Showing papers in "Tohoku Mathematical Journal in 1994"

Spectral convergence of riemannian manifolds, ii

Abstract: We prove a precompactness theorem concerning the spectral distance on the set of isometry classes of compact Riemannian manifolds and study the comple- tion of a precompact family. Introduction. For a compact connected Riemannian manifold M=(M,g), we denote by pM(t, x, y) the heat kernel of the Laplace operator of M with respect ot the normarized Riemannian measure μM (= dvg/Vol(M)). Given two compact connected Riemannian manifolds M and N, a mapping /: M->N is called an e-spectral ap- proximation if it satisfies e-{t + llt) \pM(U x, y)~PN(t, f(x)9 Ry)) I 0 and x,yeM. The spectral distance SD(M, N) between M and N is by definition the lower bound of the positive numbers e such that there exist e-spectral approximations /: M-^N and h: N^M. The distance SD gives a uniform structure on the set Mc of isometry classes of compact connected Riemannian manifolds. Riemannian manifolds are considered as metric spaces endowed with Riemannian distances. From this point of view, the set Mc has another uniform structure introduced by Gromov (18), called the Hausdorff distance HD. In (18), the conditions for a family of Jic to be HD-precompact are described and it is shown that the boundaries of such a family consist of certain metric spaces, called length spaces. This decade has seen intensive activities around the convergence theory of Riemannian manifolds with respect to the Gromov-Hausdorff distance. These includes some works from the viewpoint of spectral geometry, for instance, (14), (4), and (23). In (25), motivated by these results, we introduced the spectral distance SD mentioned above and discussed some basic properties of the distance on a set of compact connected Riemannian manifolds of the same dimension with diameters uniformly bounded from above and Ricci curvatures uniformly bounded from below. In the present paper, we are concerned with a certain precompact family of Jίc and its compactification with respect to the spectral distance. More precisely, the main results are stated as follows.

Oscillating multipliers on Lie groups and Riemannian manifolds

Abstract: are given by \"strongly singular kernels\" (cf. [9]). Oscillating multipliers have already been studied extensively in the context of Rn (cf. [9], [10], [21], [22], [23], [26]). Some of these results have been generalised to stratified nilpotent Lie groups (cf. [19]) and to rank one noncompact symmetric spaces (cf. [11]). In this article we study the oscillating multipliers in the context of connected Lie groups of polynomial volume growth and Riemannian manifolds of nonnegative Ricci curvature. More nreciselv:

AN EXPLICIT INTEGRAL REPRESENTATION OF WHITTAKER FUNCTIONS ON Sp(2; R) FOR THE LARGE DISCRETE SERIES REPRESENTATIONS

Abstract: We obtain an explicit integral representation of Siegel-Whittaker functions on Sp(2,R) for the large discrete series representations. We have another integral expression different from that of Miyazaki [7].

Left cells in affine Weyl groups

Abstract: We prove a property of left cells in certain crystallographic groups W , by which we formulate an algorithm to find a representative set of left cells of W in any given two-sided cell As an illustration, we make some applications of this algorithm to the case where W is the affine Weyl group of type e F4 The cells of affine Weyl groups W , as defined by Kazhdan and Lusztig in [6], have been described explicitly in certain special cases: for W of rank 2, see [11]; for W of type An, see [16], [10]; for W of rank 3, see [1], [4]; for the cells with a-values 1, 2 and |Φ|/2 in a general W , see [2], [8], [9], [18], [19], where Φ is the root system determined by W It is known that there exists a bijection between the set of two-sided cells in an affine Weyl group W and the set of unipotent classes in a certain complex reductive group G associated with W It is also known that the value of the a-function on a two-sided cell of W is equal to the dimension of the variety of Borel subgroups of G containing an element of the corresponding unipotent class (see [14]) Thus for an affine Weyl group W , the two-sided cells of W are relatively well understood to certain extent But the classification of left cells in a given two-sided cell of W is not known in general, even the number of these left cells In the present paper, we shall introduce an algorithm to find a representative set of left cells of W ′ in a given two-sided cell, where W ′ is a group belonging to certain family of crystallographic

On the class number one problem for non-normal quartic cm-fields

Abstract: We give explicit upper bounds for the discriminants of the non-normal quartic CM-fields with class number one, and for the discriminants of the dihedral octic CM-fields with class number one These upper bounds are too large to enable us to achieve the determination of these number fields Nevertheless, whenever a real quadratic number field k is fixed, we can explain how to determine the non-normal quartic CM- fields or the dihedral octic CM-fields with class number one and with real quadratic subfield k

The mordell-weil groups of unirational quasi-elliptic surfaces in characteristic 2

Abstract: Abstract. We continue to study the Mordell-Weil groups of unirational quasi-elliptic surfaces. We classify them in the case of rational quasi-elliptic surfaces in characteristic 2 and show how to construct them from the projective plane. In the classification, a key role is played by a theorem which guarantees that the relevant properties of unirational quasi-elliptic surfaces are determined explicitly by the equations of the surfaces as affine hypersurface.

