Minimal Surfaces- A proof of Bernstein´s theorem

Abstract

This thesis is meant as an introduction to the subject of minimal surfaces, i.e. surfaces having mean curvature zero everywhere. In a physical sense, minimal surfaces can be thought of as soap lms spanning a given wire frame. The main object will be to prove Bernstein's theorem, which states that a minimal surface in R3 which is de ned in the whole parameter plane is linear, meaning it is a plane. We will give two proofs of this theorem, both involving methods from complex analysis, and relying on a proposition stating that we can always reparametrize the surface into so called isothermal parameters.