Examples of area-minimizing surfaces in 3-manifolds

Abstract

In this paper, we give some examples of area-minimizing surfaces to clarify some wellknown features of these surfaces in more general settings. The first example is about Meeks–Yau’s result on the embeddedness of the solution to the Plateau problem. We construct an example of a simple closed curve in R3 which lies in the boundary of a mean convex domain in R3, but the area-minimizing disk in R3 bounding this curve is not embedded. Our second example shows that White’s boundary decomposition theorem does not extend when the ambient space has nontrivial homology. Our last examples show that there are properly embedded absolutely area-minimizing surfaces in a mean convex 3-manifold M such that, while their boundaries are disjoint, they intersect each other nontrivially, unlike the area-minimizing disks case.