007
"... 2. Carnot groups 7 3. Two basic models 10 4. The subbundle of horizontal planes 13 ..."
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Cited by 25 (2 self)
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2. Carnot groups 7 3. Two basic models 10 4. The subbundle of horizontal planes 13
An intrinsic measure for submanifolds in stratified groups
 J. Reine Angew. Math
"... Abstract. For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hau ..."
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Cited by 20 (3 self)
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Abstract. For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hausdorff measure of the submanifold with respect to the CarnotCarathéodory distance, respectively. Our main technical tool is an intrinsic blowup at points of maximum degree. We also show that the intrinsic tangent cone to the submanifold at these points is always a subgroup. Finally, by direct computations in the Engel group, we show how our results can be extended to higher step
SMOOTHNESS OF LIPSCHITZ MINIMAL INTRINSIC GRAPHS IN HEISENBERG GROUPS H n, n> 1
"... Abstract. We prove that Lipschitz intrinsic graphs in the Heisenberg groups H n, with n> 1, which are vanishing viscosity solutions of the minimal surface equation are smooth. 1. ..."
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Cited by 19 (4 self)
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Abstract. We prove that Lipschitz intrinsic graphs in the Heisenberg groups H n, with n> 1, which are vanishing viscosity solutions of the minimal surface equation are smooth. 1.
Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H¹
, 2006
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A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing
 Amer. J. Math
"... Abstract. One of the main objectives of this paper is to unravel a new interesting phenomenon of the subRiemannian Bernstein problem with respect to its Euclidean ancestor, with the purpose of also indicating a possible line of attack toward its solution. We show that the global intrinsic graphs (1 ..."
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Cited by 17 (2 self)
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Abstract. One of the main objectives of this paper is to unravel a new interesting phenomenon of the subRiemannian Bernstein problem with respect to its Euclidean ancestor, with the purpose of also indicating a possible line of attack toward its solution. We show that the global intrinsic graphs (1.2) are unstable critical points of the horizontal perimeter. As a consequence of this fact, the study of the stability acquires a central position in the problem itself.
Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space
, 2007
"... We classify the entire minimal vertical graphs in the Heisenberg group Nil3 endowed with a Riemannian leftinvariant metric. This classification, which provides a solution to the Bernstein problem in Nil3, is given in terms of the AbreschRosenberg holomorphic differential for minimal surfaces in Ni ..."
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Cited by 15 (3 self)
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We classify the entire minimal vertical graphs in the Heisenberg group Nil3 endowed with a Riemannian leftinvariant metric. This classification, which provides a solution to the Bernstein problem in Nil3, is given in terms of the AbreschRosenberg holomorphic differential for minimal surfaces in Nil3.
AREASTATIONARY SURFACES INSIDE THE SUBRIEMANNIAN THREESPHERE
, 2006
"... ABSTRACT. We consider the subRiemannian metric g h on S 3 provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the CarnotCarathéodory distance and we show that, depending on their c ..."
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Cited by 14 (2 self)
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ABSTRACT. We consider the subRiemannian metric g h on S 3 provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the CarnotCarathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus. We study areastationary surfaces with or without a volume constraint in (S 3, g h). By following the ideas and techniques in [RR2] we introduce a variational notion of mean curvature, characterize stationary surfaces, and prove classification results for complete volumepreserving areastationary surfaces with nonempty singular set. We also use the behaviour of the CarnotCarathéodory geodesics and the ruling property of constant mean curvature surfaces to show that the only C 2 compact, connected, embedded surfaces in (S 3, g h) with empty singular set and constant mean curvature H such that H / √ 1 + H 2 is an irrational number, are Clifford tori. Finally we describe which are the complete rotationally invariant surfaces with constant mean curvature in (S 3, g h). 1.