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1999 | 48 | 1 | 43-53
Tytuł artykułu

What is a disk?

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EN
Abstrakty
EN
This paper should be considered as a companion report to F.W. Gehring's survey lectures "Characterizations of quasidisks" given at this Summer School [7]. Notation, definitions and background results are given in that paper. In particular, D is a simply connected proper subdomain of $R^2$ unless otherwise stated and D* denotes the exterior of D in $\overline{R}^2$. Many of the characterizations of quasidisks have been motivated by looking at properties of euclidean disks. It is therefore natural to go back and ask if any of the original properties in fact characterize euclidean disks. We follow the procedure in Gehring's lectures and look at four different categories of properties: 1. Geometric properties, 2. Conformal invariants, 3. Injectivity criteria, 4. Extension properties. As we shall see, the answers are not equally easy to obtain and not always positive. There are, in fact, still many interesting open questions.
Słowa kluczowe
Rocznik
Tom
48
Numer
1
Strony
43-53
Opis fizyczny
Daty
wydano
1999
Twórcy
autor
  • Norwegian University of Science and Technology, N-7491 Trondheim, Norway
Bibliografia
  • [1] L. V. Ahlfors, Complex analysis, McGraw-Hill 1979.
  • [2] L. V. Ahlfors, Conformal invariants, McGraw-Hill 1973.
  • [3] L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291-301.
  • [4] K. Astala and F. W. Gehring, Injectivity, the BMO norm and the universal Teichmüller space, J. d'Analyse Math. 46 (1986), 16-57.
  • [5] J. Becker and Ch. Pommerenke, Schlichtheitskriterien und Jordangebiete, J. reine angew. Math. 354 (1984), 74-94,
  • [6] M. Berger, Geometry I and II, Springer-Verlag 1980.
  • [7] F. W. Gehring, Characterizations of quasidisks, this volume.
  • [8] F. W. Gehring, Univalent functions and the Schwarzian derivative, Comm. Math. Helv. 52 (1977), 561-572.
  • [9] F. W. Gehring, Injectivity of local quasi-isometries, Comm. Math. Helv. 57 (1982), 202-220.
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  • [11] F. W. Gehring and B. G. Osgood, Uniform domains and the quasihyperbolic metric, J. d'Analyse Math. 36 (1979), 50-74.
  • [12] V. M. Gol'dstein and S. K. Vodop'janov, Prolongement des fonctions de classe $L^p_1$ et applications quasi conformes, C. R. Acad. Sc. Paris 290 (1980), 453-456.
  • [13] F. John, On quasi-isometric mappings II, Comm. Pure Appl. Math. 22 (1969), 265-278.
  • [14] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71-88.
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  • [17] O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag 1973.
  • [18] O. Martio and J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. 4 (1978-79), 383-401.
  • [19] Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551.
  • [20] M. H. A. Newman, The topology of plane point sets, Cambridge University Press 1954.
  • [21] Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag 1992.
  • [22] S. Rickman, Extensions over quasiconformally equivalent curves, Ann Acad. Sci. Fenn. 10 (1985), 511-514.
  • [23] D. Stowe, Injectivity and the pre-Schwarzian derivative, Mich. Math. J. 45 (1998), 537-546.
  • [24] M. Walker, Linearly locally connected sets and quasiconformal mappings, Ann. Acad. Sci. Fenn. 11 (1986), 77-86.
  • [25] S. Yang, QED domains and NED sets in $\overline{R}^n$, Trans. Amer. Math. Soc. 334 (1992), 97-120.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-bcpv48i1p43bwm
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