Curvature estimate on an open Riemann surface with the induced metric
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Mathematische Zeitschrift
Curvature estimate on an open Riemann surface with the induced metric X. Chen1 · Y. Li2 · Z. Liu2 · M. Ru3 Received: 13 December 2019 / Accepted: 11 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, motivated by the result of Osserman and Ru [12], we give a Gauss curvature estimate on an open Riemann surface M whose metric is induced from a nonconstant holomorphic map G : M → Pn (C). Keywords Holomorphic maps · Gauss curvature · Conformal metric · Riemann surfaces · Normal family Mathematics Subject Classification 32H30 · 32A30 · 53A10 · 32H02
1 Introduction In 1983, Nochka [6] solved the longstanding Cartan’s conjecture by introducing the so-called Nochka weights. As a consequence, he proved the following result as an extension of the classical Little Picard theorem. For recent development, see [14]. Theorem A (Nochka [6]) Let f : C → Pn (C) be a holomorphic map. Assume that f is knondegenerate for some k with 1 ≤ k ≤ n, i.e., its image is contained in some k-dimensional projective subspace of Pn (C) but not in any subspace whose dimension is lower than k. Then f can omit at most 2n −k +1 hyperplanes in Pn (C) located in general position. In particular,
B
M. Ru [email protected] ; [email protected] X. Chen [email protected] Y. Li [email protected] Z. Liu [email protected]
1
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, Fujian, People’s Republic of China
2
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of China
3
Department of Mathematics, University of Houston, Houston, TX 77204, USA
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if a holomorphic map f : C → Pn (C) omits more than 2n hyperplanes in general position, then f must be constant. The purpose of this paper is to generalize the above theorem. The following result extends the theorem of Nochka. Theorem 1 Let M be an open Riemann surface and let G : M → Pn (C) be a holomorphic map. Consider the conformal metric on M given by ˜ 2m |ω|2 , ds 2 = G where G˜ is a reduced representation of G, ω is a holomorphic 1-form, and m ∈ Z≥0 . Assume that ds 2 is complete. If G is k-nondegenerate for some k with 1 ≤ k ≤ n, then G can omit n at most mk(n − k−1 2 ) + 2n − k + 1 hyperplanes in P (C) located in general position. Here a holomorphic vector-valued function G˜ : U → Cn+1 is said to be a representation ˜ = G, where P is the projection map. The representation is said of G on U ⊂ M if P(G) ˜ to be reduced if G(x) = 0 for all x ∈ U , i.e., when we write G˜ = (g0 , . . . , gn ), the functions g0 , . . . , gn are holomorphic on U and have no common zeros. Obviously, every holomorphic map G : M → Pn (C) has a local reduced representation around each point p ∈ M. When M is simply connected, G has a global reduced representation. The expression ˜ 2m |ω|2 is independent of the choice of the representations in the of the metric ds 2 = G following sense: if G˜ and G˜ are two reduced representations of G, then G˜ = h G˜ where ˜
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