Topologically Minimal Surfaces in 3-manifolds

Pitzer College, Claremont CA

Investigators

Abstract

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Motivated by analogies with geometrically minimal surfaces, the PI recently introduced the notion of a topologically minimal surface in a 3-manifold. Such a surface has an associated disk complex that is either empty, or is non-contractible. Topologically minimal surfaces generalize other well known classes, such as incompressible surfaces and strongly irreducible Heegaard splittings. They also generalize critical surfaces, a class that was instrumental in the PI's solutions to a conjecture of Gordon's and to the Stabilization Conjecture. Future work concerns the application of topologically minimal surfaces to understand the persistence of the set of Heegaard surfaces after Dehn filling. A second project is to apply a relative version of the theory to study knots and links with multiple bridge presentations. Finally, the study of topologically minimal surfaces has itself generated many natural questions that reveal information about the topology of 3-manifolds. Just as the surface of the Earth seems like a plane to those confined to local observations, a 3-manifold is an object that is locally indistinguishable from 3-dimensional Euclidean space, such as our universe. A classical way to study 3-manifolds is to utilize surfaces they contain that are non-trivial in some suitable sense. Topological minimality is a very general notion of non-triviality for such surfaces that was introduced by the PI to solve several problems in topology. These surfaces are the analogue of geometrically minimal surfaces, such as soap films. The PI will both develop the theory of topologically minimal surfaces further and use it to study the ambiguity inherent in various common constructions of 3-manifolds.