  We consider a free boundary problem on cones depending on a parameter c and
study when the free boundary is allowed to pass through the vertex of the cone.
We show that when the cone is three-dimensional and c is large enough, the free
boundary avoids the vertex. We also show that when c is small enough but still
positive, the free boundary is allowed to pass through the vertex. This
establishes 3 as the critical dimension for which the free boundary may pass
through the vertex of a right circular cone. In view of the well-known
connection between area-minimizing surfaces and the free boundary problem under
consideration, our result is analogous to a result of Morgan that classifies
when an area-minimizing surface on a cone passes through the vertex.
