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 Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms with at Most Two Distinct Principal Curvatures

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chao Yang ◽  
Jiancheng Liu
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

In this paper, we show that biharmonic hypersurfaces with at most two distinct principal curvatures in pseudo-Riemannian space form Nsn+1c with constant sectional curvature c and index s have constant mean curvature. Furthermore, we find that such biharmonic hypersurfaces Mr2k−1 in even-dimensional pseudo-Euclidean space Es2k, Ms−12k−1 in even-dimensional de Sitter space Ss2kcc>0, and Ms2k−1 in even-dimensional anti-de Sitter space ℍs2kcc<0 are minimal.

Biharmonic hypersurfaces in pseudo-Riemannian space forms

2016 ◽  
Vol 13 (07) ◽  
pp. 1650094 ◽  
Author(s):  
Dan Yang ◽  
Yu Fu
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Constant Sectional Curvature ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

Let [Formula: see text] be a nondegenerate biharmonic pseudo-Riemannian hypersurface in a pseudo-Riemannian space form [Formula: see text] with constant sectional curvature [Formula: see text]. We show that [Formula: see text] has constant mean curvature provided that it has three distinct principal curvatures and the Weingarten operator can be diagonalizable.

scholarly journals A note on Weingarten hypersurfaces in the warped product manifold

2014 ◽  
Vol 25 (14) ◽  
pp. 1450121 ◽  
Author(s):  
Haizhong Li ◽  
Yong Wei ◽  
Changwei Xiong
Keyword(s):  
Mean Curvature ◽  
Warped Product ◽  
Constant Mean Curvature ◽  
Higher Order ◽  
Product Manifold ◽  
Weingarten Surfaces ◽  
Warped Product Manifold ◽  
Higher Order Mean Curvature ◽  
Special Case ◽  
Differential Geom

In this paper, we consider the closed embedded hypersurface Σ in the warped product manifold [Formula: see text] equipped with the metric g = dr2 + λ(r)2 gN. We give some characterizations of slice {r} × N by the condition that Σ has constant weighted higher-order mean curvatures (λ′)αpk, or constant weighted higher-order mean curvature ratio (λ′)αpk/p1, which generalize Brendle's [Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 247–269] and Brendle–Eichmair's [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] results. In particular, we show that the assumption convex of Brendle–Eichmair's result [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] is unnecessary. Here pk is the kth normalized mean curvature of the hypersurface Σ. As a special case, we also give some characterizations of geodesic spheres in ℝn, ℍn and [Formula: see text], which generalize the classical Alexandrov-type results.

ON THE UMBILICITY OF HYPERSURFACES IN THE HYPERBOLIC SPACE

2016 ◽  
Vol 103 (1) ◽  
pp. 45-58
Author(s):  
C. P. AQUINO ◽  
M. BATISTA ◽  
H. F. DE LIMA
Keyword(s):  
Hyperbolic Space ◽  
Mean Curvature ◽  
Warped Product ◽  
Gauss Map ◽  
Higher Order ◽  
Totally Umbilical ◽  
Totally Umbilical Hypersurfaces ◽  
Product Models ◽  
Higher Order Mean Curvature ◽  
Complete Hypersurfaces

In this paper, we establish new characterization results concerning totally umbilical hypersurfaces of the hyperbolic space$\mathbb{H}^{n+1}$, under suitable constraints on the behavior of the Lorentzian Gauss map of complete hypersurfaces having some constant higher order mean curvature. Furthermore, working with different warped product models for$\mathbb{H}^{n+1}$and supposing that certain natural inequalities involving two consecutive higher order mean curvature functions are satisfied, we study the rigidity and the nonexistence of complete hypersurfaces immersed in$\mathbb{H}^{n+1}$.

Curvatures of complete hypersurfaces in space forms

2004 ◽  
Vol 134 (1) ◽  
pp. 55-68 ◽  
Author(s):  
Qing-Ming Cheng
Keyword(s):  
Scalar Curvature ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Three Dimensional ◽  
Space Forms ◽  
Totally Geodesic ◽  
Minimal Hypersurfaces ◽  
Totally Umbilical Hypersurfaces ◽  
Complete Hypersurfaces

In this paper we investigate three-dimensional complete minimal hypersurfaces with constant Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0). We prove that if the scalar curvature of a such hypersurface is bounded from below, then its Gauss-Kronecker curvature vanishes identically. Examples of complete minimal hypersurfaces which are not totally geodesic in the Euclidean space E4 and the hyperbolic space H4(c) with vanishing Gauss-Kronecker curvature are also presented. It is also proved that totally umbilical hypersurfaces are the only complete hypersurfaces with non-zero constant mean curvature and with zero quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature is bounded from below. In particular, we classify complete hypersurfaces with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature r satisfies r≥ ⅔c.

GOLDEN- AND PRODUCT-SHAPED HYPERSURFACES IN REAL SPACE FORMS

2013 ◽  
Vol 10 (04) ◽  
pp. 1320006 ◽  
Author(s):  
MIRCEA CRASMAREANU ◽  
CRISTINA-ELENA HREŢCANU ◽  
MARIAN-IOAN MUNTEANU
Keyword(s):  
Hyperbolic Space ◽  
Space Form ◽  
Shape Operator ◽  
Real Space ◽  
Clifford Torus ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Real Space Forms ◽  
Unit Hypersphere

We define two classes of hypersurfaces in real space forms, golden- and product-shaped, respectively, by imposing the shape operator to be of golden or product type. We obtain the whole families of above hypersurfaces, based on the classification of isoparametric hypersurfaces, as follows: in the golden case all are hyperspheres, a hyperbolic space and a generalized Clifford torus, while for the product case we obtain the unit hypersphere, the hyperplane, a hypersphere and its associated Clifford torus, respectively, according to the type of the ambient space form namely parabolic, hyperbolic or elliptic, respectively.

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