Stability results for rotationally invariant constant mean curvature surfaces in hyperbolic space

2012 ◽  
Vol 126 (2) ◽  
pp. 269-280 ◽  
Author(s):  
Mohamed Jleli
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Constant Mean Curvature Surfaces ◽  
Rotationally Invariant ◽  
Stability Results

scholarly journals SINGULARITIES OF HYPERBOLIC GAUSS MAPS

2003 ◽  
Vol 86 (2) ◽  
pp. 485-512 ◽  
Author(s):  
SHYUICHI IZUMIYA ◽  
DONGHE PEI ◽  
TAKASI SANO
Keyword(s):  
Differential Geometry ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Singular Set ◽  
Constant Mean Curvature Surfaces ◽  
Zero Sets ◽  
Intrinsic Form ◽  
Hyperbolic Gauss Map

In this paper we adopt the hyperboloid in Minkowski space as the model of hyperbolic space. We define the hyperbolic Gauss map and the hyperbolic Gauss indicatrix of a hypersurface in hyperbolic space. The hyperbolic Gauss map has been introduced by Ch. Epstein [J. Reine Angew. Math. 372 (1986) 96–135] in the Poincaré ball model, which is very useful for the study of constant mean curvature surfaces. However, it is very hard to perform the calculation because it has an intrinsic form. Here, we give an extrinsic definition and we study the singularities. In the study of the singularities of the hyperbolic Gauss map (indicatrix), we find that the hyperbolic Gauss indicatrix is much easier to calculate. We introduce the notion of hyperbolic Gauss–Kronecker curvature whose zero sets correspond to the singular set of the hyperbolic Gauss map (indicatrix). We also develop a local differential geometry of hypersurfaces concerning their contact with hyperhorospheres.2000 Mathematical Subject Classification: 53A25, 53A05, 58C27.

Nontrivial minimal surfaces in a class of Finsler 3-spheres

2014 ◽  
Vol 25 (04) ◽  
pp. 1450034 ◽  
Author(s):  
Ningwei Cui
Keyword(s):  
Minimal Surfaces ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Arbitrary Real Number ◽  
Constant Mean Curvature Surfaces ◽  
New Class ◽  
Constant Flag Curvature ◽  
Berger Spheres ◽  
Rotationally Invariant ◽  
Randers Metrics

In this paper, we study the rotationally invariant minimal surfaces in the Bao–Shen's spheres, which are a class of 3-spheres endowed with Randers metrics [Formula: see text] of constant flag curvature K = 1, where [Formula: see text] are Berger metrics, [Formula: see text] are one-forms and k > 1 is an arbitrary real number. We obtain a class of nontrivial minimal surfaces isometrically immersed in the Bao–Shen's spheres, which is the first class of nontrivial minimal surfaces with respect to the Busemann–Hausdorff measure in Finsler spheres. Moreover, we also obtain a new class of explicit minimal surfaces in the classical Berger spheres [Formula: see text], which was expected to get in [F. Torralbo, Rotationally invariant constant mean curvature surfaces in homogeneous 3-manifolds, Differential Geom. Appl.28(5) (2010) 593–607].

Constant Mean Curvature Surfaces in Hyperbolic Space

1992 ◽  
Vol 114 (1) ◽  
pp. 1 ◽  
Author(s):  
Nicholas J. Korevaar ◽  
Rob Kusner ◽  
William H. Meeks ◽  
Bruce Solomon
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Constant Mean Curvature Surfaces
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