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Foliated Plateau problem. I: Minimal varieties. (English) Zbl 0768.53011
On a compact Riemannian manifold without boundary the Plateau foliation \(Plt_ kX\) consists of minimal submanifolds of given dimension \(k \leq \dim X\). The case \(k = 1\) is represented by the closed geodesics in \(X\). The foliated Plateau problem studies the subfoliations in \(Plt_ kX\) and the transversal measures for the canonical foliation in \(Plt_ kX\). For a comparison with the ordinary Plateau problem notice that the latter one concerns the existence of closed leaves in \(Plt_ kX\). The author’s approach can be described as the deformation of a compact foliation of \(k\)-dimensional submanifolds of \(X\) to a subfoliation in \(Plt_ kX\) and the deformation of a transversal measure on the set of \(k\)-dimensional submanifolds to a transversal measure on \(Plt_ kX\). In order to include singular solutions of the Plateau problem one allows the use of more general \(k\)-dimensional subvarieties than just submanifolds with an appropriate topology in the space of subvarieties. First, the author shows how one can find some Plateau foliations by solving the asymptotic Plateau problem in the universal covering \(Y\) of \(X\). Then, the author solves the Plateau problem by deforming measures to \(Plt_ kX\) as mentioned above. This leads to new existence results for minimal subvarieties in \(Y\). The arguments in the paper rely on variational methods inspired from the geometric measure theory and the theory of harmonic mappings.
[Part II, see the review below ( Zbl 0768.53012)].
Reviewer: D.Motreanu (Iaşi)

MSC:
53C12 Foliations (differential geometric aspects)
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58E20 Harmonic maps, etc.
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