scholarly journals On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms

2004 ◽  
Vol 48 (1) ◽  
pp. 219-240 ◽  
Author(s):  
Luis J. Alías ◽  
J. Miguel Malacarne
Keyword(s):  
Mean Curvature ◽  
Higher Order ◽  
First Eigenvalue ◽  
Space Forms ◽  
The First Eigenvalue ◽  
Higher Order Mean Curvature ◽  
Closed Hypersurfaces ◽  
Linearized Operator

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.

scholarly journals Totally Umbilical Lightlike Hypersurfaces in Robertson-Walker Spacetimes

ISRN Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Junhong Dong ◽  
Ximin Liu
Keyword(s):  
Mean Curvature ◽  
Higher Order ◽  
Totally Umbilical ◽  
Lightlike Hypersurface ◽  
Higher Order Mean Curvature ◽  
Lightlike Hypersurfaces ◽  
The Relationship

We study the problem of lightlike hypersurface immersed into Robertson-Walker (RW) spacetimes in this paper, where the screen bundle of the hypersurface has constant higher order mean curvature. We consider the following question: under what conditions is the compact lightlike hypersurface totally umbilical? Our approach is based on the relationship between the lightlike hypersurface with its screen bundle and the Minkowski formulae for the screen bundle.

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