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

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