Hypersurfaces with radial mean curvature in space forms (original) (raw)

A new characterization of complete linear Weingarten hypersurfaces in real space forms

Pacific Journal of Mathematics, 2013

We apply the Hopf's strong maximum principle in order to obtain a suitable characterization of the complete linear Weingarten hypersurfaces immersed in a real space form ‫ޑ‬ n+1 c of constant sectional curvature c. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a Clifford torus, if c = 1, a circular cylinder, if c = 0, or a hyperbolic cylinder, if c = −1.

2003

We completely classify constant mean curvature hypersurfaces (CMC) with con-stant δ-invariant in the unit 4-sphere S4 and in the Euclidean 4-space E4. 2000 Mathematics Subject Classification: 53C40, 53B25, 53C42. 1. Introduction. A hypersurface in the unit round sphere Sn+1 is called isoparametric if it has constant principal curvatures. It is known from [1] that an isoparametric hypersurface in S4 is either an open portion of a 3-sphere or an open portion of the product of a circle and a 2-sphere, or an open por-tion of a tube of constant radius over the Veronese embedding. Because every

Hypersurfaces with constant scalar or mean curvature in a unit sphere

Balkan Journal of Geometry and Its Applications

Let M be an n(n ≥ 3)-dimensional complete connected hypersurface in a unit sphere S n+1 (1). In this paper, we show that (1) if M has non-zero mean curvature and constant scalar curvature n(n − 1)r and two distinct principal curvatures, one of which is simple, then M is isometric to the Riemannian product S 1 (√ 1 − c 2) × S n−1 (c), c 2 = n−2 nr if r ≥ n−2 n−1 and S ≤ (n − 1) n(r−1)+2 n−2 + n−2 n(r−1)+2. (2) if M has non-zero constant mean curvature and two distinct principal curvatures, one of which is simple, then M is isometric to the Riemannian product S 1 (√ 1 − c 2) × S n−1 (c), c 2 = n−2 nr if one of the following conditions is satisfied: (i) r ≥

Hypersurfaces in Non-Flat Pseudo-Riemannian Space Forms Satisfying a Linear Condition in the Linearized Operator of a Higher Order Mean Curvature

Taiwanese Journal of Mathematics, 2013

We study hypersurfaces either in the pseudo-Riemannian De Sitter space S n+1 t ⊂ R n+2 t or in the pseudo-Riemannian anti De Sitter space H n+1 t ⊂ R n+2 t+1 whose position vector ψ satisfies the condition L k ψ = Aψ + b, where L k is the linearized operator of the (k+1)-th mean curvature of the hypersurface, for a fixed k = 0, . . ., n−1, A is an (n+2)×(n+2) constant matrix and b is a constant vector in the corresponding pseudo-Euclidean space. For every k, we prove that when H k is constant, the only hypersurfaces satisfying that condition are hypersurfaces with zero (k +1)-th mean curvature and constant k-th mean curvature, open pieces of a totally umbilical hypersurface in S n+1 t (S n t−1 (r), r > 1; S n t (r), 0 < r < 1;

Hypersurfaces with null higher order mean curvature

Bulletin of the Brazilian Mathematical Society, New Series, 2010

A hypersurface M n immersed in a space form is r-minimal if its (r + 1) thcurvature (the (r + 1) th elementary symmetric function of its principal curvatures) vanishes identically. Let W be the set of points which are omitted by the totally geodesic hypersurfaces tangent to M. We will prove that if an orientable hypersurface M n is r-minimal and its r th-curvature is nonzero everywhere, and the set W is nonempty and open, then M n has relative nullity n − r. Also we will prove that if an orientable hypersurface M n is r-minimal and its r th-curvature is nonzero everywhere, and the ambient space is euclidean or hyperbolic and W is nonempty, then M n is r-stable.

Hypersurfaces in a hyperbolic space with constant k-th mean curvature

2009

Let M n be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H n+1 (c) with constant k-th mean curvature H k > 0(k < n) and with two distinct principal curvatures, one of which is simple. In this paper, we show that M n is isometric to the Riemannian product H 1 (c1) × S n−1 (c2) or H n−1 (c1) × S 1 (c2), 1 c 1 + 1 c 2 = 1 c , c1 < 0, c2 > 0 if S ≥ (n − 1)t 2 2 + c 2 t −2 2 on M n , or S ≤ (n − 1)t 2 2 + c 2 t −2 2 on M n , where t2 is the positive real root of (1.6). We extend recent result of Z.Hu et al. [6].

Generalized Weingarten surfaces of harmonic type in hyperbolic 3-space

Differential Geometry and Its Applications, 2018

In this paper we study a large class of Weingarten surfaces M with prescribed hyperbolic Gauss map in the Hyperbolic 3-space, which are the analogous to the Laguerre minimal surfaces in Euclidean space, these surfaces will be called Generalized Weingarten surfaces of harmonic type (HGW-surfaces), this class includes the surfaces of mean curvature one and the linear Weingarten surfaces of Bryant type (BLW-surfaces). We obtain a Weierstrass type representation for this surfaces which depend of three holomorphic functions. As applications we classify the HGW-surfaces of rotation and we obtain a Weierstrass type representation for surfaces of mean curvature one with prescribed hyperbolic Gauss map which depend of two holomorphic functions. Moreover, we classify a class of complete mean curvature one surfaces parametrized by lines of curvature whose coordinates curves has the same geodesic curvature up to sign.

On a class of hypersurfaces in n × ℝ and ℍ n × ℝ

Bull Braz Math Soc, 2010

We give a complete description of all hypersurfaces of the product spaces S n × R and H n × R that have flat normal bundle when regarded as submanifolds with codimension two of the underlying flat spaces R n+2 ⊃ S n × R and L n+2 ⊃ H n × R. We prove that any such hypersurface in S n × R (respectively, H n × R) can be constructed by means of a family of parallel hypersurfaces in S n (respectively, H n) and a smooth function of one variable. Then we show that constant mean curvature hypersurfaces in this class correspond to an isoparametric family in the base space and a smooth function that is explicitly determined in terms of the mean curvature function of the isoparametric family. As another consequence of our general result, we classify the constant angle hypersurfaces of S n × R and H n × R, that is, hypersurfaces with the property that its unit normal vector field makes a constant angle with the unit vector field spanning the second factor R. This extends previous results by Dillen, Fastenakels, Van der Veken, Vrancken and Munteanu for surfaces in S 2 × R and H 2 × R. Our method also yields a classification of all Euclidean hypersurfaces with the property that the tangent component of a constant vector field in the ambient space is a principal direction, in particular of all Euclidean hypersurfaces whose unit normal vector field makes a constant angle with a fixed direction.

Hypersurfaces of revolution with proportional principal curvatures

Advances in Geometry, 2000

We classify hypersurfaces of revolution for which the principal curvatures are proportional. These hypersurfaces form a one-parameter family indexed by the reals. The subsequent classification unites at least two important families of hypersurfaces of revolution: the generalized catenoids and the equizonal ovaloids.