William Thurston proposed regarding the map induced from two circle packings with the same tangency pattern as a discrete holomorphic function. A discrete analogue of the Riemann mapping is deduced from Koebe-Andreev-Thurston theorem. A natural question is how to extend this theory to Riemann surfaces and relate classical conformal structures to discrete conformal structures. Since circles are preserved under complex projective transformations, one can consider circle packings on surfaces with complex projective structures. Kojima, Mizushima and Tan conjectured that for a given combinatorics the deformation space of circle packings is diffeomorphic to the Teichmueller space via a natural projection. In this talk, we will report progress on the torus case.

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Last updated: 17 May 2024