Certain hyperbolic regular polygonal tiles are isoperimetric

20 February 2021 • Jack Hirsch, Kevin Li, Jackson Petty and Christopher Xue

[arXiv] [Geometriae Dedicata]

Abstract. 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 $\mathbb{R}^n$ with density $r^p$, 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.