SLANT IMMERSIONS OF COMPLEX SPACE FORMS AND CHEN＇S INEQUALITY

A submanifold in a complex space form is called slant it it has constant Wirtinger angles. B, Y, Chen and Y. Tazawa proved that there do not exist minimal proper slant surfaces in CP^2 and CH^2. So it seems that the slant immersion has some interesting properties. The authors have great interest to consider slant immersions satisfying some additional conditions, such as unfull first normal bundles or Chen‘s equality holding. They prove that there do not exist n-dimensional Kaehlerian slant immersion sin CP^n and CH^n with unfull first normal bundles. Next, it is seen that every Kaehlerian slant submanifold satisfying an equality of Chen is minimal which is similar to that of Lagrangian immersions. But in contrast, it is shown that a large class of slant immersions do not exist thoroughly. Finally, they give an application of Chen‘s inequality to general slant immersions in a complex projective space, which generalizes a result of Chen.     

Eigenmaps and minimal and bandlimited immersions of graphs into Euclidean spaces

We introduce concepts of minimal immersions and bandlimited (Paley-Wiener) immersions of combinatorial weighted graphs (finite or infinite) into Euclidean spaces. The notion of bandlimited immersions generalizes the known concept of eigenmaps of graphs. It is shown that our minimal immersions can be used to perform interpolation, smoothing and approximation of immersions of graphs into Euclidean spaces. It is proved that under certain conditions minimal immersions converge to bandlimited immersions. Explicit expressions of minimal immersions in terms of eigenmaps are given. The results can find applications for data dimension reduction, image processing, computer graphics, visualization and learning theory.     

New Construction for Spherical Minimal Immersions

We describe a general method of manufacturing new minimal immersions between round spheres out of old ones. The resulting spherical minimal immersions are given analytically in terms of the harmonic projection operator and have higher source dimensions. Applied to classical examples, this gives an abundance of new minimal immersions of even-dimensional spheres.     

Orbits of SU(2)-representations and minimal isometric immersions

In contrast to all known examples, we show that in the case of minimal isometric immersions of into the smallest target dimension is almost never achieved by an -equivariant immersion. We also give new criteria for linear rigidity of a fixed minimal isometric immersion of into . The minimal isometric immersions arising from irreducible SU(2)-representations are linearly rigid within the moduli space of SU(2)-equivariant immersions. Hence the question arose whether they are still linearly rigid within the full moduli space. We show that this is false by using our new criteria to construct an explicit SU(2)-equivariant immersion which is not linearly rigid. Various authors [GT], [To3], [W1] have shown that minimal isometric immersions of higher isotropy order play an important role in the study of the moduli space of all minimal isometric immersions of into . Using a new necessary and sufficient condition for immersions of isotropy order , we derive a general existence theorem of such immersions. Received: 13 May 1999 / in final form: 13 July 1999     

Spherical Minimal Immersions with Prescribed Codimension

We describe a general construction of manufacturing new spherical minimal immersions between round spheres out of old ones. The new immersions have higher domain dimension and degree and the construction has a precise control on the codimension. Applied to classified and recent examples, the construction gives an abundance of new spherical minimal immersions with prescribed codimensions.Mathematics Subject Classifications (2000). 53C42     

Embedding problems: Geometric and topological aspects

A survey of results obtained after 1976 on questions of the isometric immersions of Riemannian spaces in a Euclidean space, the immersions and embeddings of differential manifolds, and the immersions with minimal absolute curvature is presented.     

Some explicit examples of Hamiltonian minimal Lagrangian submanifolds in complex space forms

In this paper, we find some new explicit examples of Hamiltonian minimal Lagrangian submanifolds among the Lagrangian isometric immersions of a real space form in a complex space form.     

Universal constraints on the range of eigenmaps and spherical minimal immersions

The purpose of this paper is to give lower estimates on the range dimension of spherical minimal immersions in various settings. The estimates are obtained by showing that infinitesimal isometric deformations (with respect to a compact Lie group acting transitively on the domain) of spherical minimal immersions give rise to a contraction on the moduli space of the immersions and a suitable power of the contraction brings all boundary points into the interior of the moduli space.

     

Semi-parallel surfaces in Euclidean space

Semi-parallel immersions are defined as extrinsic analogue for semi-symmetric spaces and as a direct generalization of parallel immersions. Using results of Backes on Euclidean Jordan triple systems, the totally geodesic immersions are shown to be the only minimal semi-parallel immersions into a Euclidean space. Semi-parallel immersions of surfaces into E^m are studied and a classification of semi-parallel immersions with pointwise planar normal sections of surfaces in E^m is given.Research Assistant of the National Fund of Scientific Research     

ON DEGREES OF MINIMAL IMMERSIONS OF SYMMETRIC SPACES INTO SPHERES

This paper considers the degrees of minimal immersions of symmetric spaces intospheres.A practical way to count the degree of a given regular minimal immersion of acompact irreducible symmetric space into sphere S_1~n is given.As an application,degrees ofregular minimal immersions of all rank one compact symmetric spaces into spheres arecounted.     

