ON THE COMPLETE 2-DIMENSIONALλ-TRANSLATORS WITH A SECOND FUNDAMENTAL FORM OF CONSTANT LENGTH

In this article we study the two-dimensional completeλ-translators immersed in the Euclidean space R^3 and Minkovski space R1^ 3.We obtain two classification theorems:one for two-dimensional completeλ-translators x:M 2→R^3 and one for two-dimensional complete space-likeλ-translators x:M 2→R1^3,with a second fundamental form of constant length.

A Formula for Submanifolds in S^n and Its Applications in Moebius Geometry

In this paper, we obtain a formula for submanifolds in Sn+p by calculating the Laplacian of the Moebius second fundamental form. Using this formula, we obtain some pinching theorems about the minimal eigenvalue of the Blaschke tensor.

Spectral Characterizations of Veronese Surface in S^4

In this paper, we prove that the Veronese surface can be determined by the 1 spectrum of the Laplace operator.

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.

Helicoidal Surfaces and Their Relationship to Bonnet Surfaces

An important question that arises is which surfaces in three-space admit a mean curvature preserving isometry which is not an isometry of the whole space. This leads to a class of surface known as a Bonnet surface in which the number of noncongruent immersions is two or infinity. The intention here is to present a proof of a theorem using an approach which is based on differential forms and moving frames and states that helicoidal surfaces necessarily fall into the class of Bonnet surfaces. Some other results are developed in the same manner.

Hypersurfaces with Constant Mean Curvature in Space Forms

In this paper,we study the pinching problem for a hypersurface with constant mean curvature in space forms to be totally umbilical by osing the relationship between the square of the length of the second fundamental form and the mean curvature. We obtained a best pinching interval and decided the complete classification of hypersurfaces at the terminal of the interval.This improved the relative results of M. Okumura,Shen Yibihg and Sun Ziqi,etc.

THE DISCUSSION ON THE NORMAL FUNDAMENTAL REGION OF FUCHS GROUP WITH NONEUCLIDEAN GEOMETRY

We define the fundamental region homeomorphic to the corresponding Riemann surface according to the methods on form-conserved circle of the fractional linear transformation in explaining that the form-conserved circle is the perpendicular bisector of noneuclidean segment limited by the end points both the origin and the equivalent point by the same transformation just mentioned and, consequently, its sense on noneuclidean geometry is clarified.The result does not appear in current literatures and is useful for the research of superstring.

Classification of hypersurfaces with parallel Mobius second fundamental form in S^(n+1)

Let Mn(n≥2) be an immersed umbilic-free hypersurface in the(n+1)-dimensional unit sphere Sn+1. Then Mn is associated witha so-called M(o)bius metric g, and a M(o)bius second fundamental form Bwhich are invariants of Mn under the M(o)bius transformation groupof Sn+1.In this paper, we classify all umbilic-free hypersurfaces withparallel M(o)bius second fundamental form.

Second fundamental forms of holomorphic isometries of the Poincar disk into bounded symmetric domains and their boundary behavior along the unit circle

Motivated by problems arising from Arithmetic Geometry,in an earlier article one of the authors studied germs of holomorphic isometries between bounded domains with respect to the Bergman metric.In the case of a germ of holomorphic isometry f:(Δ,λ ds 2Δ ;0) → (Ω,ds 2Ω ;0) of the Poincar disk Δ into a bounded symmetric domain Ω C N in its Harish-Chandra realization and equipped with the Bergman metric,f extends to a proper holomorphic isometric embedding F:(Δ,λ ds 2Δ) → (Ω,ds 2Ω) and Graph(f) extends to an affine-algebraic variety V C × C N.Examples of F which are not totally geodesic have been constructed.They arise primarily from the p-th root map ρ p:H → H p and a non-standard holomorphic embedding G from the upper half-plane to the Siegel upper half-plane H 3 of genus 3.In the current article on the one hand we examine second fundamental forms σ of these known examples,by computing explicitly σ 2.On the other hand we study on the theoretical side asymptotic properties of σ for arbitrary holomorphic isometries of the Poincar disk into polydisks.For such mappings expressing via the inverse Cayley transform in terms of the Euclidean coordinate τ=s + it on the upper half-plane H,we have φ(τ)=t 2 u(τ),where u t=0 ≡ 0.We show that u must satisfy the first order differential equation u t | t=0 ≡ 0 on the real axis outside a finite number of points at which u is singular.As a by-product of our method of proof we show that any non-standard holomorphic isometric embedding of the Poincar disk into the polydisk must develop singularities along the boundary circle.The equation φuφt | t=0 ≡ 0 along the real axis for holomorphic isometries into polydisks distinguishes the latter maps from holomorphic isometries into Siegel upper half-planes arising from G.Towards the end of the article we formulate characterization problems for holomorphic isometries suggested both by the theoretical and the computational results of the article.

Complete Space-like λ-surfaces in the Minkowski Space R1^3 with the Second Fundamental Form of Constant Length

In this paper we study the complete space-like λ-surfaces in the three dimensional Minkowski space R1^3.As the result,we obtain a complete classification theorem for all the complete space-like λ-surfaces x:M^2→R1^3 with the second fundamental form of constant length.This is a natural extension to the λ-surfaces in R1^3 of a recent interesting classification theorem by Cheng and Wei forλ-surfaces in the Euclidean space R^3.

Pointwise characterizations of curvature and second fundamental form on Riemannian manifolds

Let M be a complete Riemannian manifold possibly with a boundary?M.For any C^1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it exists.These characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified.

