On Plateau's Problem for Soap Films with a Bound on Energy

Details On Plateau's Problem for Soap Films with a Bound on Energy On Plateau's Problem for Soap Films with a Bound on Energy

2004-03-20

arxiv.org/abs/math/0403333v1

Mathematics

Differential Geometry

13 pages, 1 figure

Scientific paper

We prove existence and a.e. regularity of an area minimizing soap film with a bound on energy spanning a given Jordan curve in R^3. The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of surfaces over which area is minimized includes images of disks, integral currents, nonorientable surfaces and soap films as observed by Plateau with a bound on energy. Our area minimizing solution is shown to be a smooth surface away from its branched set which is a union of Lipschitz Jordan curves of finite total length.

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