1.

ON COMPLETE SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE IN NEGATIVE PINCHED MANIFOLDS





Leng Yan Xu Hongwei《高校应用数学学报(英文版)》,2007年第22卷第2期


A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian (n p)dimensional manifold Nn p with negative sectional curvature is proved. For given positive integers n(≥ 2), p and for a constant H satisfying H ＞ 1 there exists a negative number τ(n,p, H) ∈ (1, 0) with the property that if the sectional curvature of N is pinched in [1, τ(n,p, H)], and if the squared length of the second fundamental form is in a certain interval, then Nn p is isometric to the hyperbolic space Hn p(1). As a consequence, this submanifold M is congruent to Sn(1/ H21) or theVeronese surface in S4(1/√H21).

2.

On the Volume Formulas of Cones and Orthogonal Multicones in S~n(1) and H~n(1)





WuYi HSIANG 《数学年刊B辑(英文版)》,2006年第27卷第1期


In the study of ndimensional spherical or hyperbolic geometry, n ≥3, the volume of various objects such as simplexes, convex polytopes, etc. often becomes rather difficult to deal with. In this paper, we use the method of infinitesimal symmetrization to provide a systematic way of obtaining volume formulas of cones and orthogonal multiple cones in Sn(l) and Hn(1).

3.

A survey of the homotopy properties of inclusion of certain types of configuration spaces into the Cartesian product





Daciberg Lima GONCALVES John GUASCHI《数学年刊B辑(英文版)》,2017年第38卷第6期


Let X be a topological space.In this survey the authors consider severaltypes of configuration spaces,namely,the classical (usual) configuration spaces Fn(X)and Dn(X),the orbit configuration spaces FGn(X) and FGn(X)/Sn with respect to a freeaction of a group G on X,and the graph configuration spaces FΓn(X) and FΓn(X)/H,where F is a graph and H is a suitable subgroup of the symmetric group Sn.The orderedconfiguration spaces Fn (X),FGn (X),FΓn(X) are all subsets of the nfold Cartesian productnП1 X of X with itself,and satisfy FGn(X) (C) Fn(X) (C) Frn(X) (C) nП1 X.If A denotes one of these configuration spaces,the authors analyse the difference between A and nП1 X from a topological and homotopical point of view.The principal results known in the literature concern the usual configuration spaces.The authors are particularly interested in the homomorphism on the level of the homotopy groups of the spaces induced by the inclusion (ι):A → nП1 X,the homotopy type of the homotopy fibre I(ι) of the map (ι) via certain constructions on various spaces that depend on X,and the long exact sequence in homotopy of the fibration involving I(ι) and arising from the inclusion (ι).In this respect,if X is either a surface without boundary,in particular if X is the 2sphere or the real projective plane,or a space whose universal covering is contractible,or an orbit space Sk/G of the kdimensional sphere by a free action of a Lie group G,the authors present recent results obtained by themselves for the first case,and in collaboration with Golasi(n)ski for the second and third cases.The authors also briefly indicate some older results relative to the homotopy of these spaces that are related to the problems of interest.In order to motivate various questions,for the remaining types of configuration spaces,a few of their basic properties are described and proved.A list of open questions and problems is given at the end of the paper.

4.

Global Poincaré Inequalities on the Heisenberg Group and Applications





Yu Xin DONG Guo Zhen LU Li Jing SUN《数学学报(英文版)》,2007年第23卷第4期


Let f be in the localized nonisotropic Sobolev space Wloc^1,p （H^n） on the ndimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n ＋ 2 is the homogeneous dimension of H^n. Suppose that the subelliptic gradient is gloablly L^p integrable, i.e., fH^n ｜△H^n f｜^p du is finite. We prove a Poincaré inequality for f on the entire space H^n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C0^∞（H^n） under the norm of （∫H^n ｜f｜ Qp/Qp）^Qp/Qp ＋ （∫ H^n ｜△H^n f｜^p）^1/p. We will also prove that the best constants and extremals for such Poincaré inequalities on H^n are the same as those for Sobolev inequalities on H^n. Using the results of Jerison and Lee on the sharp constant and extremals for L^2 to L（2Q/Q2） Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H^n when p=2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H^n.

5.

ISOMETRIES ON THE SPACE s





傅小红《数学物理学报(B辑英文版)》,2006年第26卷第3期


In this article, the author presents some results of the isometric linear extension from some spheres in the finite dimensional space S(n). Moreover, the author presents the representation for the onto isometric mappings in the space a. It is obtained that if V is a surjective isometry from the space s onto s with V(0)=0, then V must be real linear.

