Acta Mathematica

Sharpness of Rickman’s Picard theorem in all dimensions

David Drasin and Pekka Pankka

Full-text: Open access

Abstract

We show that given n3, q1, and a finite set {y1,,yq} in Rn there exists a quasiregular mapping RnRn omitting exactly points y1,,yq.

Note

P.P. was partly supported by NSF grant DMS-0757732 and the Academy of Finland projects 126836 and 256228.

Article information

Source
Acta Math., Volume 214, Number 2 (2015), 209-306.

Dates
Received: 28 April 2013
Revised: 13 November 2014
First available in Project Euclid: 30 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485802422

Digital Object Identifier
doi:10.1007/s11511-015-0125-x

Mathematical Reviews number (MathSciNet)
MR3372169

Zentralblatt MATH identifier
1326.30025

Subjects
Primary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations

Rights
2015 © Institut Mittag-Leffler

Citation

Drasin, David; Pankka, Pekka. Sharpness of Rickman’s Picard theorem in all dimensions. Acta Math. 214 (2015), no. 2, 209--306. doi:10.1007/s11511-015-0125-x. https://projecteuclid.org/euclid.acta/1485802422


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References

  • Astala, K., Iwaniec, T. & Martin, G., Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Mathematical Series, 48. Princeton University Press, Princeton, NJ, 2009.
  • Drasin D.: Picard’s theorem and the Rickman construction. Sci. China Math., 53, 523–532 (2010)
  • Eremenko, A. & Lewis, J. L., Uniform limits of certain A-harmonic functions with applications to quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I Math., 16 (1991), 361–375.
  • Heinonen J., Rickman S.: Geometric branched covers between generalized manifolds. Duke Math. J., 113, 465–529 (2002)
  • Holopainen, I. & Rickman, S., A Picard type theorem for quasiregular mappings of Rn into n-manifolds with many ends. Rev. Mat. Iberoamericana, 8 (1992), 131–148.
  • Holopainen I., Rickman S.: Quasiregular mappings of the Heisenberg group. Math. Ann., 294, 625–643 (1992)
  • Holopainen, I. & Rickman, S., Ricci curvature, Harnack functions, and Picard type theorems for quasiregular mappings, in Analysis and Topology, pp. 315–326. World Scientific, River Edge, NJ, 1998.
  • Hudson, J. F. P., Piecewise Linear Topology. University of Chicago Lecture Notes. Benjamin, New York–Amsterdam, 1969.
  • Lewis, J. L., Picard’s theorem and Rickman’s theorem by way of Harnack’s inequality. Proc. Amer. Math. Soc., 122 (1994), 199–206.
  • Martio O., Sarvas J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math., 4, 383–401 (1979)
  • Martio, O. & Väisälä, J., Elliptic equations and maps of bounded length distortion. Math. Ann., 282 (1988), 423–443.
  • Piergallini, R., Four-manifolds as 4-fold branched covers of S4. Topology, 34 (1995), 497–508.
  • Rajala K.: Mappings of finite distortion: the Rickman–Picard theorem for mappings of finite lower order. J. Anal. Math., 94, 235–248 (2004)
  • Rickman, S., On the number of omitted values of entire quasiregular mappings. J. Anal. Math., 37 (1980), 100–117.
  • Rickman, S., The analogue of Picard’s theorem for quasiregular mappings in dimension three. Acta Math., 154 (1985), 195–242.
  • Rickman, S., Quasiregular Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete, 26. Springer, Berlin–Heidelberg, 1993.
  • Rickman S.: Simply connected quasiregularly elliptic 4-manifolds. Ann. Acad. Sci. Fenn. Math., 31, 97–110 (2006)
  • Rourke, C. P. & Sanderson, B. J., Introduction to Piecewise-Linear Topology. Springer Study Edition. Springer, Berlin–New York, 1982.
  • Semmes, S., Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. Selecta Math., 2 (1996), 155–295.
  • Väisälä, J., A survey of quasiregular maps in Rn, in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 685–691. Acad. Sci. Fennica, Helsinki, 1980.
  • Väisälä J.: Unions of John domains. Proc. Amer. Math. Soc., 128, 1135–1140 (2000)
  • Zorich, V. A., A theorem of M. A. Lavrent′ev on quasiconformal space maps. Mat. Sb., 74 (1967), 417–433 (Russian); English translation in Math. USSR–Sb, 3 (1967), 389–403.