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Fredholm Eigenvalues of Jordan Curves: Geometric, Variational and Computational Aspects

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Analysis and Mathematical Physics

Part of the book series: Trends in Mathematics ((TM))

Abstract

The Fredholm eigenvalues of closed Jordan curves L on the Riemann sphere (especially their least nontrivial values pL = p 1 are intrinsically connected with conformai and quasiconformal maps and have various applications. These values have been investigated by many authors from different points of view.

We provide completely different quantitative and qualitative approaches which involve the complex Finsler geometry of the universal Teichmüller space, the metrics of generalized negative curvature and holomorphic motions.

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References

  1. L.V. Ahlfors, Remarks on the Neumann-Poincaréintegral equation, Pacific J. Math. 2 (1952), 271–280.

    MATH  MathSciNet  Google Scholar 

  2. L.V. Ahlfors, Lectures on Quasiconformal Mappings, Van Nostrand, Princeton, 1966.

    MATH  Google Scholar 

  3. J. Becker, Conformal mappings with quasiconformal extensions, Aspects of Contemporary Complex Analysis, Proc. Confer. Durham 1979 (D.A. Brannan and J.G. Clunie, eds.), Academic Press, New York, 1980, pp. 37–77.

    Google Scholar 

  4. E.M. Chirka, On the extension of holomorphic motions, Doklady Mathematics 70 (2004), 516–519.

    Google Scholar 

  5. S. Dineen, The Schwarz Lemma, Clarendon Press, Oxford, 1989.

    MATH  Google Scholar 

  6. C.J. Earle, I. Kra and S.L. Krushkal, Holomorphic motions and Teichmüller spaces, Trans. Amer. Math. Soc. 944 (1994), 927–948.

    Article  MathSciNet  Google Scholar 

  7. C.J. Earle and S. Mitra, Variation of moduli under holomorphic motions, In the tradition of Ahlfors and Bers (Stony Brook, NY, 1998), Contemp. Math. 256, Amer. Math. Soc, Providence, RI, 2000, pp. 39–67.

    Google Scholar 

  8. D. Galer, Konstruktive Methoden der konformen Abbildung, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964.

    Google Scholar 

  9. F.P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, Amer. Math. Soc., 2000.

    Google Scholar 

  10. A.Z. Grinshpan, Univalent functions and regularly measurable mappings, Siberian Math. J. 27 (1986), 825–837.

    Article  MATH  MathSciNet  Google Scholar 

  11. A.Z. Grinshpan and Ch. Pommerenke, The Grunsky norm and some coefficient estimates for bounded functions, Bull. London Math. Soc. 29 (1997), 705–712.

    Article  MathSciNet  Google Scholar 

  12. H. Grunsky, Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen, Math. Z. 45 (1939), 29–61.

    Article  MathSciNet  Google Scholar 

  13. M. Heins, A class of conformal metrics, Nagoya Math. J. 21 (1962), 1–60.

    MATH  MathSciNet  Google Scholar 

  14. S. Kobayashi, Hyperbolic Complex Spaces, Springer, New York, 1998.

    MATH  Google Scholar 

  15. S.L. Krushkal, Grunsky coefficient inequalities, Carathéodory metric and extremal quasiconformal mappings, Comment. Math. Helv. 64 (1989), 650–660.

    Article  MATH  MathSciNet  Google Scholar 

  16. S.L. Krushkal, Quasiconformal reflections across arbitrary planar sets, Sci. Ser. A Math. Sci. 8 (2002), 57–62.

    MATH  MathSciNet  Google Scholar 

  17. S.L. Krushkal, A bound for reflections across Jordan curves, Georgian Math. J. 10 (2003), 561–572.

    MATH  MathSciNet  Google Scholar 

  18. S.L. Krushkal, Plurisubharmonic features of the Teichmüller metric, Publications de l’Institut Mathématique-Beograd, Nouvelle série 75(89) (2004), 119–138.

    MathSciNet  Google Scholar 

  19. S.L. Krushkal, Univalent holomorphic functions with quasiconformal extensions (variational approach), Ch. 5 in: Handbook of Complex Analysis: Geometric Function Theory, Vol. II (R. Kühnau, ed.), Elsevier Science, Amsterdam, 2005, pp. 165–241.

    Chapter  Google Scholar 

  20. S.L. Krushkal, Quasiconformal extensions and reflections, Ch. 11 in: Handbook of Complex Analysis: Geometric Function Theory, Vol. II (R. Kühnau, ed.), Elsevier Science, Amsterdam, 2005, pp. 507–553.

    Chapter  Google Scholar 

  21. S.L. Krushkal, Complex Finsler metrics and quasiconformal characteristics of maps and curves, Revue Roumalne Pure et Appl. 51 (2006), 665–682.

    MATH  MathSciNet  Google Scholar 

  22. S.L. Krushkal, Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm, eigenvalues, Central European J. Math 5(3) (2007), 551–580.