6.

Isometries in hyperbolic spaces





HUANG ManZi WANG XianTao & WANG YueFei《中国科学 数学(英文版)》,2010年第1期


Suppose that f:Hn → Hn (n≥2) maps any rdimensional hyperplane (1≤r

7.

MAXIMAL ELEMENTS OF A FAMILY OF GBMAJORIZED MAPPINGS IN PRODUCT FCSPACES AND APPLICATIONS 被引次数：2





丁协平《应用数学和力学(英文版)》,2006年第27卷第12期


A new family of GBmajorized mappings from a topological space into a finite continuous topological spaces （in short, FCspace） involving a better admissible setvalued mapping is introduced. Some existence theorems of maximal elements for the family of GBmajorized mappings are proved under noncompact setting of product FCspaces. Some applications to fixed point and system of minimax inequalities are given in product FCspaces. These theorems improve, unify and generalize many important results in recent literature.

8.

A differentiable sphere theorem with positive Ricci curvature and reverse volume pinching





WANG PeiHe & WEN YuLiang School of Mathematical Sciences Qufu Normal University Qufu China《中国科学 数学(英文版)》,2011年第3期


Let Mn be a compact, simply connected n (≥3)dimensional Riemannian manifold without boundary and Sn be the unit sphere Euclidean space Rn+1. We derive a differentiable sphere theorem whenever themanifold concerned satisfies that the sectional curvature KM is not larger than 1, while Ric(M)≥n+2 4 and the volume V (M) is not larger than (1 + η)V (Sn) for some positive number η depending only on n.

9.

Energy identity of the heat flow of Hsystems at finite singular time





Tao HUANG~ Zhong TAN School of Mathematical Sciences Xiamen University Xiamen 361005 China《中国科学A辑(英文版)》,2007年第50卷第11期


It is well known that there exists a global solution to the heat flow of Hsystems.If the solution satisfies a certain energy inequality,it is global regular with at most finitely many singularities. Under the same energy inequality,we can show the energy identity of the heat flow of Hsystems at finite singular time.The most interesting thing in our proof is that we find the singular points can only occur in the interior of the set in some sense.

10.

Weierstrass Type Representation of Willmore Surfaces in S^n





QiaoLingXIA YiBingSHEN《数学学报(英文版)》,2004年第20卷第6期


In this paper, we reformulate the EulerLagrange equations of Willmore surfaces in S^n as the flatness of a family of certain loop algebravalued 1forms. Therefore we can give the Weierstrass type representation of conformal Willmore surfaces. We also discuss the relations between conformal Willmore surfaces in S^n and minimal surfaces in constant curvature spaces S^n, R^n, H^n, and prove that some special Willmore surfaces can be derived from minimal surfaces in S^n, R^n, H^n.

11.

E1ementary Bifurcations of Non—Critical but N0n—Hyperbolic Invariant Tori





JianHuaSUN《数学学报(英文版)》,2003年第19卷第1期


Consider the timeperiodic peturbations of ndimensional autonomous systems with nonhyperbolic but noncritical closed orbits in the phase space.The elementary bifurcations,such as the saddlenode,transcritical,pitchfork bifurcation to a nonhyperbolic but noncritical invariant torus of the unperturbed systems in the extended phase space(x,t),are sutdied.Some conditions which depend only on ithe original systems and can be used to determine the bifurcation structures of these problems are obtained.The theory is applied to two concrete examples.

12.

On the Uniqueness Theorems for the Closed Convex Hypersurfaces in a Space of Constant Curvature





李安民《数学研究与评论》,1984年第4期


In this paper we generalized the uniqueness theorem of. AlexadroffFenchelJessen, the cohnVossen theorem and the HilbertLiebmannHsiung theorem to hypersurfaces in a sphere S~(n+1) or a hyperbolic space H~(n+1)

13.

HYPERSURFACES IN SPACE FORMS WITH SCALAR CURVATURE CONDITIONS





徐森林 张运涛《数学物理学报(B辑英文版)》,2004年第24卷第1期


Let f : M^n→S^n 1真包含于R^n 2 be an ndimensional complete oriented Riemannian manifold minimally immersed in an (n 1)dimensional unit sphere S^n 1. Denote by S^n 1 the upper closed hemisphere. If f(M^n)包含于S ^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.

14.

