Constant angle surfaces in the Heisenberg group (original) (raw)
MINIMAL SURFACES IN PSEUDOHERMITIAN GEOMETRY AND THE BERNSTEIN PROBLEM IN THE HEISENBERG GROUP
We develop a surface theory in pseudohermitian geometry. We define a notion of (p-)mean curvature and the associated (p-)minimal surfaces. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate the go through theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and hence solves the analogue of the Bernstein problem in the Heisenberg group H 1 . In H 1 , identified with the Euclidean space R 3 , the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set. We interpret the p-mean curvature: as the curvature of a characteristic curve, as the tangential sublaplacian of a defining function, and as a quantity in terms of calibration geometry. We also show that there are no closed, connected, C 2 smoothly embedded constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard S 3 . This fact continues to hold when S 3 is replaced by a general spherical pseudohermitian 3manifold.
Steiner Formula and Gaussian Curvature in the Heisenberg Group
Bruno Pini Mathematical Analysis Seminar, 2017
The classical Steiner formula expresses the volume of the ∈-neighborhood Ω ∈ of a bounded and regular domain Ω⊂R n as a polynomial of degree n in ∈. In particular, the coefficients of this polynomial are the integrals of functions of the curvatures of the boundary ∂Ω. The aim of this note is to present the Heisenberg counterpart of this result. The original motivation for studying this kind of extension is to try to identify a suitable candidate for the notion of horizontal Gaussian curvature. The results presented in this note are contained in the paper [4] written in collaboration with Zoltan Balogh, Fausto Ferrari, Bruno Franchi and Kevin Wildrick
The Hessian of the distance from a surface in the Heisenberg group
2006
We compute the horizontal Hessian of the signed Carnot-Charatheodory distance from a surface S in the Heisenberg group H. The expression for the Hessian is in terms of the surface's intrinsic curvatures. As an application, we compute the horizontal Hessian of the Carnot-Charatheodory distance from a point in H.
Annali di Matematica Pura ed Applicata (1923 -)
In this paper, we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space {\mathbb {R}}^3$$ R 3 to the case of helicoidal surfaces in the Bianchi–Cartan–Vranceanu (BCV) spaces, i.e., in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension 4 or 6, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.
Geometry of left-invariant Randers metric on the Heisenberg groups
Arab Journal of Mathematical Sciences, 2021
Purpose In this paper, we consider the Heisenberg groups which play a crucial role in both geometry and theoretical physics. Design/methodology/approach In the first part, we retrieve the geometry of left-invariant Randers metrics on the Heisenberg group H2n+1, of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H2n+1 have constant negative scalar curvature. Findings In the second part, we present our main results. We show that the Heisenberg group H2n+1 cannot admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, the flag curvature of Z-Randers metrics in some special directions is obtained which shows that there exist flags of strictly negative and strictly positive curvatures. Originality/value In this work, we present complete Reimannian geometry of left invarint-metrics on Heisenberg groups. Also, some geometric properties of...
Geometry of left invariant Randers metric on the Heisenberg group
arXiv (Cornell University), 2019
In this paper, we investigate the geometry of left-invariant Randers metrics on the Heisenberg group H 2n+1 , of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H 2n+1 have constant negative scalar curvature. Also, we show the Heisenberg group H 2n+1 can not admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, an explicit formula for computing flag curvature is obtained which shows that there exist flags of strictly negative and strictly positive curvatures.
Generalized Weierstrass representation for surfaces in Heisenberg spaces
Differential Geometry and its Applications, 2012
We establish a spinorial representation for surfaces immersed with prescribed mean curvature in Heisenberg space. This permits to obtain minimal immersions starting with a harmonic Gauss map whose target is either the Poincaré disc or a hemisphere of the round sphere.
A new approach on constant angle surfaces in the special linear group
Journal of Geometry, 2019
We give a new approach to a global classification of surfaces for which the unit normal makes a constant angle with a fixed direction in the special linear group. In particular, we give an description of these surfaces by means of two suitable curves and investigate geometric properties such as completeness, extensibility and invariance under a one-parameter subgroup of isometries.