Applications of Geometric Measure Theory to Variational Problems with Surfaces

Abstract : Geometric measure theory offers compactness theorems for candidate objects in variational problems, giving existence results in variational problems. Additionally it offers regularity information about limits of regular objects.

I will introduce the objects of geometric measure theory; rectifiable sets, currents and varifolds. Key examples of each and their associated topologies will be given.

The primary application of geometric measure theory was minimal surface theory, but I will also explain my own research into geometric measure theory techniques and my motivating applications.