A deformation theorem for the Kobayashi metric

Abstract

Let M 0 {M_0} be a compact hyperbolic complex manifold. It is shown that the infinitesimal Kobayashi metric is upper semicontinuous in a C ∞ {C^\infty } deformation parameter t ∈ U ⊆ R k t \in U \subseteq {R^k} . This is accomplished by proving deformation theorems for holomorphic maps.