Conformal Geometry and Dynamics

Author(s): Jeremy T. Tyson
Journal: Conform. Geom. Dyn. 5 (2001), 21-73.
MSC (2000): Primary 30C65; Secondary 28A78, 46E35, 43A85
Posted: August 8, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Recent developments in geometry have highlighted the need for abstract formulations of the classical theory of quasiconformal mappings. We modify Pansu's generalized modulus to study quasiconformal geometry in spaces with metric and measure-theoretic properties sufficiently similar to Euclidean space. Our basic objects of study are locally compact metric spaces equipped with a Borel measure which is Ahlfors-David regular of dimension

, and satisfies the Loewner condition of Heinonen-Koskela. For homeomorphisms between open sets in two such spaces, we prove the equivalence of three conditions: a version of metric quasiconformality, local quasisymmetry and geometric quasiconformality.

We derive from these results several corollaries. First, we show that the Loewner condition is a quasisymmetric invariant in locally compact Ahlfors regular spaces. Next, we show that a proper

-regular Loewner space,

, is not quasiconformally equivalent to any subdomain. (In the Euclidean case, this result is due to Loewner.) Finally, we characterize products of snowflake curves up to quasisymmetric/bi-Lipschitz equivalence: two such products are bi-Lipschitz equivalent if and only if they are isometric and are quasisymmetrically equivalent if and only if they are conformally equivalent.