Pointwise convergence of Hermite-Fejér interpolation of higher order for Freud weights

Abstract: This paper is concerned with the approximation by Hermite-Fejer interpolation of higher order based at the zeros of orthogonal polynomials with respect to the typical Freud weight. We will prove a convergence result for even order and a divergence result for odd order.

Liapunov functionals and periodicity in integral equations

Abstract: Theodore A. Burton Abstract. Liapunov’s direct method has been used very effectively for a hundred years on various types of differential equations. It has not, however, been used with much success on non-differentiated equations. In this paper we construct a Liapunov function for a nonlinear integral equation with an infinite delay which is of nonconvolution type. From that Liapunov function we deduce conditions for boundedness, stability, and the existence of periodic solutions. The kernel of the integral equation is a perturbation of a positive kernel and there are estimates showing how large the perturbation can be. The advantage of the Liapunov approach over classical methods for integral equations is the simplicity of analysis, once a Liapunov function is constructed. 1991 Mathematics Subject classification: 45M10

Classification of pluriharmonic maps from compact complex manifolds with positive first Chern class into complex Grassmann manifolds

Abstract: We prove that any pluriharmonic map from a compact complex manifold with positive first Chern class (defined outside a certain singularity set of codimension at least two) into a complex Grassmann manifold of rank two is explicitly constructed from a rational map into a complex projective space. Under some restrictions on dimension and rank of the domain manifold and the target manifold, respectively, we also prove that similar results hold for other complex Grassmann manifolds as targets. Introduction. Let φ: M -> N be a smooth map from a complex manifold into a Riemannian manifold. Then, φ is said to be pluriharmonic if the (0, l)-exterior covariant derivative D"dφ of the (1, O)-differential dφ of φ vanishes identically. Let V be the pull-back connection on the pull-back bundle φ~^TN. We have (0.1) (P"dφ){X, Y) = V$dφ(Y)-dφ(dχY), X, where Γ M 1 0 is the holomorphic tangent bundle of M. If φ~TN has the Koszul-Malgrange holomorphic structure, that is, the (0, l)-part of V coincides with the δ-operator, we may say that φ is pluriharmonic if and only if φ sends any holomorphic section of TM to a holomorphic section of φ~TN. It is easily seen that if φ is holomorphic and TV is a Kahler manifold then φ~TN' has the Koszul-Malgrange holomorphic structure, hence any holomorphic map is pluriharmonic. Note that an anti-holomorphic map is also pluriharmonic if N is a Kahler manifold. Conversely, the existence of the Koszul-Malgrange holomorphic structure OIK/)" TN is ensured if φ is pluriharmonic and Nhas nonnegative or nonpositive curvature operator. In this case, if N is a Kahler manifold, then φ~TN' has the Koszul-Malgrange holomorphic structure (cf. [O-U2]). From the point of view of Riemannian geometry, the most interesting property of pluriharmonic maps is that it Partially supported by the Grants-in-Aid for Scientific Research, The Ministry of Education, Science and Culture, Japan. 1991 Mathematics Subject Classification. Primary 58E20; Secondary 53C42.

A differential geometric property of big line bundles

Abstract: A holomorphic line bundle over a compact complex manifold is shown to be big if it has a singular Hermitian metric whose curvature current is smooth on the complement of some proper analytic subset, strictly positive on some tubular neighborhood of the analytic subset, and satisfies a condition on its integral. In particular, we obtain a sufficient condition for a compact complex manifold to be a Moishezon space.

Bounds of automorphism groups of genus $2$ fibrations

Abstract: For a complex surface of general type with a relatively minimal genus 2 fibration, the bounds of the orders of the automorphism group of the fibration, of its abelian subgroups and of its cyclic subgroups are determined as linear functions of $c^2_1$. Most of them are the best.