Isotropic Lagrangian submanifolds in the homogeneous nearly Khler S~3× S~3

We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Khler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly Khler S~3× S~3 is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function λ, such that g((▽h)(v, v, v), J v) = λ holds for all unit tangent vector v.     

EQUIVARIANT MINIMAL IMMERSIONS FROM S~3 INTO ■

Associated with an immersion φ : S~3→■, we can define a canonical bundle endomorphism F : TS~3→ TS~3 by the pull back of the K?hler form of ■. In this article,related to F we study equivariant minimal immersions from S~3 into ■ under the additional condition(?_XF)X = 0 for all X ∈ ker(F). As main result, we give a complete classification of such kinds of immersions. Moreover, we also construct a typical example of equivariant non-minimal immersion φ : S~3→■ satisfying(?_XF)X = 0 for all X ∈ ker(F), which is neither Lagrangian nor of CR type.     

On 1-connected 8-manifolds with the Same Homology as S~3×S~5

In this article, we classify 1-connected 8-dimensional Poincaré complexes, topological manifolds and smooth manifolds whose integral homology groups are isomorphic to those of S~3× S~5. A topic related to a paper of Escher and Ziller is also discussed.     

Nearly Kaehler流形S~3×S~3上的切触拉格朗日子流形

对于Nearly Kaehler流形S~3×S~3上的一个拉格朗日子流形,将给出由M上的一个单位向量场典范引出的殆切触度量结构是α-Sasakian,β-Kenmotsu以及cosymplectic的充要条件.另外,当这个殆切触度量结构为正规时,找出在什么条件下这个殆切触度量结构是3~(1/2)/3-Sasakian,3~(1/2)/3-Kenmotsu或cosymplectic结构.     

S~3中等参曲面的两个特征

本文研究了三维球空间S~3中等参曲面.利用活动标架法,获得了S~3中关于等参曲面特征的两个结果.推广了三维欧氏空间R~3和三维双曲空间中等参曲面的相关结果.     

Maximum orders of extendable actions on surfaces

We determine the maximum order E_g of finite groups G acting on the closed surface Σ_g of genus g which extends over(S~3, Σ_g) for all possible embeddings Σ_g → S~3, where g 1.     

Incompressible pairwise incompressible surfaces in link complements

The central subject of studying in this paper is incompressible pairwise incompressible surfaces in link complements. Let L be a non-split prime link and let F be an incompressible pairwise incompressible surface in S~3 - L. We discuss the properties that the surface F intersects with 2-spheres in S~3 - L. The intersection forms a topological graph consisting of a collection of circles and saddle-shaped discs. We introduce topological graphs and their moves (R-move and S~2-move), and define the characteristic number of the topological graph for F∩S~2±. The characteristic number is unchanged under the moves. In fact, the number is exactly the Euler Characteristic number of the surface when a graph satisfies some conditions. By these ways, we characterize the properties of incompressible pairwise incompressible surfaces in alternating (or almost alternating) link complements. We prove that the genus of the surface equals zero if the component number of F∩S~2+(or F∩S~2-) is less than five and the graph is simple for alternating or almost alternating links. Furthermore, one can prove that the genus of the surface is zero if #(F) ≤8.     

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori, Ⅱ

We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S~2×S~1 or RP~3#RP~3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni(2014) in an earlier paper.     

Hamiltonian Gromov–Witten Invariants on C~(n+1) with S~1-action

Consider a Hamiltonian action of S~1 on(C~(n+1), ω_(std)), we shown that the Hamiltonian Gromov–Witten invariants of it are well-defined. After computing the Hamiltonian Gromov–Witten invariants of it, we construct a ring homomorphism from H*_(S~1,CR)(X, R) to the small orbifold quantum cohomology of X //_τ S~1 and obtain a simpler formula of the Gromov–Witten invariants for weighted projective space.     

S~3内具规定平均曲率的广义Clifford环面

设H是S~3内一个实值函数,在H上给出两个条件,确信能找到S~3内一个闭曲面,它同胚于Clifford环面,平均曲率由H给定。     

A Characterization of the Ejiri Torus in S~5

We conjecture that a Willmore torus having Willmore functional between 2π~2 and 2π~23~(1/2) is either conformally equivalent to the Clifford torus,or conformally equivalent to the Ejiri torus.Ejiri's torus in S~5 is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form.Li and Vrancken classified all Willmore surfaces of tensor product in S n by reducing them into elastic curves in S~3,and the Ejiri torus appeared as a special example.In this paper,we first prove that among all Willmore tori of tensor product,the Willmore functional of the Ejiri torus in S~5 attains the minimum 2π~23~(1/2),which indicates our conjecture holds true for Willmore surfaces of tensor product.Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough.We also show that the Ejiri torus is unstable even in S~5.Moreover,similar to Li and Vrancken,we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves in S~3.All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.     

