Harmonic morphisms on $(\mathbb{S}^4

• Main Authors: Makki, Ali, Soret, Marc, Ville, Marina
• Other Authors: Laboratoire de Mathématiques et Physique Théorique (LMPT), Université de Tours-Centre National de la Recherche Scientifique (CNRS)
• Format: Report
• Language: English
• Published: HAL CCSD 2018
• Subjects: [MATH]Mathematics [math]; [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]; North Pole; South Pole; South pole
• Online Access: https://hal.archives-ouvertes.fr/hal-01912983; https://hal.archives-ouvertes.fr/hal-01912983/document; https://hal.archives-ouvertes.fr/hal-01912983/file/article%202.pdf

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 $k$).