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Existence of twisted harmonic maps with values in negatively curved manifolds. (Existence d’applications harmoniques tordues à valeurs dans les variétés à courbure négative.) (French) Zbl 0783.58016

This paper gives a generalization of a theorem of K. Corlette. Let \(\tilde N\) be the universal covering of a manifold \(N\), and \(M\) be a simply connected manifold of nonpositive curvature. Let \(\rho\) be a representation of \(\pi_ 1(N)\) into the isometry group of \(I(M)\) of \(M\). A \(\rho\)-twisted map is a \(\rho\)-equivariant map, \(f:\tilde N \to M\), i.e. a section of the flat \(M\)-fiber space based on \(N\), naturally associated to \(\rho\). When \(M\) is a Riemannian symmetric space of noncompact type, then Corlette’s theorem is as follows: There exists a \(\rho\)-twisted harmonic map with values in \(G/K\) if and only if \(\text{Im}(\rho)\) is a reductive subgroup of \(G\).
The theorem of this paper, here proved by methods involving a geometric definition of the reductivity condition, is the following generalization: If \(M\) has no flat half-strip, then there exists a \(\rho\)-twisted harmonic map if and only if \(\text{Im}(\rho)\) is a reductive subgroup of \(I(M)\).
\(M\) is said to have no flat half-strip if any Jacobi field along a geodesic which has constant norm for every \(t\) greater than or equal to a given \(t_ 0\) has constant norm for any \(t\). Negatively curved manifolds and analytic manifolds (for example, symmetric spaces) possess this latter property.
A subgroup \(G\) of \(I(M)\) is said to be reductive if there exists a closed convex set \(C\) of \(M\) globally fixed by \(G\) such that \(C \cong C_ 1 \times E\), where \(E\) is Euclidean, \(G=G_ 1 \times G_ 2\), \(G_ 1 \subset I(C_ 1)\), \(G_ 2 \subset I(E)\) and \(G_ 1\) has no infinite fixed point.
The proof of the necessary condition is nicely geometrical; the proof of the sufficient condition is based on the evolution equation \(\partial f_ 1/ \partial t=-d^*_ \nabla(Df_ t)\).

MSC:

58E20 Harmonic maps, etc.
58J35 Heat and other parabolic equation methods for PDEs on manifolds
58D25 Equations in function spaces; evolution equations
53C20 Global Riemannian geometry, including pinching
53C35 Differential geometry of symmetric spaces
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