A characterization of the Clifford torus

In this paper, we prove that an -dimensional closed minimal hypersurface with Ricci curvature of a unit sphere is isometric to a Clifford torus if , where is the squared norm of the second fundamental form of .

     

Different definitions of homogeneity of real hypersurfaces in ℂ2

The coincidence of two definitions of local homogeneity for real-analytic hypersurfaces in two-dimensional complex spaces is proved. It is shown that if any two germs of a Levi nondegenerate nonspherical surfaceM are equivalent, then this surface has a local Lie group structure:M then acts transitively on itself by left shifts, and each such shift is a local holomorphic transformation of ².Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 881–887, December, 1998.This paper was conceived and largely written during author's visits to the Universities of West Ontario and Regina (Canada) under the kind invitation of Professors B. Gilligan and A. Bowen. The author wishes to express his great indebtedness to them for providing excellent conditions for work and gratitude for useful discussions.     

Compact Null Hypersurfaces and Collapsing Riemannian Manifolds

Restrictions are obtained on the topology of a compact divergence - free null hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is zero or satisfies some weaker conditions. This is done by showing that each null hypersurface of this type can be used to construct a family of three - dimensional Riemannian metrics which collapses with bounded curvature and applying known results on the topology of manifolds which collapse. The result is then applied to general relativity, where it implies a restriction on the topology of smooth compact Cauchy horizons in spacetimes with various types of reasonable matter content.     

Log pairs on hypersurfaces of degreeN in {N{\text{ in }}\mathbb{P}^N

The objective of this paper is to study the birational structure of smooth hypersurfaces of degreeN in by examining properties of moving log pairs on them. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp 131–138, July, 2000.     

Congruence of hypersurfaces inS 6 and inC P n

The invariants needed to decide when a pair of hypersurfaces ofS ⁶ orCP ⁿ are respectivelyG ₂-congruent or holomorpic congruent are determined and this result is used to characterize the hypersurfaces of these spaces whose Hopf vector fields are also Killing fields.     

EQUIFOCAL HYPERSURFACES IN SYMMETRIC SPACES

This note investigates the multiplicity problem of principal curvatures of equifocal hypersurfaces in simply connected rank 1 symmetric spaces. Using Clifford representation theory, and the author also constructs infinitely many equifocal hypersurfaces in the symmetric spaces. Project supported by the National Natural Science Foundation of China (No. 19925104), RFDP and the Qiu-Shi Science and Technology Foundation.     

双曲空间形式中具有常平均曲率的超曲面(英文)

本文研究了双曲空间形式H~(n+1)(—1)中具有常平均曲率及两个离散主曲率(其中一个主曲率是1-重)的完备连通可定向的n-维超曲面M~n.利用活动标架,得到如果M~n的基本形式的模长满足刚性条件(1.3),那么M~n同构双曲柱面.     

Isoparametric foliation and Yau conjecture on the first eigenvalue,II

This is a continuation of [23], which investigated the first eigenvalues of minimal isoparametric hypersurfaces with g=4$g=4$ distinct principal curvatures and focal submanifolds in unit spheres. For the focal submanifolds with g=6$g=6$, the present paper obtains estimates on all the eigenvalues, among others, giving an affirmative answer in one case to the problem posed in [23], which may be regarded as a generalization of Yau's conjecture. In two of the four unsettled cases in [23] for focal submanifolds _M1$M_1$ of OT-FKM-type, we prove the first eigenvalues to be their respective dimensions.     

Real hypersurfaces in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor

In this paper we prove that there does not exist any Hopf real hypersurface in complex hyperbolic two‐plane Grassmannians with parallel Ricci tensor.