Displaying similar documents to “Minimal surfaces in sub-riemannian manifolds and structure of their singular sets in the ( 2 , 3 ) case”

Minimal surfaces in pseudohermitian geometry

Jih-Hsin Cheng, Jenn-Fang Hwang, Andrea Malchiodi, Paul Yang (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface...

On almost-Riemannian surfaces

Roberta Ghezzi (2010-2011)

Séminaire de théorie spectrale et géométrie

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An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in general it has rank 1 on a nonempty set which is generically a smooth curve. In this paper we provide a short introduction to 2-dimensional almost-Riemannian geometry highlighting...

The Laplace-Beltrami operator in almost-Riemannian Geometry

Ugo Boscain, Camille Laurent (2013)

Annales de l’institut Fourier

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We study the Laplace-Beltrami operator of generalized Riemannian structures on orientable surfaces for which a local orthonormal frame is given by a pair of vector fields that can become collinear. Under the assumption that the structure is 2-step Lie bracket generating, we prove that the Laplace-Beltrami operator is essentially self-adjoint and has discrete spectrum. As a consequence, a quantum particle cannot cross the singular set (i.e., the set where the vector fields...

Projective Reeds-Shepp car on with quadratic cost

Ugo Boscain, Francesco Rossi (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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Fix two points x , x ¯ ∈ S 2 and two directions (without orientation) η , η ¯ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost J [ γ ] = ∫ 0 T γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) + K γ ( t ) 2 γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) d t along all smooth curves starting from x with direction η and ending in x ¯ with direction η ¯ . Here g is the standard Riemannian metric on S 2...

On superminimal surfaces

Thomas Friedrich (1997)

Archivum Mathematicum

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Using the Cartan method O. Boruvka (see [B1], [B2]) studied superminimal surfaces in four-dimensional space forms. In particular, he described locally the family of all superminimal surfaces and classified all of them with a constant radius of the indicatrix. We discuss the mentioned results from the point of view of the twistor theory, providing some new proofs. It turns out that the superminimal surfaces investigated by geometers at the beginning of this century as well...