Two remarks on the Poincaré metric on a singular Riemann surface foliation

Abstract

Let be a smooth Riemann surface foliation on , where M is a complex manifold and is a closed set. Fix a hermitian metric g on and assume that all leaves of are hyperbolic. For each leaf , the ratio of , the restriction of g to L, and the Poincaré metric on L defines a positive function η that is known to be continuous on under suitable conditions on M, E. For a domain , we consider , the restriction of to U and the corresponding positive function by considering the ratio of g and the Poincaré metric on the leaves of . First, we study the variation of as U varies in the Hausdorff sense motivated by the work of Lins Neto–Martins. Secondly, Minda had shown the existence of a domain Bloch constant for a hyperbolic Riemann surface S, which in other words shows that every holomorphic map from the unit disc into S, whose distortion at the origin is bounded below, must be locally injective in some hyperbolic ball of uniform radius. We show how to deduce a version of this Bloch constant for .