Stein extensions of Riemannian symmetric spaces and dualities of orbits on flag manifolds

It is known [M4] that K_ℂ-orbits S and G_ℝ-orbits S' on a complex flag manifold are in one-to-one correspondence by the condition that S ∩ S' is nonempty and compact. It is possible to replace K_ℂ by some conjugate xK_ℂx⁻¹ so that the correspondence is preserved. We investigate the sets C(S) of such x for various orbits S and their relations with each other. We prove that for classical groups the intersection C = ∩_S C(S) equals D₀Z where D₀ = D₀/K_ℂ is the universal domain in G_ℂ/K_ℂ introduced in [AG] and Z is the center of G. As a corollary we prove that D₀ is Stein for classical groups. Moreover we conjecture that C(S)₀ = D₀ for generic S where C(S)₀ is the connected component of C(S) containing the identity.     

Complex extensions of semisimple symmetric spaces

Let G/H be a pseudo-Riemannian semisimple symmetric space. The tangent bundle T(G/H) contains a maximal G-invariant neighbourhood Ω of the zero section where the adapted-complex structure exists. Such Ω is endowed with a canonical G-invariant pseudo-Kähler metric of the same signature as the metric on G/H. We use the polar map to define a G-invariant pseudo-Kähler metric on distinguished G-invariant domains in or on coverings of principal orbit strata in . In the rank-one case, we show that the polar map is globally injective and the domain is an increasing union of q-complete domains.     

On the quantum flag manifold

Rostov State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 3, pp. 90–92, July–September, 1992.     

Limits on the real line of symmetric spaces on segments

In the same way as the known spacesM _p ,M _p , andI _p are constructed on the basis of the spaceL _p (–1, 1), we construct the corresponding limit spacesM _E ,M _E , andI _E on the real line on the basis of a symmetric function spaceE on a segment and study some of their Banach properties.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 46–55, January, 1995.     

Interpolation problems,extensions of symmetric operators and reproducing kernel spaces II

The aim of part I and this paper is to study interpolation problems for pairs of matrix functions of the extended Nevanlinna class using two different approaches and to make explicit the various links between them. In part I we considered the approach via the Kreîn-Langer theory of extensions of symmetric operators. In this paper we adapt Dym's method to solve interpolation problems by means of the de Branges theory of Hilbert spaces of analytic functions. We also show here how the two solution methods are connected.     

Interpolation problems,extensions of symmetric operators and reproducing kernel spaces II

By an oversight on the part of the authors this section was not included in the paper previously published in Integral Equations Operator Theory, volume 14/4 (1991), 466–500. Present address:Department of Mathematics Ben-Gurion University of the Negev Beersheva Israel     

Differential geometry of real submanifolds in a Kähler manifold

LetN be a real submanifold in a complex manifoldM. If the maximal complex subspaces of the tangent spaces ofM contained in the tangent spaces ofN are of constant dimension and they define a differentiable distribution, thenN is called a generic submanifold. The class of generic submanifold includes all real hypersurfaces, complex submanifolds, totally real submanifolds andCR-submanifolds. In this paper we initiate a study of generic submanifolds in a Kähler manifold from differential geometric point of view. Some fundamental results in this respect will be obtained.     

Embedding of Stein spaces into sequence spaces

In this note we show that a connected, reduced Stein space X of arbitrary dimension admits a holomorphic embedding into various sequence spaces, for example into s,s',0(ⁿ) or 1,T₂,...,T_n>, and also into infinite dimensional complex Banach spaces. As an application we prove that the Fréchet space 0 (X) of holomorphic functions on X is a quotient of s.     

Pattern formation and geometry of the manifold

The objective of the present work is to investigate how pattern formation in the Cahn–Hilliard system can be influenced by geometry of the manifold. This is in contrast to control methods in which the physical field is modified and the pattern formation of the original system changes in response to control inputs. The idea begins with the cylindrical manifold symmetry leading to circumferential rolls while the torus manifold can be used to produce and control helical rolls. The next step is to search for a weaker restriction on the geometry of the manifold in order to reduce its dimension. In particular a short amplitude sinusoidal modulation on a flat surface is studied. At the final step a sequential pattern formation is presented.