Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Snowballs are quasiballs

Author(s): Daniel Meyer
Journal: Trans. Amer. Math. Soc. 362 (2010), 1247-1300.
MSC (2000): Primary 30C65
Posted: October 5, 2009
MathSciNet review: 2563729
Retrieve article in: PDF

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 -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:

[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 and .
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)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|