Area minimizing surfaces in mean convex 3-manifolds

Abstract

In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in partial derivative M. Our main result is that for any g >= 0, the space of simple closed curves in partial derivative M where all the absolutely area minimizing surfaces they bound in M has genus >= g is open and dense in the space A of nullhomologous simple closed curves in partial derivative M. For showing this we prove a bridge principle for absolutely area minimizing surfaces. Moreover, we show that for any g >= 0, there exists a curve gamma(g) in A such that the minimum genus of the absolutely area minimizing surfaces gamma(g) bounds is exactly g. As an application of these results, we further prove that the simple closed curves in partial derivative M bounding more than one minimal surface in M is an open and dense subset of A. We also show that there are disjoint simple closed curves in partial derivative M bounding minimal surfaces in M which are not disjoint. This allows us to answer a question of Meeks, by showing that for any strictly mean convex 3-manifold M, there exists a simple closed curve Gamma in partial derivative M which bounds a stable minimal surface which is not embedded. We also gave some applications of these results to the simple closed curves in R-3.