Sg Circle Patterns On Hyperbolic And Complex Affine Geometries

A circle packing is a configurations of circles with specified patterns of tangencies ina constant curvature surface. Its theory is a fast developing field of research on borderof complex analysis and discrete di?erential geometry. In the past years, the progressin this area was initiated by W.Thurston's idea in 1985 about approximation of theRiemann mapping by hexagonal circle packings. For the study of circle configurations,classical circle packings consisting of disjoint circles were generalized to circle patterns,where the circles may intersect. In this thesis, our main work is as follows. First, we dis-cuss the existence and uniqueness of SG circle patterns in the hyperbolic plane. Givenany finite simply connected quad-decompoition K, it is proved that there is a SG circlepattern PK for K in hyperbolic plane D such that its each boundary circle is horocycle.Moreover, PK is unique up to a conformal automorphism of D. Next, we discuss theexistence and density of SG circle patterns on complex a?ne tori. It is proved that thereis a family of complex a?ne tori that admit SG circle patterns and the family covers thespace of ?at tori up to similarity by two to one manner except one point. Furthermore,the set of a?ne tori that admit circle pattern is dense in the Teichmu¨ller space of a?netori.