scholarly journals Gauss map harmonicity and mean curvature of a hypersurface in a homogeneous manifold

2006 ◽  
Vol 224 (1) ◽  
pp. 45-63 ◽  
Author(s):  
Fidelis Bittencourt ◽  
Jaime Ripoll
Keyword(s):  
Mean Curvature ◽  
Gauss Map ◽  
Homogeneous Manifold

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.

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 Ruled Surfaces of Generalized Self-Similar Solutions of the Mean Curvature Flow

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Rafael López
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Position Vector ◽  
Translation Surfaces ◽  
Self Similar ◽  
The Mean ◽  
Similar Solutions

AbstractIn Euclidean space, we investigate surfaces whose mean curvature H satisfies the equation $$H=\alpha \langle N,{\mathbf {x}}\rangle +\lambda $$ H = α ⟨ N , x ⟩ + λ , where N is the Gauss map, $${\mathbf {x}}$$ x is the position vector, and $$\alpha $$ α and $$\lambda $$ λ are two constants. There surfaces generalize self-shrinkers and self-expanders of the mean curvature flow. We classify the ruled surfaces and the translation surfaces, proving that they are cylindrical surfaces.

scholarly journals Rotational hypersurfaces with $L_r$-pointwise 1-type Gauss map

2018 ◽  
Vol 36 (3) ◽  
pp. 207-217
Author(s):  
Akram Mohammadpouri
Keyword(s):  
Euclidean Space ◽  
Smooth Function ◽  
Mean Curvature ◽  
Gauss Map ◽  
Constant Vector ◽  
Newton Transformation ◽  
Linearized Operator

In this paper, we study hypersurfaces in $\E^{n+1}$ which Gauss map $G$ satisfies the equation $L_rG = f(G + C)$ for a smooth function $f$ and a constant vector $C$, where $L_r$ is the linearized operator of the $(r + 1)$th mean curvature of the hypersurface, i.e., $L_r(f)=tr(P_r\circ\nabla^2f)$ for $f\in \mathcal{C}^\infty(M)$, where $P_r$ is the $r$th Newton transformation, $\nabla^2f$ is the Hessian of $f$, $L_rG=(L_rG_1,\ldots,L_rG_{n+1}), G=(G_1,\ldots,G_{n+1})$. We show that a rational hypersurface of revolution in a Euclidean space $\E^{n+1}$ has $L_r$-pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.

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.

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