Isoperimetric surfaces with boundary

Dorff, Rebecca; Johnson, Drew; Lawlor, Gary; Sampson, Donald

doi:10.1090/S0002-9939-2011-10872-4

Isoperimetric surfaces with boundary
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by Rebecca Dorff, Drew Johnson, Gary R. Lawlor and Donald Sampson

Proc. Amer. Math. Soc. 139 (2011), 4467-4473

DOI: https://doi.org/10.1090/S0002-9939-2011-10872-4

Published electronically: April 26, 2011

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Abstract:

We prove that many common combinations of soap films and soap bubbles that result from dipping polyhedral wire frames in a soap solution are minimizing with respect to their boundary and bubble volume. This can be thought of as a combination of the Plateau problem of least area for surfaces spanning a given boundary and the isoperimetric problem of least area for surfaces enclosing a given volume. Proof is given in arbitrary dimension using a combination of the mapping of Gromov, after Knothe, and the paired calibrations of Lawlor and Morgan.

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