On the Gauss map of complete surfaces of constant mean curvature in R3 and R4

1982 ◽  
Vol 57 (1) ◽  
pp. 519-531 ◽  
Author(s):  
D. A. Hoffman ◽  
R. Osserman ◽  
R. Schoen
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Complete Surfaces

scholarly journals Gauss map and the topology of constant mean curvature hypersurfaces of $$\mathbb {S}^{7}$$ and $$\mathbb {CP}^{3}$$

2019 ◽  
Vol 163 (1-2) ◽  
pp. 279-290
Author(s):  
Fidelis Bittencourt ◽  
Pedro Fusieger ◽  
Eduardo R. Longa ◽  
Jaime Ripoll
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Constant Mean Curvature Hypersurfaces

Hyperbolic submanifolds with finite type hyperbolic Gauss map

2015 ◽  
Vol 26 (02) ◽  
pp. 1550014 ◽  
Author(s):  
Uğur Dursun ◽  
Rüya Yeğin
Keyword(s):  
Scalar Curvature ◽  
Finite Type ◽  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Constant Scalar Curvature ◽  
Gauss Map ◽  
Hyperbolic Spaces ◽  
Nonzero Constant ◽  
Hyperbolic Gauss Map

We study submanifolds of hyperbolic spaces with finite type hyperbolic Gauss map. First, we classify the hyperbolic submanifolds with 1-type hyperbolic Gauss map. Then we prove that a non-totally umbilical hypersurface Mn with nonzero constant mean curvature in a hyperbolic space [Formula: see text] has 2-type hyperbolic Gauss map if and only if M has constant scalar curvature. We also classify surfaces with constant mean curvature in the hyperbolic space [Formula: see text] having 2-type hyperbolic Gauss map. Moreover we show that a horohypersphere in [Formula: see text] has biharmonic hyperbolic Gauss map.

scholarly journals On the Gauss map of ruled surfaces

1992 ◽  
Vol 34 (3) ◽  
pp. 355-359 ◽  
Author(s):  
Christos Baikoussis ◽  
David E. Blair
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Connected Surface ◽  
Image Position ◽  
The Laplace Operator ◽  
Special Case

Let M2 be a (connected) surface in Euclidean 3-space E3, and let G:M2→S2(1) ⊂ E3 be its Gauss map. Then, according to a theorem of E. A. Ruh and J. Vilms [3], M2 is a surface of constant mean curvature if and only if, as a map from M2 to S2(1), G is harmonic, or equivalently, if and only ifwhere δ is the Laplace operator on M2 corresponding to the induced metric on M2 from E3 and where G is seen as a map from M2to E3. A special case of (1.1) is given byi.e., the case where the Gauss map G:M2→E3 is an eigenfunction of the Laplacian δ on M2.

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.

Complete surfaces inE 3 with constant mean curvature

1966 ◽  
Vol 41 (1) ◽  
pp. 313-318 ◽  
Author(s):  
Tilla Klotz ◽  
Robert Osserman
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Complete Surfaces

A REALIZATION OF W GRAVITIES IN THE CONFORMAL GAUGE THROUGH GENERALIZED GAUSS MAPS

1992 ◽  
Vol 07 (02) ◽  
pp. 317-337 ◽  
Author(s):  
R. PARTHASARATHY ◽  
K. S. VISWANATHAN
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Extrinsic Curvature ◽  
Gauss Map ◽  
World Sheet ◽  
Hidden Symmetries ◽  
Extrinsic Geometry ◽  
Conformal Gauge ◽  
Conformal Immersion

String dynamics in ℝn with extrinsic geometry is studied in order to understand their hidden symmetries. String world sheet, realized as a conformal immersion in ℝn, is mapped into the Grassmannian G2, n through the Gauss map. This enables us to study the role of the extrinsic curvature in determining the WSO (n) gravities in the conformal gauge. It is shown that, classically, in ℝ3 and ℝ4 the geometry of surfaces of constant mean curvature densities is equivalent to WSO (n) (n = 3, 4) gravities, the corresponding W algebras being Virasoro (Vir) and Vir ⊕ Vir, respectively.

scholarly journals TIME-INDEPENDENT SOLUTIONS TO THE TWO-DIMENSIONAL NONLINEAR O(3) SIGMA MODEL AND SURFACES OF CONSTANT MEAN CURVATURE

1995 ◽  
Vol 10 (03) ◽  
pp. 337-364 ◽  
Author(s):  
MICHAEL S. ODY ◽  
LEWIS H. RYDER
Keyword(s):  
Mean Curvature ◽  
Sigma Model ◽  
Constant Mean Curvature ◽  
Three Dimensional ◽  
Gauss Map ◽  
Field Vector ◽  
Dimensional Euclidean Space ◽  
Normal Vector ◽  
Constant Mean Curvature Surface ◽  
Spherical Images

It is shown that time-independent solutions to the (2+1)-dimensional nonlinear O(3) sigma model may be placed in correspondence with surfaces of constant mean curvature in three-dimensional Euclidean space. The tools required to establish this correspondence are provided by the classical differential geometry of surfaces. A constant-mean-curvature surface induces a solution to the O(3) model through the identification of the Gauss map, or normal vector, of the surface with the field vector of the sigma model. Some explicit solutions, including the solitons and antisolitons discovered by Belavin and Polyakov, and a more general solution due to Purkait and Ray, are considered and the surfaces giving rise to them are found explicitly. It is seen, for example, that the Belavin-Polyakov solutions are induced by the Gauss maps of surfaces which are conformal to their spherical images, i.e. spheres and minimal surfaces, and that the Purkait-Ray solution corresponds to the family of constant-mean-curvature helicoids first studied by do Carmo and Dajczer in 1982. A generalization of this method to include time dependence may shed new light on the role of the Hopf invariant in this model.

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