Advertisement

Critical Points, the Gauss Curvature Equation and Blaschke Products

  • Daniela Kraus
  • Oliver RothEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 65)

Abstract

In this survey paper we discuss the problem of characterizing the critical sets of bounded analytic functions in the unit disk of the complex plane. This problem is closely related to the Berger–Nirenberg problem in differential geometry as well as to the problem of describing the zero sets of functions in Bergman spaces. It turns out that for any non-constant bounded analytic function in the unit disk there is always a (essentially) unique “maximal” Blaschke product with the same critical points. These maximal Blaschke products have remarkable properties similar to those of Bergman space inner functions and they provide a natural generalization of the class of finite Blaschke products.

Keywords

Blaschke products Elliptic PDEs Bergman spaces 

Mathematics Subject Classification

30H05 30J10 35J60 30H20 30F45 53A30 

Notes

Acknowledgements

The authors received support from the Deutsche Forschungsgemeinschaft (Grants: Ro 3462/3–1 and Ro 3462/3–2).

This paper is based on lectures given at the workshop on Blaschke products and their Applications (Fields Institute, Toronto, July 25–29, 2011). The authors would like to thank the organizers of this workshop, Javad Mashreghi and Emmanuel Fricain, as well as the Fields Institute and its staff, for their generous support and hospitality.

