  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.
