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Harmonic maps and representations of non-uniform lattices of $$PU(m,1)$$. (English) Zbl 1147.22009
The authors make use of harmonic maps techniques to study representations of lattices of $$PU(m,1)$$ into $$PU(n,1)$$ in order to provide rigidity results. This goes as follows: first one shows the existence of a harmonic map between the corresponding symmetric spaces. Then one must prove, under certain constraints, that the harmonic map is pluriharmonic, holomorphic, totally geodesic, isometric, depending on the situation. The existence is proved of a harmonic map from $$\mathbb H^m_C$$ to $$\mathbb H^n_C$$ making use of Corlette’s results. Finally the authors study a new invariant associated to representations of fundamental groups of finite topological type and negative Euler characteristic into $$PU(n,1)$$.

MSC:
 22E40 Discrete subgroups of Lie groups 32Q05 Negative curvature complex manifolds 32Q20 Kähler-Einstein manifolds 53C24 Rigidity results 53C35 Differential geometry of symmetric spaces 53C43 Differential geometric aspects of harmonic maps
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