论文信息 - Schwarz's lemma for circle packings

Schwarz's lemma for circle packings

Burt Rodin* Department of Mathematics, University of California, La Jolla, CA 92093, USA 1. Introduction Conformal mappings can be approximated by circle packing isomorphisms; this is proved in [-10]. Roughly speaking (see Sect. 5 for a more detailed description), a bounded region f2 is almost filled by e-circles 7 from the regular hexagonal e-circle packing of the plane. Denote this approximate circle pack- ing of Q by Q=. By results of Andreev and Thurston, there is a combinatorially isomorphic circle packing D+ of the unit disk D. Denote this isomorphism, suitably normalized, by 7~--,7': f2=--*D=. Let f= be the piecewise linear quasicon- formal mapping determined by the associated triangulations. As e~0, f~ con- verges uniformly on compacta to the Riemann mapping function f: f2~D. Let

B. Rodin | Burt Rodin

[1] Dennis V. Lindley,et al. Principles of Random Walk. , 1965 .

[2] W. Mccrea,et al. XXII.—Random Paths in Two and Three Dimensions. , 1940 .

[3] Olli Lehto,et al. Quasiconformal mappings in the plane , 1973 .

[4] J. Pach,et al. Discrete Convex Functions and Proof of the Six Circle Conjecture of Fejes Tóth , 1984, Canadian Journal of Mathematics.

[5] L. Ahlfors,et al. RIEMANN'S MAPPING THEOREM FOR VARIABLE METRICS* , 1960 .

[6] S. E. Warschawski. On the degree of variation in conformal mapping of variable regions , 1950 .

[7] E. Stein,et al. Hp spaces of several variables , 1972 .

[8] Peter G. Doyle,et al. Random Walks and Electric Networks: REFERENCES , 1987 .

[9] W. Wasow,et al. Finite-Difference Methods for Partial Differential Equations , 1961 .

[10] W. Thurston. The geometry and topology of 3-manifolds , 1979 .

[11] W. Wasow. The accuracy of difference approximations to plane Dirichlet problems with piecewise analytic boundary values , 1957 .

[12] B. Rodin,et al. The convergence of circle packings to the Riemann mapping , 1987 .

[13] R. J. Duffin,et al. Discrete potential theory , 1953 .