Monogenic vector fields on the poincaré manifold

Abstract

In this paper a study is made of vector valued functions satisfying the equation , where D E is the Euclidean Dirac operator, and is the (anti-Euclidean) inner product. These functions, called hypermonogenic, were introduced by Leutwiler (see [10-13]). In the first part of this paper we explore the relation between these functions and the Poincaré metric on half space . Indeed, up to a scalar correction factor, these functions are monogenic, in the sense that they are solutions of the Hodge Dirac operator. As a result, they are a logical generalisation of classical holomorphic functions on C - clearly the equation considered reduces to the Cauchy-Riemann equation for n= 2 – in a way similar to, but not equal to, the theory of monogenic functions on Euclidean space. Furthermore, a natural isomorphism between hypermonogenic functions in a half space and in a ball (the second model of hyperbolic space) is given. This isomorphism shows that the series expansion of a hyperharmonic function in the neighbourhood of a point of the manifold can be most easily expressed in terms of the spherical harmonics of Euclidean space. Then two important results for these functions are given. First, a result by Leutwiler, stating that a hypermonogenic function near the ‘boundary’ can be described by its values on and its ‘characteristic function’ on , is generalised from the case n = 3 to arbitrary n> 2. Second, it is proved that a hypermonogenic function can be developed in a neighbourhood of a boundary point in a series of cyhndrical waves. Notice that both results are remarkable in that they deal with an inhomogeneous space, because of the singular hyperplane . Finally, we introduce a similar theory for a metric associated with a subspace of lower dimension, and indicate how it may be Hnked with a higher dimensional analogue of the theory of Padé approximation.