Conformal welding and Koebe's theorem

Annals of Mathematics

Volumn 166, Issue 3, 2007, Pages 613-656

  Bishop, Christopher J ^a  

^a STONY BROOK UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 41549159387     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.4007/annals.2007.166.613     Document Type: Article

References (47)

1. L. AHLFORS and A. BEURLING, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101-129.
2. LARS V. AHLFORS, Lectures on Quasiconformal Mappings, D. Van Nostrand Co., Inc., Toronto, Ontario-New York-London, 1966.
3. Z. BALOGH and M. BONK, Lengths of radii under conformal maps of the unit disc, Proc. Amer. Math. Soc. 127 (1999), 801-804.
4. A. BEURLING and L. AHLFORS, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125-142.
5. C. J. BISHOP, Constructing continuous functions holomorphic off acurve, J. Funct. Anal. 82 (1989), 113-137.
6. _, Boundary interpolation sets for conformal maps, Bull. London Math. Soc. 38 (2006), 607-616.
7. _, Some homeomorphisms of the sphere conformal off a curve, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), 323-338.
8. C. J. BISHOP, L. CARLESON, J. B. GARNETT, and P. W. JONES, Harmonic measures supported on curves, Pacific J. Math. 138 (1989), 233-236.
9. A. BROWDER, Introduction to Function Algebras, W. A. Benjamin, Inc., New York-Amsterdam, 1969.
10. A. BROWDER and J. WERMER, Some algebras of functions on an arc, J. Math. Mech. 12 (1963), 119-130.
11. _, A method for constructing Dirichlet algebras, Proc. Amer. Math. Soc. 15 (1964), 546-552.
12. L. CARLESON, Representations of continuous functions, Math. Z. 66 (1957), 447-451.
13. _, Selected Problems on Exceptional Sets, Van Nostrand, 1967.
14. R. J. DAVERMAN, Decompositions of Manifolds, Volume 124 of Pure and Applied Mathematics, Academic Press Inc., Orlando, FL, 1986.
15. G. DAVID, Solutions de l'équation de Beltrami avec |μ|∞ = 1, Ann. Acad. Sci. Fenn. Ser. A I Math. 13 (1988), 25-70.
16. PETER DUREN, HAROLD M. EDWARDS, and UTA C. MERZBACH, editors, A Century of Mathematics in America. Part III, American Mathematical Society, Providence, RI, 1989.
17. B. FITZPATRICK, JR, Some aspects of the work and influence of R. L. Moore, in Handbook of the History of General Topology 1 (1997), 41-61, Kluwer Acad. Publ., Dordrecht.
18. F. W. GEHRING and W. K. HAYMAN, An inequality in the theory of conformal mapping, J. Math. Pures Appl. 41 (1962) 353-361.
19. D. H. HAMILTON, Generalized conformal welding, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), 333-343.
20. _, Simultaneous uniformisation, J. Reine Angew. Math. 455 (1994), 105-122.
21. _, Length of Julia curves, Pacific J. Math. 169 (1995), 75-93.
22. _, Conformal welding, in The Handbook of Geometric Function Theory. North Holland, 2002.
23. Z.-X. HE and O. SCHRAMM, Fixed points, Koebe uniformization and circle packings, Ann. of Math. 137 (1993), 369-406.
24. _, Koebe uniformization for "almost circle domains", Amer. J. Math. 117 (1995), 653-667.
25. R. KAUFMAN, Fourier-Stieltjes coefficients and continuation of functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 27-31.
26. _, Plane curves and removable sets, Pacific J. Math. 125 (1986), 409-413.
27. P. KOEBE, Uber die uniformisierung beliebiger analytisher kurven, III, Nachr. Ges. Wiss. Gott (1908), 337-358.
28. _, Abhandlungen zur theorie der konformen abbildung: Vi abbildung mehrfach zusammenhängender bereiche auf kreisbereiche etc., Math. Z. 7 (1920), 235-301.
29. O. LEHTO, Homeomorphisms with a given dilatation, In Proceedings of the Fifteenth Scandinavian Congress (Oslo, 1968, 58-73, Berlin, 1970, Springer.
30. O. LEHTO and K. I. VIRTANEN, On the existence of quasiconformal mappings with prescribed complex dilatation, Ann. Acad. Sci. Fenn. Ser. A I No. 274 (1960), 24.
31. _, Quasiconformal Mappings in the Plane, Springer-Verlag, New York, second edition, 1973, Translated from the German by K. W. Lucas, Die Grundlehren der mathematischen Wissenschaften, Band 126.
32. K. LUNDBERO, A theorem on the boundary behavior of a uniformly convergent sequence of conformal maps on the disk, PhD thesis, State University of New York at Stony Brook, Stony Brook, NY, USA, August 2005.
33. R. L. MOORE, Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc. 27 (1925), 416-428.
34. _, Concerning triods in the plane and the junction points of plane continua, Proc. Nat. Acad. Sci. 14 (1928), 85-88.
35. K. OIKAWA, Welding of polygons and the type of Riemann surfaces, Kōdai Math. Sem. Rep. 13 (1961), 37-52.
36. ALBERT PFLUGER, Ueber die Konstruktion Riemannscher Flächen durch Verheftung, J. Indian Math. Soc'(N.S.) 24 (1960), 401-412.
37. C. R. PITTMAN, An elementary proof of the triod theorem, Proc. Amer. Math. Soc. 25 (1970), 919.
38. CH. POMMERENKE, On the boundary continuity of conformal maps, Pacific J. Math. 120 (1985), 423-430.
39. CH. POMMERENKE, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
40. W. RUDIN, Boundary values of continuous analytic functions, Proc. Amer. Math. Soc. 7 (1956), 808-811.
41. L. SARIO and K. OIKAWA, Capacity Functions, Springer-Verlag New York Inc., New York, 1969.
42. J. V. VAINIO, Conditions for the possibility of conformal sewing, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 53 (1985), 43.
43. _, On the type of sewing functions with a singularity, Ann. Acad. Sci. Fenn. Ser. A 1 Math. 14 (1989), 161-167.
44. _, Properties of real sewing functions, Ann. Acad. Sci. Fenn. Ser. A 1 Math. 20 (1995), 87-95.
45. R. L. WILDER, The mathematical work of R. L. Moore: its background, nature and influence, Arch. Hist. Exact Sci. 26 (1982), 73-97.
46. G. B. WILLIAMS, Approximation of quasisymmetries using circle packings, Discrete Comput. Geom. 25 (2001), 103-124.
47. _, Discrete conformal welding, Indiana Univ. Math. J. 53 (2004), 765-804.