References

  1. 1.
    Ahlfors, L.: An extension of Schwarz’s lemma. Trans. Am. Math. Soc. 43, 359–364 (1938) MathSciNetGoogle Scholar
  2. 2.
    Ahlfors, L.: Bounded analytic functions. Duke Math. J. 14, 1–11 (1947) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Aubin, T.: Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin (1998) zbMATHGoogle Scholar
  4. 4.
    Aviles, P.: Conformal complete metrics with prescribed non-negative Gaussian curvature in ℝ2. Invent. Math. 83, 519–544 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Beardon, A., Minda, D.: The hyperbolic metric and geometric function theory. In: Ponnusamy, S., Sugawa, T., Vuorinen, M. (eds.) Quasiconformal Mappings and Their Applications. Narosa, New Delhi (2007) Google Scholar
  6. 6.
    Bland, J., Kalka, M.: Complete metrics conformal to the hyperbolic disc. Proc. Am. Math. Soc. 97(1), 128–132 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Blaschke, W.: Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen. S.-B. Sächs. Akad. Wiss. Leipz. Math.-Natur. Kl. 67, 194–200 (1915) Google Scholar
  8. 8.
    Bieberbach, L.: Δu=e u und die automorphen Funktionen. Math. Ann. 77, 173–212 (1916) MathSciNetCrossRefGoogle Scholar
  9. 9.
    Chang, S.Y.A.: Non-linear Elliptic Equations in Conformal Geometry. Eur. Math. Soc., Zurich (2004) CrossRefGoogle Scholar
  10. 10.
    Cheng, K.S., Ni, W.M.: On the structure of the conformal Gaussian curvature equation on ℝ2. Duke Math. J. 62(3), 721–737 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chou, K.S., Wan, T.: Asymptotic radial symmetry for solutions of Δu+e u=0 in a punctured disc. Pac. J. Math. 163(2), 269–276 (1994) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chou, K.S., Wan, T.: Correction to “Asymptotic radial symmetry for solutions of Δu+e u=0 in a punctured disc”. Pac. J. Math. 171(2), 589–590 (1995) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Colwell, P.: Blaschke Products. University of Michigan Press, Ann Arbor (1985) Google Scholar
  14. 14.
    Duren, P.: On the Bloch–Nevanlinna conjecture. Colloq. Math. 20, 295–297 (1969) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Duren, P.: Theory of H p Spaces. Dover, New York (2000) Google Scholar
  16. 16.
    Duren, P., Khavinson, D., Shapiro, H.S.: Extremal functions in invariant subspaces of Bergman spaces. Ill. J. Math. 40, 202–210 (1996) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Duren, P., Khavinson, D., Shapiro, H.S., Sundberg, C.: Contractive zero-divisors in Bergman spaces. Pac. J. Math. 157(1), 37–56 (1993) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Duren, P., Khavinson, D., Shapiro, H.S., Sundberg, C.: Invariant subspaces in Bergman spaces and the biharmonic equation. Mich. Math. J. 41(2), 247–259 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Duren, P., Schuster, A.: Bergman Spaces. Am. Math. Soc., Providence (2004) zbMATHGoogle Scholar
  20. 20.
    Ebenfelt, P., Khavinson, D., Shapiro, H.S.: Two–dimensional shapes and lemniscates. Complex Anal. Dyn. Syst. IV, Contemp. Math., 553, 45–59 (2011) MathSciNetGoogle Scholar
  21. 21.
    Forster, O.: Lectures on Riemann Surfaces. Springer, Berlin (1999) Google Scholar
  22. 22.
    Garnett, J.B.: Bounded Analytic Functions, revised 1st edn. Springer, Berlin (2007) Google Scholar
  23. 23.
    Grunsky, H.: Lectures on Theory of Functions in Multiply Connected Domains. Vandenhoeck & Rupprecht, Gortingen (1978) zbMATHGoogle Scholar
  24. 24.
    Hedenmalm, H.: A factorization theorem for square area-integrable analytic functions. J. Reine Angew. Math. 442, 45–68 (1991) MathSciNetGoogle Scholar
  25. 25.
    Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Springer, Berlin (2000) zbMATHCrossRefGoogle Scholar
  26. 26.
    Heins, M.: A class of conformal metrics. Bull. Am. Math. Soc. 67, 475–478 (1961) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Heins, M.: On a class of conformal metrics. Nagoya Math. J. 21, 1–60 (1962) MathSciNetzbMATHGoogle Scholar
  28. 28.
    Heins, M.: Some characterizations of finite Blaschke products of positive degree. J. Anal. Math. 46, 162–166 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Horowitz, C.: Zeros of functions in the Bergman spaces. Duke Math. J. 41, 693–710 (1974) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Horowitz, C.: Factorization theorems for functions in the Bergman spaces. Duke Math. J. 44, 201–213 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Hulin, D., Troyanov, M.: Prescribing curvature on open surfaces. Math. Ann. 293(2), 277–315 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Jensen, J.: Sur un nouvel et important théorème de la théorie des fonctions. Acta Math. 22, 359–364 (1899) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Kazdan, J.: Prescribing the Curvature of a Riemannian Manifold. CMBS Regional Conf. Ser. in Math., vol. 57 (1985) Google Scholar
  34. 34.
