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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Some new constructions and estimates in the problem of least area


Author: Harold Parks
Journal: Trans. Amer. Math. Soc. 248 (1979), 311-346
MSC: Primary 49F22
DOI: https://doi.org/10.1090/S0002-9947-1979-0522264-X
MathSciNet review: 522264
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Abstract: Surfaces of least k dimensional area in $ {\textbf{R}^n}$ are constructed by minimization of the n dimensional volume of suitably thickened sets subject to a homological constraint. Specifically, let $ 1 \,\, \leqslant \,\,k\,\, \leqslant \,n$ be integers and $ B\, \subset \,{\textbf{R}^n}$ be compact and $ k\, - \,1$ rectifiable. Let G be a compact abelian group and L be a subgroup of the Čech homology group $ {H_{k - 1}}\left( {B;\,\,G} \right)$ (in case $ k = \,1$, suppose, additionally, L is contained in the kernel of the usual augmentation map). J. F. Adams has defined what it means for a compact set $ {\rm X}\, \subset \,{\textbf{R}^n}$ to span L. Using also a natural notion of what it means for a compact set to be $ \varepsilon $-thick, we show that, for each $ \varepsilon \, > \,0$, there exists an $ \varepsilon $-thick set which minimizes n dimensional volume subject to the requirement that it span L. Our main result is that as $ \varepsilon $ approaches 0 a subsequence of the above volume minimizing sets converges in the Hausdorff distance topology to a set, X, which minimizes k dimensional area subject to the requirement that it span L. It follows, of course, from the regularity results of Reifenberg or Almgren that, except for a compact singular set of zero k dimensional measure, X is a real analytic minimal submanifold of $ {\textbf{R}^n}$.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0522264-X
Keywords: Homological constraint, Hausdorff measure, Hausdorff distance, polyhedral complexes
Article copyright: © Copyright 1979 American Mathematical Society

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