Quasispheres and metric doubling measures

Lohvansuu, Atte; Rajala, Kai; Rasimus, Martti

doi:10.1090/proc/13971

Quasispheres and metric doubling measures
HTML articles powered by AMS MathViewer

by Atte Lohvansuu, Kai Rajala and Martti Rasimus PDF

Proc. Amer. Math. Soc. 146 (2018), 2973-2984 Request permission

Abstract:

Applying the Bonk-Kleiner characterization of Ahlfors $2$-regular quasispheres, we show that a metric two-sphere $X$ is a quasisphere if and only if $X$ is linearly locally connected and carries a weak metric doubling measure, i.e., a measure that deforms the metric on $X$ without much shrinking.

References

Christopher J. Bishop, A quasisymmetric surface with no rectifiable curves, Proc. Amer. Math. Soc. 127 (1999), no. 7, 2035–2040. MR 1610908, DOI 10.1090/S0002-9939-99-04900-X
Mario Bonk, Quasiconformal geometry of fractals, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1349–1373. MR 2275649
Mario Bonk, Uniformization of Sierpiński carpets in the plane, Invent. Math. 186 (2011), no. 3, 559–665. MR 2854086, DOI 10.1007/s00222-011-0325-8
Mario Bonk and Bruce Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math. 150 (2002), no. 1, 127–183. MR 1930885, DOI 10.1007/s00222-002-0233-z
Mario Bonk and Bruce Kleiner, Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geom. Topol. 9 (2005), 219–246. MR 2116315, DOI 10.2140/gt.2005.9.219
Mario Bonk and Sergei Merenkov, Quasisymmetric rigidity of square Sierpiński carpets, Ann. of Math. (2) 177 (2013), no. 2, 591–643. MR 3010807, DOI 10.4007/annals.2013.177.2.5
Mario Bonk, Mikhail Lyubich, and Sergei Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, Adv. Math. 301 (2016), 383–422. MR 3539379, DOI 10.1016/j.aim.2016.06.007
Mario Bonk and Daniel Meyer, Expanding Thurston maps, Mathematical Surveys and Monographs, vol. 225, American Mathematical Society, Providence, RI, 2017. MR 3727134, DOI 10.1090/surv/225
Michel Coornaert, Mesures de Patterson-Sullivan sur le bord d’un espace hyperbolique au sens de Gromov, Pacific J. Math. 159 (1993), no. 2, 241–270 (French, with French summary). MR 1214072, DOI 10.2140/pjm.1993.159.241
Guy David and Stephen Semmes, Strong $A_\infty$ weights, Sobolev inequalities and quasiconformal mappings, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 101–111. MR 1044784
Guy David and Stephen Semmes, Fractured fractals and broken dreams, Oxford Lecture Series in Mathematics and its Applications, vol. 7, The Clarendon Press, Oxford University Press, New York, 1997. Self-similar geometry through metric and measure. MR 1616732
G. David and T. Toro, Reifenberg flat metric spaces, snowballs, and embeddings, Math. Ann. 315 (1999), no. 4, 641–710. MR 1731465, DOI 10.1007/s002080050332
Peter Haïssinsky and Kevin M. Pilgrim, Coarse expanding conformal dynamics, Astérisque 325 (2009), viii+139 pp. (2010) (English, with English and French summaries). MR 2662902
Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
Juha Heinonen and Jang-Mei Wu, Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum, Geom. Topol. 14 (2010), no. 2, 773–798. MR 2602851, DOI 10.2140/gt.2010.14.773
Bruce Kleiner, The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 743–768. MR 2275621
A. Lytchak and S. Wenger, Canonical parametrizations of metric discs, preprint.
Sergei Merenkov and Kevin Wildrick, Quasisymmetric Koebe uniformization, Rev. Mat. Iberoam. 29 (2013), no. 3, 859–909. MR 3090140, DOI 10.4171/RMI/743
Daniel Meyer, Quasisymmetric embedding of self similar surfaces and origami with rational maps, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 461–484. MR 1922201
M. H. A. Newman, Elements of the topology of plane sets of points, Cambridge, at the University Press, 1951. 2nd ed. MR 0044820
Pekka Pankka and Jang-Mei Wu, Geometry and quasisymmetric parametrization of Semmes spaces, Rev. Mat. Iberoam. 30 (2014), no. 3, 893–960. MR 3254996, DOI 10.4171/RMI/802
Kai Rajala, Uniformization of two-dimensional metric surfaces, Invent. Math. 207 (2017), no. 3, 1301–1375. MR 3608292, DOI 10.1007/s00222-016-0686-0
Stephen Semmes, Chord-arc surfaces with small constant. II. Good parameterizations, Adv. Math. 88 (1991), no. 2, 170–199. MR 1120612, DOI 10.1016/0001-8708(91)90007-T
Stephen Semmes, Good metric spaces without good parameterizations, Rev. Mat. Iberoamericana 12 (1996), no. 1, 187–275. MR 1387590, DOI 10.4171/RMI/198
S. Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Selecta Math. (N.S.) 2 (1996), no. 2, 155–295. MR 1414889, DOI 10.1007/BF01587936
P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 97–114. MR 595180, DOI 10.5186/aasfm.1980.0531
Jussi Väisälä, Quasisymmetric embeddings in Euclidean spaces, Trans. Amer. Math. Soc. 264 (1981), no. 1, 191–204. MR 597876, DOI 10.1090/S0002-9947-1981-0597876-7
Vyron Vellis and Jang-Mei Wu, Quasisymmetric spheres over Jordan domains, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5727–5751. MR 3458397, DOI 10.1090/tran/6634
K. Wildrick, Quasisymmetric parametrizations of two-dimensional metric planes, Proc. Lond. Math. Soc. (3) 97 (2008), no. 3, 783–812. MR 2448247, DOI 10.1112/plms/pdn023
Jang-Mei Wu, Geometry of Grushin spaces, Illinois J. Math. 59 (2015), no. 1, 21–41. MR 3459626