HARMONIC MORPHISMS AND DEFORMATION OF MINIMAL SURFACES IN MANIFOLDS OF DIMENSION 4

• Main Author: Makki, Ali
• Other Authors: Laboratoire de Mathématiques et Physique Théorique (LMPT), Université de Tours-Centre National de la Recherche Scientifique (CNRS), LMPT - Laboratoire de Mathématiques et Physique Théorique, FRDP - Fédération de recherche Denis Poisson, Marc Soret, Marina Ville
• Format: Doctoral or Postdoctoral Thesis
• Language: English
• Published: HAL CCSD 2014
• Subjects: Harmonic Morphism; Minimal surfaces; Almost complex structure; Morphismes harmoniques; Surfaces minimales; Structure presque complexe; [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG]; North Pole; South Pole; South pole
• Online Access: https://tel.archives-ouvertes.fr/tel-01078598; https://tel.archives-ouvertes.fr/tel-01078598/document; https://tel.archives-ouvertes.fr/tel-01078598/file/These.pdf

In this thesis, we investigate the structure of harmonic morphism F from Riemannian 4-manifold M4 to a 2-surface N2 near critical point m0. If m0is an isolated critical point or if M4 is compact without boundary, we show that F is pseudo-holomorphic w.r.t. an almost Hermitian structure defined in a neighbourhood of m0. If M4 is compact without boundary, the singular fibres of F are branched minimal surfaces. We study examples of harmonic morphisms due to Burel Φk,l from (S4,gk,l) into S2 where (gk,l) is a family of metrics which are conformal to the canonical metric. To do this construction we define the two maps, F from (S4,gk,l) to S3 and φk,l from S3 to S2; these two maps are both horizontally conformal and harmonic. it follows from Baird-Eells that the regular fibres of Φk,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 6= 1 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. Finally, we investigate a construction by Baird-ou of harmonic morphisms from open sets of (S2 ×S2, can) to a 2-surface S2. We check that they are holomorphic with respect to one of the four canonical Hermitian complex structures. Dans cette thèse, nous étudions la structure d’un morphisme harmonique F d’une varriété riemannienne M4 dans une surface N2 au voisinage d’un point critique m0. Si m0 est un point critique isolé ou si M4 est compact sans bord, nous montrons que F est pseudo-holomorphe par rapport à une structure presque hermitienne definie dans un voisinage de m0. Si M4 est compact sans bord, les fibres singuliers de F sont des surfaces minimales avec points de branchement.Ensuite, nous étudions des exemples de morphismes harmoniques due a Burel de (S4, gk,l) dans S2 où (gk,l) est une famille de métriques conforme à la métrique canonique. ...