Properties of Ahlfors constant in Ahlfors covering surface theory

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 E_q of q (≥3) distinct points in the extended complex plane \overline{ℂ}; there is a minimal positive constant H₀ (E_q) (called Ahlfors constant with respect to E_q), such that the inequality

(q-2)A(\sum )-4\pi ♯(f^{-1}(E_q)\cap U)\le H_0(E_q)L(\partial \sum )

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

Keywords

Nevanlinna Theory / value distribution / Ahlfors theory of covering surfaces

References

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

[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

[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

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

[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

[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

[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

RIGHTS & PERMISSIONS

Higher Education Press