Soap films and covering spaces |
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Authors: | Kenneth A Brakke |
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Institution: | 1. The Geometry Center, University of Minnesota, 1300 South Second Street, 55454, Minneapolis, MN
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Abstract: | A new mathematical model of soap films is proposed, called the “covering space model.” The two sides of a film are modeled
as currents on different sheets of a covering space branching along the film boundary. Hence a film may be seen as the minimal
cut separating one sheet of the covering space from the others. The film is thus the oriented boundary of one sheet, which
represents the exterior of the film. As oriented boundaries, films may be calibrated with differential forms on the covering
space, a version of the min-cut, max-flow duality of network theory. This model applies to unoriented films, films with singularities,
films touching only part of a knotted curve, films that deformation retract to their boundaries, and other examples that have
proved troublesome for previous soap film models.
Communicated by Frederick Almgren |
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Keywords: | Math Subject Classification" target="_blank">Math Subject Classification 49Q20 |
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