Least-area way to enclose and separate two regions of given volumes in R^3

You need to know: Sphere, spherical cap, angle at which spherical caps intersect, volume, surface area.

Background: Standard double bubble in is a construction which consists on three spherical caps meeting along a common circle at 120-degree angles (If 2 of caps has equal radii, the third one becomes a flat disc). It divides the space into 3 regions: infinite one and 2 finite ones with some volumes and .

The Theorem: On 22nd March 2000 Michael Hutchings, Frank Morgan, Manuel Ritoré, and Antonio Ros submitted to Annals of Mathematics a paper in which they proved that the standard double bubble provides the way to enclose and separate two regions of prescribed volumes and in with the least possible total surface area of the boundary.

Short context: The problem of finding area-minimising way to enclose and separate 2 given volumes was studied by Plateau in the 19th century, and the conjecture that the standard double bubble provides the solution was known as the double bubble conjecture. The 2-dimensional version of the conjecture was proved in 1991 by the team of undergraduate students. Later, Hass and Schlafly proved the case in . The Theorem confirms the conjecture in general.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.5 of this book for an accessible description of the Theorem.

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