Planar Soap Bubbles
  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. 
Rick Vaughn