Baird, Paul; Wood, John C. Bernstein theorems for harmonic morphisms from \({\mathbb R}\) 3 and S 3. (English) Zbl 0621.58011 Math. Ann. 280, No. 4, 579-603 (1988). 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 B. Fuglede [Ann. Inst. Fourier 28, No.2, 107-144 (1978; Zbl 0339.53026)] and T. Ishihara [J. Math. Kyoto Univ. 19, 215-229 (1979; Zbl 0421.31006)], and as Brownian path preserving mappings by A. Bernard, E. A. Campbell and 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 C. G. J. Jacobi [J. Reine Angew. Math. 36, 113-134 (1847)], whose problem we thereby solve completely. Cited in 6 ReviewsCited in 28 Documents MSC: 58E20 Harmonic maps, etc. 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 60J45 Probabilistic potential theory 57R30 Foliations in differential topology; geometric theory Citations:Zbl 0369.53044; Zbl 0339.53026; Zbl 0386.30029; Zbl 0394.30040; Zbl 0421.31006 × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] Baird, P.: Harmonic maps with symmetry, harmonic morphisms and deformations of metrics. Research Notes in Mathematics, Vol. 87, Boston London Melbourne: Pitman 1983 · Zbl 0515.58010 [2] Baird, P.: Harmonic morphisms onto Riemann surfaces and generalized analytic functions. Ann. Inst. Fourier, Grenoble37, 135-173 (1987) · Zbl 0608.58015 [3] Baird, P., Eells, J.: A conservation law for harmonic maps. In: Looijenga, E., Siersma, D., Takens, F. (eds.). Geometry Symposium Utrecht 1980, Proceedings. Lecture Notes Mathematics, Vol. 894, pp. 1-25. Berlin Heidelberg New York: Springer 1981 [4] Bernard, A., Campbell, E.A., Davie, A.M.: Brownian motion and generalized analytic and inner functions. Ann. Inst. Fourier, Grenoble29, 207-228 (1979) · Zbl 0386.30029 [5] Chern, S.S.: Minimal surfaces in an Euclidean space ofN dimensions. In: Cairns, S.S. (ed.) Differential and combinatorial topology, pp. 187-198 Princeton: Princeton University Press 1965 [6] Conway, J.B.: Functions of one complex variable. Berlin Heidelberg New York: Springer 1983 · Zbl 0508.20023 [7] Eells, J., Polking, J.C.: Removable singularities of harmonic maps. Indiana Univ. Math. J.33, 859-871 (1984) · Zbl 0559.58011 · doi:10.1512/iumj.1984.33.33046 [8] Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math.86, 109-160 (1964) · Zbl 0122.40102 · doi:10.2307/2373037 [9] Fuglede, B.: Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier Grenoble28, 107-144 (1978) · Zbl 0339.53026 [10] Greene, R.E., Wu, H.: Embeddings of open Riemannian manifolds by harmonic functions. Ann. Inst. Fourier, Grenoble12, 415-571 (1962) [11] Helms, L.L.: Introduction to potential theory. New York London Sydney Toronto: Wiley 1969 · Zbl 0188.17203 [12] Hoffman, D.A., Osserman, R.: The geometry of the generalized Gauss map. Memoirs of the American Math. Soc., Vol. 28, No. 236. Providence: Am. Math. Soc. 1980 · Zbl 0469.53004 [13] Ishihara, T.: A mapping of Riemannian manifolds which preserves harmonic functions. J. Math. Kyoto Univ.19, 215-229 (1979) · Zbl 0421.31006 [14] Jacobi, C.G.J.: Über eine particuläre Lösung der partiellen Differentialgleichung \(\frac{{\partial ^2 V}}{{\partial x^2 }} + \frac{{\partial ^2 V}}{{\partial y^2 }} + \frac{{\partial ^2 V}}{{\partial z^2 }} = 0\) . Crelle Reine Angew. Math.36, 113-134 (1847) · ERAM 036.1002cj [15] Kamber, F.W., Tondeur, Ph.: The Bernstein problem for foliations. In: Global differential geometry and global analysis. Proceedings Berlin 1984. Lecture Notes Mathematics, Vol. 1156, pp. 216-218 Berlin Heidelberg New York: Springer 1985 [16] Osserman, R.: A survey of minimal surfaces. New York: Van Nostrand Reinhold 1969. Secd. edit.: New York: Dover 1986 · Zbl 0209.52901 [17] Reinhart, B.L.: Differential geometry of foliations. Ergebnisse Math., Vol. 99. Berlin Heidelberg New York: Springer 1983 · Zbl 0506.53018 [18] Ruh, E.A., Vilms, J.: The tension field of the Gauss map. Trans. Am. Math. Soc.149, 569-573 (1970) · Zbl 0199.56102 · doi:10.1090/S0002-9947-1970-0259768-5 [19] Siegel, C.L.: Topics in complex function theory. New York London Sydney Toronto: Wiley 1969 · Zbl 0184.11201 [20] Steenrod, N.: The topology of fibre bundles. Princeton: Princeton University Press 1951 · Zbl 0054.07103 [21] Vaisman, I.: Conformal foliations. Kodai Math. J.2, 26-37 (1979) · Zbl 0402.57014 · doi:10.2996/kmj/1138035963 [22] Wolf, J.: Spaces of constant curvature. New York St. Louis San Francisco Toronto London Sydney: McGraw-Hill 1967 · Zbl 0162.53304 [23] Wood, J.C.: The Gauss map of a harmonic morphism. In: Cordero, L.A. (ed.) Differential geometry. Research Notes Mathematics, Vol. 131, pp. 149-155. Boston London Melbourne: Pitman 1985 · Zbl 0648.58010 [24] Wood, J.C.: Harmonic morphisms, foliations and Gauss maps. In: Siu, Y.T. (ed.) Complex differential geometry and nonlinear differential equations. Contemporary Mathematics, Vol. 49, pp. 145-183. Providence: Am. Math. 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