The Planar Soap Bubble Problem with Six Equal Pressure Regions

The planar soap bubble problem is the mathematically analogous problem in two dimensions to search for the least-perimeter way to enclose and separate regions $R_1, \ldots ,R_m$ of given areas $A_1, \ldots ,A_m$ on the plane. In this work, we study the possible configurations for perimeter minimizing enclosures for more than three regions. In particular, we focus on the case of equal pressure regions. For four and five regions, in 2007, we proved that a perimeter minimizing enclosure with equal pressure regions and without empty chambers must have connected regions. In this paper, we show that for six equal pressure regions, the solutions must have connected regions.