Holomorphic mappings of complex manifolds

Introduction Schwarz's lemma, as formulated by Pick, can be stated as follows: Every holomorphic map / of the open unit disk D into itself is distance-decreasing with respect to the Poincare-Bergman metric ds 2 , i.e. /*(ds 2 ) < ds 2 , where the equality holding at one point of D, implies that / is an isometry. Bochner and Martin proved in their book [2] the following generalization of Schwarz's lemma to higher dimensions. Let D n be the n-dimensional open unit ball. If / is a holomorphic map of D m into D n such that /(0) = 0, then f(z) < z for all zeD m . In other words, every holomorphic map of D m into D n is distance-decreasing with respect to the Bergman metric ds 2 Dv and ds 2 Dn of D m and D n respectively. Koranyi proved [9] that if M is a henitian symmetric space of non-compact type with the Bergman metric ds 2 , and / is a holomorphic map of M into itself, then f*(ds 2 ) < kds 2 , where k denotes the rank of M. This is another generalized Schwarz's lemma. Ahlfors was the first to generalize Schwarz's lemma by essentially considering the curvature; his result can be stated as the following: Let M be a Riemann surface with hermitian metric ds 2 M whose Gaussian curvature is bounded above by a negative constant -, and D the unit disk in C with an invariant metric ds 2 D whose Gaussian curvature is a negative constant -A, then every holomorphic A map f:D->M satisfies /*(<s 2 ) <ds 2 D . Kobayashi generalized this result B to higher dimensional case in his recent paper [6].