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 Poincare 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) Teichmiiller space. As we show in Proposition 2 of this note, these coordinates admit a natural extension to harmonic coordinates on the upper sheet HEE+ of the unit hyperboloid in Minkowski space; furthermore, the harmonic conjugate is computable in these coordinates, and the associated analytic function on JEE+ 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 JEE+, 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 JEE+ 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 3OCxx, 3OFxx. The second author was partially supported by the National Science Foundation. (D1997 American Mathematical Society 1659 This content downloaded from 157.55.39.180 on Tue, 27 Sep 2016 04:08:21 UTC All use subject to http://about.jstor.org/terms 1660 P. F. GLEZEN AND R. C. PENNER 1. NOTATION The reader desiring further information on (some of) the material in this section is referred to ?2 of [P1]. Let M denote Minkowski three-space, so M is identified with R' (with its standard coordinates x, y, and z) and is endowed with the pairing ((xi, Y1, Z1), (X2, Y2, Z2)) = XlX2 + Y1Y2 ZlZ2. The "height" of a point J = (x, y, z) E M is defined to be simply h(Q) = z. Let us also define the following distinguished subsets of M: L ={P jE M: (P'P) =oQ}, L+ = {jpE L: h(j7) > 0}, E=P PE M: ('') = -1}, E+ = {p E H: h(j) > 0},