Complex Analysis and Operator Theory

, Volume 4, Issue 3, pp 541–587

Geometry Behind Chordal Loewner Chains

  • Manuel D. Contreras
  • Santiago Díaz-Madrigal
  • Pavel Gumenyuk


Loewner Theory is a deep technique in Complex Analysis affording a basis for many further important developments such as the proof of famous Bieberbach’s conjecture and well-celebrated Schramm’s stochastic Loewner evolution. It provides analytic description of expanding domains dynamics in the plane. Two cases have been developed in the classical theory, namely the radial and the chordal Loewner evolutions, referring to the associated families of holomorphic self-mappings being normalized at an internal or boundary point of the reference domain, respectively. Recently there has been introduced a new approach (Bracci F et al. in Evolution families and the Loewner equation I: the unit disk. Preprint 2008. Available on ArXiv 0807.1594; Bracci F et al. in Math Ann 344:947–962, 2009; Contreras MD et al. in Loewner chains in the unit disk. To appear in Revista Matemática Iberoamericana; preprint available at arXiv:0902.3116v1 [math.CV]) bringing together, and containing as quite special cases, radial and chordal variants of Loewner Theory. In the framework of this approach we address the question what kind of systems of simply connected domains can be described by means of Loewner chains of chordal type. As an answer to this question we establish a necessary and sufficient condition for a set of simply connected domains to be the range of a generalized Loewner chain of chordal type. We also provide an easy-to-check geometric sufficient condition for that. In addition, we obtain analogous results for the less general case of chordal Loewner evolution considered in (Aleksandrov IA et al. in Complex Analysis. PWN, Warsaw, pp 7–32, 1979; Bauer RO in J Math Anal Appl 302: 484–501, 2005; Goryainov VV and Ba I in Ukrainian Math J 44:1209–1217, 1992).


Univalent functions Loewner chains Loewner evolution Evolution families Chordal Loewner equation Parametric representation 

Mathematics Subject Classification (2000)

