Discrete Uniformization Theorem For Inversive Distance Circle Packings

The uniformization theorem is an important result in complex analysis,which claims that every simply connected Riemann surface conformally identifies with either S~2,C or D.Ricci flow is one strategy for establishing the uniformization theorem.In numerical computational contexts,the discretized version of Ricci flow implemented on circle pack-ings may be used to solve for discretized uniformization maps.Inversive distance is one common method for quantifying the positional relationship between two positive genus,does discrete Ricci flow with respect to inversive distance converge to a unique constant Gaussian curvature surface?This thesis resolves this problem in the positive,the central technique of this proof lies in the technical computation of the transformation of inversive distances with respect to flips on triangulations.During the course of the“existence”part of the proof,we also establish the exis-tence and uniqueness of weighted Voronoi cell decompositions and their dual Delaunay triangulations for three types of surfaces.Moreover,we construct a type of real analytic cell decomposition and a C~1diffeomorphism between two distinct families of Teichmüller spaces.