Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues

Samuel Krushkal

Open Mathematics (2007)

  • Volume: 5, Issue: 3, page 551-580
  • ISSN: 2391-5455

Abstract

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The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = z: |z| > 1 can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.

How to cite

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Samuel Krushkal. "Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues." Open Mathematics 5.3 (2007): 551-580. <http://eudml.org/doc/269263>.

@article{SamuelKrushkal2007,
abstract = {The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to \[\widehat\{\mathbb \{C\}\}\] are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = z: |z| > 1 can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.},
author = {Samuel Krushkal},
journal = {Open Mathematics},
keywords = {quasiconformal; univalent function; Grunsky coefficient inequalities; universal Teichmüller space; subharmonic function; Strebel’s point; Kobayashi metric; generalized Gaussian curvature; holomorphic curvature; Fredholm eigenvalues; univalent functions; quasiconformal mappings; Strebel's point},
language = {eng},
number = {3},
pages = {551-580},
title = {Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues},
url = {http://eudml.org/doc/269263},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Samuel Krushkal
TI - Strengthened Moser’s conjecture, geometry of Grunsky coefficients and Fredholm eigenvalues
JO - Open Mathematics
PY - 2007
VL - 5
IS - 3
SP - 551
EP - 580
AB - The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to \[\widehat{\mathbb {C}}\] are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = z: |z| > 1 can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality.
LA - eng
KW - quasiconformal; univalent function; Grunsky coefficient inequalities; universal Teichmüller space; subharmonic function; Strebel’s point; Kobayashi metric; generalized Gaussian curvature; holomorphic curvature; Fredholm eigenvalues; univalent functions; quasiconformal mappings; Strebel's point
UR - http://eudml.org/doc/269263
ER -

