Some Remarks on Complex Crowns of Riemannian Symmetric Spaces

We discuss several results and problems in which complex geometric constructions appear in the geometry of real pseudo Riemannian symmetric spaces in connection with analysis on such spaces.     

Geometry of Infinitesimal Harmonic Transformations

One of the present authors defined the infinitesimal harmonictransformation in a Riemannian manifold. This paperis devoted to the study of the local and global geometryinfinitesimal harmonic transformations.     

On the global theory of some classes of mappings

This paper is devoted to the study of the global theory of certain mappings between Riemannian manifolds. We generalize results by Vilms, Yano and Ishihara, and study in detail projective, umbilical and harmonic maps.The work of the author was supported by RBRF, grant 94-01-01595 (Russia).     

Universal Plane Curve and Moduli Spaces of 1-dimensional Coherent Sheaves

We show that the universal plane curve M of fixed degree d ≥ 3 can be seen as a closed subvariety in a certain Simpson moduli space of 1-dimensional sheaves on ?₂ contained in the stable locus. The universal singular locus of M coincides with the subvariety M′ of M consisting of sheaves that are not locally free on their support. It turns out that the blow up Bl _M′ M may be naturally seen as a compactification of M _B  = M?M′ by vector bundles (on support).     

Compact Homogeneous Einstein 7-Manifolds

This paper is devoted to the classification of seven-dimensional homogeneous Einstein manifolds with positive scalar curvature.Mathematics Subject Classifications (2000). 53C25, 53C30.The author was supported by RFBR (codes 02-01-01071, 01-01-06224, 00-15-96165).     

调和Bergman度量

We have constructed the positive definite metric matrixes for the bounded domains of Rn and proved an inequality which is about the Jacobi matrix of a harmonic mapping on a bounded domain of Rn and the metric matrix of the same bounded domain.     

On harmonic and asymptotically harmonic homogeneous spaces

We classify noncompact homogeneous spaces which are Einstein and asymptotically harmonic. This completes the classification of Riemannian harmonic spaces in the homogeneous case: Any simply connected homogeneous harmonic space is flat, or rank-one symmetric, or a nonsymmetric Damek–Ricci space. Independently, Y. Nikolayevsky has obtained the latter classification under the additional assumption of nonpositive sectional curvatures [N2]. Supported in part by DFG priority program “Global Differential Geometry” (SPP 1154). Received: September 2004; Revision: June 2005; Accepted: September 2005     

Sobolev spaces with variable exponents on Riemannian manifolds

In this article we study variable exponent Sobolev spaces on Riemannian manifolds. The spaces are examined in the case of compact manifolds. Continuous and compact embeddings are discussed. The paper contains an example of the application of the theory to elliptic equations on compact manifolds.     

一类完备Riemann流形上的有界调和函数

本文我们将对一类完备Ｒｉｅｍａｎｎ流形上的有界调和函数所组成的线性空间的维数的上界进行估计，同时给出了一个关于测地球体积的Ｂｉｓｈｏｐ－Ｇｒｏｍｏｖ型体积比较定理。     

Proximal subgradient and a characterization of Lipschitz function on Riemannian manifolds

A characterization of Lipschitz behavior of functions defined on Riemannian manifolds is given in this paper. First, it is extended the concept of proximal subgradient and some results of proximal analysis from Hilbert space to Riemannian manifold setting. A technique introduced by Clarke, Stern and Wolenski [F.H. Clarke, R.J. Stern, P.R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition, and convexity, Canad. J. Math. 45 (1993) 1167-1183], for generating proximal subgradients of functions defined on a Hilbert spaces, is also extended to Riemannian manifolds in order to provide that characterization. A number of examples of Lipschitz functions are presented so as to show that the Lipschitz behavior of functions defined on Riemannian manifolds depends on the Riemannian metric.     

Concerning Nikodym-type sets in 3-dimensional curved spaces

We investigate maximal functions involving averages over geodesics in three-dimensional Riemannian manifolds. We first show that one can easily extend the Euclidean results of Bourgain and Wolff if one assumes constant curvature. These results need not hold if this assumption is dropped. Nonetheless, we formulate a generic geometric condition which allows favorable estimates. Curiously, this condition ensures that one is in some sense as far as possible from the constant curvature case. Assuming this, one can prove dimensional estimates for Nikodym-type sets which are essentially optimal. Optimal estimates for the related maximal functions are still open though.

     

Dini derivative and a characterization for Lipschitz and convex functions on Riemannian manifolds

Dini derivatives in Riemannian manifold settings are studied in this paper. In addition, a characterization for Lipschitz and convex functions defined on Riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the Dini derivative are given.     

Noncompact Homogeneous Einstein 5-Manifolds

This article is devoted to the classification of noncompact homogeneous Einstein 5-manifolds. In particular, we prove that each noncompact homogeneous Einstein 5-manifolds is locally isometric to some standard Einstein solvmanifoldMathematics Subject Classifications (2000). 53C25, 53C30     

Path and quasi-homotopy for Sobolev maps between manifolds

We study the relationship between quasi-homotopy and path homotopy for Sobolev maps between manifolds. By employing singular integrals on manifolds we show that, in the critical exponent case, path homotopic maps are quasi-homotopic – and observe the rather surprising fact that quasi-homotopic maps need not be path homotopic. We also study the case where the target is an aspherical manifold, e.g. a manifold with non-positive sectional curvature, and the contrasting case of the target being a sphere.     

Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds

This paper extends the full convergence of the steepest descent method with a generalized Armijo search and a proximal regularization to solve minimization problems with quasiconvex objective functions on complete Riemannian manifolds. Previous convergence results are obtained as particular cases and some examples in non-Euclidian spaces are given. In particular, our approach can be used to solve constrained minimization problems with nonconvex objective functions in Euclidian spaces if the set of constraints is a Riemannian manifold and the objective function is quasiconvex in this manifold.     

Hardy Spaces of Differential Forms on Riemannian Manifolds

Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H ^p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H ^p -boundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ^∞ functional calculus and Hodge decomposition, are given.      

Three-step Harmonic Solvmanifolds

The Lichnerowicz conjecture asserted that every harmonic Riemannian manifold is locally isometric to a two-point homogeneous space. In 1992, E. Damek and F. Ricci produced a family of counter-examples to this conjecture, which arise as abelian extensions of two-step nilpotent groups of type-H. In this paper we consider a broader class of Riemannian manifolds: solvmanifolds of Iwasawa type with algebraic rank one and two-step nilradical. Our main result shows that the Damek–Ricci spaces are the only harmonic manifolds of this type.     

Busemann Functions in a Harmonic Manifold

For a noncompact complete and simply connected harmonic manifold M, we prove the analyticity of Busemann functionson M. This is the main result of this paper. An application of it shows that the harmonic spaces having minimal horospheres have the bi-asymptotic property. Finally, we prove that the total Busemann functionis continuous in C topology. As a consequence, we show that the uniform divergence of geodesics holds in these spaces.     

Harmonic Analysis for Real Spherical Spaces

We give an introduction to basic harmonic analysis and representation theory for homogeneous spaces Z = G/H attached to a real reductive Lie group G. A special emphasis is made to the case where Z is real spherical.