Rotational hypersurfaces of periodic mean curvature

In this paper we discuss rotational hypersurfaces in and more specifically rotational hypersurfaces with periodic mean curvature function. We show that, for a given real analytic function H(s) on , every rotational hypersurface M in with mean curvature H(s) can be extended infinitely in the sense that all coordinate functions of the generating curve of M are defined on all of as well. For rotational hypersurfaces with periodic mean curvature we present a criterion characterizing the periodicity of such hypersurfaces in terms of their mean curvature function. We also discuss a method to produce families of periodic rotational hypersurfaces where each member of the family has the same mean curvature function. In fact, given any closed planar curve with curvature κ, we prove that there is a family of periodic rotational hypersurfaces such that the mean curvature of each element of the family is explicitly determined by κ. Delaunay's famous result for surfaces of revolution with constant mean curvature is included here as the case where n=3 and κ is constant.     

L2调和形式与具有常平均曲率的稳定超曲面

设M为(n 1)维流形Ⅳ中完备、非紧、定向的、具有常平均曲率H的强稳定超曲面,文中证明了若Ⅳ的双Ricci曲率沿M不小于-n^2H^2,则M上不存在非平凡的L^2调和1-形式．     

Stable complete noncompact hypersurfaces with constant mean curvature

This article concerns the structure of complete noncompact stable hypersurfaces M ⁿ with constant mean curvature H > 0 in a complete noncompact oriented Riemannian manifold N ⁿ⁺¹. In particular, we show that a complete noncompact stable constant mean curvature hypersurface M ⁿ , n = 5, 6, in the Euclidean space must have only one end. Any such hypersurface in the hyperbolic space with , respectively, has only one end.     

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.     

Complete spacelike hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we investigate the complete spacelike hypersurfaces with constant mean curvature and two distinct principal curvatures in an anti-de Sitter space. We give a characterization of hyperbolic cylinder and prove the conjecture in a paper by L. F. Cao and G. X. Wei [J. Math. Anal. Appl., 2007, 329(1): 408–414].     

A Bernstein theorem for complete spacelike hypersurfaces of constant mean curvature in Minkowski space

In this paper we prove a general Bernstein theorem on the complete spacelike constant mean curvature hypersurfaces in Minkowski space. The result generalizes the previous result of Cao-Shen-Zhu (1998) and Xin (1991). The proof again uses the fact that the Gauss map of a constant mean curvature hypersurface is harmonic, which was proved by K. T. Milnor (1983), and the maximum principle of S. T. Yau (1975).

     

Total mean curvature of positively curved hypersurfaces

In this article, we prove that every positively curved, complete non-compact hypersurface in Rn has infinite total mean curvature.     

Bernstein type result for constant mean curvature hypersurface

We prove a Bernstein type theorem for constant mean curvature hypersurfaces in ℝ ⁿ⁺¹ under certain growth conditions for n ⩽ 3. Our result extends the case when M is a minimal hypersurface in the same condition.      

Hypersurfaces with constant mean curvature in a hyperbolic space

Let Mⁿ be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space Hⁿ⁺¹（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that （1） if the multiplicities of the two distinct principal curvatures are greater than 1,then Mⁿ is isometric to the Riemannian product S^k（r）×H^n-k（-1/（r² + ρ²））, where r > 0 and 1 < k < n - 1;（2）if H² > -c and one of the two distinct principal curvatures is simple, then Mⁿ is isometric to the Riemannian product S^n-1（r） × H¹（-1/（r² +ρ²）） or S¹（r） × H^n-1（-1/（r² +ρ²））,r > 0, if one of the following conditions is satisfied （i） S≤（n-1）t²2+c²t^-22 on Mⁿ or （ii）S≥ （n-1）t²1+c²t^-21 on Mⁿ or（iii）（n-1）t²2+c²t^-22≤ S≤（n-1）t²1+c²t^-21 on Mⁿ, where t₁ and t₂ are the positive real roots of （1.5）.     

L 2 harmonic forms and stability of hypersurfaces with constant mean curvature

We prove that a complete noncompact oriented strongly stable hypersurfaceM ⁿ with cmc (constant mean curvature)H in a complete oriented manifoldN ⁿ⁺¹ with bi-Ricci curvature, satisfying alongM, admits no nontrivialL ² harmonic 1-forms. This implies ifM ⁿ (2n4) is a complete noncompact strongly stable hypersurface in hyperbolic spaceH ⁿ⁺¹(–1) with cmc , there exist no nontrivialL ² harmonic 1-forms onM. We also classify complete oriented strongly stable surfaces with cmcH in a complete oriented manifoldN ³ with scalar curvature satisfying .     

On the scalar curvature of constant mean curvature hypersurfaces in space forms

In this paper we study the behavior of the scalar curvature S of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of S. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli, Setti (2005) [17].     

局部对称黎曼流形中的超曲面

研究了局部对称黎曼流形中具有常平均曲率超曲面的几何性质,得到了关于其第二基本形式模长平方的拼挤定理.     

Stability of constant mean curvature hypersurfaces of revolution in hyperbolic space

In this article, by solving a nonlinear differential equation, we prove the existence of a one parameter family of constant mean curvature hypersurfaces in the hyperbolic space with two ends. Then, we study the stability of these hypersurfaces.     

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.     

Non-preserved curvature conditions under constrained mean curvature flows

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our examples is that they disprove some statements of the previous literature, overshadow a widespread folklore conjecture about the behaviour of these flows and bring out the discouraging news that a traditional singularity analysis is not possible for constrained versions of the mean curvature flow.     

On the stability index of hypersurfaces with constant mean curvature in spheres

Barbosa, do Carmo and Eschenburg characterized the totally umbilical spheres as the only weakly stable compact constant mean curvature hypersurfaces in the Euclidean sphere . In this paper we prove that the weak index of any other compact constant mean curvature hypersurface in n+1 which is not totally umbilical and has constant scalar curvature is greater than or equal to , with equality if and only if is a constant mean curvature Clifford torus with radius .

     

The Riemannian product structures of spacelike hypersurfaces with constant k-th mean curvature in the de Sitter spaces

In this paper, we investigate complete spacelike hypersurfaces in the de Sitter space with constant k-th mean curvature and two distinct principal curvatures one of which is simple. We obtain some characterizations of the Riemannian product H¹(c₁)×Sn−1(c₂) or Hn−1(c₁)×S¹(c₂) in the de Sitter space .     

First eigenvalue of a Jacobi operator of hypersurfaces with a constant scalar curvature

Let be an -dimensional compact hypersurface with constant scalar curvature , , in a unit sphere . We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral of the mean curvature . In this paper, we first study the eigenvalue of the Jacobi operator of . We derive an optimal upper bound for the first eigenvalue of , and this bound is attained if and only if is a totally umbilical and non-totally geodesic hypersurface or is a Riemannian product , .