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

Un morphisme harmonique f:M→N 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:M→N 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.

@article{AIF_1978__28_2_107_0,
     author = {Fuglede, Bent},
     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},
     zbl = {0339.53026},
     mrnumber = {80h:58023},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.691/}
}
TY  - JOUR
AU  - Fuglede, Bent
TI  - Harmonic morphisms between riemannian manifolds
JO  - Annales de l'Institut Fourier
PY  - 1978
SP  - 107
EP  - 144
VL  - 28
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.691/
DO  - 10.5802/aif.691
LA  - en
ID  - AIF_1978__28_2_107_0
ER  - 
%0 Journal Article
%A Fuglede, Bent
%T Harmonic morphisms between riemannian manifolds
%J Annales de l'Institut Fourier
%D 1978
%P 107-144
%V 28
%N 2
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.691/
%R 10.5802/aif.691
%G en
%F AIF_1978__28_2_107_0
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. https://aif.centre-mersenne.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

Cité par Sources :