Labourie, François 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 Proc. Am. Math. Soc. 111, No. 3, 877-882 (1991). 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 \(\partial f_ 1/ \partial t=-d^*_ \nabla(Df_ t)\). Cited in 1 ReviewCited in 15 Documents 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 Keywords:twisted harmonic maps; negatively curved manifolds; theorem of K. Corlette PDFBibTeX XMLCite \textit{F. Labourie}, Proc. Am. Math. Soc. 111, No. 3, 877--882 (1991; Zbl 0783.58016) Full Text: DOI