Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Snowballs are quasiballs

Author: Daniel Meyer
Journal: Trans. Amer. Math. Soc. 362 (2010), 1247-1300
MSC (2000): Primary 30C65
Published electronically: October 5, 2009
MathSciNet review: 2563729
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce snowballs, which are compact sets in $ \mathbb{R}^3$ homeomorphic to the unit ball. They are $ 3$-dimensional analogs of domains in the plane bounded by snowflake curves. For each snowball $ \mathcal{B}$ a quasiconformal map $ f\colon \mathbb{R}^3\to \mathbb{R}^3$ is constructed that maps $ \mathcal{B}$ to the unit ball.

References [Enhancements On Off] (What's this?)

  • [Ahl63] Lars V. Ahlfors.
    Quasiconformal reflections.
    Acta Math., 109:291-301, 1963. MR 0154978 (27:4921)
  • [Ahl73] Lars V. Ahlfors.
    Conformal invariants: Topics in geometric function theory.
    McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Company, New York, Düsseldorf-Johannesburg, 1973. MR 0357743 (50:10211)
  • [Bis99] Christopher J. Bishop.
    A quasisymmetric surface with no rectifiable curves.
    Proc. Amer. Math. Soc., 127(7):2035-2040, 1999. MR 1610908 (99j:30023)
  • [BK02] Mario Bonk and Bruce Kleiner.
    Quasisymmetric parametrizations of two-dimensional metric spheres.
    Invent. Math., 150(1):127-183, 2002. MR 1930885 (2004k:53057)
  • [Can94] James W. Cannon.
    The combinatorial Riemann mapping theorem.
    Acta Math., 173(2):155-234, 1994. MR 1301392 (95k:30046)
  • [Car54] Constantin Carathéodory.
    Theory of functions of a complex variable. Vol. 2.
    Chelsea Publishing Company, New York, 1954.
  • [CFP01] James W. Cannon, William J. Floyd, and Walter R. Parry.
    Finite subdivision rules.
    Conform. Geom. Dyn., 5:153-196 (electronic), 2001. MR 1875951 (2002j:52021)
  • [CG93] Lennart Carleson and Theodore W. Gamelin.
    Complex Dynamics.
    Springer, New York, 1993. MR 1230383 (94h:30033)
  • [DT99] Guy David and Tatiana Toro.
    Reifenberg flat metric spaces, snowballs, and embeddings.
    Math. Ann., 315(4):641-710, 1999. MR 1731465 (2001c:49067)
  • [DT02] Tobin A. Driscoll and Lloyd N. Trefethen.
    Schwarz-Christoffel mapping.
    Cambridge University Press, 2002. MR 1908657 (2003e:30012)
  • [Hei01] Juha Heinonen.
    Lectures on analysis on metric spaces.
    Universitext. Springer-Verlag, New York, 2001. MR 1800917 (2002c:30028)
  • [HK98] Juha Heinonen and Pekka Koskela.
    Quasiconformal maps in metric spaces with controlled geometry.
    Acta Math., 181(1):1-61, 1998. MR 1654771 (99j:30025)
  • [Mey02] Daniel Meyer.
    Quasisymmetric embedding of self similar surfaces and origami with rational maps.
    Ann. Acad. Sci. Fenn. Math, 27(2):461-484, 2002. MR 1922201 (2003g:52037)
  • [Mil99] John Milnor.
    Dynamics in one complex variable. Introductory lectures.
    Friedr. Vieweg & Sohn, Braunschweig, 1999. MR 1721240 (2002i:37057)
  • [Moi77] Edwin E. Moise.
    Geometric topology in dimensions $ 2$ and $ 3$.
    Springer-Verlag, New York, 1977.
    Graduate Texts in Mathematics, Vol. 47. MR 0488059 (58:7631)
  • [Roh01] Steffen Rohde.
    Quasicircles modulo bilipschitz maps.
    Rev. Mat. Iberoamericana, 17(3):643-659, 2001. MR 1900898 (2003b:30022)
  • [Tuk80] Pekka Tukia.
    The planar Schönflies theorem for Lipschitz maps.
    Ann. Acad. Sci. Fenn. Ser. A I Math., 5:49-72, 1980. MR 595177 (82e:57003)
  • [TV80] Pekka Tukia and Jussi Väisälä.
    Quasisymmetric embeddings of metric spaces.
    Ann. Acad. Sci. Fenn. Ser. A I Math., 5(5):97-114, 1980. MR 595180 (82g:30038)
  • [Väi71] Jussi Väisälä.
    Lectures on n-dimensional quasiconformal mappings.
    Springer-Verlag., Berlin-Heidelberg-New York, 1971. MR 0454009 (56:12260)
  • [Väi99] Jussi Väisälä.
    The free quasiworld. Freely quasiconformal and related maps in Banach spaces.
    In Quasiconformal geometry and dynamics (Lublin, 1996), volume 48 of Banach Center Publ., pages 55-118. Polish Acad. Sci., Warsaw, 1999. MR 1709974 (2000h:58017)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 30C65

Retrieve articles in all journals with MSC (2000): 30C65

Additional Information

Daniel Meyer
Affiliation: Department of Mathematics, University of Helsinki, P.O. Box 68, Gustaf Hällströmin katu 2b, FI-00014 University of Helsinki, Finland

Keywords: Quasiconformal maps, quasiconformal uniformization, snowball
Received by editor(s): August 16, 2007
Published electronically: October 5, 2009
Additional Notes: This research was partially supported by an NSF postdoctoral fellowship and by NSF grant DMS-0244421.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society