On the complete linear Weingarten spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

We deal with complete linear Weingarten spacelike hypersurfaces immersed in a Lorentzian space form, having two distinct principal curvatures. In this setting, we show that such a spacelike hypersurface must be isometric to a certain isoparametric hypersurface of the ambient space, under suitable restrictions on the values of the mean curvature and of the norm of the traceless part of its second fundamental form. Our approach is based on the use of a Simons type formula related to an appropriated Cheng–Yau modified operator jointly with some generalized maximum principles.     

Complete Linear Weingarten Spacelike Hypersurfaces Immersed in a Locally Symmetric Lorentz Space

In this paper, we establish a characterization theorem concerning the complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space, whose sectional curvature is supposed to obey certain appropriated conditions. Under a suitable restriction on the length of the second fundamental form, we prove that a such spacelike hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures one of which is simple.     

复双曲空间中的曲率齐性实超曲面

本文讨论了复双曲空间中的连通定向实超曲面M的一些性质,利用正交标架法的建立,证明了在适当条件下,M等参与M具有常主曲率等价,M等参与M曲率齐性等价.     

On Cartan’s identities of equiaffine isoparametric hypersurfaces

The purpose of the present paper is to obtain Gartan’s identities for affine principal curvatures of an equiaffine isoparametric hypersurface under certain conditions. The result can be applied to a pseudo-Riemannian isoparametric hypersurface and a Blaschke isoparametric hypersurface.     

Möbius hypersurfaces in S S+1with three distinct principal curvatures

On a hypersurface of a unit sphere without umbilical points, we know that three Möbius invariants can be defined and analogous to Euclidean case, we have the concepts of Möbius isoparametric and isotropic hypersurfaces. In this paper, we study the relationship between Euclidean geometry and Möbius geometry, and prove that a hypersurface in a sphere with constant length of the second fundamental form is Euclidean isoparametric if and only if it is Möbius isoparametric. When restricting to the case of three distinct principal curvatures, we show that such a hypersurface is either Möbius isoparametric or isotropic if the Blaschke tensor has constant eigenvalues.     

Isoparametric hypersurfaces in Funk spaces

Funk metrics are a kind of important Finsler metrics with constant negative flag curvature. In this paper, it is proved that any isoparametric hypersurface in Funk spaces has at most two distinct principal curvatures. Moreover, a complete classification of isoparametric families in a Funk space is given.     

Null Screen Isoparametric Hypersurfaces in Lorentzian Space Forms

In this paper, we develop the notion of screen isoparametric hypersurface for null hypersurfaces of Robertson–Walker spacetimes. Using this formalism we derive Cartan identities for the screen principal curvatures of null screen isoparametric hypersurfaces in Lorentzian space forms and provide a local characterization of such hypersurfaces.     

R~6中的Laguerre等参超曲面的分类

若超曲面的Laguerre形式为零且Laguerre第二基本形式的特征值(称为Laguerre主曲率)为常数,则称超曲面为Laguerre等参超曲面.对R~6中的Laguerre等参超曲面进行了研究,得到了分类定理.     

de Sitter空间中具有常均曲率的类空超曲面

设M是deSitter空间中具有常均曲率的类空超曲面,本文给出M是全脐的或是等参的一些条件．     

Clifford systems, algebraically constant second fundamental form and isoparametric hypersurfaces

In this paper we prove that a submanifold with parallel mean curvature of a space of constant curvature, whose second fundamental form has the same algebraic type as the one of a symmetric submanifold, is locally symmetric. As an application, using properties of Clifford systems, we give a short and alternative proof of a result of Cartan asserting that a compact isoparametric hypersurface of the sphere with three distinct principal curvatures is a tube around the Veronese embedding of the real, complex, quaternionic or Cayley projective planes. Received: 22 April 1998     

On Laguerre form and Laguerre isoparametric hypersurfaces

Let x : Mn-1→ Rnbe an umbilical free hypersurface with non-zero principal curvatures.M is called Laguerre isoparametric if it satisfies two conditions, namely, it has vanishing Laguerre form and has constant Lauerre principal curvatures. In this paper, under the condition of having constant Laguerre principal curvatures, we show that the hypersurface is of vanishing Laguerre form if and only if its Laguerre form is parallel with respect to the Levi–Civita connection of its Laguerre metric.     

The convex-concave principle and uniqueness for hypersurfaces in space Forms

We establish very general Weyl identities for pairs of symmetric functions of the invariants of the shape operator of a hypersurface in a space form or a general Codazzi tensor, respectively. They are used to characterize certain isoparametric hypersurfaces by assuming constancy or extremal properties of the functions, provided they fulfill ellipticity and/or convexity (concavity) properties. This way, many wellknown results are generalized. Finally, a chain rule for Weyl identities offers additional extension of some results.     

