Research on rigidity and finiteness for holomorphic mappings on complex hyperbolic manifolds

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We consider holomorphic mappings from an m-dimensional complex hyperbolic manifold M to an n-dimensional complex manifold N represented by a quotient space of a bounded domain N' of the complex space via a discrete automorphism group of N'. We show that if M is of divergence type, then the dimension of the space of the holomorphic mappings from M to N is bounded by the essential boundary dimension of N'. In particular, we may show a finiteness theorem for dominant holomorphic mappings from M to N. We also consider holomorphic mappings on compact Riemann surfaces and their dynamics.