Harmonic maps from compact Kähler manifolds to exceptional hyperbolic spaces (Q1177615)

Let \(M\) be a compact Kaehler manifold and consider a quotient \(H_{\mathbb{Q}}^ 2/\Gamma\) of the Cayley hyperbolic plane. Theorem. If \(f: M\to H_{\mathbb{Q}}^ 2/\Gamma\) is harmonic then either (a) \(f\) has rank 1 and factors through a geodesically embedded circle, (b) \(f\) has rank 2 and factors as \(g\cdot h\), where \(h\) is a holomorphic map to a compact Riemann surface and \(g\) is harmonic, or (c) \(f\) has rank 4 and factors as \(g\cdot h\), where \(h\) is a holomorphic map to a compact quotient of the 2-ball and \(g\) is a geodesic immersion. Then, proceeding just like in the case of \(SO(1,n)\) the authors prove Theorem 1.1. Let \(\Gamma\) be a co- compact discrete torsion free subgroup of \(F_{4(-20)}\). Then \(\Gamma\) is non-Kaehler.