Harmonic morphisms and circle actions on 3- and 4-manifolds

Baird, Paul

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 $\phi :M\to N$ 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 $\phi :M\to N$ défini sur une variété fermée à $n+1$ dimensions et à valeurs dans une variété à $n$ dimensions ($n\ge 2$, 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 $\phi :M\to N$ 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 $\phi :M\to N$ from a closed $(n+1)$-dimensional manifold to an $n$-dimensional manifold (n$\ge 2$, with $M$, $N$ analytic in the case $n=2)$ determines a locally smooth circle action on $M$.

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