Uniformization of two-dimensional metric surfaces. (English) Zbl 1367.30044
The author studies metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. The main result gives a necessary and sufficient condition for such spaces to be QC equivalent to the Euclidean plane, disk, or sphere. This result gives a new approach to the Bonk-Kleiner theorem on parametrizations of Ahlfors 2-regular spheres by quasisymmetric maps.
Reviewer: Matti Vuorinen (Turku)
MSC:
| 30L10 | Quasiconformal mappings in metric spaces |
| 30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |
| 28A75 | Length, area, volume, other geometric measure theory |
References:
| [1] | Ambrosio, L., Kirchheim, B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318, 527-555 (2000) · Zbl 0966.28002 · doi:10.1007/s002080000122 |
| [2] | Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces. Oxford University Press, Oxford (2004) · Zbl 1080.28001 |
| [3] | Astala, K., Bonk, M., Heinonen, J.: Quasiconformal mappings with Sobolev boundary values. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1(3), 687-731 (2002) · Zbl 1170.30307 |
| [4] | Azzam, J., Badger, M., Toro, T.: Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps. Adv. Math. 275, 195-259 (2015) · Zbl 1320.30043 · doi:10.1016/j.aim.2015.02.008 |
| [5] | Ball, K.: Volumes of sections of cubes and related problems. In: Geometric aspects of functional analysis (1987-1988), Lecture Notes in Math., 1376, pp. 251-260. Springer, Berlin (1989) · Zbl 0858.46017 |
| [6] | Balogh, Z., Koskela, P., Rogovin, S.: Absolute continuity of quasiconformal mappings on curves. Geom. Funct. Anal. 17, 645-664 (2007) · Zbl 1134.30014 · doi:10.1007/s00039-007-0607-x |
| [7] | Barthe, F.: Inégalités de Brascamp-Lieb et convexité. C. R. Acad. Scis. Paris 324, 885-888 (1997) · Zbl 0904.26011 · doi:10.1016/S0764-4442(97)86963-7 |
| [8] | Behrend, F.: Über einige Affininvarianten konvexer Bereiche. Math. Ann. 113(1), 713-747 (1937) · JFM 62.0834.03 · doi:10.1007/BF01571662 |
| [9] | Bonk, M.: Quasiconformal geometry of fractals. In: International Congress of Mathematicians. vol. II, pp. 1349-1373. European Mathematical Society, Zürich (2006) · Zbl 1102.30016 |
| [10] | Bonk, M.: Uniformization of Sierpinski carpets in the plane. Invent. Math. 186(3), 559-665 (2011) · Zbl 1242.30015 · doi:10.1007/s00222-011-0325-8 |
| [11] | Bonk, M., Kleiner, B.: Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150(1), 127-183 (2002) · Zbl 1037.53023 · doi:10.1007/s00222-002-0233-z |
| [12] | Bonk, M., Kleiner, B.: Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary. Geom. Topol. 9, 219-246 (2005) · Zbl 1087.20033 · doi:10.2140/gt.2005.9.219 |
| [13] | Bonk, M., Lang, U.: Bi-Lipschitz parametrization of surfaces. Math. Ann. 327(1), 135-169 (2003) · Zbl 1042.53044 · doi:10.1007/s00208-003-0443-8 |
| [14] | Bonk, M., Merenkov, S.: Quasisymmetric rigidity of square Sierpinski carpets. Ann. Math. 177(2), 591-643 (2013) · Zbl 1269.30027 · doi:10.4007/annals.2013.177.2.5 |
| [15] | Bonk, M., Meyer, D.: Expanding Thurston maps. In: AMS Mathematical Surveys and Monographs (to appear) (2010) · Zbl 1430.37001 |
| [16] | Cannon, J.W.: The combinatorial Riemann mapping theorem. Acta Math. 173, 155-234 (1994) · Zbl 0832.30012 · doi:10.1007/BF02398434 |
| [17] | David, G., Semmes, S.: Quantitative rectifiability and Lipschitz mappings. Trans. Am. Math. Soc. 337(2), 855-889 (1993) · Zbl 0792.49029 · doi:10.1090/S0002-9947-1993-1132876-8 |
| [18] | David, G., Toro, T.: Reifenberg flat metric spaces, snowballs, and embeddings. Math. Ann. 315(4), 641-710 (1999) · Zbl 0944.53004 · doi:10.1007/s002080050332 |
| [19] | Derrick, W.R.: Inequalities for \[p\] p-modules of curve families on Lipschitz surfaces. Math. Z. 119, 1-10 (1971) · Zbl 0193.01002 · doi:10.1007/BF01110937 |
