Unstable minimal surfaces and Heegaard splittings

Abstract

In recent years minimal surfaces have played an important role in the study of three-dimensional manifolds. Least area surfaces, which minimize area in their isotopy [15], homotopy [23] or homology [10] classes, have been used to make progress towards the understanding of three-manifolds which contain such surfaces. Such surfaces can be shown to exist and to have strong regularity properties under the assumption that the homotopy, isotopy or homology class in which they minimize area is non-trivial. However these techniques tend not to be applicable to the study of surfaces which are not essential in one of the above classes, or to the study of manifolds which contain only such nonessential surfaces. Recently developed techniques open the way for the use of a new class of surfaces in three-dimensional manifolds, namely the class of unstable embedded minimal surfaces. Pitts and Rubinstein have established the existence of unstable minimal submanifolds which arise from a Heegaard splitting in such a way that their genus is less than or equal to that of the Heegaard splitting [19]. In this paper we extend their result to obtain greater topological control of the resulting minimal surface, and apply the resulting theorem to purely topological problems relating to Heegaard splittings of a three-manifold. A compact manifold is a handlebody if it is homeomorphic to a regular neighborhood of a graph in R 3. A Heegaard splitting of a three-manifold M is a triple (F, H1, H2) where F is an embedded connected orientable surface in M, H1 and H 2 are handlebodies embedded in M, M = H 1 w H 2 and H~ c~H2 = F = O H I = O H 2. F is called a Heegaard surface. We say that two Heegaard splittings of M, (F, H1, H2) and (F', H'I, H~), are equivalent if there is a homeomorphism h: M ~ M such that f (F ) = F', f ( H O = H'I and f (H2) = H~. By considering neighborhoods of triangulations, one can show that any closed orientable three-manifold admits a Heegaard splitting. Of particular interest are the splittings with Heegaard surface of smallest possible genus. The genus of such a splitting is called the Heegaard genus of the three-manifold.