1.: P. Alestalo and J. Väisälä, Quasisymmetric embeddings of products of cells into the Euclidean space, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), 375-392. MR 95h:30023
2.: Z. Balogh, Hausdorff dimension distortion of QC mappings on the Heisenberg group, J. Anal. Math. 83 (2001), 289-312.
3.: Z. Balogh and P. Koskela, Quasiconformality, quasisymmetry and removability in Loewner spaces, Duke Math. J. 101 (2000), no. 3, 554-577, with an appendix by J. Väisälä. MR 2001d:30029
4.: Z. Balogh and J. T. Tyson, Polar coordinates and regularity of quasiconformal mappings in Carnot groups, preprint, 2001.
5.: Z. M. Balogh, I. Holopainen, and J. T. Tyson, Singular solutions, homogeneous norms and quasiconformal mappings on Carnot groups, Preprint #269, University of Helsinki, September 2000.
6.: N. Benakli, Polyèdres hyperboliques, passage du local au global, Thèse, Université Paris-Sud, 1992.
7.: C. J. Bishop and J. T. Tyson, Locally minimal sets for conformal dimension, Ann. Acad. Sci. Fenn. Ser. A I Math. 26 (2001), 361-373.
8.: B. Bojarski, Remarks on Sobolev imbedding inequalities, Complex analysis, Joensuu 1987, Lecture Notes in Mathematics, no. 1351, Springer-Verlag, Berlin, 1988, pp. 52-68. MR 90b:46068
9.: M. Bourdon, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal. 7 (1997), 245-268. MR 98c:20056
10.: M. Bourdon and H. Pajot, Poincaré inequalities and quasiconformal structure on the boundaries of some hyperbolic buildings, Proc. Amer. Math. Soc. 127 (1999), no. 8, 2315-2324. MR 99j:30024
11.: A. Bruckner, J. Bruckner, and B. Thomson, Real analysis, Prentice-Hall, N.J., 1997.
12.: J. W. Cannon, The combinatorial Riemann mapping theorem, Acta Math. 173 (1994), 155-234. MR 95k:30046
13.: J. W. Cannon, W. J. Floyd, and W. R. Parry, Squaring rectangles: the finite Riemann mapping theorem, The mathematical legacy of Wilhelm Magnus: groups, geometry and special functions (Brooklyn, NY, 1992), Contemp. Math., no. 169, Amer. Math. Soc., Providence, RI, 1994, pp. 133-212. MR 95g:20045
14.: J. W. Cannon and E. L. Swenson, Recognizing constant curvature discrete groups in dimension , Trans. Amer. Math. Soc. 350 (1998), 809-849. MR 98i:57023
15.: L. Capogna, Regularity of quasilinear equations in the Heisenberg group, Comm. Pure Appl. Math. 50 (1997), no. 9, 867-889.
16.: -, Regularity for quasilinear equations and -quasiconformal maps in Carnot groups, Math. Ann. 313 (1999), no. 2, 263-295. MR 2000a:35027
17.: L. Capogna and P. Tang, Uniform domains and quasiconformal mappings on the Heisenberg group, Man. Math. 86 (1995), 267-281. MR 96f:30019
18.: C. Champetier, Propriétés statistiques des groupes de présentation finie, Adv. Math. 116 (1995), 197-262. MR 96m:20056
19.: J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), 428-517. MR 2000g:53043
20.: W.-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98-105. MR 1:313d
21.: M. Coornaert, T. Delzant, and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Mathematics, no. 1441, Springer-Verlag, Berlin, 1990, Les groupes hyperboliques de Gromov. [Gromov hyperbolic groups], With an English summary. MR 92f:57003
22.: G. David and S. Semmes, Fractured fractals and broken dreams: self-similar geometry through metric and measure, Oxford Lecture Series in Mathematics and its Applications, vol. 7, Clarendon Press Oxford University Press, 1997. MR 99h:28018
23.: G. A. Edgar, Packing measure as a gauge variation, Proc. Amer. Math. Soc. 122 (1994), 167-174. MR 94k:28009
24.: -, Integral, probability, and fractal measures, Springer-Verlag, New York, 1998. MR 99c:28024
25.: -, Packing measure in general metric spaces, Real Anal. Exchange (to appear).
26.: H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153, Springer-Verlag, New York, 1969. MR 41:1976
27.: B. Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171-219. MR 20:4187
28.: F. W. Gehring, Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499-519. MR 24:A2677
29.: -, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353-393. MR 25:3166
30.: -, The ${L}\sp{p}$ -integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. MR 53:5861
31.: -, Quasiconformal mappings, pp. 213-268, Internat. Atomic Energy Agency, Vienna, 1976. MR 58:1144
32.: P. Haj $\l$ asz and P. Koskela, Sobolev met Poincaré, Memoirs Amer. Math. Soc. 145 (2000), no. 688, 101 pp. MR 2000j:46063
33.: B. Hanson and J. Heinonen, An -dimensional space that admits a Poincaré inequality but has no manifold points, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3379-3390. MR 2001e:43004
34.: J. Heinonen, Calculus on Carnot groups, Fall School in Analysis (Jyväskylä, 1994), vol. 68, Ber. Univ. Jyväskylä Math. Inst., Jyväskylä, 1995, pp. 1-31. MR 96j:22015
