Examples of Area Minimizing Surfaces in 3-manifolds

TL;DR

This paper provides specific examples illustrating complex behaviors of area minimizing surfaces in 3-manifolds, challenging some existing theorems and highlighting new phenomena.

Contribution

It constructs explicit examples demonstrating limitations of known results on embeddedness and boundary behavior of area minimizing surfaces in 3-manifolds.

Findings

An example of a non-embedded area minimizing disk in a mean convex domain.

Counterexample to White's boundary decomposition theorem in nontrivial homology.

Existence of intersecting, disjoint boundary curves with nontrivially intersecting minimal surfaces.

Abstract

In this paper, we give some examples of area minimizing surfaces to clarify some well-known features of these surfaces in more general settings. The first example is about Meeks-Yau's result on embeddedness of solution to the Plateau problem. We construct an example of a simple closed curve in R^3 which lies in the boundary of a mean convex domain in R^3, but the area minimizing disk in R^3 bounding this curve is not embedded. Our second example shows that Brian 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.