Results 1  10
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23
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.
Maximum and comparison principles for convex functions on the Heisenberg group
 Comm. Part. Diff. Eqns
"... The purpose in this paper is to establish pointwise estimates for a class of convex functions on the Heisenberg group. An integral estimate for classical convex functions in terms of the Monge–Ampère operator det D 2 u was proved by Aleksandrov, see [3, Theorem 1.4.2]. Such estimate is of great impo ..."
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Cited by 20 (1 self)
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The purpose in this paper is to establish pointwise estimates for a class of convex functions on the Heisenberg group. An integral estimate for classical convex functions in terms of the Monge–Ampère operator det D 2 u was proved by Aleksandrov, see [3, Theorem 1.4.2]. Such estimate is of great importance in the theory of weak solutions for
Viscosity convex functions on Carnot groups
, 2003
"... We prove that any locally bounded from below, upper semicontinuous vconvex function in any Carnot group is hconvex. Convex functions have played very important roles in PDEs, especially fully nonlinear elliptic PDEs in Euclidean spaces (see CaffarelliCabré [CC] and CrandallIshiiLions [CIL]). Mo ..."
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Cited by 15 (1 self)
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We prove that any locally bounded from below, upper semicontinuous vconvex function in any Carnot group is hconvex. Convex functions have played very important roles in PDEs, especially fully nonlinear elliptic PDEs in Euclidean spaces (see CaffarelliCabré [CC] and CrandallIshiiLions [CIL]). Motivated by this fact and the aim to develop an intrinsic theory of subelliptic fully nonlinear PDEs on Carnot groups, there have been works towards the theory of convex
On the second order derivatives of convex functions on the Heisenberg group
 Ann. Scuola Norm. Sup. Pisa Cl. Sci
"... A classical result of Aleksandrov asserts that convex functions inR n are twice differentiable a.e., and a first step to prove it is to show that these functions have second order distributional derivatives which are measures, see [3, pp. 239245]. On the Heisenberg group, and more generally in Carn ..."
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Cited by 14 (0 self)
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A classical result of Aleksandrov asserts that convex functions inR n are twice differentiable a.e., and a first step to prove it is to show that these functions have second order distributional derivatives which are measures, see [3, pp. 239245]. On the Heisenberg group, and more generally in Carnot groups, several notions of convexity have been introduced
CONVEXITY AND HORIZONTAL SECOND FUNDAMENTAL FORMS FOR HYPERSURFACES IN CARNOT GROUPS
"... Abstract. We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli–Garofalo– Nhieu or Lu–Manfredi–Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph. 1. ..."
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Cited by 8 (1 self)
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Abstract. We use a Riemannian approximation scheme to give a characterization for smooth convex functions on a Carnot group (in the sense of Danielli–Garofalo– Nhieu or Lu–Manfredi–Stroffolini) in terms of the positive semidefiniteness of the horizontal second fundamental form of their graph. 1.
Generalized mean curvature flow in Carnot groups
"... Abstract. In this paper we study the generalized mean curvature flow of sets in the subRiemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [29] and [14]. We establish two special cases of the ..."
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Cited by 8 (5 self)
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Abstract. In this paper we study the generalized mean curvature flow of sets in the subRiemannian geometry of Carnot groups. We extend to our context the level sets method and the weak (viscosity) solutions introduced in the Euclidean setting in [29] and [14]. We establish two special cases of the comparison principle, existence, uniqueness and basic geometric properties of the flow. 1.
Mannucci; Comparison principles for subelliptic equations of MongeAmpère type
 Boll. Unione Mat. Ital. 9
"... Sunto. – We present two comparison principles for viscosity sub and supersolutions of MongeAmpèretype equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvatu ..."
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Cited by 7 (7 self)
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Sunto. – We present two comparison principles for viscosity sub and supersolutions of MongeAmpèretype equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvature in a Carnot group.
THE HESSIAN OF THE DISTANCE FROM A SURFACE IN THE HEISENBERG GROUP
, 2006
"... Abstract. Given a smooth surface S in the Heisenberg group, we compute the Hessian of the function measuring the CarnotCharathéodory distance from S in terms of the Mean Curvature of S and of an “imaginary curvature ” which was introduced in [2] in order to find the geodesics which are metrically n ..."
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Cited by 4 (2 self)
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Abstract. Given a smooth surface S in the Heisenberg group, we compute the Hessian of the function measuring the CarnotCharathéodory distance from S in terms of the Mean Curvature of S and of an “imaginary curvature ” which was introduced in [2] in order to find the geodesics which are metrically normal to S. Explicit formulae are given when S is a plane or the metric sphere. Contents
Characterizations of differentiability for H–convex functions in stratified groups
, 2010
"... Abstract. Using the notion of hsubdifferential, we characterize both first and second order differentiability of hconvex functions in stratified groups. Besides some new results involving the hsubdifferential of hconvex functions, we show that at all hdifferentiability points of an hconvex fu ..."
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Cited by 3 (0 self)
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Abstract. Using the notion of hsubdifferential, we characterize both first and second order differentiability of hconvex functions in stratified groups. Besides some new results involving the hsubdifferential of hconvex functions, we show that at all hdifferentiability points of an hconvex function, the existence of a second order expansion coincides with a suitable differentiability of its horizontal gradient.
GEOMETRIC SECOND DERIVATIVE ESTIMATES IN CARNOT GROUPS AND CONVEXITY
, 803
"... Abstract. We prove some new a priori estimates for H2convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group G. Such estimates are global and are geometric in nature as they involve the horizontal mean curvature H of ∂Ω. As a consequence of our bounds we show ..."
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Cited by 3 (1 self)
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Abstract. We prove some new a priori estimates for H2convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group G. Such estimates are global and are geometric in nature as they involve the horizontal mean curvature H of ∂Ω. As a consequence of our bounds we show that if G has step two, then for any smooth H2convex function in Ω ⊂ G vanishing on ∂Ω one has mX Z ([Xi, Xj]u) 2 dg ≤ 4