Skip to main content
SpringerLink
Log in

Geometry Behind Chordal Loewner Chains

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

An Erratum to this article was published on 06 July 2012

Abstract

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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abate M.: Iteration theory of holomorphic maps on taut manifolds. Mediterranean, Rende (1989)

    MATH  Google Scholar 

  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, 1993

  3. Aleksandrov, I.A.: Parametric continuations in the theory of univalent functions (Russian), Izdat. “Nauka”, Moscow, 1976

  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)

  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)

    MathSciNet  MATH  Google Scholar 

  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)

  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)

  8. Bauer R.O.: Chordal Loewner families and univalent Cauchy transforms. J. Math. Anal. Appl. 302, 484–501 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bracci, F., Contreras, M.D., Díaz-Madrigal, S.: Evolution Families and the Loewner Equation I: the unit disk, Preprint 2008. Available on ArXiv 0807.1594

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. de Branges L.: A proof of the Bieberbach conjecture. Acta Math. 154, 137–152 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Collingwood E.F., Lohwater A.J.: The theory of cluster sets. Cambridge University Press, Cambridge (1966)

    Book  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  16. Contreras M.D., Díaz-Madrigal S., Pommerenke Ch.: On boundary critical points for semigroups of analytic functions. Math. Scand. 98, 125–142 (2006)

    MathSciNet  MATH  Google Scholar 

  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]

  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. Donoghue W.F. Jr: Monotone matrix functions and analytic continuation. Springer, New York, Heidelberg (1974)

    MATH  Google Scholar 

  20. Duren P.L.: Univalent functions. Springer, New York (1983)

    MATH  Google Scholar 

  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)

  22. Goryainov V.V.: Semigroups of conformal mappings. Math. USSR Sbornik 57, 463–483 (1987)

    Article  Google Scholar 

  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)

  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)

  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)

  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)

    Article  MathSciNet  Google Scholar 

  27. Gustafsson B., Vasil’ev A.: Conformal and potential analysis in Hele-Shaw cells. Birkhäuser, Basel (2006)

    MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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. 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)

    MathSciNet  Google Scholar 

  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)

  32. Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187, 237–273 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lawler G.F., Schramm O., Werner W.: Values of Brownian intersection exponents. II. Plane exponents. Acta Math. 187, 275–308 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  36. Löwner K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. Math. Ann. 89, 103–121 (1923)

    Article  MathSciNet  MATH  Google Scholar 

  37. Markina I., Prokhorov D., Vasil’ev A.: Sub-Riemannian geometry of the coefficients of univalent functions. J. Funct. Anal. 245, 475–492 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pommerenke Ch.: Über dis subordination analytischer funktionen. J. Reine Angew Math. 218, 159–173 (1965)

    MathSciNet  MATH  Google Scholar 

  39. Pommerenke Ch.: Univalent functions. With a chapter on quadratic differentials by Gerd Jensen. Vandenhoeck & Ruprecht, Göttingen (1975)

    MATH  Google Scholar 

  40. Pommerenke Ch.: Boundary behaviour of conformal Maps. Springer, Berlin (1992)

    MATH  Google Scholar 

  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. 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. Prokhorov D., Vasil’ev A.: Univalent functions and integrable systems. Comm. Math. Phys. 262, 393–410 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Schramm O.: Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118, 221–288 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  45. Shapiro J.H.: Composition operators and classical function theory. Springer, New York (1993)

    MATH  Google Scholar 

  46. Shoikhet D.: Semigroups in geometrical function theory. Kluwer Academic Publishers, Dordrecht (2001)

    MATH  Google Scholar 

  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, RI

  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. Valiron G.: Fonctions analytiques. Presses Univ. France, Paris (1954)

    MATH  Google Scholar 

  50. Vinogradov Yu.P., Kufarev P.P.: On a problem of filtration. Akad. Nauk SSSR. Prikl. Mat. Meh. 12, 181–198 (1948)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Camino de los Descubrimientos, s/n, Departamento de Matemática Aplicada II, Escuela Técnica Superior de Ingenieros, Universidad de Sevilla, Sevilla, 41092, Spain

    Manuel D. Contreras & Santiago Díaz-Madrigal

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

    Pavel Gumenyuk

Authors
  1. Manuel D. Contreras
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Santiago Díaz-Madrigal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pavel Gumenyuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manuel D. Contreras.

Additional information

Communicated by Dr. Alexander Vasiliev.

M. D. Contreras and S. Díaz-Madrigal were partially supported by the Ministerio de Ciencia e Innovación and the European Union (FEDER), projects MTM2006-14449-C02-01 and MTM2009-14694-C02-02, by La Consejería de Educación y Ciencia de la Junta de Andalucía and by the ESF Networking Programme “Harmonic and Complex Analysis and its Applications”.

P. Gumenyuk was partially supported by the ESF Networking Programme “Harmonic and Complex Analysis and its Applications”, by the Scandinavian Network “Analysis and Applications” (NordForsk), project #080151, and the Research Council of Norway, project #177355/V30.

An erratum to this article can be found at http://dx.doi.org/10.1007/s11785-012-0246-6

Rights and permissions

Reprints and permissions

About this article

Cite this article

Contreras, M.D., Díaz-Madrigal, S. & Gumenyuk, P. Geometry Behind Chordal Loewner Chains. Complex Anal. Oper. Theory 4, 541–587 (2010). https://doi.org/10.1007/s11785-010-0057-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-010-0057-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation