Hypersurfaces with constant mean curvature in unit sphere

The pinching of n-dimensional closed hypersurface Mwith constant mean curvature H in unit sphere S^（n＋1）（ 1） is considered. Let A = ∑i,j,k h（ijk）~2（ λi＋ nH）~2,B = ∑i,j,k h（ijk）~2（ λi＋ nH） ·（ λj＋ nH）,S = ∑i（ λi＋ nH）~2, where h（ij）= λiδ（ij）. Utilizing Lagrange＇s method, a sharper pointwise estimation of 3（A- 2B） in terms of S and ｜▽h｜~2 is obtained, here ｜▽h｜~2= ∑i,j,k h（ijk）~2. Then, with the help of this, it is proved that Mis isometric to the Clifford hypersurface if the square norm of the second fundamental form of Msatisfies certain conditions. Hence, the pinching result of the minimal hypersurface is extended to the hypersurface with constant mean curvature case.

COMPLETE HYPERSURFACES WITH CONSTANT MEAN CURVATURE AND FINITE INDEX IN HYPERBOLIC SPACES

In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4（-1）（H5（-1）） with constant mean curvature H satisfying H2 6634 （H2 114785 respectively） must be compact. Specially, we verify that any complete and stable hypersurface in the hyperbolic space H4（-1） （resp. H5（-1）） with constant mean curvature H satisfying H2 6643 （resp. H2 114785 ） must be compact. It shows that there is no manifold satisfying the conditions of some theorems in [7, 9].

RIGIDITY OF COMPACT SURFACES IN HOMOGENEOUS 3-MANIFOLDS WITH CONSTANT MEAN CURVATURE

In this paper, we establish a rigidity theorem for compact constant mean curva- ture surfaces of the Berger sphere in terms of the surfaces＇ geometric invariants. This extends the previous similar result on compact minimal surfaces of the Berger sphere.

Weakly stable constant mean curvature hypersurfaces

Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an （n ＋ 1）-dimensional Riemannian manifold N^n＋1 whose （n - 1）th Ricci curvature satisfying Ric^N（n-1） （n - 1）c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If ｜φ｜^2 - （n- 2）√n（n- 1）｜H｜｜φ｜＋ n（2n - 1）（H^2＋ c） 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c ＋ H^2 〉 0, then M must have only one end.

On the generalized Chern conjecture for hypersurfaces with constant mean curvature in a sphere

Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).

On Complete Hypersurfaces with Constant Mean Curvature and Finite L^p-norm Curvature in R^(n+1)

By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1＋1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n＋1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1＋1 are considered.

A Global Pinching Theorem for Compact Surfaces in S^3 with Constant Mean Curvature

Let M be a compact minimal surface in S3.Y.J.Hsu proved that if ‖S‖2≤2(21/2π, then M is either the equatorial sphere or the Clifford torus,where 5" is the square of the length of the second fundamental form of M,‖·‖2 denotes the L2-norm on M.In this paper,we generalize Hsu’s result to any compact surfaces in S3 with constant mean curvature.

A STABILITY RESULT FOR TRANSLATINGSPACELIKE GRAPHS IN LORENTZ MANIFOLDS

In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.

On Submanifolds Whose Tubular Hypersurfaces Have Constant Higher Order Mean Curvatures

Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k^v（k ≥ 1） of a submanifold M^n-is defined as the k-th power sum of the principal curvatures,or equivalently,of the shape operator with respect to the unit normal vector v.We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures,then the submanifold M itself has some constant higher order mean curvatures Q_k^v independent of the choice of v.Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained.In particular,we generalize several classical results in isoparametric theory given by E.Cartan,K.Nomizu,H.F.Miinzner,Q.M.Wang,et al.As an application,we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.

曲面上的一些整体定理

A、Svec[1]用积分公式,极大值原理和广义解析函数三种方法研究了Weingarten曲面,主要是研究球面的特征。本文用广义解析函方法,得到一些其它曲面的特征。定理2得到旋转曲面,定理3—4得到常中曲率曲面。

COMPLETE SPACE-LIKE SUBMANIFOLDS IN LOCALLY SYMMETRIC SEMI-DEFINITE SPACES

The purpose of this paper is to study complete space-like submanifolds with parallel mean curvature vector and flat normal bundle in a locally symmetric semi-defnite space satisfying some curvature conditions. We first give an optimal estimate of the Laplacian of the squared norm of the second fundamental form for such submanifold. Furthermore, the totally umbilical submanifolds are characterized.

Totally Umbilical Subrnanifolds in a Locally Symmetric Manifold

In this paper we obtain some formulas for totally umbilical submanifolds in a localiy symmetric manifold, and dcrivc some local rcsults on the submanifolds from these formulas.

Regular Space-like Hypersurfaces in S1^m+1 with Parallel Para-Blaschke Tensors

In this paper, we give a complete conformal classification of the regular space-like hyper- surfaces in the de Sitter Space S~＋1 with parallel para-Blaschke tensors.

On Twistor Gauss Maps of Surfaces in 4-Spheres

In this paper we study surfaces in S^4 and their twistor Gauss maps.Some necessary and sufficient conditions that the twistor Gauss map is harmonic are given.We find many examples of nonisotropic harmonic maps from a surface to(?)P^3.

Some Theorems for Hypersurface of Randers Spaces

In this paper,we consider the hypersurfaces of Randers space with constant flag curvature.(1)Let(M^n+1,F)be a Randers–Minkowski space.If(M^n,F)is a hypersurface of(M^n+1,F)with constant flag curvature K=1,then we can prove that M is Riemannian.(2)Let(M^n+1,F)be a Randers space with constant flag curvature.Assume(M,F)is a compact hypersurface of(M^n+1,F)with constant mean curvature|H|.Then a pinching theorem is established,which generalizes the result of[Proc.Amer.Math.Soc.,120,1223–1229(1994)]from the Riemannian case to the Randers space.