References

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  1. [1] H. Grunsky: “Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen”, Math. Z., Vol. 45, (1939), pp. 29–61. http://dx.doi.org/10.1007/BF01580272 Zbl0022.15103
  2. [2] R. Kühnau: “Verzerrungssätze und Koeffizientenbedingungen vom Grunskyschen Typ für quasikonforme Abbildungen”, Math. Nachr., Vol. 48, (1971), pp. 77–105. http://dx.doi.org/10.1002/mana.19710480107 Zbl0226.30021
  3. [3] C. Pommerenke: Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975. 
  4. [4] I.V. Zhuravlev: “Univalent functions and Teichmüller spaces”, Preprint: Inst. of Mathematics, Novosibirsk, 1979, p. 23 (Russian). Zbl0467.30037
  5. [5] S.L. Krushkal and R. Kühnau: Quasikonforme Abbildungen-neue Methoden und Anwendungen, Teubner-Texte zur Math., Bd. 54, Teubner, Leipzig, 1983. Zbl0539.30001
  6. [6] R. Kühnau: “Möglichst konforme Spiegelung an einer Jordankurve”, Jber. Deutsch. Math. Verein., Vol. 90, (1988), pp. 90–109. Zbl0638.30021
  7. [7] S.L. Krushkal: “Grunsky coefficient inequalities, Carathéodory metric and extremal quasiconformal mappings”, Comment. Math. Helv., Vol. 64, (1989), pp. 650–660. http://dx.doi.org/10.1007/BF02564699 Zbl0697.30016
  8. [8] S.L. Krushkal and R. Kühnau: “Grunsky inequalities and quasiconformal extension”, Israel J. Math., Vol. 152, (2006), pp. 49–59. Zbl1126.30013
  9. [9] R. Kühnau: “Zu den Grunskyschen Coeffizientenbedingungen”, Ann. Acad. Sci. Fenn. Ser. A. I. Math., Vol. 6, (1981), pp. 125–130. Zbl0454.30016
  10. [10] R. Kühnau: “Wann sind die Grunskyschen Koeffizientenbedingungen hinreichend für Q-quasikonforme Fortsetzbarkeit?”, Comment. Math. Helv., Vol. 61, (1986), pp. 290–307. http://dx.doi.org/10.1007/BF02621917 Zbl0605.30023
  11. [11] S.L. Krushkal: “Beyond Moser’s conjecture on Grunsky inequalities”, Georgian Math. J., Vol. 12, (2005), pp. 485–492. Zbl1087.30042
  12. [12] Y.L. Shen: “Pull-back operators by quasisymmetric functions and invariant metrics on Teichmüller spaces”, Complex Variables, Vol. 42, (2000), pp. 289–307. Zbl1023.30024
  13. [13] L. Ahlfors: “An extension of Schwarz’s lemma”, Trans. Amer. Math. Soc., Vol. 43, (1938), pp. 359–364. http://dx.doi.org/10.2307/1990065 Zbl64.0315.04
  14. [14] D. Gaier: Konstruktive Methoden der konformen Abbildung, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964. Zbl0132.36702
  15. [15] R. Kühnau: “Zur Berechnung der Fredholmschen Eigenwerte ebener Kurven”, Z. Angew. Math. Mech., Vol. 66, (1986), pp. 193–200. http://dx.doi.org/10.1002/zamm.19860660602 Zbl0608.65092
  16. [16] M. Schiffer: “Fredholm eigenvalues and Grunsky matrices”, Ann. Polon. Math., Vol. 39, (1981), pp. 149–164. 
  17. [17] G. Schober: “Estimates for Fredholm eigenvalues based on quasiconformal mapping.” In: Numerische, insbesondere approximationstheoretische Behandlung von Funktiongleichungen, Lecture Notes in Math., Vol. 333, Springer-Verlag, Berlin, (1973), pp. 211–217. http://dx.doi.org/10.1007/BFb0060699 
  18. [18] S.E. Warschawski: “On the effective determination of conformal maps”, In: L. Ahlfors, E. Calabi et al. (Eds.): Contribution to the Theory of Riemann surfaces, Princeton Univ. Press, Princeton, (1953), pp. 177–188. Zbl0052.35403
  19. [19] L. Ahlfors: “Remarks on the Neumann-Poincaré integral equation”, Pacific J. Math. Vol. 2, (1952), pp. 271–280. Zbl0047.07907
  20. [20] S.L. Krushkal: “Quasiconformal extensions and reflections”, In: R. Kühnau (Ed.): Handbook of Complex Analysis: Geometric Function Theory, Vol. II, Elsevier Science, Amsterdam, 2005, pp. 507–553. 
  21. [21] K. Strebel: “On the existence of extremal Teichmueller mappings”, J. Anal. Math, Vol. 30, (1976), pp. 464–480. Zbl0334.30013
  22. [22] F.P. Gardiner and N. Lakic: Quasiconformal Teichmüller Theory, Amer. Math. Soc., 2000. 
  23. [23] S. Dineen: The Schwarz Lemma, Clarendon Press, Oxford, 1989. 
  24. [24] S. Kobayayshi: Hyperbolic Complex Spaces, Springer, New York, 1998. 
  25. [25] C.J. Earle, I. Kra and S.L. Krushkal: “Holomorphic motions and Teichmüller spaces”, Trans. Amer. Math. Soc., Vol. 944, (1994), pp. 927–948. http://dx.doi.org/10.2307/2154750 Zbl0812.30018
  26. [26] C.J. Earle and S. Mitra: “Variation of moduli under holomorphic motions”, In: Stony Brook, NY, 1998: The tradition of Ahlfors and Bers, Contemp. Math. Vol. 256, Amer. Math. Soc., Providence, RI, 2000, pp. 39–67. Zbl0972.30005
  27. [27] 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. 
  28. [28] S.L. Krushkal: “Plurisubharmonic features of the Teichmüller metric”, Publications de l’Institut Mathématique-Beograd, Nouvelle série, Vol. 75, (2004), pp. 119–138. Zbl1079.30019
  29. [29] N.A. Lebedev: The Area Principle in the Theory of Univalent Functions, Nauka, Moscow, 1975 (Russian). Zbl0747.30015
  30. [30] I.M. Milin: “Univalent Functions and Orthonormal Systems”, Transl. of mathematical monographs, vol. 49, Transl. of Odnolistnye funktcii i normirovannie systemy, Amer. Math. Soc., Providence, RI, 1977. 
  31. [31] M. Schiffer and D. Spencer: Functionals of finite Riemann Surfaces, Princeton Univ. Press, Princeton, 1954. 
  32. [32] S. L. Krushkal: “Schwarzian derivative and complex Finsler metrics”, Contemporary Math., Vol. 382, (2005), pp. 243–262. Zbl1085.30009
  33. [33] D. Minda: “The strong form of Ahlfors’ lemma”, Rocky Mountain J. Math., Vol. 17, (1987), pp. 457–461. http://dx.doi.org/10.1216/RMJ-1987-17-3-457 Zbl0628.30031
  34. [34] M. Abate and G. Patrizio: “Isometries of the Teichmüller metric”, Ann. Scuola Super. Pisa Cl. Sci., Vol. 26, (1998), pp. 437–452. Zbl0933.32027
  35. [35] V. Božin, N. Lakic, V. Markovic and M. Mateljević: “Unique extremality”, J. Anal. Math., Vol. 75, (1998), pp. 299–338. http://dx.doi.org/10.1007/BF02788704 Zbl0929.30017
  36. [36] C.J. Earle and Zong Li: “Isometrically embedded polydisks in infinite dimensional Teichmüller spaces”, J. Geom. Anal., Vol. 9, (1999), pp. 51–71. Zbl0963.32004
  37. [37] M. Heins: “A class of conformal metrics”, Nagoya Math. J., Vol. 21, (1962), pp. 1–60. Zbl0113.05603
  38. [38] S.L. Krushkal: Quasiconformal Mappings and Riemann Surfaces, Wiley, New York, 1979. 
  39. [39] S.L. Krushkal: “A bound for reflections across Jordan curves”, Georgian Math. J., Vol. 10, (2003), pp. 561–572. Zbl1061.30041
  40. [40] R. Kühnau: “Über die Grunskyschen Koeffizientenbedingungen”, Ann. Univ. Mariae Curie-Sklodowska, Sect. A, Vol. 54, (2000), pp. 53–60. 
  41. [41] O. Lehto: Univalent Functions and Teichmüller Spaces, Springer-Verlag, New York, 1987. 
  42. [42] Yu.G. Reshetnyak: “Two-dimensional manifolds of bounded curvature”, Geometry, IV, Encyclopaedia Math. Sci., Vol. 70, Springer, Berlin, 1993, pp. 3–163, 245–250. Engl. transl. from: iTwo-dimensional manifolds of bounded curvature, Geometry, 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1989, pp. 189, 273–277, 279 (Russian). 
  43. [43] H.L. Royden: “The Ahlfors-Schwarz lemma: the case of equality”, J. Anal. Math., Vol. 46, (1986), pp. 261–270. Zbl0604.30027

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