A Basic Study On Ahlfors' Constant In Ahlfors' Theory Of Covering Surfaces

For a set Eq=?a1,…,aq} consisting of q distinct extended complex numbers(q?3),#12 is called the Ahlfors' constant of Eq.where ? denotes the open unit disk,and M is the set of all non-constant meromorphic functions on ?.The Ahlfors' constant H0(Eq)is the minimal possible value of the constant h in Ahlfors' second fundamental theorem(Theorem 1.1),which depends on a1,…,aq in a rather complicated way.Few prop-erties of H0(Eq)could be found in published articles.Roughly speaking,the closer a1,…,aq are to each other,the larger H0(Eq)is.The optimal surface is an effective tool to determine H0(Eq).In this paper,we provide some good properties of optimal surfaces,and apply this tool to some sym-metric Eq.For example,we determine H0(0,±1)?4.03416,H0((?))?3.1098,H0(±1,±i)?4.13676,H0(0,(?))?3.58518,H0(0,(?),?)?4.3087,H0((?))?6.136724,H0(0,±1,±i,?)?4.55412,H0((?))?8.123132,H0(0,(?),?)?6.30738,and H0((?)(±1±i))?5.62976.Here,(?)mean all unit roots,not only 1.The extended complex plane C could be identified with the unit sphere S in R3,and so {0,(?)} is the vertices set of a regular tetrahedron,and {(?)(±1±i)} is the vertices set of a cube.We also prove that H0(E3)?H0((?)),H0(E4)?H0(0,(?)),H0(E5)?H0(0,(?),?),and H0(E6)?H0(0,±1,±i,?)for all E3,E4,E5,E6,and each equality holds iff Eq is a rotation of {(?)},{0,(?)},{0,(?),?}.or {0,±1,±i,?} respectively.These inequalities could be intuitively explained as follows.For a fixed q=3,4,5,6,the points in these four sets are as far away from each other as possible,and so their Ahlfors' constants should be minimal.There are many other results in this paper,most of which coincide with our intuition.