Hypersurfaces with isotropic paraBlaschke tensor





Jian Bo Fang Kun Zhang《数学学报(英文版)》,2014年第30卷第7期


Let Mn be an ndimensional submanifold without umbilical points in the （n ＋ 1）dimen sional unit sphere Sn＋l. Four basic invariants of Mn under the Moebius transformation group of Sn＋1 are a 1form Ф called moebius form, a symmetric （0, 2） tensor A called Blaschke tensor, a symmetric （0, 2） tensor B called Moebius second fundamental form and a positive definite （0, 2） tensor g called Moebius metric. A symmetric （0,2） tensor D = A ＋ μB called paraBlaschke tensor, where μ is constant, is also an Moebius invariant. We call the paraBlaschke tensor is isotropic if there exists a function ,λ such that D = λg. One of the basic questions in Moebius geometry is to classify the hypersurfaces with isotropic paraBlaschke tensor. When λ is not constant, all hypersurfaces with isotropic paraBlaschke tensor are explicitly expressed in this paper.

15.

ADM Mass for Asymptotically de Sitter SpaceTime





黄仕明 岳瑞宏 贾冬燕《理论物理通讯》,2010年第9期


In this paper, an ADM mass formula for asymptotically de Sitter（dS） spacetime is derived from the energymomentum tensor. We take the vacuum dS space as the background and investigate the ADM mass of the （d ＋ 3）dimensional spheresymmetric space with a positive cosmological constant, and find that the ADM mass of asymptotically dS space is based on the ADM mass of Schwarzschild field and the cosmological background brings some small mass contribution as well.

16.

On the Extension of SmaleBirkhoff Homoclinic Theorem





郑志明《数学进展》,1991年第2期


In this paper the SB homoclinic theorem is extended to the following situation, If the stable and the unstable manifolds of some zerodimensional hyperbolic basic sets for f∈ Diff (M, M) , dim M≥2, form a "transversal" circle, then there exists a "bigger" zerodimensional hyperbolic basic set of f in M which includes all those original ones together with the points of intersection of their stable and unstable manifolds.

17.

The Crossratio Compactification of the Configuration Space of Ordered Points on





Risako Funahashi Masahiko Taniguchi《数学学报(英文版)》,2012年第28卷第10期


A natural compactification of the virtual configuration space of N points on the Riemann sphere is constructed by using crossratios. We show that this compactification is homeomorphic to the Bers' compactification of the virtual moduli space of a punctured Riemann sphere of type N . In particular, the system of global and explicit coordinates of this standard compactification is given by crossratios.

18.

Minkowski空间中带超平面边界的紧致类空超曲面





徐森林 寿乐丽《应用数学》,2004年第17卷第2期


In this paper, we study the relations between a compact spacelike hypersurface with hyperplanar boundary in the (n 1) dimensional Minkowski spacetime L^n 1 being totally umbilicaland its hyperplanar boundary in a fixed hyperplane π of L^(n 1) being totally umbilical under certainconditions. We give the sufficient conditions for such hypersurface and its hyperplanar boundary tobe totally umbilical in their respective ambients.

19.

AN ESTIMATE FOR THE MEAN CURVATURE OF SUBMANIFOLDS CONTAINED IN A HOROBALL





邱红兵《数学物理学报(B辑英文版)》,2013年第6期


We obtain the OmoriYau maximum principle on complete properly immersed submanifolds with the mean curvature satisfying certain condition in complete Riemannian manifolds whose radial sectional curvature satisfies some decay condition, which generalizes our previous results in [17]. Using this generalized maximum principle, we give an estimate on the mean curvature of properly immersed submanifolds in H^n × R^e with the image under the projection on H^n contained in a horoball and the corresponding situation in hyperbolic space. We also give other applications of the generalized maximum principle.

20.

THE UNCONDITIONAL CONVERGENT DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR QUASILINEAR PARABOLIC SYSTEMS WITH TWO DIMENSIONS





LongjunShen GuangweiYuan《计算数学(英文版)》,2003年第21卷第1期


In the present work we are going to solve the boundary value problem for the quasilinear parabolic systems of partial differential equations with two space dimensions by the finite difference method with intrinsic parallelism.Some fundamental behaviors of general finite difference schemes with intrinsic parallelism for the mentioned problems are studied.By the method of a priori estimation of the discrete solutions of the nonlinear difference systems,and the interpolation formulas of the various norms of the discrete functions and the fixedpoint technique in finite dimensional Euclidean space,the existennce of the discrete vector solutions of the nonliear difference system with intrinsic parallelism are proved .Moreover the convergence of the discrete vector solutions of these difference schemes to the unique generalizd solution of the original quasilinear parabolic problem is proved.
