Circle Patterns And Discrete Integrable Systems

The theory of circle patterns is a rich fascinating area having its origin in the clas-sical theory of circle packings. Its fast development in recent years is caused by themutual in?uence and interplay of ideas and concepts from discrete di?erential geometry,complex analysis and the theory of integrable systems. Circle packings are configura-tions of circles in a constant curvature surface with prescribed pattern of tangencies.The progress in this area was initiated by W.Thurston in 1985 when he suggested amethod for approximation of the Riemann mapping by hexagonal circle packings. In1987, Rodin and Sullivan proved the convergence of Thurston's scheme, which gives anew discrete geometry view of the Riemann mapping. For the study of circle configura-tions, classical circle packings consisting of disjoint open disks were generalized to circlepatterns, where the disks may overlap. In this thesis, our main work is as follows. First,we investigate the SG circle patterns with a negative constant cross-ratio, that is, forevery circle the cross-ratio of its four intersection points with neighboring circles is equalto a negative constant. The existence of such circle patterns is obtained by solving asuitable Cauchy problem. The relation between SG circle patterns and integrable sys-tems on the square grid is established. A class of isomonodromic solutions of integrablesystem on the square grid is discussed. Discrete analogous of analytic functions zαandlog z are presented in terms of SG circle patterns. Next, the quasicrystallic circle pat-terns with constant angles are defined by using the relationship between quasicrystallicrhombic embeddings and multi-dimensional regular square lattice Z+d. The cross-ratiosystems on Z+d are established, and their zero curvature conditions are given. Also aclass of isomonodromic solutions determined by the cross-ratio equations and a non-autonomous constraint on Zd+ are discussed. The existence of the quasicrystallic circlepatterns with constant angles is obtained by solving some suitable Cauchy problems forthe cross-ratio systems.