Minimal Surfaces

Minimal surfaces are an object within differential geometry. Differential geometry is a field of mathematics which studies geometric objects that can be described by smooth, i.e. infinitely differentiable maps. These geometric objects are called manifolds. Within this context, minimal surfaces are manifolds which minimize area among all surfaces with the same boundary. This leads to minimal surfaces playing a role in the study of partial differential equations, as this minimal area corresponds to ideas of stability. In fact, physical minimal surfaces can be generated using soap films and wire frames. In this thesis I will open the topic of minimal surfaces to an advanced undergraduate level reader, with the objective of proving Bernstein's Theorem providing direction to the topics discussed.