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.      

Real hypersurfaces with constant totally real bisectional curvature in complex space forms

In this paper we classify real hypersurfaces with constant totally real bisectional curvature in a non flat complex space form M _m (c), c ≠ 0 as those which have constant holomorphic sectional curvature given in [6] and [13] or constant totally real sectional curvature given in [11].     

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.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter space S ^{n +1} ₁(c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvature n(n-1)r is isometric to a sphere if r << c. Received: 18 December 1996 / Revised version: 26 November 1997     

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.     

Hypersurfaces with constant inner curvature of the second fundamental form,and the non-rigidity of the sphere

We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of classC ² where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233, (1972)) saying that such a hypersurface of classC ⁴ has to be a round sphere. In particular, the sphere is notII-rigid in the class of all convexC ²-hypersurfaces.     

Hypersurfaces with constant inner curvature of the second fundamental form,and the non-rigidity of the sphere

We classify the hypersurfaces of revolution in euclidean space whose second fundamental form defines an abstract pseudo-Riemannian metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of class C ² where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them there are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider (Proc. AMS 35, 230–233, (1972)) saying that such a hypersurface of class C ⁴ has to be a round sphere. In particular, the sphere is not II-rigid in the class of all convex C ² -hypersurfaces. Received 11 October 1994; in final form 26 April 1995     

Complete hypersurfaces of R4 with constant mean curvature

In this paper, we completely classify complete hypersurfaces inR ⁴ with constant mean curvature and constant scalar curvature.The project was supported by NNSFC, FECC, and CPF.     

Spacelike hypersurfaces with constant scalar curvature

In this paper, we shall give an integral equality by applying the operator □ introduced by S.Y. Cheng and S.T. Yau [7] to compact spacelike hypersurfaces which are immersed in de Sitter spaceS ₁ ⁿ⁺¹ (c) and have constant scalar curvature. By making use of this integral equality, we show that such a hypersurface with constant scalar curvaturen(n−1)r is isometric to a sphere ifr<c. Research partially Supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Science and Culture.     

Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces

We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.

     

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.     

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.     

The affine complete hypersurfaces of constant Gauss-Kronecker curvature

Given a bounded convex domain Ω with C∞ boundary and a function ψ∈C∞（δΩ）, Li-Simon-Chen can construct an Euclidean complete and W-complete convex hypersurface M with constant affine Gauss-Kronecker curvature, and they guess the M is also affine complete. In this paper, we give a confirmation answer.     

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.     

Gluing constructions amongst constant mean curvature hypersurfaces in $${\mathbb {S}^{n+1}}$$

Four constructions of constant mean curvature (CMC) hypersurfaces in \mathbb Sⁿ⁺¹{\mathbb {S}^{n+1}} are given, which should be considered analogues of ‘classical’ constructions that are possible for CMC hypersurfaces in Euclidean space. First, Delaunay-like hypersurfaces, consisting roughly of a chain of hyperspheres winding multiple times around an equator, are shown to exist for all the values of the mean curvature. Second, a hypersurface is constructed which consists of two chains of spheres winding around a pair of orthogonal equators, showing that Delaunay-like hypersurfaces can be fused together in a symmetric manner. Third, a Delaunay-like handle can be attached to a generalized Clifford torus of the same mean curvature. Finally, two generalized Clifford tori of equal but opposite mean curvature of any magnitude can be attached to each other by symmetrically positioned Delaunay-like ‘arms’. This last result extends Butscher and Pacard’s doubling construction for generalized Clifford tori of small mean curvature.     

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.     

Two-dimensional Finsler metrics with constant flag curvature

 We construct infinitely many two-dimensional Finsler metrics on 𝕊² and 𝔻² with non-zero constant flag curvature. They are all not locally projectively flat. Received: 10 October 2001 / Revised version: 15 July 2002     

Complete hypersurfaces with constant mean curvature and finite L p norm curvature in Euclidean spaces

In this paper, we consider complete hypersurfaces in R ⁿ⁺¹ with constant mean curvature H and prove that M ⁿ is a hyperplane if the L ² norm curvature of M ⁿ satisfies some growth condition and M ⁿ is stable. It is an improvement of a theorem proved by H. Alencar and M. do Carmo in 1994. In addition, we obtain that M ⁿ is a hyperplane (or a round sphere) under the condition that M ⁿ is strongly stable (or weakly stable) and has some finite L ^p norm curvature. Received: 14 July 2007     

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.