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

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 ; there is a minimal positive constant H0 (Eq) (called Ahlfors constant with respect to Eq), such that the inequality

(q2)A()4π(f1(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.

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

1
Ahlfors L V. Zur Theorie derÜherlagerung-Sächen. Acta Math, 1935, 65: 157–194

DOI

2
Ahlfors L V. Complex Analysis. 3rd ed. New York: McGraw-Hill, 1979

3
Bernstein F. Uber die isoperimetrische eigenschaft des kreises auf der kugeloberache und in der ebene. Math Ann, 1905, 60: 117–136

DOI

4
Drasin D. The impact of Lars Ahlfors' work in value-distribution theory. Ann Acad Sci Fenn Ser A I Math, 1988, 13(3): 329–353

DOI

5
Dufresnoy J. Sur les domaines couverts par les valeurs dune fonction méromorphe ou algébroïde. Ann Sci Éc Norm Supér, 1941, 58: 179–259

DOI

6
Eremenko A. Ahlfors contribution to the theory of meromorphic functions. In: Lectures in Memory of Lars Ahlfors (Haifa, 1996). Israel Math Conf Proc, 14. Ramat Gan: Bar-Ilan Univ, 2000, 41–63

7
Hayman W K. Meromorphic Functions. Oxford: Oxford Univ Press, 1964

8
Nevanlinna R. Zur theorie der meromorphen funktionen. Acta Math, 1925, 46: 1–99

DOI

9
Stoilow S. Lecons sur les Principes Topologiques de la Theorie des Fonctions Analytiques. Paris: Gauthier-Villars, 1956

10
Sun Z H, Zhang G Y. Branch values in Ahlfors' theory of covering surfaces. Sci China Math, 2020, 63: 1535–1558

DOI

11
Yang L. Value Distribution Theory. Berlin: Springer, 1993

12
Zhang G Y. The precise bound for the area-length ratio in Ahifors' theory of covering surfaces. Invent Math, 2013, 191: 197–253

DOI

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