Codimension one minimal cycles with coefficients inZ orZp, and variational functionals on fibered spaces

Given a compact, oriented Riemannian manifold M, without boundary, and a codimension-one homology class in H* (M, Z) (or, respectively, in H* (M, Zp) with p an odd prime), we consider the problem of finding a cycle of least area in the given class: this is known as the homological Plateau’s problem....

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Bibliographic Details
Published in:The Journal of geometric analysis 1999-12, Vol.9 (4), p.547-568
Main Authors: Sisto, Baldo, Orlandi Giandomenico
Format: Article
Language:English
Subjects:
Geometry
Homology
Parameters
Regularization
Riemann manifold
Online Access:Get full text
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Summary:Given a compact, oriented Riemannian manifold M, without boundary, and a codimension-one homology class in H* (M, Z) (or, respectively, in H* (M, Zp) with p an odd prime), we consider the problem of finding a cycle of least area in the given class: this is known as the homological Plateau’s problem.We propose an elliptic regularization of this problem, by constructing suitable fiber bundles ξ (resp. ζ) on M, and one-parameter families of functionals defined on the regular sections of ξ, (resp. ζ), depending on a small parameter ε.As ε → 0, the minimizers of these functionals are shown to converge to some limiting section, whose discontinuity set is exactly the minimal cycle desired.
ISSN:1050-6926
1559-002X
DOI:10.1007/BF02921972