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

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.

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.

@article{AIF_1978__28_2_107_0,
     author = {Fuglede, Bent},
     title = {Harmonic morphisms between riemannian manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {28},
     number = {2},
     year = {1978},
     pages = {107-144},
     doi = {10.5802/aif.691},
     zbl = {0339.53026},
     mrnumber = {80h:58023},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1978__28_2_107_0}
}
Harmonic morphisms between riemannian manifolds. Annales de l'Institut Fourier, Volume 28 (1978) no. 2, pp. 107-144. doi : 10.5802/aif.691. http://www.numdam.org/item/AIF_1978__28_2_107_0/

[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 19,1056c | Zbl 0084.30402

[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 36 #4012 | Zbl 0164.14003

[3] N. Cioranesco, Sur les fonctions harmoniques conjuguées, Bull. Sc. Math., 56 (1932), 55-64. | JFM 58.0502.04 | Zbl 0003.34701

[4] C. Constantinescu and A. Cornea, Compactifications of harmonic spaces. Nagoya Math. J., 25 (1965), 1-57. | MR 30 #4960 | Zbl 0138.36701

[5] C. Constantinescu and A. Cornea, Potential Theory on Harmonic Spaces, Berlin-Heidelberg-New York : Springer 1972. | MR 54 #7817 | Zbl 0248.31011

[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 0074.08002

[7] J. Eells, jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109-160. | MR 29 #1603 | Zbl 0122.40102

[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 52 #3583 | Zbl 0307.31003

[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 25 #3186 | Zbl 0101.08103

[10] O. D. Kellogg, Foundations of potential theory, Berlin, Springer, 1929 (re-issued 1967). | Zbl 0152.31301

[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 0099.37901

[13] A. Sard, Images of critical sets, Ann. Math., 68 (1958), 247-259. | MR 20 #6499 | Zbl 0084.05204

[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 40 #395 | Zbl 0182.15002