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- Title
Picard-Type Theorem and Curvature Estimate on an Open Riemann Surface with Ramification.
- Authors
Liu, Zhixue; Li, Yezhou
- 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 d s 2 = ∣ ∣ G ˜ ∣ ∣ 2 m ∣ ω ∣ 2 , where 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, ⋯, Hq ⊂ ℙn(ℂ) located in general position such that G is ramified over Hj with multiplicity at least γj(> k) for each j ∈ {1, 2, ⋯, q}, and it holds that ∑ j = 1 q (1 − k γ j ) > (2 n − k + 1) ( m k 2 + 1) , 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.
- Subjects
CURVATURE; HOLOMORPHIC functions; RIEMANN surfaces; HYPERPLANES
- Publication
Chinese Annals of Mathematics, 2023, Vol 44, Issue 4, p533
- ISSN
0252-9599
- Publication type
Article
- DOI
10.1007/s11401-023-0030-0