Applications of Topologically Minimal Surfaces

Pitzer College, Claremont CA

Investigators

Abstract

A classical way to study 3-manifolds is to begin with an embedded surface in it that satisfies some useful property. For example, Haken solved many important problems by building hierarchies of incompressible surfaces that decompose large classes of 3-manifolds into balls. Casson and Gordon introduced the idea of strongly irreducible surfaces to study Heegaard splittings of 3-manifolds. This proposal concerns the further development and application of the PI's theory of topologically minimal surfaces, which generalize both incompressible and strongly irreducible surfaces. Such surfaces have been used by the PI to solve several long standing conjectures, "Gordon's Conjecture" and the "Stabilization Conjecture." The specific goals of this proposal are to: (1) Further explore the relationships between topologically minimal, PL minimal, and geometrically minimal surfaces; (2) Use topologically minimal surfaces to explore what happens to the set of Heegaard surfaces under Dehn filling; and (3) Understand the behavior of topologically minimal surfaces under finite coverings. 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 more classical objects, the so-called "geometrically minimal" surfaces. Familiar examples of such surfaces include soap films. Conjecturally, topologically minimal surfaces have all of the same properties as geometrically minimal surfaces. The goal of the present work is to elucidate those properties that are shared by these two kinds of surfaces.