Min-max constructions of 2d-minimal surfaces

Abstract

In this thesis we will present a proof of the existence of closed embedded minimal surfaces in a closed 3-dimensional manifold constructed via min-max arguments and we will prove genus bounds for the produced surfaces. A stronger estimate was announced by Pitts and Rubinstein but to our knowledge its proof has never been published. Our proof follows ideas of Simon and uses an extension of a famous result of Meeks, Simon and Yau on the convergence of minimizing sequences of isotopic surfaces.

Eine Minimalfläche ist eine Fläche im Raum, die lokal minimalen Flächeninhalt hat. In dieser Doktorarbeit studiere ich ähnlichen Probleme nicht in Raum, sondern zum Beispiel in einer 3-dimensionale Sphere. Es wird gezeigt, dass 2-dimensionale minimale Flächen in einer 3-Sphere existieren. Danach analysiere ich deren Geometrie.