Hypersurfaces with constant mean curvature in hyperbolic space form

1996 ◽  
Vol 59 (2) ◽  
Author(s):  
Jean-Marie Morvan ◽  
Wo Bao-Qiang
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Hyperbolic Space Form

Biharmonic hypersurfaces in 5-dimensional non-flat space forms

2019 ◽  
Vol 19 (2) ◽  
pp. 235-250
Author(s):  
Ram Shankar Gupta ◽  
Deepika ◽  
A. Sharfuddin
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Flat Space ◽  
Higher Order ◽  
Principal Curvatures ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Higher Order Mean Curvature

Abstract We prove that every biharmonic hypersurface having constant higher order mean curvature Hr for r > 2 in a space form M5(c) is of constant mean curvature. In particular, every such biharmonic hypersurface in 𝕊5(1) has constant mean curvature. There exist no such compact proper biharmonic isoparametric hypersurfaces M in 𝕊5(1) with four distinct principal curvatures. Moreover, there exist no proper biharmonic hypersurfaces in hyperbolic space ℍ5 or in E5 having constant higher order mean curvature Hr for r > 2.

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.

Sign in / Sign up

Export Citation Format

Share Document