CLASSIFICATIONS OF DUPIN HYPERSURFACES IN LIE SPHERE GEOMETRY

This is a survey of local and global classification results concerning Dupin hypersurfaces in S^(n)(or R^(n))that have been obtained in the context of Lie sphere geometry.The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres.Along with these classification results,many important concepts from Lie sphere geometry,such as curvature spheres,Lie curvatures,and Legendre lifts of submanifolds of S^(n)(or R^(n)),are described in detail.The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

CONFORMALLY FLAT AFFINE HYPERSURFACES WITH SEMI-PARALLEL CUBIC FORM

In this paper,we study locally strongly convex affine hypersurfaces with the vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of the affine metric.As a main result,we classify these hypersurfaces as not being of a flat affine metric.In particular,2 and 3-dimensional locally strongly convex affine hypersurfaces with semi-parallel cubic forms are completely determined.

ON A CURVATURE CHARACTERIZATION OF RICCI-PSEUDOSYMMETRIC HYPERSURFACES

The author presents a curvature characterization of Ricci-pseudosymmetric hypersurfaces in semi-Riemannian space forms Nsn+1(c), n>4.

The Ricci Operator and Shape Operator of Real Hypersurfaces in a Non-Flat 2-Dimensional Complex Space Form

In this paper, we study a real hypersurface M in a non-at 2-dimensional complex space form M2(c) with η-parallel Ricci and shape operators. The characterizations of these real hypersurfaces are obtained.

On Laguerre Isopararmetric Hypersurfaces in R^7

Let x : M → R n be an umbilical free hypersurface with non-zero principal curvatures, then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. A classical theorem of Laguerre geometry states that M(n > 3) is characterized by g and B up to Laguerre equivalence. A Laguerre isopararmetric hypersurface is defined by satisfying the conditions that C = 0 and all the eigenvalues of B with respect to g are constant. It is easy to see that all Laguerre isopararmetric hypersurfaces are Dupin hypersurfaces. In this paper, we established a complete classification for all Laguerre isopararmetric hypersurfaces with three distinct principal curvatures in R7.

On spectral characterizations of Willmore hypersurfaces in a sphere

Let M be a closed Willmore hypersurface in the sphere S^n＋1（1） （n ≥ 2） with the same mean curvature of the Willmore torus Wm,n-m, if SpecP（M） = Spec^P（Wm,n-m ） （p = 0, 1,2）, then M is Wm,n-m.

HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN A HYPERBOLIC SPACE

Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H（n＋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 Sk（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（-2）2 on Mn or （ii）S≥ （n-1）t21＋c2t（-2）1 on Mn or（iii）（n-1）t22＋c2t（-2）2≤ S≤（n-1）t21＋c2t（-2）1 on Mn, where t_1 and t_2 are the positive real roots of （1.5）.

ASYMPTOTIC EXPANSION OF DIRAC-TYPE DISTRIBUTION ASSOCIATED WITH A CLASS OF HYPERSURFACES WITH DEGENERATED CRITICAL POINTS

In order to generalize Hadamard's theory of fundamental solutions to the case of degenerate holomorphic PDE, this paper studies the asymptotic expansion of Dirac-type distribution associated with a class of hypersurfaces F(x) with degenerate critical points and proves that [F(x)](+)(lambda) is a distribution-valued meromorphic of lambda is an element of C under some assumptions on F(x). Next, the authors use the Normal form theory of Arnold and prove that for a hypersurface F(x) = 0 with A(mu) type degenerate critical point at x = 0, F-+(lambda) is a distribution-valued meromorphic function of lambda.

ISOPARAMETRIC HYPERSURFACES AND COMPLEX STRUCTURES

The main purpose of this note is to construct almost complex or complex structures on certain isoparametric hypersurfaces in unit spheres.As a consequence,complex structures on S^(1)×S^(7)×S^(6),and on S^(10)×S^(3)×S(2)with vanishing first Chern class,are built.

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.

ON STATIONARY HYPERSURFACES IN EUCLIDEAN SPACES

Some techniques in the Geometric Measure Theory are used to study the hypersurfaces in Euclidean spaces and some fundamental properties with this subject are discussed in this article.

NORMAL CRITERION FOR FAMILIES OF HOLOMORPHIC MAPS OF SEVERAL COMPLEX VARIABLES INTO P^N(C)WITH MOVING HYPERSURFACES

This article gives a normal criterion for families of holomorphic mappings of several complex variables into P N(C)for moving hypersurfaces in pointwise general position,related to an Eremenko’s theorem.

DEFORMING CONVEX HYPERSURFACES BY THE LINEAR COMBINATION OF THE MEAN CURVATURE AND THE N-TH ROOT OF THE GAUSS-KRONECKER CURVATURE

In this paper a flow of convex hypersurfaces in the Euclidean space by the linear-combination of the mean curvature and the n-th root of the Gauss-Kronecker curvature is considered. It is proved that such deforming convex hypersurfaces converge to a round sphere in the Huisken's sense.

ROUGH MULTIPLE SINGULAR INTEGRALS ALONG HYPERSURFACES

In this paper, the authors study the mapping properties of singular integrals on product domains with kernels in L（log＋L）ε（Sm-1 × Sn-1） （ε = 1 or 2） supported by hyper-surfaces. The Lp bounds for such singular integral operators as well as the related Marcinkiewicz integral operators are established, provided that the lower dimensional maximal function is bounded on Lq（R3） for all q 1. The condition on the integral kernels is known to be optimal.

THE GEOMETRY OF HYPERSURFACES IN A KAEHLER MANIFOLD

The geometry of hypersurfaces of a Kaehler manifold are studied. Some well-known formulas and theorems in theory of surfaces of Euclidean 3-space are generalized to the hypersurfaces in a Kaehler manifold, such as Gauss's formulae, second fundamental form, the equation of Gauss and Codazzi and so forth.

ON PSEUDOSYMMETRY TYPE HYPERSURFACES OF SEMI-EUCLIDEAN SPACES I

In this paper the authors investigate hypersurfaces M of a semi-Euclidean space E9n+1, n > 4, satisfying (aC + BR) .H = LkQ(g, Hk), k = 1,2, 3. Using obtained results they show additional curvature properties of investigated hypersurfaces.

NEW SPECTRAL CHARACTERIZATIONS OF EXTREMAL HYPERSURFACES

Let M be a closed extremal hypersurface in S^n＋1 with the same mean curvature of the Willmore torus Wm,n-m.We proved that if Spec^p（M） = Spec^p（Wm,n-m ） for p = 0, 1, 2, then M is Wm,m.

HYPERSURFACES IN SPACE FORMS WITH SCALAR CURVATURE CONDITIONS

Let f: Mn(?)Sn+1 C Rn+2 be an n-dimensional complete oriented Rieman-nian manifold minimally immersed in an (n+1)-dimensional unit sphere Sn+1. Denote by S_+~n+1 the upper closed hemisphere. If f(Mn)(?)_+~n+1, then under some curvature conditions the authors can get that the isometric immersion is a totally embedding. They also generalize a theorem of Li Hai Zhong on hypersurface of space form with costant scalar curvature.

On Translation Hypersurfaces with Constant Mean Curvature in (n+1)-Dimensional Spaces

The translation hypersurfaces with nonzero constant mean curvature in (n+1) dimensional spaces are studied. It is the generalization of the classical Scherk theorem. The classification of translation hypersurfaces with nonzero constant mean curvature in Euclidean and Lorentz spaces is completely given.

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.