Properties of Ahlfors constant in Ahlfors covering surface theory

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 C,there is a minimal positive constant H_(0)(E_(q))(called Ahlfors constant with respect to E_(q)),such that the inequality(q-2)A(Σ)-4π#(f^(-1)(E_(q))∩U)≤H_(0)(E_(q))L(ЭΣ)holds for any simply-connected surfaceΣ=(f,U),where A(Σ)is the area ofΣ,L(ЭΣ)is the perimeter ofΣ,and#denotes the cardinality.It is difficult to compute H_(0)(E_(q))explicitly for general set E_(q),and only a few properties of H_(0)(E_(q))are known.The goals of this paper are to prove the continuity and differentiability of H_(0)(E_(q)),to estimate H_(0)(E_(q)),and to discuss the minimum of H_(0)(E_(q))for fixed q.