Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1

Danielli, Donatella; Garofalo, Nicola; Nhieu, Duy-Minh; Pauls, Scott D.

doi:10.4310/jdg/1231856262

Cited by 47 publications

(67 citation statements)

References 29 publications

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“…In [4] it was proved that the Euclidean vertical planes are the unique entire stable area-stationary intrinsic graphs in M(0). The same characterization was obtained in [21] for entire area-stationary Euclidean graphs with empty singular set. These results suggest that any complete stable area-stationary surface with empty singular set in M(0) must be a vertical plane.…”

Section: Introductionmentioning

confidence: 49%

Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

Rosales

2011

Calc. Var.

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For constant mean curvature surfaces of class C 2 immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classification results for complete surfaces with empty singular set which are stable, i.e., second order minima of the area under a volume constraint, inside the 3-dimensional sub-Riemannian space forms. In the first Heisenberg group we show that such a surface is a vertical plane. In the sub-Riemannian hyperbolic 3-space we give an upper bound for the mean curvature of such surfaces, and we characterize the horocylinders as the unique ones with squared mean curvature 1. Finally we deduce that any complete surface with empty singular set in the sub-Riemannian 3-sphere is unstable.

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Section: Introductionmentioning

confidence: 49%

Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

Rosales

2011

Calc. Var.

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“…Under the assumptions of the Theorem 1.2, as ǫ → 0 the solutions u ǫ converge uniformly (with all its derivatives) on compact subsets of Q to the unique, smooth Regularity of minimal surfaces in the special case of Heisenberg group has been investigated in [23,36,16,15,8,9,18,33,37,39].…”

Section: 2mentioning

confidence: 99%

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

2015

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Abstract. We consider (smooth) solutions of the mean curvature flow of graphs over bounded domains in a Lie group free up to step two (and not necessarily nilpotent), endowed with a one parameter family of Riemannian metrics σ ǫ collapsing to a subRiemannian metric σ 0 as ǫ → 0. We establish C k,α estimates for this flow, that are uniform as ǫ → 0 and as a consequence prove long time existence for the subRiemannian mean curvature flow of the graph. Our proof extend to the setting of every step two Carnot group (not necessarily free) and can be adapted following our previous work in [10] to the total variation flow. IntroductionThe mean curvature flow is the motion of a surface where each points is moving in the direction of the normal with speed equal to the mean curvature In the case where the evolution of graphs S t = {(x, u(x, t))} ⊂ R n × R is considered, then, provided enough regularity is assumed, the function u satisfies the equationGiven appropriate boundary/initial conditions, global in time solutions asymptotically converge to minimal graphs.In this paper we study long time existence of graph solutions of the mean curvature flow in a special class of degenerate Riemannian ambient spaces: The so-called sub-Riemannian Hörmander type setting [22], [40]. In particular we will focus on a class of Lie groups endowed with a metric structure (G, σ 0 ) that arises as limit of collapsing left-invariant tame Riemannian structures (G, σ ǫ ).Our approach to the existence of global (in time) smooth solutions is based on a Riemannian approximation scheme. We study graph solutions of the mean curvature flow in the Riemannian spaces (G, σ ǫ ) where G is a group and σ ǫ is a family of Riemannian metrics that 'collapse' as ǫ → 0 to a sub-Riemannian metric σ 0 in G.

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“…Then the constants appearing in (13) and (16) can be taken to be independent of z as it varies in a compact neighborhood S x ⊂ S x of x in the relatively open set A x . This concludes the proof.…”

Section: Then In a Neighborhoodmentioning

confidence: 99%

Blow-Up Estimates at Horizontal Points and Applications

Magnani

2010

J Geom Anal

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Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathéodory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure at all horizontal points of C 1,1 smooth submanifolds. As an application, we establish an integral formula to compute the spherical Hausdorff measure of any C 1,1 submanifold. Our technique also shows that C 2 smooth submanifolds everywhere admit an intrinsic blow-up and that the limit set is an intrinsically homogeneous algebraic variety.

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Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1

Cited by 47 publications

References 29 publications

Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

Regularity of mean curvature flow of graphs on Lie groups free up to step 2

Blow-Up Estimates at Horizontal Points and Applications

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