The Double Bubble Problem on the Flat Two Torus
by Joseph Cornelli, Paul Holt, George Lee, Nicholas Leger, Eric Schoenfeld, and S.
Published in The Transactions of the American Mathematical Society vol. 356 no. 9 (2005).

On a torus the idea is to contain and separate two areas \( A_1\) and \( A_2\). What are the best shapes to do this with? In \( \mathbb{ R}^{n}\) this can be accomplished using a two chambered soap bubble. But there is something that is different in the torus case than in the \( \mathbb{ R}^{n}\) case. The torus is bounded and \( \mathbb{ R}^{n}\) is not. So on the torus when two areas are contained and separated there is a finite third piece that is the complement. This consideration helps present some symmetry arguments to reduce the number of cases that need to be coo spidered but also generates some new potentially minimizing configurations. This paper described the various candidates for perimeter minimizing configurations and phase diagrams showing for which pairs of areas which candidate is optimal.