Some Harmonic Functions on Minkowski Space

This note presents elementary geometric descriptions of several simple families of harmonic functions on the upper sheet of the unit hyperboloid in Minkowski three-space. As is briefly discussed here, these calculations grew out of an earlier attempt to construct Poincaré series on punctured surfaces using Minkowski geometry. Introduction The material in this note grew out of an attempt (discussed briefly in §3 below) some time ago to construct harmonic functions on a punctured surface in terms of Minkowski geometry. Indeed (in [EP] and [P1]), we associate a convex body in Minkowski three-space to a punctured Riemann surface, and Minkowski lengths of extreme edges of this body were seen (in [P1]) to give a coordinatization of the (decorated) Teichmüller space. As we show in Proposition 2 of this note, these coordinates admit a natural extension to harmonic coordinates on the upper sheet H of the unit hyperboloid in Minkowski space; furthermore, the harmonic conjugate is computable in these coordinates, and the associated analytic function on H is the one naturally associated to a suitable pair of points in the light-cone as we shall see. There also arose another family of harmonic functions generalizing the usual Green’s function on H, where this family is parametrized by suitable pairs of points in Minkowski space. These harmonic functions are described in Proposition 1, the proof of which depends upon a family of solutions to the partial differential equation on conformal factors in the solution to the Yamabe problem; this last family is parametrized by points of Minkowski three-space itself. Finally, in Proposition 3, we describe a picture (which is certainly standard but anyway deserves to be better known) of Poisson kernel for H in terms of the geometry of Minkowski space. It is not that we are constructing “new” families of harmonic functions here, but rather that “known” harmonic functions admit essentially linear expressions in our coordinates on Minkowski space. The techniques of this paper are entirely elementary and largely computational, and there evolves a (presumably well-known) interplay between hyperbolic geometry and harmonic analysis. The second-named author thanks Itzhak Bars and Paul Yang for helpful discussions. Received by the editors April 7, 1995. 1991 Mathematics Subject Classification. Primary 30Cxx, 30Fxx. The second author was partially supported by the National Science Foundation. c ©1997 American Mathematical Society