  The generalized soap bubble problem seeks the least perimeter way to enclose
and separate n given volumes in R^m. We study the possible configurations for
perimeter minimizing bubble complexes enclosing more than two regions. We prove
that perimeter minimizing planar bubble complexes with equal pressure regions
and without empty chambers must have connected regions. As a consequence, we
show that the least perimeter planar graph that encloses and separates three
equal areas in R^2 using convex cells and without empty chambers is a "standard
triple bubble" with connected regions.
