EMBEDDED MINIMAL SURFACES IN THREE AND FOUR DIMENSIONAL MANIFOLDS

Recent results of Schoen-Yau and Sacks-Uhlenbeck have shown the existence of immersed minimal surfaces of positive genus in 3-manifolds, but said nothing about the self-intersections of these surfaces. Conjectures about the existence of area minimizing embeddings homotopic to these surfaces were made by Meeks and by Uhlenbeck. These conjectures are shown to be false as originally stated and a modified conjecture is presented. It is demonstrated that in many cases, the area minimizing surface homotopic to an incompressible embedding is itself imbedded. In addition, the idea of using minimal surfaces to study foliations is examined. A new proof of Novikov's theorem about the existence of compact leaves in foliations of certain 3-manifolds is obtained, with the hypothesis of Novikov's theorem weakened somewhat. The nature of the intersection of minimal surfaces and totally geodesic submanifolds of 4-manifolds is used to prove an analogue of the sphere theorem for certain 4-manifolds. Finally, minimal disks bounded by curves on the unit 3-sphere in R('4) are examined. This topic is shown to be closely related to the slice-ribbon problem of knot theory.