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．     

Curved flats,pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudo-Riemannian space forms. As a corollary, we obtain a non-immersibility theorem for spheres into certain pseudo-Riemannian spheres and hyperbolic spaces. The second aspect pursued is to clarify the relationship between the loop group formulation of isometric immersions of space forms and that of pluriharmonic maps into symmetric spaces. We show that the objects in the first class are, in the real analytic case, extended pluriharmonic maps into certain symmetric spaces which satisfy an extra reality condition along a totally real submanifold. We show how to construct such pluriharmonic maps for general symmetric spaces from curved flats, using a generalised DPW method.      

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.     

On singly-periodic minimal surfaces with planar ends

The spaces of nondegenerate properly embedded minimal surfaces in quotients of by nontrivial translations or by screw motions with nontrivial rotational part, fixed finite topology and planar type ends, are endowed with natural structures of finite dimensional real analytic manifolds. This nondegeneracy is defined in terms of Jacobi functions. Riemann's minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces. We also give natural immersions of these spaces into certain complex Euclidean spaces which turn out to be Lagrangian immersions with respect to the standard symplectic structures.

     

Codimension of immersions with parallel pluri-mean curvature

Immersions with parallel pluri-mean curvature into euclidean n-space generalize constant mean curvature immersions of surfaces to Kähler manifolds of complex dimension m. Examples are the standard embeddings of Kähler symmetric spaces into the Lie algebra of its transvection group. We give a lower bound for the codimension of arbitrary ppmc immersions. In particular we show that M is locally symmetric if the codimension is minimal.     

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     

Minimal isometric immersions of inhomogeneous spherical space forms into spheres--- a necessary condition for existence

Although much is known about minimal isometric immersions into spheres of homogeneous spherical space forms, there are no results in the literature about such immersions in the dominant case of inhomogeneous space forms. For a large class of these, we give a necessary condition for the existence of such an immersion of a given degree. This condition depends only upon the degree and the fundamental group of the space form and is given in terms of an explicitly computable function. Evaluating this function shows that neither nor admit a minimal isometric immersion into any sphere if the degree of the immersion is less than , or less than , respectively.

     

Immersions of graphs into projective plane

Immersions of graphs into the projective plane are studied. A classification of immersions up to a regular homotopy is obtained. A complete invariant of immersions up to a regular homotopy is constructed. The case of immersions of graphs into any compact surface differing from the projective plane was known previously.     

Does negative type characterize the round sphere?

We discuss the measure-theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact Riemannian manifold with symmetry ratio must be a round sphere was put forward by the author in 2004. We resolve this conjecture in the class of Riemannian symmetric spaces by showing that a Riemannian manifold with symmetry ratio must be of negative type and that the only compact Riemannian symmetric spaces of negative type are the round spheres.

     

Associated families of pluriharmonic maps and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are Kähler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic Kähler symmetric spaces.     

Topological obstructions for submanifolds in low codimension

We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for \(\delta \)-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second fundamental form.     

Stability of certain minimal submanifolds in compact symmetric spaces of rank two

In [T. Kimura, M.S. Tanaka, Totally geodesic submanifold in compact symmetric spaces of rank two, Tokyo J. Math., in press], the authors obtained the global classification of the maximal totally geodesic submanifolds in compact connected irreducible symmetric spaces of rank two. In this paper, we determine their stability as minimal submanifolds in compact symmetric spaces of rank two.     

Associated families of pluriharmonic maps¶and isotropy

Like minimal surface immersions in 3-space, pluriharmonic maps into symmetric spaces allow a one-parameter family of isometric deformations rotating the differential (“associated family”); in fact, pluriharmonic maps are characterized by this property. We give a geometric proof of this fact and investigate the “isotropic” case where this family is constant. It turns out that isotropic pluriharmonic maps arise from certain holomorphic maps into flag manifolds. Further, we also consider higher dimensional generalizations of constant mean curvature surfaces which are K?hler submanifolds with parallel (1,1) part of their soecond fundamental form; under certain restrictions there are also characterized by having some kind of (“weak”) associated family. Examples where this family is constant arise from extrinsic K?hler symmetric spaces. Received: 8 July 1997     

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.

     

On the Charshiladze-Lozinski theorem for compact topological groups and homogeneous spaces

This paper is concerned with an extension of the Charshiladze-Lozinski theorem to compact (not necessarily abelian) topological groups G and symmetric compact homogeneous spaces G/H. The proof is based on a generalized Marcinkiewicz — Berman formula. As an application, some divergence theorems for expansions of continuous resp. integrable complex — valued functions on Euclidean spheres and projective spaces in series of polynomial functions on these spaces are established.     

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 in R-closed spaces

Some problems in the theory of R-closed spaces are solved by showing that every regular space can be embedded in a minimal regular space and there is an R-closed space with no coarser minimal regular topology. A class of spaces is found so that when fed into the Jone's machinery for producing non-Tychonoff, regular spaces, the output is non-tychonoff R-closed and minimal regular spaces. Also, an example of a strongly minimal regular space that is not locally R-closed is given.