Picard-Type Theorem and Curvature Estimate on an Open Riemann Surface with Ramification

Abstract

Let M be an open Riemann surface and G: M → ℙⁿ(ℂ) be a holomorphic map. Consider the conformal metric on M which is given by ${\rm{d}}{s^2} = ||\tilde G|{|^{2m}}|\omega {|^2}$, where ${\tilde G}$ is a reduced representation of G, ω is a holomorphic 1-form on M and m is a positive integer. Assume that ds² is complete and G is k-nondegenerate(0 ≤ k ≤ n). If there are q hyperplanes H₁, H₂, ⋯, H_q ⊂ ℙⁿ(ℂ) located in general position such that G is ramified over H_j with multiplicity at least γ_j(> k) for each j ∈ {1, 2, ⋯, q}, and it holds that

$$\sum\limits_{j = 1}^q {\left( {1 - {k \over {{\gamma _j}}}} \right) > (2n - k + 1)\left( {{{mk} \over 2} + 1} \right),} $$

then M is flat, or equivalently, G is a constant map. Moreover, one further give a curvature estimate on M without assuming the completeness of the surface.