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Quasisymmetry and rectifiability of quasispheres


Authors: Matthew Badger, James T. Gill, Steffen Rohde and Tatiana Toro
Journal: Trans. Amer. Math. Soc. 366 (2014), 1413-1431
MSC (2010): Primary 30C65; Secondary 28A75, 30C62
DOI: https://doi.org/10.1090/S0002-9947-2013-05926-0
Published electronically: September 12, 2013
MathSciNet review: 3145736
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Abstract: We obtain Dini conditions that guarantee that an asymptotically conformal quasisphere is rectifiable. In particular, we show that for any $ \epsilon >0$ integrability of $ ( {\rm ess}\sup _{1-t<\vert x\vert<1+t} K_f(x)-1 )^{2-\epsilon } dt/t$ implies that the image of the unit sphere under a global quasiconformal homeomorphism $ f$ is rectifiable. We also establish estimates for the weak quasisymmetry constant of a global $ K$-quasiconformal map in neighborhoods with maximal dilatation close to 1.


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  • 1. G.D. Anderson, M.K. Vamanamurthy, and M.K. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. MR 1462077 (98h:30033)
  • 2. J.M. Anderson, J. Becker, and F.D. Lesley, On the boundary correspondence of asymptotically conformal automorphisms, J. London Math. Soc. (2) 38 (1988), no. 3, 453-462. MR 972130 (90g:30022)
  • 3. C.J. Bishop, V.Ya. Gutlyanskiĭ, O. Martio and M. Vuorinen, On conformal dilatation in space, Int. J. Math. Math. Sci. 2003, no. 22, 1397-1420. MR 1980177 (2004c:30038)
  • 4. G. David and T. Toro, Reifenberg flat metric spaces, snowballs, and embeddings, Math. Ann. 315 (1999), no. 4, 641-710. MR 1731465 (2001c:49067)
  • 5. G. David and T. Toro, Reifenberg parameterizations for sets with holes, Mem. Amer. Math. Soc. 215 (2012), no. 1012. MR 2907827
  • 6. V.Ya. Gutlyanskiĭ and A. Golberg, On Lipschitz continuity of quasiconformal mappings in space, J. Anal. Math. 109 (2009), 233-251. MR 2585395 (2011f:30045)
  • 7. J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917 (2002c:30028)
  • 8. P.W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1-15. MR 1069238 (91i:26016)
  • 9. P. Mattila and M. Vuorinen, Linear approximation property, Minkowski dimension, and quasiconformal spheres, J. London Math. Soc. (2) 42 (1990), no. 2, 249-266. MR 1083444 (92e:30011)
  • 10. D. Meyer, Quasisymmetric embedding of self similar surfaces and origami rational maps, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 461-484. MR 1922201 (2003g:52037)
  • 11. D. Meyer, Snowballs are quasiballs, Trans. Amer. Math. Soc. 362 (2010), no. 3, 1247-1300. MR 2563729 (2011a:30067)
  • 12. I. Prause, Flatness properties of quasispheres, Comput. Methods Func. Theory 7 (2007), no. 2, 527-541. MR 2376688 (2009d:30054)
  • 13. I. Prause, X. Tolsa, I. Uriarte-Tuero, Hausdorff measure of quasicircles, Adv. Math. 229 (2012), no. 2, 1313-1328. MR 2855095
  • 14. E. Reifenberg, Solution of the Plateau problem for $ m$-dimensional surface of varying topological type, Acta Math. 104 (1960), 1-92. MR 0114145 (22:4972)
  • 15. Yu.G. Reshetnyak, Stability theorems in geometry and analysis, Mathematics and Its Applications, vol. 304, Kluwer Academic Publishers Group, Dordrecht, 1994. MR 1326375 (96i:30016)
  • 16. S. Rohde, Quasicircles modulo bilipschitz maps, Rev. Mat. Iberoamicana 17 (2001), no. 3, 643-659. MR 1900898 (2003b:30022)
  • 17. P. Seittenranta, Linear dilitation of quasiconformal maps in space, Duke Math. J. 91 (1998), no. 1, 1-16. MR 1487976 (99f:30036)
  • 18. S. Smirnov, Dimension of quasicircles, Acta Math. 205 (2010), no. 1, 189-197. MR 2736155 (2011j:30027)
  • 19. T. Toro, Geometric conditions and existence of bi-Lipschitz parameterizations, Duke Math. J. 77 (1995), no. 1, 193-227. MR 1317632 (96b:28006)
  • 20. J. Väisälä, Lectures on $ n$-dimensional quasiconformal mappings, Lecture Notes in Math., vol. 229, Springer-Verlag, Berlin, 1971. MR 0454009 (56:12260)
  • 21. M. Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Math., vol. 1319, Springer, Berlin, 1988. MR 950174 (89k:30021)
  • 22. M. Vuorinen, Quadruples and spatial quasiconformal mappings, Math. Z. 205 (1990), no. 4, 617-628. MR 1082879 (92a:30021)

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Additional Information

Matthew Badger
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
Email: badger@math.sunysb.edu

James T. Gill
Affiliation: Department of Mathematics and Computer Science, Saint Louis University, St. Louis, Missouri 63103
Email: jgill5@slu.edu

Steffen Rohde
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
Email: rohde@math.washington.edu

Tatiana Toro
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
Email: toro@math.washington.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05926-0
Keywords: Quasisymmetry, quasisphere, asymptotically conformal, rectifiable, Hausdorff measure, Reifenberg flat, linear approximation property, Jones $\beta$-number, modulus
Received by editor(s): January 18, 2012
Published electronically: September 12, 2013
Additional Notes: The first author was partially supported by NSF grant #0838212
The second author was partially supported by NSF grant #1004721
The third author was partially supported by NSF grant #0800968
The fourth author was partially supported by NSF grant #0856687
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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