*(English)*Zbl 1009.53007

The double bubble conjecture (now theorem) is that, in ${\mathbb{R}}^{3}$, the unique perimeter-minimizing double bubble enclosing and separating regions ${R}_{1}$ and ${R}_{2}$ of prescribed volumes ${v}_{1}$ and ${v}_{2}$ are a standard double bubble consisting of three spherical caps meeting along a common circle at 120-degree angles (for equal volumes, the middle cap is a flat disc). The physical fact expressed by the double bubble conjecture was observed and published by Plateau in 1873 (see JFM 06.0516.03). One of the difficulties is that existence proofs rely on allowing the regions ${R}_{1}$ and ${R}_{2}$ to be disconnected. A priori even the exterior region complementary to ${R}_{1}$ and ${R}_{2}$ might be disconnected.

In this paper, an estimate of Hutchings is used to prove that the larger region is connected. A stability argument is used to show that the smaller region has at most two components. A second crucial part of the argument involves the consideration of rotations about an axis orthogonal to the axis of symmetry of the double bubble. The proper choice of axis allows the construction of variations that respect both volume constraints. Stability implies that the variation satisfies a nice differential equation leading to sufficient information to conclude that the surface is made up of pieces of spheres.