Potential Analysis 2: 295-298, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands.

295

A Note on the Classification of Holomorphic Harmonic Morphisms

S I G M U N D U R G U D M U N D S S O N * and R A G N A R S I G U R D S S O N 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 I d~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.

296 SIGMUNDUR GUDMUNDSSON AND RAGNAR SIGURDSSON

every holomorphic function ¢: U---~ C defined on an open subset of C m is a harmonic morphism. This is generalized in [4] where the authors show that every holomorphic map from a Kiihler manifold to a Riemann surface is necessarily a harmonic morphism. This provides us with a rich class of examples when mapping into a surface. In this note we prove that in the case of higher codomains the situation is drastically different. Namely, if ¢: U--~ C" is a non-constant holomorphic map defined on an open and connected subset U of C m and n > 1, then ¢ is a harmonic morphism if and only if it is an orthogonal projection followed by a homothety.

1. A Classification Theorem

The following theorem gives a complete classification of holomorphic harmonic morphisms between open and connected subsets of complex euclidean vector spaces,

THEOREM. Let ¢: U - * C " be a non-constant holomorphic map from an open and

connected subset U of C m. (a) I f n = 1, then 4) is a harmonic morphism. (b) I f n > 1, then ~) is a harmonic morphism if and only if q5 is an orthooonal projection

followed by a homothety. Proof (a) This was proved in [9]. (b) It is well known that if ¢: U---,C" is an orthogonal projection followed by a

homothety then ¢ is a harmonic morphism, see for example [7]. On the other hand let us assume that ~b = (¢1 . . . . . ¢,): U-~ C" is a holomorphic

harmonic morphism. Then it is a direct consequence of Theorem p. 112 of [9] that the horizontal conformality of ¢ is equivalent to

(grad(¢k),grad(¢~)) =-- 0 and tgrad(¢k)] 2 -- [grad(~b~)[ 2 ---- 0 for all k ¢ l, (1)

where grad is the operator given by grad := (O/Ozl . . . . . O/Oz,), ( , ) is the Hermitian inner product with (z, w) := Z z j ~ and [ " I the corresponding norm.

It immediately follows from the definition of harmonic morphisms that the composition of two such maps is again a harmonic morphism. This means that the map (~b k, Cz): U---~ C 2 is a holomorphic harmonic morphism for all k ~ l, being the composition of ¢ with the orthogonal projection zc: (z ~,..., z,) ~ (z k, z~). It then follows from the Lemma below that all the gradients grad(¢k) are constant. The horizontal space Jog of ¢ at each point is given by

= span~ {grad(~k),igrad(¢k) I 1 ~< k ~< n}, (2)

i.e. constant on the whole of U. This means that ¢ must be an orthogonal projection followed by a homothety. []

H O L O M O R P H I C H A R M O N I C M O R P H I S M S 297

LEMMA. Let U be an open and connected subset of C m and (f,g):U--~C 2 be a

holomorphic map. I f (1) (grad(f) , grad(g)) -= 0, and (2) lgrad(f)l 2 - [grad(g)[ 2 - 0,

then the gradients grad(f), grad(g): U--* C m are constant.

Proof. For 1 ~<j ~< m put f j := 8f/azj and gj:= 8g/~zj. By differentiating (1) with respect to z~ and ~j we obtain

(grad(f3), grad(g)) - (grad(f) , grad(gj)) - 0 for all j. (3)

Now define ~k: U-~ • by ~ := [grad(f)[ 2 - [grad(g)[ 2. Then it follows from (2) that for all z6 U and w6C m we have

~2 ~/(Z + #w) = ~, I(grad(fj),~)[ 2 - ~ [(grad(gj),~)[ 2. (4)

0 -- ~-'~-~- -- R=0 j = l j = l

If we now substitute v9 = grad(f) into equation (4) we obtain by using (3)

0 -- ](grad(~), grad(f))[2 = ~ [grad(f)[2

for all j. Similarly we get ~/~zj(]grad(g)[ 2) = 0 for all j. This means that the gradients grad (f), grad(g): U ~ C m are holomorDhic and have constant norms, hence constant.

[]

The following Example shows that the assumption of holomorphy is needed in part (b) of our Theorem.

EXAMPLE. It is shown in [9] that the multiplication of quaternions 4: H × H--~ H with ~ : ( p , q ) ~ , p . q is a harmonic morphism. The quatemions (H, -) can be seen as C 2 with the following multiplication (zl, z2) • (za, z4) := (zlz a - z2£ 4, z lz 4 + z22a). This means that the non-holomorphic map ~:C4---~C 2 with ~:(zl , zE,z3,z4)~--~ (zlz 3 - z 2 £ 4 , z l z 4 + z2~3) is a harmonic morphism without being an orthogonal projection followed by a homothety.

Acknowledgement

The au thors are grateful to J o h n C. W o o d for having suggested this problem.

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298 SIGMUNDUR GUDMUNDSSON AND RAGNAR StGURDSSON

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