Solving Beltrami equations by circle packing
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- Trans. Amer. Math. Soc. 322 (1990), 657-670 Request permission
Abstract:
We use Andreev-Thurston’s theorem on the existence of circle packings to construct approximating solutions to the Beltrami equations on Riemann surfaces. The convergence of the approximating solutions on compact subsets will be shown. This gives a constructive proof of the existence theorem for Beltrami equations.References
- Lars V. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand Mathematical Studies, No. 10, D. Van Nostrand Co., Inc., Toronto, Ont.-New York-London, 1966. Manuscript prepared with the assistance of Clifford J. Earle, Jr. MR 0200442
- Lars Ahlfors and Lipman Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385–404. MR 115006, DOI 10.2307/1970141 L. Bers, Riemann surfaces (mimeographed lectures notes), New York University, 1957-58.
- L. Bers and L. Nirenberg, On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications, Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali, Trieste, 1954, Edizioni Cremonese, Roma, 1955, pp. 111–140. MR 0076981
- B. V. Boyarskiĭ, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients, Mat. Sb. N.S. 43(85) (1957), 451–503 (Russian). MR 0106324
- Shiing-shen Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc. 6 (1955), 771–782. MR 74856, DOI 10.1090/S0002-9939-1955-0074856-1
- G. M. Golusin, Geometrische Funktionentheorie, Hochschulbücher für Mathematik, Band 31, VEB Deutscher Verlag der Wissenschaften, Berlin, 1957 (German). MR 0089896
- Zheng-Xu He, An estimate for hexagonal circle packings, J. Differential Geom. 33 (1991), no. 2, 395–412. MR 1094463 A. Korn, Zwei Anwendungen der Methode der sukzessiven Annäherungen, Schwarz Festschrift, Berlin, 1919, pp. 215-229. M. A. Lavrentiev, Sur une class des représentations continues, Mat. Sb. (N. S.) 42 (1935), 407-434.
- O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463 L. Lichtenstein, Zur Theorie der konformen Abbildungen; Konforme Abbildungen nichtanalytischer singularitätenfreir Flächenstücke auf ebene Gebiete, Bull. Acad. Sci. Cracovie (1916), 192-217. A. Marden and B. Rodin, On Thurston’s formulation and proof of Andreev’s theorem, preprint.
- Olli Martio, Boundary values and injectiveness of the solutions of Beltrami equations, Ann. Acad. Sci. Fenn. Ser. A I No. 402 (1967), 27. MR 0214917
- Charles B. Morrey Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166. MR 1501936, DOI 10.1090/S0002-9947-1938-1501936-8
- Burt Rodin and Dennis Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), no. 2, 349–360. MR 906396 W. Thurston, The geometry and topology of $3$-manifolds, Princeton University Notes, Princeton, N. J., 1979.
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 322 (1990), 657-670
- MSC: Primary 30C62; Secondary 30C35, 52C15
- DOI: https://doi.org/10.1090/S0002-9947-1990-0974518-5
- MathSciNet review: 974518