Foliated Plateau problem. II: Harmonic maps of foliations
type: Item
instance of: scholarly article
title: Foliated Plateau problem. II: Harmonic maps of foliations
zbMATH Open document ID: 0768.53012
DOI: 10.1007/BF01896204
published in: Geometric and Functional Analysis. GAFA
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/58124
review text: [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.
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 58E20
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 56883
zbMATH Keywords: Dirichlet boundary value problem
zbMATH Keywords: asymptotic Dirichlet problem
zbMATH Keywords: harmonic foliations
zbMATH Keywords: energy functional
zbMATH Keywords: transversal measure
zbMATH Keywords: pinching
zbMATH Keywords: rigidity
reviewed by: Dumitru Motreanu
MaRDI profile type: Publication
cites work: Complete minimal varieties in hyperbolic space
cites work: Complete minimal hypersurfaces in hyperbolic n-manifolds
cites work: The Dirichlet problem at infinity for manifolds of negative curvature
cites work: Existence of flats in manifolds of nonpositive curvature
cites work: The existence of gaps in minimal foliations
cites work: A uniqueness theorem for \(Z^ n\)-periodic variational problems
cites work: On minimal laminations of the torus
cites work: Curvature operators: pinching estimates and geometric examples
cites work: Compact Manifolds and Hyperbolicity
cites work: Q3909666
cites work: Harmonic mappings of Kähler manifolds to locally symmetric spaces
cites work: Q3721357
cites work: New examples of harmonic diffeomorphisms of the hyperbolic plane onto itself
cites work: Twisted Harmonic Maps and the Self-Duality Equations
cites work: A Report on Harmonic Maps
cites work: Another Report on Harmonic Maps
cites work: Q3662316
cites work: Harmonic Mappings of Riemannian Manifolds
cites work: Filling Riemannian manifolds
cites work: Homotopical effects of dilatation
cites work: Almost flat manifolds
cites work: Foliated Plateau problem. I: Minimal varieties
cites work: Spin and scalar curvature in the presence of a fundamental group. I
cites work: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds
cites work: Pinching constants for hyperbolic manifolds
cites work: Smoothness of horocycle foliations
cites work: Harmonic mappings and Kähler manifolds
cites work: On the rigidity of certain discrete groups and algebraic varieties
cites work: Minimal measures
cites work: Q3992565
cites work: Strong Rigidity of Locally Symmetric Spaces. (AM-78)
cites work: Carnot-Carathéodory metrics and quasiisometries of symmetric spaces of rank 1
cites work: Applications of harmonic maps to Kähler geometry
cites work: Harmonic maps of the moduli space of compact Riemann surfaces
cites work: Existence of incompressible minimal surfaces and the topology of three dimensional manifolds with non-negative scalar curvature
cites work: The complex-analyticity of harmonic maps and the strong rigidity of compact Kaehler manifolds
cites work: Q3744839
cites work: Geodesic length functions and the Nielsen problem
author: Misha Gromov
description: scientific article; zbMATH DE number 56883