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| Picard-Type Theorem and Curvature Estimate on an Open Riemann Surface with Ramification* |
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Citation： |
Zhixue LIU,Yezhou LI.Picard-Type Theorem and Curvature Estimate on an Open Riemann Surface with Ramification*[J].Chinese Annals of Mathematics B,2023,44(4):533~548 |
| Page view： 1303
Net amount： 699 |
Authors： |
Zhixue LIU; Yezhou LI |
Foundation： |
This work was supported by the National Natural Science Foundation of China (Nos. 12101068,12261106, 12171050). |
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| 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 ds2 = k e Gk2m|ω|2, where eG 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 Xj=1 1 -k/γj> (2n - k + 1) （mk/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. |
Keywords： |
Picard-type theorem, Holomorphic map, Riemann surface, Curvature estimate |
Classification： |
32H25, 32A22, 32H02 |
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