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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Goffman-Liu area and Plateau's problem


Authors: David Bindschadler and Togo Nishiura
Journal: Proc. Amer. Math. Soc. 82 (1981), 66-70
MSC: Primary 28A75; Secondary 49F25
MathSciNet review: 603603
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Abstract: Goffman and Liu defined a lower semicontinuous area $ A(f)$ for linearly continuous maps $ f$ from the disk into $ {{\mathbf{R}}^n}$, showed that $ A(f)$ is the Lebesgue area when $ f$ is continuous and thereby extended the notion of area to some discontinuous maps. With the aid of a simple retraction of the punctured disk onto its boundary, a canonical linearly continuous map $ {f_\Gamma }$ of the disk is associated with each continuous closed curve $ \Gamma $ defined on the boundary of the disk. When $ \Gamma $ is a Jordan curve, it is shown that the discontinuous map $ {f_\Gamma }$ has the property that $ A({f_\Gamma }) = A(\sigma )$ where $ \sigma $ is a continuous map of least area spanning $ \Gamma $ from the classical Plateau problem. Finally, the corresponding least area problem in the class of linearly continuous maps is shown to be trivial, that is, the least area is zero in the class of linearly continuous maps.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0603603-2
PII: S 0002-9939(1981)0603603-2
Keywords: Lebesgue area, Fréchet curve, Plateau problem, minimal area
Article copyright: © Copyright 1981 American Mathematical Society