Horospherical transform on real symmetric varieties: Kernel and cokernel

In this paper, we define a horospherical transform for a semisimple symmetric space Y. A natural double fibration is used to assign a more geometrical space Ξ of horospheres to Y. The horospherical transform relates certain integrable analytic functions on Y to analytic functions on Ξ by fiber integration. We determine the kernel of the horospherical transform and establish that the transform is injective on functions belonging to the most continuous spectrum of Y.     

Symmetric subvarieties in compactifications and the Radon transform on Riemannian symmetric spaces of noncompact type

We investigate the totally geodesic Radon transform which assigns a function to its integration over totally geodesic symmetric submanifolds in Riemannian symmetric spaces of noncompact type. Our main concern is focused on the injectivity and support theorem. Our approach is based on the projection slice theorem relating the totally geodesic Radon transform and the Fourier transforms on symmetric spaces. Our approach also uses the study of geometry concerned with the totally geodesic symmetric subvarieties in Riemannian symmetric spaces in terms of the cell structure of the Satake compactifications.     

Harmonic analysis on symmetric Stein manifolds from the point of view of complex analysis

The classical theory of finite dimensional representations of compact and complex semisimple Lie groups is discussed from the perspective of multidimensional complex geometry and analysis. The key tool is the complex horospherical transform which establishes a duality between spaces of holomorphic functions on symmetric Stein manifolds and dual horospherical manifolds. Communicated by: Toshiyuki Kobayashi     

Eigenvalue estimate on complete noncompact Riemannian manifolds and applications

We obtain some sharp estimates on the first eigenvalues of complete noncompact Riemannian manifolds under assumptions of volume growth. Using these estimates we study hypersurfaces with constant mean curvature and give some estimates on the mean curvatures.

     

Isometry theorem for the Segal-Bargmann transform on a noncompact symmetric space of the complex type

We consider the Segal-Bargmann transform on a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the dual compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities.     

Riesz transform on locally symmetric spaces and Riemannian manifolds with a spectral gap

In this paper we study the Riesz transform on complete and connected Riemannian manifolds M with a certain spectral gap in the L² spectrum of the Laplacian. We show that on such manifolds the Riesz transform is Lp bounded for all p∈(1,∞). This generalizes a result by Mandouvalos and Marias and extends a result by Auscher, Coulhon, Duong, and Hofmann to the case where zero is an isolated point of the L² spectrum of the Laplacian.     

The horospherical duality

We discuss the horospherical duality as a geometrical background of harmonic analysis on semisimple symmetric spaces.     

The Segal-Bargmann transform for noncompact symmetric spaces of the complex type

We consider the generalized Segal-Bargmann transform, defined in terms of the heat operator, for a noncompact symmetric space of the complex type. For radial functions, we show that the Segal-Bargmann transform is a unitary map onto a certain L² space of meromorphic functions. For general functions, we give an inversion formula for the Segal-Bargmann transform, involving integration against an “unwrapped” version of the heat kernel for the dual compact symmetric space. Both results involve delicate cancellations of singularities.     

局部对称的黎曼流形中的极小子流形

主要研究了局部对称的黎曼流形中的定向紧致无边极小子流形的内蕴刚性问题,利用一个矩阵不等式,得到了这类子流形的一个刚性定理．所得结果部分改进了已有的一个结论．     

On quasi-conformally flat weakly Ricci symmetric manifolds

The object of the present paper is to study quasi-conformally flat weakly Ricci symmetric manifolds.      

有界对称域上的加权Plancherel公式

本文给出加权 Ｐｌａｎｃｈｅｒｅｌ公式与Ｈｅｒｍｉｔｅ对称空间上的齐性线从上Ｐｌａｎｃｈｅｒｅｌ公式的关系，由此导出一般有界对称域上的加权Ｐｌａｎｃｈｅｒｅｌ公式．     

Logarithmic Sobolev inequalities on noncompact Riemannian manifolds

Summary. This paper presents a dimension-free Harnack type inequality for heat semigroups on manifolds, from which a dimension-free lower bound is obtained for the logarithmic Sobolev constant on compact manifolds and a new criterion is proved for the logarithmic Sobolev inequalities (abbrev. LSI) on noncompact manifolds. As a result, it is shown that LSI may hold even though the curvature of the operator is negative everywhere. Received: 24 July 1996 / In revised form: 25 June 1997     

The uncertainty principle on Riemannian symmetric spaces of the noncompact type

The uncertainty principle in says that it is impossible for a function and its Fourier transform to be simultaneously very rapidly decreasing. A quantitative assertion of this principle is Hardy's theorem. In this article we prove various generalisations of Hardy's theorem for Riemannian symmetric spaces of the noncompact type. In the case of the real line these results were obtained by Morgan and Cowling-Price.

     

Vanishing theorem for irreducible symmetric spaces of noncompact type

We prove the following vanishing theorem. Let M be an irreducible symmetric space of noncompact type whose dimension exceeds 2 and M ≠SO0（2, 2）/SO（2） × SO（2）. Let π ： E →＊ M be any vector bundle. Then any E-valued L2 harmonic 1-form over M vanishes. In particular we get the vanishing theorem for harmonic maps from irreducible symmetric spaces of noncompact type.     

On the hyperbolicity criterion for noncompact Riemannian manifolds of special type

In this paper we study the behavior of bounded harmonic functions on complete Riemannian manifolds (of a certain special type) depending on the geometry of the manifold. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 557–563, April, 1996.     

Some inequalities and asymptotic formulas for eigenvalues on Riemannian manifolds

In this paper, we establish sharp inequalities for four kinds of classical eigenvalues in bounded domains of Riemannian manifolds. We also obtain the Weyl-type asymptotic formulas for the eigenvalues of the buckling and clamped plate problems in bounded domains of Riemannian manifolds. In addition, we give a negative answer to the Payne conjecture for the one-dimensional case.     

Generalized Einstein conditions on holomorphic Riemannian manifolds

Summary At first, a necessary and sufficient condition for a K?hler-Norden manifold to be holomorphic Einstein is found. Next, it is shown that the so-called (real) generalized Einstein conditions for K?hler-Norden manifolds are not essential since the scalarcurvature of such manifolds is constant. In this context, we study generalized holomorphic Einstein conditions. Using the one-to-one correspondence between K?hler-Norden structures and holomorphic Riemannian metrics, we establish necessary and sufficient conditions for K?hler-Norden manifolds to satisfy the generalized holomorphic Einstein conditions. And a class of new examples of such manifolds is presented. Finally, in virtue of the obtained results, we mention that Theorems 1 and 2 of H. Kim and J. Kim [10] are not true in general.