Josef Dorfmeister

Homogeneous Siegel domains

In 1935 E. Cartan classified all symmetric bounded domains [6]. At that time he proved that a bounded symmetric domain is homogeneous with respect to its group of holomorphic automorphisms. Thus the more general problem of investigating homogeneous bounded domains arose. It was known to E. Cartan that all homogeneous bounded domains of dimension ≤3 are symmetric [6]. For domains of higher dimension little was known. The first example of a 4-dimensional, homogeneous, non-symmetric bounded domain was provided by I. Piatetsky-Shapiro [41]. In several papers he investigated homogeneous bounded domains [20], [21], [41], [42], [43]. One of the main results is that all such domains have an unbounded realization of a certain type, as a so-called Siegel domain. But many questions still remained open. Amongst them the question for the structure and explicit form of the infinitesimal automorphisms of a homogeneous Siegel domain.

Fine structure of reductive pseudo-Khlerian spaces

We continue the investigation of homogeneous pseudo-Kählerian manifolds (M, θ) admitting a reductive transitive group G of automorphisms. We give a detailed description of the pseudo-Kähler algebras associated with M, θ and G.

On the meromorphic potential for a harmonic surface in a k-symmetric space

Over the past few years there has been significant progress in the study of harmonic maps of compact Riemann surfaces into compact symmetric spaces. Whereas the case of harmonic 2-spheres in various symmetric spaces and Lie groups can be completely understood in terms of certain holomorphic curves into associated twistor spaces, the case of harmonic 2-tori is considerably more involved and is closely tied to certain integrable systems on loop spaces. Besides being motivated by field theories of mathematical physics, considerable motivation for studying this case was provided by the question of the existence of constant mean curvature tori in 3-space (posed by H. Hopf and resolved much later, to the affirmative, by H. Wente). By now it is fair to say we understand how ‘the generic’ harmonic 2-tori (in the sense of [5]) in compact symmetric spaces arise: they are obtained by solving a hierarchy of completely integrable ODE in Lax form on certain loop algebras, whose solutions can be expressed in terms of theta functions on certain algebraic curves, the spectral curves of the flows [6].

Automorphisms of the KdV-subvariety

The purpose of this note is to study the KdV-subvariety X (n) of the KP-Grassmannian X considered in [12] and [3], together with natural groups of automorphisms. As a main result (see Theorem 2.13), we show that G (n) {g e (Aut X) 0 = gX (n) = X (n)} ~s the semidirect product of the groups ~(n) = {g e G1(n,A) ; det g e A ~(0)} and SU(1,1) where A is a Banach algebra closely related to the Banach structure of X and SU(I,I) acts via dJffeomorphisms of the c~rcle. In the last section, we d~scuss briefly topological aspects of the relations between X, X (n) ~(n) and Aut X. In 9 9 partJclllar, we point out how to construct central extensions of ~n) for all ~, thus extending a construction of Pressley and SegaLL (see 2.15).

Groups associated with a Grassmannian modelled on the Wiener algebra

In this paper we investigate a Grassmann-like manifold X, modelled on the Wiener algebraA. We determine the group Aut(X) of biholomorphic maps of X and the structure of some Banach Lie groups associated with Aut(X). In particular, we establish Gohberg decompositions and triangular decompositions for our groups. Moreover, an application of a distributional Plemelj formula yields for X a complete description of all holomorphic vector fields.