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Quasiconformal geometry and removable sets for conformal mappings

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Abstract

We study metric spaces defined via a conformal weight, or more generally a measurable Finsler structure, on a domain Ω ⊂ ℝ2 that vanishes on a compact set E ⊂ Ω and satisfies mild assumptions. Our main question is to determine when such a space is quasiconformally equivalent to a planar domain. We give a characterization in terms of the notion of planar sets that are removable for conformal mappings. We also study the question of when a quasiconformal mapping can be factored as a 1-quasiconformal mapping precomposed with a bi-Lipschitz map.

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References

  1. L. Ahlfors and A. Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101–129.

    Article  MathSciNet  MATH  Google Scholar 

  2. N. Albin, J. Hoppis, P. Poggi-Corradini and N. Shanmugalingam, Infinity modulus and the essential metric, J. Math. Anal. Appl. 467 (2018), 570–584.

    Article  MathSciNet  MATH  Google Scholar 

  3. K. Astala, T. Iwaniec, and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, Princeton, NJ, 2009.

    MATH  Google Scholar 

  4. L. V. Ahlfors and L. Sario, Riemann Surfaces, Princeton University Press, Princeton, NJ, 1960.

    Book  MATH  Google Scholar 

  5. L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, Oxford University Press, Oxford, 2004.

    MATH  Google Scholar 

  6. A. D. Aleksandrov and V. A. Zalgaller, Intrinsic Geometry of Surfaces, American Mathematical Society, Providence, RI, 1967.

    MATH  Google Scholar 

  7. A. L. Baisón, A. Clop and J. Orobitg, Distributional solutions of the Beltrami equation, J. Math. Anal. Appl. 470 (2019), 1081–1094.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Bonk, J. Heinonen, and S. Rohde, Doubling conformal densities, J. Reine Angew. Math. 541 (2001), 117–141.

    MathSciNet  MATH  Google Scholar 

  9. C. J. Bishop, An A1weight not comparable with any quasiconformal Jacobian, in In the Tradition of Ahlfors—Bers. IV, American Mathematical Society, Providence, RI, 2007, pp. 7–18.

    Google Scholar 

  10. M. Bonk, P. Koskela, and S. Rohde, Conformal metrics on the unit ball in Euclidean space, Proc. London Math. Soc. (3) 77 (1998), 635–664.

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Busemann and W. Mayer, On the foundations of calculus of variations, Trans. Amer. Math. Soc. 49 (1941), 173–198.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Bonk and D. Meyer, Expanding Thurston Maps, American Mathematical Society, Providence, RI, 2017.

    Book  MATH  Google Scholar 

  13. V. I. Bogachev, Measure Theory. Vol. I, II, Springer, Berlin, 2007.

    Book  MATH  Google Scholar 

  14. M. Bonk, Quasiconformal geometry of fractals, in International Congress of Mathematicians. Vol. II, European Mathematical Society, Zürich, 2006, pp. 1349–1373.

    MATH  Google Scholar 

  15. P. Creutz and E. Soultanis, Maximal metric surfaces and the Sobolev-to-Lipschitzproperty, Calc. Var. Partial Differential Equations 59 (202), Article no. 177.

  16. R. J. Daverman, Decompositions of Manifolds, Academic Press, Orlando, FL, 1986.

    MATH  Google Scholar 

  17. G. David, Unrectifiable 1-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), 369–479.

    Article  MathSciNet  MATH  Google Scholar 

  18. G. De Cecco and G. Palmieri, Intrinsic distance on a Lipschitz Riemannian manifold, Rend. Sem. Mat. Univ. Politec. Torino 46 (1990), 157–170

    MathSciNet  MATH  Google Scholar 

  19. G. De Cecco and G. Palmieri, Length of curves on Lip manifolds, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 1 (1990), 215–221.

    MathSciNet  MATH  Google Scholar 

  20. G. De Cecco and G. Palmieri, Integral distance on a Lipschitz Riemannian manifold. Math. Z. 207 (1991), 223–243.

    Article  MathSciNet  MATH  Google Scholar 

  21. G. De Cecco and G. Palmieri, LIP manifolds: from metric to Finslerian structure, Math. Z. 218 (1995), 223–237.

    Article  MathSciNet  MATH  Google Scholar 

  22. G. David and S. Semmes, Strong A∞weights, Sobolev inequalities and quasiconformal mappings, in Analysis and Partial Differential Equations, Dekker, New York, 1990, pp. 101–111.

    MATH  Google Scholar 

  23. J. Duda, Absolutely continuous functions with values in a metric space, Real Anal. Exchange 32 (2007), 569–581.

    Article  MathSciNet  MATH  Google Scholar 

  24. H. Federer, Geometric Measure Theory, Springer, New York, 1969.

    MATH  Google Scholar 

  25. F. W. Gehring and O. Martio, Quasiextremal distance domains and extension of quasieonformal mappings, J. Analyse Math. 45 (1985), 181–206.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, American Mathematical Society, Providence, RI, 1969.

    Book  MATH  Google Scholar 

  27. A. Garroni, M. Ponsiglione and F. Prinari, From 1-homogeneous supremal funetionals to difference quotients: relaxation and Γ-convergence, Calc. Var. Partial Differential Equations 27 (2006), 397–420.

