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 + 1 hyperplanes in general position in P~n, we find an explicit constant K such that for every holomorphic map f from the unit disc to the complement of these hyperplanes, we have f#(0) · K, where f# denotes the norm of the derivative measured with respect to the Fubini-Study metric. This result gives an explicit lower bound on the Royden function, i.e., the ratio of the Kobayashi metric on the hyperplane complement to the Fubini-Study metric. Our estimate is based on the potential-theoretic method of Eremenko and Sodin.