Bernstein theorems for harmonic morphisms from \({\mathbb R}\) 3 and S 3
type: Item
instance of: scholarly article
title: Bernstein theorems for harmonic morphisms from \({\mathbb R}\) 3 and S 3
zbMATH Open document ID: 0621.58011
DOI: 10.1007/BF01450078
author: Paul Baird
author: J. C. Wood
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164386
review text: A smooth mapping between Riemannian manifolds is called a harmonic morphism if, for every function f harmonic on an open set \(V\subset N\), the composition \(f\circ \phi\) is harmonic on \(\phi ^{-1}(V)\subset M\). Harmonic morphisms have been characterized as ''horizontally weakly conformal'' harmonic maps by \textit{B. Fuglede} [Ann. Inst. Fourier 28, No.2, 107-144 (1978; Zbl 0339.53026)] and \textit{T. Ishihara} [J. Math. Kyoto Univ. 19, 215-229 (1979; Zbl 0421.31006)], and as Brownian path preserving mappings by \textit{A. Bernard}, \textit{E. A. Campbell} and \textit{A. M. Davie} [Ann. Inst. Fourier 29, 207-228 (1979; Zbl 0386.30029)]. Simple examples are (i) any orthogonal projection \({\mathbb{R}}^ 3\to {\mathbb{R}}^ 2\), (ii) the Hopf map \(S^ 3\to S^ 2\). In this paper we show that these are essentially the only harmonic morphisms from their domains to a Riemann surface, namely we show that (i) any nonconstant harmonic morphism from \({\mathbb{R}}^ 3\) to a Riemann surface N is an orthogonal projection \({\mathbb{R}}^ 3\to {\mathbb{R}}^ 2\) followed by a weakly conformal map of \({\mathbb{R}}^ 2\) to N, (ii) any nonconstant harmonic morphism from \(S^ 3\) to a Riemann surface N is, after an orthogonal change of coordinates, the Hopf map \(S^ 3\to S^ 2\) followed by a weakly conformal map of \(S^ 2\) to N, hence \(N=S^ 2\). Since harmonic morphisms from \({\mathbb{R}}^ 3\) and \(S^ 3\) to manifolds of dimension \(\neq 2\) are easy to determine, these results mean that all harmonic morphisms from \({\mathbb{R}}^ 3\) and \(S^ 3\) are now known. The main tool is a Weierstrass type representation theorem for harmonic morphisms from \({\mathbb{R}}^ 3\) and \(S^ 3\) to surfaces arising from the conformal foliation by fibres which is related to the work of \textit{C. G. J. Jacobi} [J. Reine Angew. Math. 36, 113-134 (1847)], whose problem we thereby solve completely.
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 4007282
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4007282