Area minimizing surfaces in mean convex 3-manifolds

TL;DR

This paper investigates the properties of area minimizing surfaces in strictly mean convex 3-manifolds, establishing generic genus bounds, a bridge principle, and the existence of non-embedded stable minimal surfaces.

Contribution

It introduces a bridge principle for absolutely area minimizing surfaces and demonstrates that the genus of such surfaces can be arbitrarily prescribed, extending understanding of minimal surface topology.

Findings

The set of boundary curves bounding surfaces of genus >= g is open and dense.

For any g, there exists a boundary curve with minimal surface genus exactly g.

There are boundary curves bounding multiple minimal surfaces, some not embedded.

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 the boundary of M. Our main result is that for any g>=0, the space of simple closed curves in the boundary of 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 the boundary of 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 in A such that the minimum genus of the absolutely area minimizing surfaces it bounds is exactly g. As an application of these results, we further prove that the simple closed curves in the boundary of 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 the boundary of 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 the boundary of M which bounds a stable minimal surface which is not embedded.