Minimal surfaces of finite genus with two limit ends

Laurent Hauswirth (1) , Frank Pacard (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

Riemann surfaces constitute a one parameter family of embedded minimal surfaces which are periodic and have infinitely many horizontal planar ends. The surfaces in this family are foliated by circles (or straight lines). In this paper, we prove the existence of a one parameter family of embeded minimal surfaces which have infinitely many horizontal planar ends and have genus k, for k = 1, ... , 37. Riemann surfaces, as their flux is nearly vertical, can be understood as a sequence of parallel planes connected by slightly bent catenoidal neks. The surfaces we construct are obtained by replacing one of these catenoidal necks by a member of the family of minimal surfaces discovered by C. Costa, D. Hoffman and W. Meek.

Domaines: Géométrie différentielle [math.DG]

Laurent Hauswirth, Frank Pacard. Minimal surfaces of finite genus with two limit ends. Inventiones Mathematicae, 2007, 169 (3), pp.569-620. ⟨hal-00796849⟩