In 2008 Reichardt proved that the optimal Euclidean double bubble---the least-perimeter way to enclose and separate two given volumes---is three spherical caps meeting along a sphere at 120 degrees. We consider Rn with density rp, joining the surge of research on manifolds with density after their appearance in Perelman's 2006 proof of the Poincaré Conjecture. Boyer et al. proved that the best single bubble is a sphere through the origin. We conjecture that the best double bubble is the Euclidean solution with the singular sphere passing through the origin, for which we have verified equilibrium (first variation or ``first derivative'' zero). To prove the exterior of the minimizer connected, it would suffice to show that least perimeter is increasing as a function of the prescribed areas. We give the first direct proof of such monotonicity in the Euclidean plane. Such arguments were important in the 2002 Annals proof of the double bubble in Euclidean 3-space.

Author Bio

Jack Hirsch is a junior at Yale University studying ethics, politics, and economics as well as math. He is very passionate about using mathematical methods to advance political studies.

Kevin Li is a junior at Yale University. He plans to go to graduate school in math and is particularly interested in geometry and differential equations. He is also grateful for Frank Morgan's mentorship throughout the project.

Jackson is a double major in mathematics and linguistics at Yale University. He is interested in topology and information theory.

Christopher Xue is a math major who enjoys geometry and algebra and who in his free time enjoys playing Tetris.