Bernstein theorems for harmonic morphisms from \({\mathbb R}\) 3 and S 3
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.
scientific article; zbMATH DE number 4007282