Remarks on biharmonic hypersurfaces in space forms

We consider closed biharmonic hypersurfaces in a Euclidean sphere and prove a rigidity result under a suitable condition on the scalar curvature. Moreover, we establish an integral formula involving the position vector for biharmonic hypersurfaces in space forms. As an application of this formula, we reobtain a result concerning the closed biharmonic hypersurfaces in Euclidean spheres that lie in a closed hemisphere.     

On closed minimal hypersurfaces with constant scalar curvature in \mathbb{S}^7

We consider minimal closed hypersurfaces ${M \subset \mathbb{S}^7(1)}$ with constant scalar curvature. We prove that if M fulfills particular additional assumptions, then it is isoparametric. This gives a partial answer to the question made by S.-S. Chern about the pinching of the scalar curvature for closed minimal hypersurfaces in dimension 6.     

Hypersurfaces of the hyperbolic space with constant scalar curvature

We classify hypersurfaces of the hyperbolic space ?ⁿ⁺¹(c) with constant scalar curvature and with two distinct principal curvatures. Moreover, we prove that if Mⁿ is a complete hypersurfaces with constant scalar curvature n(n ? 1) R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n? 1, then R ≥ c. Additionally, we prove two rigidity theorems for such hypersurfaces.     

COMPLETE HYPERSURFACES WITH CONSTANT SCALAR CURVATURE IN A SPECIAL KIND OF LOCALLY SYMMETRIC MANIFOLD

In this paper, we investigate n-dimensional complete and orientable hypersufaces M n (n ≥ 3) with constant normalized scalar curvature in a locally symmetric manifold. Two rigidity theorems are obtained for these hypersurfaces.     

具有平行平均曲率向量子流形的曲率估计与稳定性

本文估计空间形式中具有平行平均曲率向量子流形上共形度量的数量曲率上界,并利用其研究了具有常平均曲率超曲面的稳定性．     

On spacelike hypersurfaces with constant scalar curvature in locally symmetric Lorentz spaces

The purpose of this paper is to study compact or complete spacelike hypersurfaces with constant normalized scalar curvature in a locally symmetric Lorentz space satisfying some curvature conditions. We give an optimal estimate of the squared norm of the second fundamental form of such hypersurfaces. Furthermore, the totally umbilical hypersurfaces are characterized.     

On Deformable Minimal Hypersurfaces in Space Forms

The aim of this paper is to complete the local classification of minimal hypersurfaces with vanishing Gauss–Kronecker curvature in a 4-dimensional space form. Moreover, we give a classification of complete minimal hypersurfaces with vanishing Gauss–Kronecker curvature and scalar curvature bounded from below.     

Rigidity of linear Weingarten hypersurfaces in locally symmetric manifolds

Our purpose in this paper is to study the rigidity of complete linear Weingarten hypersurfaces immersed in a locally symmetric manifold obeying some standard curvature conditions (in particular, in a Riemannian space with constant sectional curvature). Under appropriated constrains on the scalar curvature function, we prove that such a hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of them being simple. Furthermore, we also deal with the parabolicity of these hypersurfaces with respect to a suitable Cheng–Yau modified operator.     

de Sitter空间中具有常数量曲率的类空超曲面

研究了de Sitter空间中具有常数量曲率的类空超曲面,得到了曲面Mn关于截面曲率的一个刚性定理,并且额外获得关于de Sitter空间子流形的一个结论.     

单位球中完备超曲面的一个刚性定理

本文研究了单位球中的数量曲率满足r=aH+b的完备超曲面的问题.利用极值原理的方法,获得了超曲面的一个刚性结果,推广了这一类具有常中曲率或者常数量曲率超曲面的结果.     

Hypersurfaces with constant scalar curvature and constant mean curvature

Non-spherical hypersurfaces inE ⁴ with non-zero constant mean curvature and constant scalar curvature are the only hypersurfaces possessing the following property: Its position vector can be written as a sum of two non-constant maps, which are eigenmaps of the Laplacian operator with corresponding eigenvalues the zero and a non-zero constant.     

Radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric

In this paper we introduce radical transversal lightlike hypersurfaces of almost complex manifolds with Norden metric. Such class of lightlike hypersurfaces cannot exist for indefinite almost Hermitian manifolds. The considered lightlike hypersurfaces have two important properties. The first one is the uniqueness of their screen distributions, which implies that the induced geometric objects are well-defined. The second property is that the induced Ricci tensor on radical transversal lightlike hypersurface of a Kähler manifold with Norden metric is symmetric. This allows to define an induced scalar curvature of the hypersurface. We obtain new results about lightlike hypersurfaces concerning their relations with non-degenerate hypersurfaces of almost complex manifolds with Norden metric. Examples of the considered hypersurfaces are given.     

Lower bounds for eigenvalues of the Dirac-Witten operator

We get optimal lower bounds for the eigenvalues of the Dirac-Witten operator of compact(with or without boundary) spacelike hypersurfaces of Lorentian manifold satisfying certain conditions,just in terms of the mean curvature and the scalar curvature and the spinor energy-momentum tensor. In the limiting case,the spacelike hypersurface is either maximal and Einstein manifold with positive scalar curvature or Ricci-flat manifold with nonzero constant mean curvature.     

Complete minimal hypersurfaces in the hyperbolic space with vanishing Gauss-Kronecker curvature

We investigate 3-dimensional complete minimal hypersurfaces in the hyperbolic space with Gauss-Kronecker curvature identically zero. More precisely, we give a classification of complete minimal hypersurfaces with Gauss-Kronecker curvature identically zero, a nowhere vanishing second fundamental form and a scalar curvature bounded from below.

     

Curvatures properties of Lie hypersurfaces in the complex hyperbolic space

A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In this paper, we study intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures, and sectional curvatures.     

Lorentz空间型中具有常数量曲率类空超曲面

研究了Lorentz空间型中具有常数量曲率的紧致类空超曲面,得到了这类超曲面的内蕴刚性定理.     

Spacelike Hypersurfaces with Constant Scalar Curvature in the Lorentz–Minkowski Space

We show that the only compact spacelike hypersurfaces in the Lorentz–Minkowski space Lⁿ⁺¹ having nonzero constant scalar curvature and spherical boundary are the hyperbolic caps (with negative constant scalar curvature). One key ingredient in our proof will be an integral formula for the n-dimensional volume enclosed by the boundary of a compact spacelike hypersurface, in the case where the boundary is contained in a hyperplane of Lⁿ⁺¹. As a direct application of that integral formula we also derive an interesting result for the volume of spacelike hypersurfaces.     

Complete hypersurfaces in a 4-dimensional hyperbolic space

This paper gives a classification of complete hypersurfaces with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature in the hyperbolic space H4（-1）,whose scalar curvature is bounded from below.     

局部对称空间中具有常数量曲率的超曲面

利用自伴算子研究局部对称空间中具有常数量曲率的紧致超曲面,得到了这类超曲面中的某些刚性定理,推广了已有的结果.