Primary 30C80 Secondary 30D05 30C35 34M15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abate M.: Iteration theory of holomorphic maps on taut manifolds. Mediterranean, Rende (1989)MATHGoogle Scholar
  2. 2.
    Akhiezer, N.I., Glazman, I.M.: Theory of linear operators in Hilbert space, Translated from the Russian and with a preface by Merlynd Nestell, Reprint of the 1961 and 1963 translations, Dover, New York, 1993Google Scholar
  3. 3.
    Aleksandrov, I.A.: Parametric continuations in the theory of univalent functions (Russian), Izdat. “Nauka”, Moscow, 1976Google Scholar
  4. 4.
    Aleksandrov, I.A., Aleksandrov, S.T., Sobolev, V.V.: Extremal properties of mappings of a half plane into itself. In: Complex Analysis, pp. 7–32. PWN, Warsaw (1979)Google Scholar
  5. 5.
    Aleksandrov I.A., Sobolev V.V.: Extremal problems for certain classes of functions that are univalent in the half-plane. Ukrainian Math. Ž. 22, 291–307 (1970)MathSciNetMATHGoogle Scholar
  6. 6.
    Aleksandrov, S.T.: Parametric representation of functions univalent in the half plane. In: Extremal problems of the theory of functions, pp. 3–10. Tomsk. Gos. Univ., Tomsk (1979)Google Scholar
  7. 7.
    Aleksandrov, S.T., Sobolev, V.V.: Extremal problems in some classes of functions, univalent in the half plane, having a finite angular residue at infinity. Siberian Math. J. 27(2), 145–154 (1986). Translation from Sibirsk. Mat. Zh. 27(2), 3–13 (1986)Google Scholar
  8. 8.
    Bauer R.O.: Chordal Loewner families and univalent Cauchy transforms. J. Math. Anal. Appl. 302, 484–501 (2005)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution Families and the Loewner Equation I: the unit disk, Preprint 2008. Available on ArXiv 0807.1594Google Scholar
  10. 10.
    Bracci F., Contreras M.D., Díaz-Madrigal S.: Evolution families and the Loewner equation II: complex hyperbolic manifolds. Math. Ann. 344, 947–962 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    de Branges L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Burns D.M., Krantz S.G.: Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary. J. Am. Math. Soc. 7, 661–676 (1994)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Collingwood E.F., Lohwater A.J.: The theory of cluster sets. Cambridge University Press, Cambridge (1966)MATHCrossRefGoogle Scholar
  14. 14.
    Contreras M.D., Díaz-Madrigal S.: Fractional iteration in the disk algebra: prime ends and composition operators. Rev. Math. Iberoam. 21, 911–928 (2005)MATHCrossRefGoogle Scholar
  15. 15.
    Contreras M.D., Díaz-Madrigal S.: Analytic flows in the unit disk: angular derivatives and boundary fixed points. Pac. J. Math. 222, 253–286 (2005)MATHCrossRefGoogle Scholar
  16. 16.
    Contreras M.D., Díaz-Madrigal S., Pommerenke Ch.: On boundary critical points for semigroups of analytic functions. Math. Scand. 98, 125–142 (2006)MathSciNetMATHGoogle Scholar
  17. 17.
    Contreras, M.D., Díaz-Madrigal, S., Gumenyuk, P.: Loewner chains in the unit disk. To appear in Revista Matemática Iberoamericana; preprint available at arXiv:0902.3116v1 [math.CV]Google Scholar
  18. 18.
    Conway J.B.: Functions of one complex variable, II. Second edition, Graduate Texts in Mathematics, 159. Springer, New York, Berlin (1996)Google Scholar
  19. 19.
    Donoghue W.F. Jr: Monotone matrix functions and analytic continuation. Springer, New York, Heidelberg (1974)MATHGoogle Scholar
  20. 20.
    Duren P.L.: Univalent functions. Springer, New York (1983)MATHGoogle Scholar
  21. 21.
    Goluzin, G.M.: Geometric Theory of Functions of a Complex Variable. American Mathematical Society, Providence, R.I. (1969). (translated from G. M. Goluzin, Geometrical theory of functions of a complex variable (Russian), Second edition, Izdat. “Nauka”, Moscow, 1966)Google Scholar
  22. 22.
    Goryainov V.V.: Semigroups of conformal mappings. Math. USSR Sbornik 57, 463–483 (1987)CrossRefGoogle Scholar
  23. 23.
    Goryainov, V.V.: The Königs function and fractional integration of probability-generating functions (in Russian). Mat. Sb. 184, 55–74 (1993); translation in Russian Acad. Sci. Sb. Math. 79, 47–61 (1994)Google Scholar
  24. 24.
    Goryainov, V.V.: The embedding of iterations of probability-generating functions into continuous semigroups (in Russian). Dokl. Akad. Nauk 330, 539–541 (1993); translation in Russian Acad. Sci. Dokl. Math. 47, 554–557 (1993)Google Scholar
  25. 25.
