Jarolím Bureš

Fundaments of Hermitean Clifford analysis part II: splitting of h -monogenic equations

Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and mutually adjoint Dirac operators, which are invariant under the action of a Clifford realization of the unitary group. In part 1 of the article the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e., by introducing a complex structure on the underlying vector space, eventually extended to the whole complex Clifford algebra . The two Hermitean Dirac operators are then shown to originate as generalized gradients when projecting the gradient on invariant subspaces. In this part of the article, the aim is to further unravel the conceptual meaning of h-monogenicity, by studying possible splittings of the corresponding first-order system into independent parts without changing the properties of the solutions. In this way further connections with holomorphic functions of several complex variables are established. As an illustration, we give a full characterization of h-monogenic functions for the case n = 2. During the final redaction of this article, we received the sad news that our friend, colleague and co-author Jarolím Bureš died on 1 October 2006.

Symmetric Analogues of Rarita-Schwinger Equations

In this paper a generalization of the classicalRarita–Schwinger equations for spin 3/2 fields to the case of spin fieldswith values in irreducible representation spaces with weight k+1/2 isgiven. It corresponds to the study of serie of first orderconformal invariant operators, which are constructed from twisted Diracoperators. The representation character of polynomial solutions of the equations onflat space and their relations are described in details.

Generalized hypercomplex analysis and its integral formulas

The homological version of the Cauchy integral formula is formulated in the paper for solutions of corresponding equations in complexified hypercomplex analysis. Many different cases are treated in unified manner, including some higher order operators. The notion of index of n-cycles is defined in this complexified situation and its properties are studied.

The rarita—schwinger operator and spherical monogenic forms

There is a certain family of conformally invariant first order elliptic systems which include the Dirac operator as its first member, and the Rarita-Schwinger operator, as the second simplest operator in the row. Its basic properties on general spin manifolds are described there. The aim of the paper is to do first step towards a function theory for Rarita-Schwinger equation. The main result contained in the paper is a complete classification of polynomial solutions of Rarita-Schwinger equation on R n . Relations with Clifford analysis and representation theory are discussed.

Integral Formulae in Complex Clifford Analysis

In this note some integral formulae for complex left regular mappings and for solutions of the complex Laplace equation are presented.These integral formulae are of two types (elliptic and hyperbolic) and there is a general procedure (described by V.Soucek and M. Dodson for the Laplace equation in [3]) to transform one type into another, namely the Leray residue formula.In this way it is possible to derive from integral formulae in Clifford analysis e.g.Riesz’ integral formula for the solution of the wave equation in the Minkowski space and Penrose’s integral formula for a spin-1/2 massless field. There are good hopes that the new integral formula of hyperbolic type can be used to give further information on left regular mappings and spinor fields.

Spin Structures and Harmonic Spinors on Riemann Surfaces

The aim of the paper is a description of some relations between spin-structures, the Dirac operators and harmonic spinors on the one side and the complex geometry of Riemann surfaces on the other side. The case of hyperelliptic surfaces is studied in more details.

Monogenic Forms of Polynomial Type

The classical Dirac equation for spin 1/2-fields is an example of (first order) equation for spin λ-fields related with irreducible representation spaces of the group Spin(m) with weight λ on an oriented riemannian spin manifold M. For the flat space M = R m there is the Clifford analysis as a natural method for the study of properties of these fields. Their Taylor series are composed from polynomial-type fields, the elements of some finite-dimensional representation space of the group Spin(m). The representation character (decomposition into irreducible components) of polynomial-type fields were studied for Rarita Schwinger fields in [4], for symmetric analogies of Rarita-Schwinger fields in two related papers [7, 8]. In this paper the representation character of monogenic s-forms of polynomial-type with s < m/2 is described in details.

Conformally invariant first order equations, their analysis and geometry

In the paper interactions between Clifford analysis and differential geometry are presented. It is a review article on the results which I obtained together mainly with V. Souček and also with P. Van Lancken and F. Sommen. Two important applications of Clifford analysis in geometry, first for generalized Penrose transform and second for the description of the polynomial solutions of some conformally invariant differential equations on the flat space are given.

Spin Structures and Harmonic Spinors on Nonhyperelliptic Riemann Surfaces of Small Genera

The paper is a continuation of the preceding papers [4], [5] where description of spin structures and harmonic spinors on the hyperelliptic Riemann surfaces was presented. Here the case of nonhyperelliptic surfaces of genera 3 ≤ g ≤ 6 is studied in details by means of algebraic geometry.