Harmonic maps from compact Kähler manifolds to exceptional hyperbolic spaces

KÄHLER MANIFOLDS AND FUNDAMENTAL GROUPS OF NEGATIVELY Δ-Pinched MANIFOLDS

“…As a consequence of our proof, we also show that any nonuniform lattice in F 4(−20) cannot be the fundamental group of a quasi-compact Kähler manifold. The corresponding result for uniform lattices was proved by Carlson and Hernández [3]. Finally, we follow Gromov and Thurston [6] to give some examples of negatively δ-pinched manifolds (δ > 1 4 ) of finite volume which, as topological manifolds, admit no hyperbolic metric with finite volume under any smooth structure.…”

“…Completely similar to the discussion in [3], one also has a decomposition for f : f = g • h, here h is a holomorphic map from M to a quotient of the two-ball of finite volume and g is a geodesic immersion. Then, the same cohomological dimension arguments show this is also impossible.…”

“…This tangent space can be identified with the second factor of the Cartan decomposition g = t + p, here g is the Lie algebra of F 4(−20) and t is the Lie algebra of the maximal compact subgroup in F 4(−20) . By the Lie-theoretic analysis in [3], one knows that the complex dimension of an Abelian subspace in p C is at most 2. So, we have three cases to discuss for df (T 1,0 p M ).…”

On the fundamental groups of compact Sasakian manifolds

“…Now Γ is a Kähler group and also a cocompact lattice in G, where G is SO (1, n), n > 2 or F 4(−20) . This is impossible by the results of Carlson, Hernández and Toledo; see [8,9].…”

On pseudo-harmonic maps in conformal geometry

“…By Abelian subspaces theorem I, we have the inequality ν(G/K) < (1/2) dim(G/K) provided G/K is not Hermitian symmetric. For hyperbolic spaces over various division algebras the following theorem, due to [4] and [6], gives more precise information. Proof.…”