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.     

Blaschke isoparametric hypersurfaces in the conformal space ℚ 1 n+1 , I

Let x : M → Q₁ⁿ⁺¹ be a regular hypersurface in the conformal space Q₁ⁿ⁺¹. We classify all the space-like Blaschke isoparametric hypersurfaces with two distinct Blaschke eigenvalues in the conformal space up to the conformal equivalence.     

ISOPARAMETRIC HYPERSURFACES IN CP~n WITH CONSTANT PRINCIPAL CURVATURES

This paper proves that the number of distinct principal curvatures of a realisoparametric hypersurface in CP~n with constant principal curvatures can be only 2, 3 or 5.The prehnage of such hypersurface under the Hopf fibration is an isoparametrichypersarface in S~(2n+l) with 2 or 4 distinct principal curvatures. For real isoparametrichypersurfaces in CP~n with 5 distinct constant principal curvatures a local structuretheorem is given.     

Gradient map of isoparametric polynomial and its application to Ginzburg-Landau system

In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces to isoparametric hypersurfaces, but also map isoparametric hypersurfaces to focal submanifolds. In particular, it turns out to be a homogeneous polynomial automorphism on certain isoparametric hypersurface. As an immediate consequence, we get the Brouwer degree of the gradient map which was firstly obtained by Peng and Tang with moving frame method. Following Farina's construction, another immediate consequence is a counterexample of the Brézis question about the symmetry for the Ginzburg-Landau system in dimension 6, which gives a partial answer toward the Open problem 2 raised by Farina.     

等参函数及相关问题研究

著名的Yau 猜想断言单位球面中的紧致嵌入极小超曲面的Laplace 算子的第一特征值等于其维数. 近年来有许多几何学家致力于对Yau 猜想的研究, 但是到目前为止, 已有的结论只是一些关于第一特征值估计的不等式. 作为本文的一个主要结果, 本文证明了对于单位球面中的等参极小超曲面,Yau 猜想是正确的. 进一步地, 对于等参超曲面的焦流形(实际上是球面的极小子流形), 本文还证明了在一定维数条件下, 它的第一特征值也是其维数.
作为本文的第二个主要结果, 以著名的Schoen-Yau-Gromov-Lawson 的关于数量曲率的手术理论为出发点, 本文在一个Riemann 流形的嵌入超曲面处作手术, 构造了一个新的具有丰富几何性质的流形, 称为double 流形. 特别地, 本文在单位球面的极小等参超曲面处实行了这一手术, 发现得到的double 流形不仅有很复杂的拓扑(但其示性类有精确描述), 还存在数量曲率为正的度量, 更重要的是保持了等参叶状结构.
比Willmore 曲面更广泛的定义是Willmore 子流形, 即Willmore 泛函在球面中的的极值子流形.单位球面中的Willmore 子流形的例子在已有文献中是非常罕见的. 作为本文的另外两个主要结果, 通过深入挖掘单位球面上的OT-FKM- 型等参函数的焦流形的性质, 本文发现其极大值对应的焦流形是单位球面的一系列Willmore 子流形; 之后, 本文用几何办法统一证明了单位球面中具有4 个不同主曲率的等参超曲面的焦流形都是单位球面的Willmore 子流形. 这些新的Willmore 子流形是极小的,但一般不是Einstein 的.     

单位球面S~6中的Blaschke等参超曲面

单位球面中的一个无脐点浸入子流形称为Blaschke等参子流形如果它的Mbius形式恒为零并且所有的Blaschke特征值均为常数.维数m4的Blaschke等参超曲面已经有了完全的分类.截止目前,Mbius等参超曲面的所有已知例子都是Blaschke等参的.另一方面,确实存在许多不是Mbius等参的Blaschke等参超曲面,它们都具有不超过两个的不同Blaschke特征值.在已有分类定理的基础上,本文对于5维Blaschke等参超曲面进行了完全的分类.特别地,我们证明了S6中具有多于两个不同Blaschke特征值的Blaschke等参超曲面一定是Mbius等参的,给出了此前一个问题的部分解答.     

Pseudosymmetric Lightlike Hypersurfaces in Indefinite Sasakian Space Forms

We study pseudosymmetric lightlike hypersurfaces of an indefinite Sasakian space form, tangent to the structure vector field. We obtain sufficient conditions for a lightlike hypersurface to be pseudosymmetric, pseudoparallel and Ricci-pseudosymmetric in an indefinite Sasakian space form. We also find certain conditions for a pseudosymmetric lightlike hypersurface of an indefinite Sasakian space form to be totally geodesic and check the effect of Weyl projective pseudosymmetry conditions on the geometry of a lightlike hypersurface of an indefinite Sasakian space form. Moreover we give some physical interpretations of pseudo-symmetry conditions.