Advertisement

Random Conformal Welding for Finitely Connected Regions

  • Shi-Yi Lan
  • Wang Zhou
Article
  • 10 Downloads

Abstract

Given a finitely connected region \(\Omega \) of the Riemann sphere whose complement consists of m mutually disjoint closed disks \({\bar{U}}_j\), the random homeomorphism \(h_j\) on the boundary component \(\partial U_j\) is constructed using the exponential Gaussian free field. The existence and uniqueness of random conformal welding of \(\Omega \) with \(h_j\) is established by investigating a non-uniformly elliptic Beltrami equation with a random complex dilatation. This generalizes the result of Astala, Jones, Kupiainen and Saksman to multiply connected domains.

Keywords

Random welding Quasiconformal mapping Gaussian free field SLE 

Mathematics Subject Classification (2010)

30C62 60D05 

Notes

Acknowledgements

The authors would like to thank one referee for his/her helpful comments and suggestions.

References

  1. 1.
    Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  2. 2.
    Astala, K., Jones, P., Kupiainen, A., Saksman, E.: Random conformal weldings. Acta Math. 207, 203–254 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beurling, A., Ahlfors, L.: The boundary correspondence under quasiconformal mappings. Acta Math. 96, 125–142 (1956)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bishop, C.J.: Conformal welding and Koebe’s theorem. Ann. Math. 166, 613–656 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Camia, F., Newman, C.M.: Critical percolation exploration path and \(SLE_6\): a proof of convergence. Prob. Theor. Relat. Fields 139, 473–519 (2007)CrossRefGoogle Scholar
  6. 6.
    Doyon, B.: Factorisation of conformal maps on finitely connected domains. Preprint (2011). arXiv:1107.0582v1 [math.cv]
  7. 7.
    Gardiner, F.P., Lakic, N.: Quasiconformal Teichmüller Theory, Mathematical Surveys and Monographs, vol. 76. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  8. 8.
    Hamilton, D.H.: Conformal welding. In: Kühnau, R. (ed.) The Handbook of Geometric Function Theory. North Holland, Amsterdam (2002)Google Scholar
  9. 9.
    Jones, P.W., Smirnov, S.K.: Removability theorems for Sobolev functions and quasiconformal maps. Ark. Mat. 38, 263–279 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kager, W., Nienhuis, B.: A guide to stochastic Löwner evolution and its applications. J. Stat. Phys. 115, 1149–1229 (2004)CrossRefGoogle Scholar
  11. 11.
    Lawler, G.F.: Conformal Invariant Processes in the Plane. American Mathematical Society, Providence (2005)zbMATHGoogle Scholar
  12. 12.
    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)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lawler, G.F., Sheffield, S.: A natural parametrization for the Schramm–Loewner evolution. Ann. Probab. 39, 1896–1937 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lawler, G.F., Zhou, W.: SLE curves and natural parametrization. Ann. Probab. 41, 1556–1584 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lehto, O.: Homeomorphisms with a Given Dilatation. Lecture Notes in Mathematics, vol. 118, pp. 58–73. Springer, Berlin (1970)Google Scholar
  16. 16.
    Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane, 2nd edn. Springer, Berlin (1973)CrossRefGoogle Scholar
  17. 17.
    Marshall, D.E.: Conformal welding for finitely connected regions. Comput. Methods Funct. Theory 11, 655–669 (2011)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Radnell, D., Schippers, E.: Quasisymmetric sewing in rigged Teichmüller space. Commun. Contemp. Math. 8, 481–534 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math. 161, 883–924 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees. Isr. J. Math. 118, 221–288 (2000)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Schramm, O., Sheffield, S.: The harmonic explorer and its convergence to SLE(4). Ann. Probab. 33, 2127–2148 (2006)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Schramm, O., Sheffield, S.: Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202, 21–137 (2009)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139, 521–541 (2007)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sheffield, S.: Conformal weldings of random surfaces: SLE and the quantum gravity zipper. Preprint (2010). arXiv:1012.4797 [math.pr]
  25. 25.
    Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333, 239–244 (2001a)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Smirnov, S.: Critical percolation in the plane. I. Conformal invariance and Cardy’s formula. II. Continuum scaling limit. (long version of [25]) (2001b). arXiv:0909.4499
  27. 27.
    Smirnov, S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. 172, 1435–1467 (2010)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tecu, N.: Random conformal weldings at criticality. Preprint (2012). arXiv:1205.3189v1 [math.cv]
  29. 29.
    Williams, G.B.: Discrete conformal welding. Indiana Univ. Math. J. 53, 765–804 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Guangxi University for NationalitiesNanningChina
  2. 2.National University of SingaporeSingaporeSingapore

Personalised recommendations