常曲率球面到单位球面的等距极小浸入与嵌入

本文主要估计了球面到单位球面的等距极小浸入的覆盖层数，从而导出一些有关嵌入的结果。同时，在本文证明过程中，给出了一个构造指定覆盖层数的等距极小浸入的方法．     

Bifurcation for minimal surface equation in hyperbolic 3-manifolds

Initiated by the work of Uhlenbeck in late 1970s, we study existence, multiplicity and asymptotic behavior for minimal immersions of a closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions, by analyzing the Gauss equation governing the immersion. We determine when existence holds, and obtain unique stable solutions for area minimizing immersions. Furthermore, we find exactly when other (unstable) solutions exist and study how they blow-up. We prove our class of unstable solutions exhibit different blow-up behaviors when the surface is of genus two or greater. We establish similar results for the blow-up behavior of any general family of unstable solutions. This information allows us to consider similar minimal immersion problems when the total extrinsic curvature is also prescribed.     

Semi-Parallel, Semi-Symmetric Immersions and Chen’s Equality

Recently, B. Y. Chen introduced a new invariant δ(n₁,n₂,…,n_k) of a Riemannian manifold and proved a basic inequality between the invariant and the extrinsic invariant if, where H is the mean curvature of an immersion Mⁿ in a real space form R^m(ε) of constant curvature ε. He pointed out that such inequality also holds for a totally real immersion in a complex space form. The immersion is called ideal (by B. Y. Chen) if it satisfies the equality case of such inequality identically. In this paper we classify ideal semi-parallel immersions in an Euclidean space if their normal bundle is flat, and prove that every ideal semi-parallel Lagrangian immersion in a complex space form is totally geodesic, moreover this result also holds for ideal semi-symmetric Lagrangian immersions in complex projective space and hyperbolic space.     

Classification of conformal minimal immersions of constant curvature from S^2 to { HP}^2

In this paper, we study geometry of conformal minimal two-spheres immersed in quaternionic projective spaces. We firstly use Bahy-El-Dien and Wood’s results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from \(S^2\) to the quaternionic projective space \({ HP}^2\) . Then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from \(S^2\) to the quaternionic projective space \({ HP}^2\) .     

On conformal minimal 2-spheres in complex Grassmann manifold <Emphasis Type="BoldItalic">G</Emphasis>(2,<Emphasis Type="BoldItalic">n</Emphasis>)

For a harmonic map f from a Riemann surface into a complex Grassmann manifold, Chern and Wolfson [4] constructed new harmonic maps ?f\partial\!f and [`(?)]f\bar{\partial}\!f through the fundamental collineations ?\partial and [`(?)]\bar{\partial} respectively. In this paper, we study the linearly full conformal minimal immersions from S ² into complex Grassmannians G(2,n), according to the relationships between the images of ?f\partial\!f and [`(?)]f\bar{\partial}\!f. We obtain various pinching theorems and existence theorems about the Gaussian curvature, K?hler angle associated to the given minimal immersions, and characterize some immersions under special conditions. Some examples are given to show that the hypotheses in our theorems are reasonable.     

Equivariant Lagrangian Minimal S^3 in CP^3

The equivariant Lagrangian minimal immersion of Sa into C.Pa is studied. The complete classification and analytic expression for such kinds of immersions are given.     

常曲率空间中的具有共形第二基本形式的子流形

以把调和态射看作等距浸入的单位法投影的问题为背景,研究了具有共形第二基本形式的子流形,论证了具有共形第二基本形式的高维子流形,一般不是由极小点和全脐点构成,这和曲面的情形形成了鲜明的对照。也给出了常曲率空间中具有平行中曲率的奇数维子流形的一个完全分类。     

Integrable deformation of critical surfaces in spaceforms

We investigate a Goursat-type transformation for Bryant surfaces and a few of its geometric features. The basic result is that, when regarded as a non-isometric group action on the moduli space of CMC1 immersions of a fixed surface, the Lawson correspondence is equivariant with respect to the more familiar Goursat transform on minimal immersions. This generalizes certain well-known deformations of critical surfaces and enlarges the number of explicitly computable cousin pairs, visualized here in a quaternionic upper-half space.     

紧黎曼对称空间到Grassmann流形的等变等距极小浸入

在本文中，我们利用李群及其表示理论作为主要工具，　讨论了紧黎曼对称空间到Ｇｒａｓｓｍａｎｎ　流形的等变等距极小浸入问题．     

Rigidity of minimal surfaces inS 3

Isometric deformations of compact minimal surfaces in the standard three-sphere are studied. It is shown that a given surface admits only finitely many noncongruent minimal immersions intoS ³ with the same first fundamental form.