Landau's theorem for holomorphic curves in projective space and the Kobayashi metric on hyperplane complements

摘要

We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1hyperplanes in general position in n-dimensional complex projective space, wefind an explicit constant K such that for every holomorphic map f from the unitdisc to the complement of these hyperplanes, the derivative of f at the originmeasured with respect to the Fubuni-Study metric is bouned above by K. Thisresult gives an explicit lower bound on the Royden function, i.e., the ratio ofthe Kobayashi metric on the hyperplane complement to the Fubini-Study metric.Our estimate is based on the potential-theoretic method of Eremenko and Sodin.