    Article  MATH  MathSciNet  Google Scholar 

  23. S.L. Krushkal and R. Kühnau, Quasikonforme Abbildungen — neue Methoden und Anwendungen, Teubner-Texte zur Math., Bd. 54, Teubner, Leipzig, 1983.

    Google Scholar 

  24. S.L. Krushkal and R. Kühnau, Grunsky inequalities and quasiconformal extension, Israel J. Math. 152 (2006), 49–59.

    Article  MATH  MathSciNet  Google Scholar 

  25. R. Kühnau, Verzerrungssätze und, Koeffizientenbedingungen vom Grunskyschen Typ für quasikonforme Abbildungen, Math. Nachr. 48 (1971), 77–105.

    Article  MATH  MathSciNet  Google Scholar 

  26. R. Kühnau, Zur Berechnung der Fredholmschen Eigenwerte ebener Kurven, ZAMM 66 (1986), 193–201.

    Article  MATH  Google Scholar 

  27. R. Kühnau, Möglichst konforme Spiegelung an einer Jordankurve, Jber. Deutsch. Math. Verein. 90 (1988), 90–109.

    MATH  Google Scholar 

  28. R. Kühnau, Wann sind die Grunskyschen Koeffizientenbedingungen hinreichend für Q-quasikonforme Fortsetzbarkeit? Comment. Math. Helv. 61 (1986), 290–307.

    Article  MATH  MathSciNet  Google Scholar 

  29. R. Kühnau, Öber die Grunskyschen Koeffizientenbedingungen, Ann. Univ. Mariae Curie-Sklodowska, sect. A 54 (2000), 53–60.

    MATH  MathSciNet  Google Scholar 

  30. D. Minda, The strong form of Ahlfors’ lemma, Rocky Mountaln J. Math., 17 (1987), 457–461.

    Article  MATH  MathSciNet  Google Scholar 

  31. Chr. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.

    MATH  Google Scholar 

  32. H.L. Royden, Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces (Ann. of Math. Stud., vol. 66), Princeton Univ. Press, Princeton, 1971, pp. 369–383.

    Google Scholar 

  33. H.L. Royden, The Ahlfors-Schwarz lemma: the case of equality, J. Anal. Math. 46 (1986), 261–270.

    Article  MATH  MathSciNet  Google Scholar 

  34. M. Schiffer, Extremum problems and variational methods in conformal mapping, 1958 Internat. Congress Math., Cambridge Univ. Press, New York, 1958, pp. 211–231.

    Google Scholar 

  35. M. Schiffer, Fredholm, eigenvalues and Grunsky matrices, Ann. Polon. Math. 39 (1981), 149–164.

    MathSciNet  Google Scholar 

  36. G. Schober, Estimates for Fredholm eigenvalues based on quasiconformal mapping, Numerische, insbesondere approximationstheoretische Behandlung von Funktionsgleichungen. Lecture Notes in Math. 333, Springer-Verlag, Berlin, 1973, pp. 211–217.

    Chapter  Google Scholar 

  37. Y.L. Shen, Pull-back operators by quasisymmetric functions and invariant metrics on Teichmüller spaces, Complex Variables 42 (2000), 289–307.

    MATH  Google Scholar 

  38. Z. Slodkowski, Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc. 111 (1991), 347–355.

    Article  MATH  MathSciNet  Google Scholar 

  39. K. Strebel, On the existence of extremal Teichmueller mappings, J. Anal. Math 30 (1976), 464–480.

    Article  MATH  MathSciNet  Google Scholar 

  40. S.E. Warschawski, On the effective determination of conformal maps, Contribution to the Theory of Riemann surfaces (L. Ahlfors, E. Calabi et al., eds.), Princeton Univ. Press, Princeton, 1953, pp. 177–188.

    Google Scholar 

  41. S. Werner, Spiegelungskoeffizient und Fredholmscher Eigenwert für gewisse Polygone, Ann. Acad. Sci. Fenn. Ser.AI. Math. 22 (1997), 165–186.

    Google Scholar 

  42. I.V. Zhuravlev, Univalent functions and Teichmüller spaces, Inst. of Mathematics, Novosibirsk, preprint, 1979, 23 pp. (Russian).

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Bar-Ilan University, 52900, Ramat-Gan, Israel

    Samuel Krushkal

  2. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Samuel Krushkal

Authors
  1. Samuel Krushkal
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology (KTH), 100 44, Stockholm, Sweden

    Björn Gustafsson

  2. Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008, Bergen, Norway

    Alexander Vasil’ev

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Krushkal, S. (2009). Fredholm Eigenvalues of Jordan Curves: Geometric, Variational and Computational Aspects. In: Gustafsson, B., Vasil’ev, A. (eds) Analysis and Mathematical Physics. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9906-1_16

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