MSc thesis defense by Rasmus Johansen Jouttijärvi

Abstract: In this thesis, we study the Morse index of complete orientable minimal surfaces, immersed in Euclidean three-space. Based on constructions by Johan Tysk and Otis Chodosh & Davi Maximo, we show how it is possible to bound the index from above and below be a constant multiple of the total curvature of the surface. This will allow us to give a thorough description of existing rigidity results for low-index minimal surfaces.
In the course of constructing the bounds, we will prove some classical results regarding the behaviour of the stability operator and classification of stable minimal surfaces and subdomains, including uniqueness of the plane as a stable minimal surface. We will also describe how the properties of minimal surfaces affect the behaviour of the ends of the surface and give a proof of the Jorge-Meeks Formula.
In the last section of the thesis, we discuss the index of complete immersed self-shrinkers, seen as minimal surfaces in Euclidean space with respect to a weighted Gaussian metric.
Max. 30 people