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 Stable constant mean curvature hypersurfaces

2007 ◽  
Vol 135 (10) ◽  
pp. 3359-3367 ◽  
Author(s):  
Maria Fernanda Elbert ◽  
Barbara Nelli ◽  
Harold Rosenberg
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Constant Mean Curvature Hypersurfaces

scholarly journals The structure of weakly stable constant mean curvature hypersurfaces

2008 ◽  
Vol 60 (1) ◽  
pp. 101-121 ◽  
Author(s):  
Xu Cheng ◽  
Leung-fu Cheung ◽  
Detang Zhou
Keyword(s):  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Constant Mean Curvature Hypersurfaces ◽  
Weakly Stable
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