Stokes公式的自由度及其在计算积分中的应用

考察了Stokes公式中蕴含的自由度问题.基于这种自由度,得到了计算第二类曲面积分(或一般的流形上的积分)的分部积分法和更改被积曲面积分法的两个对偶公式.用具体例子说明了这两个公式的应用.

QUANTUM PHENOMENON OF THE ENERGY DENSITY OF A HARMONIC MAP TO A SPHERE

This paper proves that if the energy density of a harmonic map to a unit sphere varies between two successive half eigenvalues, then it must be one of them. Applying this result to the Gaussian maps of some submanifolds, the quantum phenomena of the square length of the second fundamental forms of these submanifolds is obtained. Some related topics are discussed in this note.

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 rigidity of Clifford torus in a unit sphere

We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S^n＋1 satisfying S f4 - f^2 3 ≤1/nS^3 where S is the squared norm of the second fundamental form of M, and fk = ∑λi^k and λi（1 ≤ i ≤ n） are the principal curvatures of M. We prove that there exists a positive constant δ（n）（≥ n/2） depending only on n such that if n ≤ S ≤ n ＋δ（n）, then S ≡ n, i.e., M is one of the Clifford torus S^K （√k/n） × S^n-k （V√n-k/n） for 1≤ k ≤ n - i. Moreover, we prove that if S is a constant, then there exists a positive constant T（n）（≥ n -2/3） depending only on n such that ifn ≤ S 〈 n ＋ τ（n）, then S ≡n, i.e.. M is a Clifford torus.

GEOMETRIC PROPERTIES FOR GAUSSIAN IMAGE OF SUBMANIFOLDS IN S^(n+p)(1)

The geometric properties for Gaussian image of submanifolds in a sphere are investigated. The computation formula, geometric equalities and inequalities for the volume of Gaussian image of certain submanifolds in a sphere are obtained.

SPACE-LIKE BLASCHKE ISOPARAMETRIC SUBMANIFOLDS IN THE LIGHT-CONE OF CONSTANT SCALAR CURVATURE

Let E_(s)^(m+p+1) ?R_(s+1)^(m+p+2)(m≥ 2,p≥ 1,0≤s≤p) be the standard(punched)light-cone in the Lorentzian space R_(s+1)^(m+p+2),and let Y:M^(m)→E_(s)^(m+p+1) be a space-like immersed submanifold of dimension m.Then,in addition to the induced metric g on Mm,there are three other important invariants of Y:the Blaschke tensor A,the conic second fundamental form B,and the conic Mobius form C;these are naturally defined by Y and are all invariant under the group of rigid motions on E_(s)^(m+p+1).In particular,g,A,B,C form a complete invariant system for Y,as was originally shown by C.P.Wang for the case in which s=0.The submanifold Y is said to be Blaschke isoparametric if its conic Mobius form C vanishes identically and all of its Blaschke eigenvalues are constant.In this paper,we study the space-like Blaschke isoparametric submanifolds of a general codimension in the light-cone E_(s)^(m+p+1) for the extremal case in which s=p.We obtain a complete classification theorem for all the m-dimensional space-like Blaschke isoparametric submanifolds in Epm+p+1of constant scalar curvature,and of two distinct Blaschke eigenvalues.

The Hypersurfaces in a Unit Sphere with Nonnegative Mobius Sectional Curvature

Let x ： M→S^n＋1 be a hypersurface in the （n ＋ 1）-dimensional unit sphere S^n＋1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n＋1 are Mobius metric, Mobius form, Mobius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem： Let x ： M→S^n＋1 （n≥2） be an umbilic free hypersurface in S^n＋1 with nonnegative Mobius sectional curvature and with vanishing Mobius form. Then x is locally Mobius equivalent to one of the following hypersurfaces： （i） the torus S^k（a） × S^n-k（√1- a^2） with 1 ≤ k ≤ n - 1; （ii） the pre-image of the stereographic projection of the standard cylinder S^k × R^n-k belong to R^n＋1 with 1 ≤ k ≤ n- 1; （iii） the pre-image of the stereographic projection of the Cone in R^n＋1 ： -↑x（u, v, t） = （tu, tv）, where （u,v, t）∈S^k（a） × S^n-k-1（ √1-a^2）× R^＋.

Complete Spacelike Hypersurfaces with Constant Scalar Curvature in Locally Symmetric Lorentz Spaces

The complete space-like hypersurfaces with constant normal saclar curvature is discussed in a locally symmetric Lorentz space. A classified theorem is obtained by the operator L1 introduced by S Y Cheng and S T Yau [3].

子流形低阶曲率泛函的变分计算与间隙现象

设φ:M^(n)→N^(n+p)是一般外围流形中的n维紧致无边子流形.φ的第二基本型模长平方S、平均曲率模长平方H^(2)和迹零第二基本型模长平方ρ=S-nH^(2)等重要的低阶曲率分别刻画了全测地、极小、全脐等重要的几何性质.本文构造低阶曲率泛函L(I,n,F)(φ)=∫_(M F)(S,H^(2))dv,L(II,n,F)(φ)=∫_(M) F(ρ,H^(2))dv,其中F:[0,+∞)×[0,+∞)→R是一个抽象的充分光滑的双变量函数.这类泛函可刻画子流形与全测地子流形、极小子流形和全脐子流形的整体差异,将多类子流形泛函囊括在统一的框架之下,且与子流形中多类著名问题,如Willmore猜想,有着密切联系.本文将计算第一变分公式,在空间形式中构造临界点的一些例子,推导泛函临界点的积分不等式,并基于此对间隙现象进行讨论.