Branch values in Ahlfors’ theory of covering surfaces

In the study of the constant in Ahlfors’second fundamental theorem involving a set Eq consisting of q points,branch values of covering surfaces outside Eq bring a lot of troubles.To avoid this situation,for a given surfaceΣ0,it is useful to construct a new surfaceΣ1,such that L(■Σ1)≤L(■Σ0),and H(Σ1,Eq)≥H(Σ0,Eq).and all branch values ofΣ1 are contained in Eq.The goal of this paper is to prove the existence of suchΣ1,which generalizes a result found by Zhang(2013).