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

MathSciNet review:
522264

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Abstract: Surfaces of least *k* dimensional area in are constructed by minimization of the *n* dimensional volume of suitably thickened sets subject to a homological constraint. Specifically, let be integers and be compact and rectifiable. Let *G* be a compact abelian group and *L* be a subgroup of the Čech homology group (in case , 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 to *span L.* Using also a natural notion of what it means for a compact set to be -thick, we show that, for each , there exists an -thick set which minimizes *n* dimensional volume subject to the requirement that it span *L*. Our main result is that as 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 .

**[BR]**Richard Bellman,*Introduction to matrix analysis*, Second edition, McGraw-Hill Book Co., New York-Düsseldorf-London, 1970 (Russian). MR**0258847****[EH]**Samuel Eilenberg and O. G. Harrold Jr.,*Continua of finite linear measure. I*, Amer. J. Math.**65**(1943), 137–146. MR**0007643****[ES]**Samuel Eilenberg and Norman Steenrod,*Foundations of algebraic topology*, Princeton University Press, Princeton, New Jersey, 1952. MR**0050886****[FH]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[RE]**E. R. Reifenberg,*Solution of the Plateau Problem for 𝑚-dimensional surfaces of varying topological type*, Acta Math.**104**(1960), 1–92. MR**0114145**

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Additional Information

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