Harmonic morphisms and circle actions on 3- and 4-manifolds
Annales de l'Institut Fourier, Tome 40 (1990) no. 1, pp. 177-212.

On considère les morphismes harmoniques comme généralisation naturelle des fonctions analaytiques qu’on rencontre dans la théorie des surfaces de Riemann. On montre que chaque variété fermée et analytique à 3 dimensions qui supporte un morphisme harmonique à valeurs dans une surface de Riemann est un espace fibré de Seifert. On étudie les morphismes harmoniques φ:MN définies sur une variété fermée à 4 dimensions et à valeurs dans une variété à 3 dimensions. Ceux-ci déterminent une action du cercle sur M qui est localement différentiable, peut-être avec des points fixes. Par conséquent la topologie de M est limitée. Dans chaque cas, un morphisme harmonique φ:MN défini sur une variété fermée à n+1 dimensions et à valeurs dans une variété à n dimensions (n2, avec M, N analytiques dans le cas où n=2) détermine une action du cercle sur M qui est localement différentiable.

Harmonic morphisms are considered as a natural generalization of the analytic functions of Riemann surface theory. It is shown that any closed analytic 3-manifold supporting a non-constant harmonic morphism into a Riemann surface must be a Seifert fibre space. Harmonic morphisms φ:MN from a closed 4-manifold to a 3-manifold are studied. These determine a locally smooth circle action on M with possible fixed points. This restricts the topology of M. In all cases, a harmonic morphism φ:MN from a closed (n+1)-dimensional manifold to an n-dimensional manifold (n2, with M, N analytic in the case n=2) determines a locally smooth circle action on M.

@article{AIF_1990__40_1_177_0,
     author = {Baird, Paul},
     title = {Harmonic morphisms and circle actions on 3- and 4-manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {40},
     number = {1},
     year = {1990},
     pages = {177-212},
     doi = {10.5802/aif.1210},
     mrnumber = {91e:57025},
     zbl = {0676.58023},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_1990__40_1_177_0/}
}
Baird, Paul. Harmonic morphisms and circle actions on 3- and 4-manifolds. Annales de l'Institut Fourier, Tome 40 (1990) no. 1, pp. 177-212. doi : 10.5802/aif.1210. https://aif.centre-mersenne.org/item/AIF_1990__40_1_177_0/

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