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

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Abstract

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.

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Authors and Affiliations

  1. Dipartimento di Matematica, Universitá della Basilicata (Potenza), Italia

    Sisto Baldo & Giandomenico Orlandi

  2. Dipartimento di Matematica, Universitá di Verona, Italia

    Sisto Baldo & Giandomenico Orlandi

Authors
  1. Sisto Baldo
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  2. Giandomenico Orlandi
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Baldo, S., Orlandi, G. Codimension one minimal cycles with coefficients inZ orZ p , and variational functionals on fibered spaces. J Geom Anal 9, 547–568 (1999). https://doi.org/10.1007/BF02921972

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