Circle Patterns with the Combinatorics of the Square Grid

Explicit families of entire circle patterns with the combinatorics of the square grid are constructed, and it is shown that the collection of entire, locally univalent circle patterns on the sphere is innnite dimensional. In Particular, Doyle's conjecture is false in this setting. MM obius invariants of circle patterns are introduced, and turn out to be discrete analogs of the Schwarzian derivative. The invariants satisfy a nonlinear discrete version of the Cauchy-Riemann equations. A global analysis of the solutions of these equations yields a rigidity theorem characterizing the Doyle spirals. It is also shown that by prescribing boundary values for the MM obius invariants, and solving the appropriate Dirichlet problem, a locally univalent meromorphic function can be approximated by circle patterns. Some of the work exposed here was done while the author visited UCSD, and the author wishes to express thanks to the UCSD mathematics department for its hospitality.