Minimal surfaces and three manifold topology

In the ’80s, Pitts and Rubinstein sketched some arguments to show that certain types of Heegaard surfaces in a three-manifold could be isotoped to be a minimal surface. I’ll describe some recent work filling in the missing pieces from their sketch. I’ll then explain some further applications of such minimal surfaces to three-manifold topology, in particular the problem of classifying the irreducible Heegaard splittings of a given three-manifold.