Potential Analysis 2: 295-298, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands. 295 A Note on the Classification of Holomorphic Harmonic Morphisms SIGMUNDUR GUDMUNDSSON* and RAGNAR SIGURDSSON Science Institute, University of Iceland, Dunhaoa 3, 107 Reykjavik, Iceland. (Received: 20 January 1993) Abstract. In this note we give a complete classification of those holomorphie maps q~: U---~C" defined on open and connected subsets of C" which are harmonic morphisms. Mathematics Subject Classifications (1991). 58E20, 58G32, 32A10. Key words. Harmonic morphisms, Brownian motions, holomorphic maps. O. Introduction Let (M", O) and (N", h) be Riemannian manifolds. A map 4~:(M, 9)----~ (N, h) is called a harmonic morphism if for any harmonic function f: U ~ ~ defined on an open subset U of N with ~b-l(U) non-empty, fo~b:~b-l(U)--~ is a harmonic function. An alternative description of non-constant harmonic morphisms is that they map Brownian motions on (M, O) to Brownian motions on (N, h), see [6]. If m < n then every harmonic morphism ~b:(M, 9)--~ (N, 9) is constant, hence not of interest. If m ~> n then a non-constant harmonic morphism ~b:(M, 9)--' (N, O) is surjeetive outside the critical set C¢~ := {x ~ M Id~b~ = 0), which has a dense complement M* := M - C a- In [9] and [10] Fuglede and Ishihara independently characterized harmonic morphisms as those harmonic maps which are horizontally conformal in the following sense: At each point p e M* let ~ be the vertical space at p given by ~ := Ker dckp c TpM and Yt],:= ~1 be the horizontal space. Here I denotes the orthogonal complement with respect to the metric g on M. The map 4~:(M, g) ~ (N, g) is said to be horizontally conformal if there exists a function 2:M*--~ + such that 22g(X,Y)= h(d4)(X), dck(Y)) for all X, Y~ oug. In recent years a substantial progress has been made in the field of harmonic morphisms, see for example El-3], [5], [7], ['8] and [11]. Concerning this work it is observed in [9], as a direct consequence of the Cauchy-Riemann equations, that * The first author was supported by the Icelandic Science Fund.