| [20] | Derrick, W.R.: Extremal length and Rayleigh’s reciprocal theorem. Appl. Anal. 6(2):119-125 (1976/1977) · Zbl 0363.31003 |
| [21] | Folland, G.B.: Real analysis: modern techniques and their applications. Wiley, New York (1984) · Zbl 0549.28001 |
| [22] | Fu, J.H.G.: Bi-Lipschitz rough normal coordinates for surfaces with an \[L^1\] L1-curvature bound. Indiana Univ. Math. J. 47(2), 439-453 (1998) · Zbl 0942.53007 · doi:10.1512/iumj.1998.47.1527 |
| [23] | Gehring, F.W.: Extremal length definitions for the conformal capacity of rings in space. Mich. Math. J. 9(2), 137-150 (1962) · Zbl 0109.04904 · doi:10.1307/mmj/1028998672 |
| [24] | Gehring, F.W.: Lower dimensional absolute continuity properties of quasiconformal mappings. Math. Proc. Camb. Philos. Soc. 78, 81-93 (1975) · Zbl 0307.30026 · doi:10.1017/S0305004100051513 |
| [25] | Gehring, F.W.: Some metric properties of quasi-conformal mappings. In: Proceedings of the International Congress of Mathematicians (Vancouver B. C. 1974), vol. 2, pp. 203-206 (1975) · Zbl 0342.30012 |
| [26] | Haïssinsky, P., Pilgrim, K.: Coarse expanding conformal dynamics, Astérisque, No. 325 (2009) · Zbl 1206.37002 |
| [27] | Heinonen, J.: Lectures on analysis on metric spaces. Springer, New York (2001) · Zbl 0985.46008 · doi:10.1007/978-1-4613-0131-8 |
| [28] | Heinonen, J., Keith, S.: Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds. Publ. Math. Inst. Hautes Études Sci. 113(1), 1-37 (2011) · Zbl 1238.30039 · doi:10.1007/s10240-011-0032-4 |
| [29] | Heinonen, J., Koskela, P.: Quasiconformal mappings in metric spaces with controlled geometry. Acta Math. 181, 1-61 (1998) · Zbl 0915.30018 · doi:10.1007/BF02392747 |
| [30] | Heinonen, J., Rickman, S.: Geometric branched covers between generalized manifolds. Duke Math. J. 113(3), 465-529 (2002) · Zbl 1017.30023 · doi:10.1215/S0012-7094-02-11333-7 |
| [31] | Heinonen, J., Semmes, S.: Thirty-three yes or no questions about mappings, measures, and metrics. Conform. Geom. Dyn. 1, 1-12 (1997) · Zbl 0885.00006 · doi:10.1090/S1088-4173-97-00012-X |
| [32] | Heinonen, J., Sullivan, D.: On the locally branched Euclidean metric gauge. Duke Math. J. 114(1), 15-41 (2002) · Zbl 1019.58002 · doi:10.1215/S0012-7094-02-11412-4 |
| [33] | Heinonen, J., Wu, J.-M.: Quasisymmetric nonparametrization and spaces associated with the Whitehead continuum. Geom. Topol. 14(2), 773-798 (2010) · Zbl 1198.30024 · doi:10.2140/gt.2010.14.773 |
| [34] | Heinonen, J., Koskela, P., Shanmugalingam, N., Tyson, J.: Sobolev spaces on metric measure spaces: an approach based on upper gradients. In: New Mathematical Monographs, p. 27. Cambridge University Press, Cambridge (2015) · Zbl 1332.46001 |
| [35] | Hurewicz, W., Wallman, H.: Dimension theory, Princeton Mathematical Series, vol. 4. Princeton University Press, Princeton (1941) · Zbl 0060.39808 |
| [36] | Kirchheim, B.: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Am. Math. Soc. 121(1), 113-123 (1994) · Zbl 0806.28004 · doi:10.1090/S0002-9939-1994-1189747-7 |
| [37] | Kleiner, B.: The asymptotic geometry of negatively curved spaces: uniformization, geometrization and rigidity, International Congress of Mathematicians. vol. II, pp. 743-768, European Mathematical Society, Zürich (2006) · Zbl 1108.30014 |
| [38] | Laakso, T.: Plane with \[A_{\infty }\] A∞-weighted metric not bilipschitz embeddable to \[R^nRn\]. Bull. Lond. Math. Soc. 34, 667-676 (2002) · Zbl 1029.30014 · doi:10.1112/S0024609302001200 |
| [39] | Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995) · Zbl 0819.28004 |
| [40] | Merenkov, S., Wildrick, K.: Quasisymmetric Koebe uniformization. Rev. Mat. Iberoam. 29(3), 859-909 (2013) · Zbl 1294.30043 · doi:10.4171/RMI/743 |