    Goryaynov, V.V.: Evolutionary families of analytic functions and time-nonhomogeneous Markov branching processes. (English. Russian original) Dokl. Math. 53, 256–258 (1996); translation from Dokl. Akad. Nauk 347, 729–731 (1996)Google Scholar
  26. 26.
    Goryainov V.V., Ba I.: Semigroups of conformal mappings of the upper half-plane into itself with hydrodynamic normalization at infinity. Ukrainian Math. J. 44, 1209–1217 (1992)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Gustafsson B., Vasil’ev A.: Conformal and potential analysis in Hele-Shaw cells. Birkhäuser, Basel (2006)MATHGoogle Scholar
  28. 28.
    Kufarev P.P.: On one-parameter families of analytic functions (in Russian. English summary). Rec. Math. [Mat. Sbornik] N.S. 13(55), 87–118 (1943)MathSciNetGoogle Scholar
  29. 29.
    Kufarev P.P.: On integrals of simplest differential equation with moving pole singularity in the right-hand side. Tomsk. Gos. Univ. Uchyon. Zapiski 1, 35–48 (1946)Google Scholar
  30. 30.
    Kufarev P.P., Sobolev V.V., Sporyševa L.V.: A certain method of investigation of extremal problems for functions that are univalent in the half-plane. Trudy Tomsk. Gos. Univ. Ser. Meh. Mat. 200, 142–164 (1968)MathSciNetGoogle Scholar
  31. 31.
    Lawler G.F.: An introduction to the stochastic Loewner evolution. In: Random Walks and Geometry, pp. 261–293. Walter de Gruyter GmbH & Co. KG, Berlin (2004)Google Scholar
  32. 32.
    Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187, 237–273 (2001)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187, 275–308 (2001)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents. III. Two-sided exponents. Ann. Inst. H. Poincaré Probab. Statist. 38, 109–123 (2002)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Lawler G.F., Schramm O., Werner W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32, 939–995 (2004)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Löwner K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. Math. Ann. 89, 103–121 (1923)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Markina I., Prokhorov D., Vasil’ev A.: Sub-Riemannian geometry of the coefficients of univalent functions. J. Funct. Anal. 245, 475–492 (2007)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Pommerenke Ch.: Über dis subordination analytischer funktionen. J. Reine Angew Math. 218, 159–173 (1965)MathSciNetMATHGoogle Scholar
  39. 39.
    Pommerenke Ch.: Univalent functions. With a chapter on quadratic differentials by Gerd Jensen. Vandenhoeck & Ruprecht, Göttingen (1975)MATHGoogle Scholar
  40. 40.
    Pommerenke Ch.: Boundary behaviour of conformal Maps. Springer, Berlin (1992)MATHGoogle Scholar
  41. 41.
    Popova N.V.: Investigation of some integrals of the equation \({\frac{dw}{dt}=\frac{A}{w-\lambda(t)}}\) . Novosibirsk. Gos. Ped. Inst. Uchyon. Zapiski 8, 13–26 (1949)Google Scholar
  42. 42.
    Popova N.V.: Dependence between Löwner’s equation and the equation \({\tfrac{dw}{dt}=\tfrac1{w-\lambda(t)}}\) . Izv. Akad. Nauk BSSR Ser. Fiz.-Mat. Nauk 6, 97–98 (1954)Google Scholar
  43. 43.
    Prokhorov D., Vasil’ev A.: Univalent functions and integrable systems. Comm. Math. Phys. 262, 393–410 (2006)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Shapiro J.H.: Composition operators and classical function theory. Springer, New York (1993)MATHGoogle Scholar
  46. 46.
    Shoikhet D.: Semigroups in geometrical function theory. Kluwer Academic Publishers, Dordrecht (2001)MATHGoogle Scholar
  47. 47.
    Siskakis, A.G.: Semigroups of composition operators on spaces of analytic functions, a review. In: Studies on composition operators (Laramie, WY, 1996), pp. 229–252. Contemporary Mathematics, vol. 213, American Mathematical Society, Providence, RIGoogle Scholar
  48. 48.
    Sobolev V.V.: Parametric representations for some classes of functions univalent in half-plane. Kemerov. Ped. Inst. Uchyon. Zapiski 23, 30–41 (1970)Google Scholar
  49. 49.
    Valiron G.: Fonctions analytiques. Presses Univ. France, Paris (1954)MATHGoogle Scholar
  50. 50.
    Vinogradov Yu.P., Kufarev P.P.: On a problem of filtration. Akad. Nauk SSSR. Prikl. Mat. Meh. 12, 181–198 (1948)MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  • Manuel D. Contreras
    • 1
  • Santiago Díaz-Madrigal
    • 1
  • Pavel Gumenyuk
    • 2
  1. 1.Camino de los Descubrimientos, s/n, Departamento de Matemática Aplicada II, Escuela Técnica Superior de IngenierosUniversidad de SevillaSevillaSpain
  2. 2.Department of MathematicsUniversity of BergenBergenNorway

Personalised recommendations