On ?-reducible 3-manifolds Which Admit Complete Surface Systems

In the present paper, we consider a class of compact orientable 3-manifolds with one boundary component, and suppose that the manifolds are ?-reducible and admit complete surface systems. One of our main results says that for a compact orientable, irreducible and ?-reducible 3-manifold M with one boundary component F of genus n 0 which admits a complete surface system S′, if D is a collection of pairwise disjoint compression disks for ?M, then there exists a complete surface system S for M, which is equivalent to S′, such that D is disjoint from S. We also obtain some properties of such 3-manifolds which can be embedded in S~3.     

Fixed points of n-valued maps on surfaces and the Wecken property—a configuration space approach

In this paper, we explore the fixed point theory of n-valued maps using configuration spaces and braid groups, focusing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane(resp. the 2-sphere S~2) has the Wecken property for n-valued maps for all n ∈ N(resp. all n 3). In the case n = 2 and S~2, we prove a partial result about the Wecken property.We then describe the Nielsen number of a non-split n-valued map ? : X■X of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering q : X → X with a subset of the coordinate maps of a lift of the n-valued split map ? ? q : X■X.     

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 Sf 4 f_3~2 ≤ 1/n S~3 , where S is the squared norm of the second fundamental form of M, and f_k =sum λ_i~k from i and λ_i (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)~1/2 ) ×S~...     

DCI-and CI-Groups of Order p~3

Let G be a finite group and S a subset of G.We define the Cayley digraphX=X(C,S)of G with respect to S bywhere V(X)and E(X)are the vertex-and edge-sets of X,respectively.S is saidto be a CI—subset of G if any graphisomorphism X(G,S)≌X(G,T),where TG,implies that there exists a group automorphism α∈ Aut G such that S~α=T.     

S~4内的常数量曲率的紧致超曲面

本文把关于 S~4内极小超曲面的一个 Pinching 定理推广到 S~4内的常中曲率及常数量曲率的起曲面的情形.设 M 为这样的超曲面,记 S 和 H 分别为 M 的第二基本形长度之平方和中曲率.证明了:如果 S≤H~2 6,则 M 只能取1/3H~2,3/4H~2±1/4 3或 H~2 6这四个数.当 H=0时,此结果即为上述的 S~4内极小超曲面的 Pinching 定理.     

Periodic Solutions of Schrdinger Flow from S~3 to S~2

This paper deals with the periodic solutions of Schrodinger flow from S3 to S2. It is shown that the periodic solution is related to the variation of some functional and there exist S1-invariant critical points to this functional. The proof makes use of the Principle of Symmetric Criticality of Palais.     

Complete Riemannian Manifold Minimally Immersed in a Unit Sphere S~(+p)(1)

Let M~n be an n-dimensional Riemannian manifold which is minimally immersed ina unit sphere S~(n+p)(1)of dimension n+p.If M~n is compact,then many authors stu-died them and obtained many beautiful results(for examples[1],[2],[3]).Inthis paper,we make use of Yau's a“maximum principle”to extend these results     

Topologically conjugate classifications of the translation actions on low-dimensional compact connected Lie groups

In this article, we focus on the left translation actions on noncommutative compact connected Lie groups with topological dimension 3 or 4, consisting of SU(2), U(2), SO(3), SO(3)×S~1 and Spin~C(3). We define the rotation vectors(numbers) of the left actions induced by the elements in the maximal tori of these groups, and utilize rotation vectors(numbers) to give the topologically conjugate classification of the left actions. Algebraic conjugacy and smooth conjugacy are also considered. As a by-product, we show that for any homeomorphism f : L(p,-1) × S~1→ L(p,-1) × S~1, the induced isomorphism(π?f?i)_* maps each element in the fundamental group of L(p,-1) to itself or its inverse, where i : L(p,-1) → L(p,-1) × S~1 is the natural inclusion and π : L(p,-1) × S~1→ L(p,-1) is the projection.     

Classification of hypersurfaces with parallel Mbius second fundamental form in S~(n+1)

Let M~n(n≥2) be an immersed umbilic-free hypersurface in the (n+1)-dimensional unit sphere S~(n+1). Then M~n is associated with a so-called Mobius metric g,and a Mobius second fundamental form B which are invariants of M~n under the Mobiustransformation group of S~(n+1). In this paper, we classify all umbilic-free hypersurfaces withparallel Mobius second fundamental form.