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Foliated Plateau problem. II: Harmonic maps of foliations. (English) Zbl 0768.53012
[Part I, see the review above ( Zbl 0768.53011).]
The author starts with illuminating characterizations of harmonic maps between Riemannian manifolds emphasizing the role of curvature in these descriptions. There are given existence results for harmonic maps with special geometric properties. The argument relies on the solvability of the Dirichlet boundary value problem for harmonic maps. Moreover, by solving the asymptotic Dirichlet problem one constructs compact harmonic foliations. A transversal measure on the foliated space of harmonic maps is found by means of the parabolic equation method. The applications concern different pinching and rigidity theorems. Generally, the results in the paper are obtained by variational methods applied to the energy functional and the search of lower bounds on the energy.
Reviewer: D.Motreanu (Iaşi)

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