Aplicaciones geométricas del Principio del Máximo para EDPs elípticas

Abstract

The aim of this work is to give a detailed proof of two results from the theory of constant mean curvature surfaces: Alexandrov theorem [Ale58] and the halfspace theorem [HofMee90]. In order to do so, we need to develop some mathematical tools from the fields of differential geometry of surfaces and the theory of absolutely elliptic partial differential equations. In the following chapter, we will start by introducing basic concepts associated with the theory of surfaces, such as tangent plane, first and second fundamental forms and the definitions of gaussian curvature and mean curvature. In the second chapter, we will start by making use of the maximum principle for linear, elliptic and homogeneus PDEs in order to develop analogous results for absolutely elliptic PDEs. By the end of the chapter, we will be able to formulate a geometric maximum principle. In the third chapter, we will prove Alexandrov’s theorem, which states that the only sufficiently regular, simple and compact surface with constant mean curvature is the sphere. Finally, we will give a proof of the halfspace theorem, as well as some generalizations. This result states that the only sufficiently regular, proper and minimal surface that can be contained in a halfspace is a plane. Both these theorems rely strongly on two elements: the geometric maximum principle and certain geometrical constructions. One of our main goals in this work will be to make these constructions as visually clear and intuitive as possible. Additionally, Bernstein theorem [Ber04] will be used to lower the restrictions on the regularity of the surfaces.