Proof of the double bubble curvature conjecture (Q881446)

The Hutchings basic estimate [\textit{M. Hutchings}, J. Geom. Anal. 97, No. 2, 285--304 (1997; Zbl 0935.53008)] provides a way to bound the number of components in each region of an area-minimizing double bubble. From this theory the following curvature conjecture emerged: In \(\mathbb R^{n}\) (with \(n\geq 2\)), let \(H_{0}\) , \(H_{1}\), \(H_{2}\), respectively, denote the mean curvature of a sphere of volume \(\omega \), a sphere of volume \(\omega +1\), and the exterior of the second region of a standard double bubble of volumes \(1,\omega \). Then \( 2H_{2}>H_{0}+H_{1}\). The curvature conjecture was proven in \(\mathbb R^{2}\) and in all dimensions for the case \(\omega \geq 1\) by David Futer. In the paper under review the author proves this conjecture in any case.