I will explain some recent work using minimal surfaces to address problems in 3-manifold topology. Given a Heegaard splitting, one can sweep out a three-manifold by surfaces isotopic to the splitting, and run the min-max procedure of Almgren-Pitts and Simon-Smith to construct a smooth embedded minimal surface. If the original splitting were strongly irreducible (as introduced by Casson-Gordon), H. Rubinstein sketched an argument in the 80s showing that the limiting minimal surface should be isotopic to the original splitting. I will explain some results in this direction and how jointly with T. Colding and D. Gabai we can use such min-max minimal surfaces to complete the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.
- Topology Seminar