Discrete Integrable Systems And M(?)bius Invariants Of Circle Packings

Discrete integrable system and circle packings are the important parts of discretedi?erential geometry. In this paper, our main work is as follow: ?rst, the Lax equationand Yang-Baxter map of discrete integrable system are discussed. The Lax equationof the discrete zero curvature condition for integrable system on quad-graphs is drived.The existence and uniqueness condition of the solution to Cauchy problem for discreteintegrable system is found. It is proved that the Lax equations of the discrete integrablesystem is equivalent to the cross-ratio system on quad-graph, and the Sym formula ofintegrable system on quad-graph is obtained. The Yang-Baxter variables are attachedto the edges of quad-graphs by sing the Yang-Baxter map and the associated adaptiveconditions for Cauchy problem are given. Then, M¨obius invariants of circle packingsare discussed. M¨obius invariants of circle packings are de?ned in terms of cross ratios.The necessary and su?cient conditions of existence of circle packings are establishedby the techniques of M¨obius invariants. It is shown that circle packings are uniquelydetermined, up to M¨obius transformations, by their M¨obius invariants. The rigidity ofin?nite circle packings with bounded degree is proved using then approach of M¨obiusinvariants..