    Article  MathSciNet  MATH  Google Scholar 

  28. J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York, 2001.

    Book  MATH  Google Scholar 

  29. H. Hakobyan and D. A. Herron, Euclidean quasieonvexity, Ann. Acad. Sci. Fenn. Math. 33 (2008), 205–230.

    MathSciNet  MATH  Google Scholar 

  30. J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev Spaces on Metric Measure Spaces, Cambridge University Press, Cambridge, 2015.

    Book  MATH  Google Scholar 

  31. Z.-X. He and O. Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. (2) 137 (1993), 369–406.

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Ikonen, Uniformization of metric surfaces using isothermal coordinates, Ann. Fenn. Math. 47 (2022), 155–180.

    Article  MathSciNet  MATH  Google Scholar 

  33. S. V. Ivanov, Volumes and areas of Lipsehitz metrics, Algebra i Analiz 20 (2008), 74–111.

    Google Scholar 

  34. B. Kirchheim, Reetifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), 113–123.

    Article  MathSciNet  MATH  Google Scholar 

  35. S. Kalmykov, L. V. Kovalev and T. Rajala, Removable sets for intrinsic metric and for holomorphic functions, J. Analyse Math. 139 (2019), 751–772.

    Article  MathSciNet  MATH  Google Scholar 

  36. T. J. Laakso, Plane with A∞-weighted metric not bi-Lipsehitz embeddable to ℝN, Bull. London Math. Soc. 34 (2002), 667–676.

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Lytchak and S. Wenger, Area minimizing discs in metric spaces, Arch. Ration. Mech. Anal. 223 (2017), 1123–1182.

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Lytchak and S. Wenger, Intrinsic structure of minimal discs in metric spaces, Geom. Topol. 22 (2018), 591–644.

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Lytchak and S. Wenger, Canonical parameterizations of metric disks, Duke Math. J. 169 (2020), 761–797.

    Article  MathSciNet  MATH  Google Scholar 

  40. K. Rajala, Uniformization of two-dimensional metric surfaces, Invent. Math. 207 (2017), 1301–1375.

    Article  MathSciNet  MATH  Google Scholar 

  41. K. Rajala, Personal communication, 2020.

  42. Yu. G. Reshetnyak, Two-dimensional manifolds of bounded curvature, in Geometry, IV, Springer, Berlin, 1993, pp. 3–163, 245–250.

    Chapter  Google Scholar 

  43. M. Romney, Quasiconformal parametrization of metric surfaces with small dilatation, Indiana Univ. Math. J. 68(2019), 1003–1011.

    Article  MathSciNet  MATH  Google Scholar 

  44. K. Rajala and M. Romney, Reciprocal lower bound on modulus of curve families in metric surfaces, Ann. Acad. Sci. Fenn. Math. 44 (2019), 681–692.

    Article  MathSciNet  MATH  Google Scholar 

  45. O. Schramm, Transboundary extremal length, J. Analyse Math. 66 (1995), 307–329.

    Article  MathSciNet  MATH  Google Scholar 

  46. S. Semmes, Bi-Lipschitz mappings and strong A∞weights, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), 211–248.

    MathSciNet  MATH  Google Scholar 

  47. S. Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A∞-weights, Rev. Mat. Iberoamericana 12 (1996), 337–410.

    Article  MathSciNet  MATH  Google Scholar 

  48. N. Tomczak-Jaegermann, Banaeh—Mazur Distances and Finite-Dimensional Operator Ideals, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989.

    MATH  Google Scholar 

  49. X. Tolsa, Painleve’s problem and the semiadditivity of analytic capacity, Acta Math. 190 (2003), 105–149.

    Article  MathSciNet  MATH  Google Scholar 

  50. J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Springer, Berlin—New York, 1971.

    Book  MATH  Google Scholar 

  51. S. K. Vodop’yanov, P-differentiability on Carnot groups in different topologies and related topics, in Proceedings on Analysis and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000, pp. 603–670.

    MATH  Google Scholar 

  52. G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, NJ, 1964.

    Book  MATH  Google Scholar 

  53. S. Willard, General Topology, Addison-Wesley, Reading, MA.-London-Don Mills, ON, 1970.

  54. M. Williams, Geometric and analytic quasiconformality in metric measure spaces, Proc. Amer. Math. Soc. 140 (2012), 1251–1266.

    Article  MathSciNet  MATH  Google Scholar 

  55. M. Younsi, On removable sets for holomorphic functions, EMS Surv. Math. Sci. 2 (2015), 219–254.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), FI-40014, Jyväskylä, Finland

    Toni Ikonen

  2. Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931, Köln, Germany

    Matthew Romney

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  1. Toni Ikonen
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  2. Matthew Romney
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Correspondence to Toni Ikonen.

Additional information

Both authors were supported by the Academy of Finland, project number 308659. The first author was also supported by the Vilho, Yrjö and Kalle Väisälä Foundation. The second author was also supported by Deutsche Forschungsgemeinschaft grant SPP 2026.

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Ikonen, T., Romney, M. Quasiconformal geometry and removable sets for conformal mappings. JAMA 148, 119–185 (2022). https://doi.org/10.1007/s11854-022-0224-5

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  • DOI: https://doi.org/10.1007/s11854-022-0224-5

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