Frontiers of Mathematics in China >

2021 , Vol. 16 >Issue 4: 957 - 977

DOI: https://doi.org/10.1007/s11464-021-0939-0

RESEARCH ARTICLE

Properties of Ahlfors constant in Ahlfors covering surface theory

  • Wennan LI 1 ,
  • Zonghan SUN 2 ,
  • Guangyuan ZHANG , 1
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  • 1. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
  • 2. Tsinghua University High School, Beijing 100084, China

Received date: 16 Jan 2021

Accepted date: 08 Apr 2021

Copyright

2021 Higher Education Press
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Abstract

This paper is a subsequent work of [Invent. Math., 2013, 191: 197-253]. The second fundamental theorem in Ahlfors covering surface theory is that, for each set Eq of q (≥3) distinct points in the extended complex plane ℂ‾; there is a minimal positive constant H0 (Eq) (called Ahlfors constant with respect to Eq), such that the inequality

(q−2)A(∑)−4π♯(f−1(Eq)∩U)≤H0(Eq)L(∂∑)

holds for any simply-connected surface∑=（f,U‾） ; where A(∑) is the area of∑; L(∂∑) is the perimeter of∑; and # denotes the cardinality. It is difficult to compute H0 (Eq) explicitly for general set Eq; and only a few properties of H0(Eq) are known. The goals of this paper are to prove the continuity and differentiability of H0 (Eq); to estimate H0 (Eq); and to discuss the minimum of H0 (Eq) for fiixed q.

Key words： Nevanlinna Theory; value distribution; Ahlfors theory of covering surfaces

Cite this article

Wennan LI , Zonghan SUN , Guangyuan ZHANG . Properties of Ahlfors constant in Ahlfors covering surface theory[J]. Frontiers of Mathematics in China, 2021 , 16(4) : 957 -977 . DOI: 10.1007/s11464-021-0939-0

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