Double bubbles minimize

Joel Hass and Roger Schlafly

Review from Zentralblatt MATH:

The authors prove the ``double bubble conjecture'' in $\bbfR^3$, in the particular case in which the two volumes are equal. Thus, the configuration in $\bbfR^3$ of smallest perimeter bounding two equal volumes consists of two spherical caps of equal radius meeting at $120^\circ$ angles, and a plane through the intersection circle of the spheres. The procedure takes as its starting point from earlier results due to Almgren, to Taylor, to White, and to Hutchings; these results greatly reduce the number of cases that need to be considered. The authors study the remaining conceivable configurations individually, ruling out extraneous cases in part by direct reasoning and in part by computer calculation with controlled error estimation. The exposition is very clear and complete, and nicely organized; the paper could serve as initial reading for some of the established techniques that are incidental to the procedure.

In the interim, a proof of the double bubble conjecture without the restriction to equal volumes was announced by {\it M. Hutchings}, {\it F. Morgan}, {\it M. Ritoré}, and {\it A. Ros} [Electron. Res. Announc. Am. Math. Soc. 6, 45-49 (2000; Zbl 0970.53009)], see the review below.

Reviewed by Robert Finn

Keywords: double bubble conjecture; equal volumes; spherical cap

Classification (MSC2000): 53A10 76B45 49Q10

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