, Volume 11, Issue 1, pp 35-49

Second derivatives of circle packings and conformal mappings

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William Thurston conjectured that the Riemann mapping functionf from a simply connected region Ω onto the unit disk \(\mathbb{D}\) can be approximated as follows. Almost fill Ω with circles of radius ɛ packed in the regular hexagonal pattern. There is a combinatorially isomorphic packing of circles in \(\mathbb{D}\) . The correspondencef ɛ of ɛ-circles in Ω with circles of varying radii in \(\mathbb{D}\) should converge tof after suitable normalization. This was proved in [RS], and in [H] an estimate was obtained which led to an approximation of |f′| in terms off ɛ ; namely, |f′| is the limit of the ratio of the radii of a target circle off ɛ to its source circle. In the present paper we show how to approximatef′ andf″ in terms off ɛ . Explicit rates for the convergence tof, f′, andf″ are obtained. In the special case of convergence to |f′|, the estimate in this paper improves the previously known estimate.