On the Validity of Failure of Gap Rigidity for Certain Pairs of Bounded Symmetric Domains

Abstract

Let Ω be a bounded symmetric domain equipped with a canonical Kahler metric. We are interested to characterize holomorphic geodesic cycles (i.e., compact complex geodesic submanifolds) S ⊂ X on Hermitian locally symmetric manifolds X uniformized by Ω, i.e., X = Ω/Γ, where Γ ⊂ Aut(Ω) is any torsion-free discrete group of automorphisms, in terms of differentialgeometric or algebro-geometric conditions. In Eyssidieux-Mok [EysMok1995] we studied almost geodesic complex submanifolds and formulated the gap phenomenon. Up to equivalence under Aut(Ω) there are only a finite number of totally geodesic complex submanifolds D ⊂ Ω which are themselves biholomorpic to bounded symmetric domains. Let 2 > 0. We say that S ⊂ X is 2-geodesic if and only if the norm of the second fundamental form of S ⊂ X is uniformly bounded by 2. Fixing Ω but letting Γ ⊂ Ω be arbitrary we showed that when 2 is sufficiently small, an 2-geodesic compact complex