| [41] | Meyer, D.: Quasisymmetric embedding of self-similar surfaces and origami with rational maps. Ann. Acad. Sci. Fenn. Math. 27(2), 461-484 (2002) · Zbl 1017.30027 |
| [42] | Meyer, D.: Snowballs are quasiballs. Trans. Am. Math. Soc. 362(3), 1247-1300 (2010) · Zbl 1190.30018 · doi:10.1090/S0002-9947-09-04635-2 |
| [43] | Moore, R.L.: Foundations of point set theory. American Mathematical Society Colloquium Publications, American Mathematical Society, Providence (1962) · Zbl 0192.28901 |
| [44] | Müller, S., Šverák, V.: On surfaces of finite total curvature. J. Differ. Geom. 42(2), 229-259 (1995) · Zbl 0853.53003 |
| [45] | Pankka, P., Wu, J.-M.: Geometry and quasisymmetric parametrization of Semmes spaces. Rev. Mat. Iberoam. 30(3), 893-970 (2014) · Zbl 1322.30021 · doi:10.4171/RMI/802 |
| [46] | Rado, T., Reichelderfer, P.V.: Continuous transformations in analysis. With an introduction to algebraic topology. Springer, Berlin (1955) · Zbl 0067.03506 |
| [47] | Schramm, O.: Transboundary extremal length (preprint version) (1993) · Zbl 0842.30006 |
| [48] | Semmes, S.: Chord-arc surfaces with small constant. II. Good parametrizations. Adv. Math. 88, 170-199 (1991) · Zbl 0733.42016 · doi:10.1016/0001-8708(91)90007-T |
| [49] | Semmes, S.: Good metric spaces without good parametrizations. Rev. Mat. Iberoam. 12(1), 187-275 (1996) · Zbl 0854.57018 · doi:10.4171/RMI/198 |
| [50] | Semmes, S.: On the nonexistence of bi-Lipschitz parametrizations and geometric problems about \[A_{\infty }\] A∞-weights. Rev. Mat. Iberoam. 12(2), 337-410 (1996) · Zbl 0858.46017 · doi:10.4171/RMI/201 |
| [51] | Semmes, S.: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. Sel. Math. (N.S.) 2, 155-295 (1996) · Zbl 0870.54031 · doi:10.1007/BF01587936 |
| [52] | Toro, T.: Surfaces with generalized second fundamental form in \[L^2\] L2 are Lipschitz manifolds. J. Differ. Geom. 39(1), 65-101 (1994) · Zbl 0806.53020 |
| [53] | Toro, T.: Geometric conditions and existence of bi-Lipschitz parametrizations. Duke Math. J. 77(1), 193-227 (1995) · Zbl 0847.42011 · doi:10.1215/S0012-7094-95-07708-4 |
| [54] | Tukia, P., Väisäl̈ä, J.: Quasisymmetric embeddings of metric spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 5(1), 97-114 (1980) · Zbl 0403.54005 |
| [55] | Tyson, J.: Quasiconformality and quasisymmetry in metric measure spaces. Ann. Acad. Sci. Fenn. Ser. A I Math. 23, 525-548 (1998) · Zbl 0910.30022 |
| [56] | Tyson, J.: Analytic properties of locally quasisymmetric mappings from Euclidean domains. Indiana Univ. Math. J. 49(3), 995-1016 (2000) · Zbl 0966.30015 · doi:10.1512/iumj.2000.49.1695 |
| [57] | Väisälä, J.: Lectures on \[n\] n-dimensional quasiconformal mappings. Lecture Notes in Mathematics, vol. 229. Springer, Berlin, New York (1971) · Zbl 0221.30031 |
| [58] | Väisälä, J.: Quasisymmetric embeddings in Euclidean spaces. Trans. Am. Math. Soc. 264(1), 191-204 (1981) · Zbl 0456.30018 · doi:10.2307/1998419 |
| [59] | Wildrick, K.: Quasisymmetric parametrizations of two-dimensional metric planes. Proc. Lond. Math. Soc. (3) 97(3), 783-812 (2008) · Zbl 1160.30010 · doi:10.1112/plms/pdn023 |
| [60] | Wildrick, K.: Quasisymmetric structures on surfaces. Trans. Am. Math. Soc. 362(2), 623-659 (2010) · Zbl 1188.30065 · doi:10.1090/S0002-9947-09-04861-2 |
| [61] | Williams, M.: Geometric and analytic quasiconformality in metric measure spaces. Proc. Am. Math. Soc. 140(4), 1251-1266 (2012) · Zbl 1256.30055 · doi:10.1090/S0002-9939-2011-11035-9 |
| [62] | Ziemer, W.P.: Extremal length and conformal capacity. Trans. Am. Math. Soc. 126(3), 460-473 (1967) · Zbl 0177.34002 · doi:10.1090/S0002-9947-1967-0210891-0 |
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