Holomorphic horospherical duality "sphere-cone"

We describe a construction of complex geometrical analysis which corresponds to the classical theory of spherical harmonics.Comment: 9 page

Complex crowns of Riemannian symmetric spaces and non-compactly causal symmetric spaces

In this paper we define a distinguished boundary for the complex crowns \Xi\subeq G_\C /K_\C of non-compact Riemannian symmetric spaces $G/K$. The basic result is that affine symmetric spaces of $G$ can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.Comment: 29 page

Invariant Stein domains in Stein symmetric spaces and a non-linear complex convexity theorem

We prove a complex version of Kostant's non-linear convexity theorem. Applications to the construction of G-invariant Grauert tubes of non-compact Riemannian symmetric G/K spaces are given.Comment: 9 page

Holomorphic horospherical transform on non-compactly causal spaces

We develop integral geometry for non-compactly causal symmetric spaces. We define a complex horospherical transform and, for some cases, identify it with a Cauchy type integral.Comment: Revised, final version; to appear in IMRN, 38

HOROSPHERICAL CAUCHY TRANSFORM ON QUADRICS

Abstract. We describe a construction of complex geometrical analysis which corresponds to the classical theory of spherical harmonics I believe that the connection of harmonic analysis and complex analysis has an universal character and is not restricted by the case of complex homogeneous manifolds. It looks as a surprise that such a connection exists and though it is quite natural for finite dimensional representations and compact Lie groups [Gi00,Gi02]. In this note we describe the complex picture which corresponds to harmonic analysis on the real sphere. The basic construction is a version of horospherical transform which in this case is a holomorphic integral transform between holomorphic functions on the complex sphere and the complex spherical cone. This situation looks quite unusual from the point of view of complex analysis and I believe presents a serious interest also in this setting. It can be considered as a version of the Penrose transform), but in a purely holomorphic situation when there is neither cohomology nor complex cycles

Solitons and admissible families of rational curves in twistor spaces

It is well known that twistor constructions can be used to analyse and to obtain solutions to a wide class of integrable systems. In this article we express the standard twistor constructions in terms of the concept of an admissible family of rational curves in certain twistor spaces. Examples of of such families can be obtained as subfamilies of a simple family of rational curves using standard operations of algebraic geometry. By examination of several examples, we give evidence that this construction is the basis of the construction of many of the most important solitonic and algebraic solutions to various integrable differential equations of mathematical physics. This is presented as evidence for a principal that, in some sense, all soliton-like solutions should be constructable in this way.Comment: 15 pages, Abstract and introduction rewritten to clarify the objectives of the paper. This is the final version which will appear in Nonlinearit