Topology Seminar (Main Talk): Local min-max surfaces and existence of minimal Heegaard splittings

Seminar | October 25 | 4-5 p.m. | 3 Evans Hall

 Antoine Song, Princeton University

 Department of Mathematics

Let M be a closed oriented 3-manifold not diffeomorphic to the 3-sphere, and suppose that there is a strongly irreducible Heegaard splitting H. Previously, Rubinstein announced that either there is a minimal surface of index at most one isotopic to H or there is a non-orientable minimal surface such that the double cover with a vertical handle attached is isotopic to H. He sketched a natural outline of a proof using min-max, however some steps are non-trivially incomplete and we will explain how to justify them. The key point is a version of min-max theory producing interior minimal surfaces when the ambient manifold has minimal boundary. Some corollaries of the theorem include the existence in any \(RP^3\) of either a minimal torus or a minimal projective plane with stable universal cover. Several consequences for metric with positive scalar curvature are also derived.