Harmonic morphisms on $(\mathbb{S}^4
Résumé
In this paper we study examples of harmonic morphisms due to Burel from $(\mathbb{S}^4,g_{k,l})$ into $\mathbb{S}^2$ where $(g_{k,l})$ is a family of conformal metrics on $\mathbb{S}^4$. To do this construction we define two maps, $F$ from $(\mathbb{S}^4,g_{k,l})$ to $(\mathbb{S}^3,\bar{g_{k,l}})$ and $\phi_{k,l}$ from $(\mathbb{S}^3,\bar{g_{k,l}})$ to $(\mathbb{S}^2,can)$; the two maps are both horizontally conformal and harmonic. Let $\Phi_{k,l}=\phi_{k,l} \circ F$. It follows from Baird-Eells that the regular fibres of $\Phi_{k,l}$ for every $k,l$ are minimal. If $|k|=|l|=1$, the set of critical points is given by the preimage of the north pole : it consists in two 2-spheres meeting transversally at 2 points. If $k,l\neq1$ the set of critical points are the preimages of the north pole (the same two spheres as for $k=l=1$ but with multiplicity $l$) together with the preimage of the south pole (a torus with multiplicity $k$).
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