  The limit of energies of a sequence of harmonic maps as their annular domains
approach the boundary of moduli space depends upon the boundary point
approached. The infinite energy case is associated with limits of images
containing ruled surfaces. The finite energy case yields a limit of images,
under a suitable topology, with a union of discs and straight line segments.
Generalization to higher numbers of boundary components shows that minimal
surfaces union straight line segments can still be achieved, and that the
configuration of straight line segments depends on the direction of approach of
domain conformal classes to the boundary point of domain moduli space. Bubbling
can occur in variable ways according to the metric representatives of the
conformal classes of domains. A method is given whereby bubbling can be
eliminated yielding a point-wise limit of maps with image limit containing a
surface union straight line segments.