    Kalka, M., Yang, D.: On conformal deformation of nonpositive curvature on noncompact surfaces. Duke Math. J. 72(2), 405–430 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Keen, L., Lakic, N.: Hyperbolic Geometry from a Local Viewpoint. Cambridge University Press, Cambridge (2007) zbMATHCrossRefGoogle Scholar
  36. 36.
    Koosis, P.: Introduction to H p Spaces. Cambridge Tracts, 2nd edn. (1998) zbMATHGoogle Scholar
  37. 37.
    Korenblum, B.: An extension of the Nevanlinna theory. Acta Math. 135, 187–219 (1975) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Krantz, St.: Complex Analysis—The Geometric Viewpoint, 2nd edn. Math. Assoc. of America, Washington (2004) zbMATHCrossRefGoogle Scholar
  39. 39.
    Kraus, D.: Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature. Proc. Lond. Math. Soc., to appear Google Scholar
  40. 40.
    Kraus, D., Roth, O., Ruscheweyh, St.: A boundary version of Ahlfors’ lemma, locally complete conformal metrics and conformally invariant reflection principles for analytic maps. J. Anal. Math. 101, 219–256 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Kraus, D., Roth, O.: Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville’s theorem. J. Lond. Math. Soc. 77(1), 183–202 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Kraus, D., Roth, O.: Conformal metrics. In: Topics in Modern Function Theory, Ramanujan Math. Soc., 41 pp., to appear Google Scholar
  43. 43.
    Kraus, D., Roth, O.: Maximal Blaschke products, submitted Google Scholar
  44. 44.
    Laine, I.: Complex differential equations. In: Battelli, F., Fečkan, M. (eds.) Handbook of Differential Equations: Ordinary Differential Equations, vol. IV. Elsevier, Amsterdam (2008) Google Scholar
  45. 45.
    Liouville, J.: Sur l’équation aux différences partielles \(\frac{d^{2} \log \lambda}{du dv}\pm \frac{\lambda}{2 a^{2}}=0\). J. Math. 16, 71–72 (1853) Google Scholar
  46. 46.
    Luecking, D.H.: Zero sequences for Bergman spaces. Complex Var. Theory Appl. 30(4), 345–362 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Mashreghi, J.: Representation Theorems in Hardy Spaces. LMS Student Texts, vol. 74 (2009) zbMATHGoogle Scholar
  48. 48.
    Minda, C.D.: The hyperbolic metric and coverings of Riemann surfaces. Pac. J. Math. 84(1), 171–182 (1979) MathSciNetzbMATHGoogle Scholar
  49. 49.
    Minda, D.: Conformal metrics. Unpublished notes Google Scholar
  50. 50.
    Moser, J.: On a nonlinear problem in differential geometry. In: Dynamical Syst., Proc. Sympos., Univ. Bahia, Salvador, vol. 1971, pp. 273–280 (1973) Google Scholar
  51. 51.
    Ng, T.W., Wang, M.-X.: Ritt’s theory on the unit disk. Preprint (2011) Google Scholar
  52. 52.
    Nehari, Z.: A generalization of Schwarz’ lemma. Duke Math. J. 14, 1035–1049 (1947) MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Nevanlinna, R.: Über die Eigenschaften analytischer Funktionen in der Umgebung einer singulären Stelle oder Linie. Acta Soc. Sci. Fenn. 50(5) (1922), 46 pp. Google Scholar
  54. 54.
    Ni, W.-M.: Recent progress on the elliptic equation Δu+Ke 2u=0 on ℝ2. Rend. Semin. Mat., Torino Fasc. Spec., 1–10 (1989) Google Scholar
  55. 55.
    Nitsche, J.: Über die isolierten Singularitäten der Lösungen von Δu=e u. Math. Z. 68, 316–324 (1957) MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    Seip, K.: On a theorem of Korenblum. Ark. Mat. 32, 237–243 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Seip, K.: On Korenblum’s density condition for zero sequences of A α. J. Anal. Math. 67, 307–322 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, Berlin (1993) zbMATHCrossRefGoogle Scholar
  59. 59.
    Smith, S.J.: On the uniformization of the n-punctured disc. Ph.D. Thesis, University of New England (1986) Google Scholar
  60. 60.
    Stephenson, K.: Introduction to Circle Packing: The Theory of Discrete Analytic Functions. Cambridge University Press, Cambridge (2005) zbMATHGoogle Scholar
  61. 61.
    Struwe, M.: A flow approach to Nirenberg’s problem. Duke Math. J. 128(1), 19–64 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Sundberg, C.: Analytic continuability of Bergman inner functions. Mich. Math. J. 44(2), 399–407 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Walsh, J.: The Location of Critical Points of Analytic and Harmonic Functions. Am. Math. Soc., Providence (1950) zbMATHGoogle Scholar
  64. 64.
    Wang, Q., Peng, J.: On critical points of finite Blaschke products and the equation Δu=e 2u. Kexue Tongbao 24, 583–586 (1979) (Chinese) MathSciNetzbMATHGoogle Scholar
  65. 65.
    Yamada, A.: Bounded analytic functions and metrics of constant curvature on Riemann surfaces. Kodai Math. J. 11(3), 317–324 (1988) MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975) zbMATHCrossRefGoogle Scholar
  67. 67.
    Yau, S.T.: A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100, 197–203 (1978) zbMATHCrossRefGoogle Scholar
  68. 68.
    Zakeri, S.: On critical points of proper holomorphic maps on the unit disk. Bull. Lond. Math. Soc. 30(1), 62–66 (1996) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WürzburgWürzburgGermany

Personalised recommendations