Soap Film Solutions to Plateau’s Problem
 J. Harrison
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Abstract
Plateau’s problem is to show the existence of an areaminimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an areaminimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau’s problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator, a Poincaré Lemma, and good compactness properties to solve Plateau’s problem for orientable, embedded surfaces. But integral currents cannot represent surfaces such as the Möbius strip or surfaces with triple junctions. In the class of varifolds, there are no existence theorems for a general Plateau problem. We use the chain complex of differential chains, a geometric Poincaré Lemma, and good compactness properties of the complex to solve Plateau’s problem in such generality as to find the first solution which minimizes area taken from a collection of surfaces that includes all previous special cases, as well as all smoothly immersed surfaces of any genus type, orientable or nonorientable, and surfaces with multiple junctions. Our result holds for all dimensions and codimensionone surfaces in ℝ^{ n }.
Inside
Within this Article
 Introduction
 Differential Chains of Type B
 Operators
 Integral Monopole and Dipole Chains
 Geometric Poincaré Lemma
 The Volume Functional Used to Compute Area
 The Part of a Chain in a Compatible Cube
 Existence of Area Minimizers for Surfaces Spanning a Smoothly Embedded Closed Curve in ℝ^{3}
 References
 References
Other actions
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 Title
 Soap Film Solutions to Plateau’s Problem
 Journal

Journal of Geometric Analysis
Volume 24, Issue 1 , pp 271297
 Cover Date
 20140101
 DOI
 10.1007/s122200129337x
 Print ISSN
 10506926
 Online ISSN
 1559002X
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Plateau’s problem
 Differential chains
 Differential forms
 Chainlets
 Dipole chains
 Dirac chains
 Poincare Lemma
 Extrusion
 Retraction
 Prederivative
 Pushforward
 Volume functional
 Soap films
 Triple branches
 Moebius strips
 Compactness
 Minimal sets
 49Q15
 49J52
 49J99
 Authors

 J. Harrison ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of California, Berkeley, USA