Harmonic morphisms between riemannian manifolds
Annales de l'Institut Fourier, Tome 28 (1978) no. 2, pp. 107-144.

Un morphisme harmonique f:MN entre variétés riemanniennes M et N est par définition une application continue qui “remonte” les fonctions harmoniques. On suppose dimM dimN, puisque autrement tout morphisme harmonique est constant. On montre qu’un morphisme harmonique n’est autre qu’une application harmonique au sens de Eells et Sampson qui, en outre est semi-conforme, c’est-à-dire est une submersion conforme hors des points ou df est nul. On montre que tout morphisme harmonique non constant est une application ouverte.

A harmonic morphism f:MN between Riemannian manifolds M and N is by definition a continuous mappings which pulls back harmonic functions. It is assumed that dimM dimN, since otherwise every harmonic morphism is constant. It is shown that a harmonic morphism is the same as a harmonic mapping in the sense of Eells and Sampson with the further property of being semiconformal, that is, a conformal submersion of the points where df vanishes. Every non-constant harmonic morphism is shown to be an open mapping.

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     title = {Harmonic morphisms between riemannian manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {107--144},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {28},
     number = {2},
     year = {1978},
     doi = {10.5802/aif.691},
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     zbl = {0339.53026},
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     url = {http://www.numdam.org/articles/10.5802/aif.691/}
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Fuglede, Bent. Harmonic morphisms between riemannian manifolds. Annales de l'Institut Fourier, Tome 28 (1978) no. 2, pp. 107-144. doi : 10.5802/aif.691. http://www.numdam.org/articles/10.5802/aif.691/

[1] N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., 36 (1957), 235-249. | MR | Zbl

[2] J.-M. Bony, Détermination des axiomatiques de théorie du potentiel dont les fonctions harmoniques sont différentiables, Ann. Inst. Fourier, 17, 1 (1967), 353-382. | Numdam | MR | Zbl

[3] N. Cioranesco, Sur les fonctions harmoniques conjuguées, Bull. Sc. Math., 56 (1932), 55-64. | JFM | Zbl

[4] C. Constantinescu and A. Cornea, Compactifications of harmonic spaces. Nagoya Math. J., 25 (1965), 1-57. | MR | Zbl

[5] C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Berlin-Heidelberg-New York : Springer 1972. | MR | Zbl

[6] H. O. Cordes, Uber die Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben, Nachr. Akad. Wiss. Göttingen, Math. Phys. Kl. IIa, Nr. 11 (1956), 239-258. | Zbl

[7] J. Eells, jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. | MR | Zbl

[8] R. E. Greene and H. Wu, Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier, Grenoble, 25, 1 (1975), 215-235. | Numdam | MR | Zbl

[9] R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier, Grenoble, 12 (1962), 415-571. | Numdam | MR | Zbl

[10] O. D. Kellogg, Foundations of potential theory, Berlin, Springer, 1929 (re-issued 1967). | Zbl

[11] J. Liouville, Note VI, p. 609-616 in G. Monge : Applications de l'Analyse à la Géométrie, 5e éd., Paris, 1850.

[12] Yu G. Rešetnjak, O konformnyk otobrazenijah prostanstva. (Russian.) (On conformal mappings in space), Dokl. Akad. Nauk SSSR, 130 (1960), 1196-1198. (Sovjet Math., 1 (1960), 153-155.) | Zbl

[13] A. Sard, Images of critical sets, Ann. Math., 68 (1958), 247-259. | MR | Zbl

[14] D. Sibony, Allure à la frontière minimale d'une classe de transformations. Théorème de Doob généralisé, Ann. Inst. Fourier, Grenoble, 18, 2 (1968), 91-120. | Numdam | MR | Zbl

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