35.: -, A capacity estimate on Carnot groups, Bull. Sci. Math. 119 (1995), 475-484. MR 96j:22011
36.: J. Heinonen and I. Holopainen, Quasiregular maps on Carnot groups, J. Geom. Anal. 7 (1997), no. 1, 109-148. MR 99i:30037
37.: J. Heinonen and P. Koskela, Definitions of quasiconformality, Invent. Math. 120 (1995), 61-79. MR 96e:30051
38.: -, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), 1-61. MR 99j:30025
39.: J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson, Sobolev classes of Banach space-valued functions and quasiconformal mappings, J. Anal. Math. 85 (2001), 87-139.
40.: J. Heinonen and S. Semmes, Thirty-three yes or no questions about mappings, measures, and metrics, Conform. Geom. Dyn. 1 (1997), 1-12. MR 99h:28012
41.: W. Hurewicz and H. Wallman, Dimension theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 3:312b
42.: A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), 309-338. MR 86m:32035
43.: -, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. 111 (1995), 1-87. MR 96c:30021
44.: P. Koskela, Removable sets for Sobolev spaces, Ark. Mat. 37 (1999), no. 2, 291-304. MR 2001g:46077
45.: P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), 1-17. MR 99e:46042
46.: T. Laakso, Ahlfors -regular spaces with arbitrary admitting weak Poincaré inequalities, Geom. Funct. Anal. 10 (2000), no. 1, 111-123.
47.: C. Loewner, On the conformal capacity in space, J. Math. Mech. 8 (1959), 411-414. MR 21:3538
48.: G. A. Margulis and G. D. Mostow, The differential of a quasi-conformal mapping of a Carnot-Carathéodory space, Geom. Funct. Anal. 5 (1995), no. 2, 402-433. MR 96m:53038
49.: P. Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR 96h:28006
50.: G. D. Mostow, Quasi-conformal mappings in -space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53-104. MR 38:4679
51.: P. Pansu, Dimension conforme et sphère à l'infini des variétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 177-212. MR 90k:53079
52.: -, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), 1-60. MR 90e:53058
53.: P. K. Rashevsky, Any two points of a totally nonholonomic space may be connected by an admissible line, Uch. Zap. Ped. Inst. im. Liebknechta, Ser. Phys. Math. 2 (1938), 83-94, in Russian.
54.: H. M. Reimann, Rigidity of -type groups, Math. Z. (to appear).
55.: -, An estimate for pseudoconformal capacities on the sphere, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), 315-324. MR 90m:32041
56.: H. M. Reimann and F. Ricci, The complexified Heisenberg group, Preprint, 2000.
57.: Yu. G. Reshetnyak, Sobolev classes of functions with values in a metric space, Sibirsk. Mat. Zh. 38 (1997), 657-675. MR 98h:46031
58.: C. A. Rogers, Hausdorff measures, Cambridge University Press, London, 1970. MR 43:7576
59.: W. Rudin, Real and complex analysis, third ed., McGraw-Hill, New York, 1987. MR 88k:00002
60.: X. Saint Raymond and C. Tricot, Packing regularity of sets in -space, Math. Proc. Cambridge Philos. Soc. 103 (1988), 133-145. MR 88m:28002
61.: S. Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Selecta Math. 2 (1996), 155-295. MR 97j:46033
62.: N. Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), no. 2, 243-279. CMP 2001:07
63.: P. Tang, Quasiconformal homeomorphisms on CR -manifolds with symmetries, Math. Z. 219 (1995), 49-69. MR 96i:32016
64.: S. J. Taylor and C. Tricot, The packing measure of rectifiable subsets of the plane, Math. Proc. Cambridge Philos. Soc. 99 (1986), 285-296. MR 87b:28008
65.: B. S. Thomson, Construction of measures in metric spaces, J. London Math. Soc. (2) 14 (1976), 21-24. MR 54:10537
66.: C. Tricot, Two definitions of fractional dimension, Math. Proc. Cambridge Philos. Soc. 91 (1982), 57-74. MR 84d:28013
67.: P. Tukia, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 149-160. MR 83b:30019
68.: P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 97-114. MR 82g:30038
69.: J. T. Tyson, Quasiconformality and quasisymmetry in metric measure spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 23 (1998), 525-548. MR 99i:30038
70.: -, Analytic properties of locally quasisymmetric mappings from Euclidean domains, Indiana Univ. Math. J. 49 (2000), no. 3, 995-1016. CMP 2001:06
71.: J. Väisälä, Lectures on -dimensional quasiconformal mappings, Lecture Notes in Mathematics, no. 229, Springer-Verlag, Berlin, 1971. MR 56:12260
72.: -, Quasisymmetric embeddings in Euclidean spaces, Trans. Amer. Math. Soc. 264 (1981), 191-204. MR 82i:30031
73.: -, Quasiconformal maps of cylindrical domains, Acta Math. 162 (1989), 201-225. MR 90f:30034
74.: S. Willard, General topology, Addison-Wesley, Reading, MA, 1970. MR 41:9173
75.: W. P. Ziemer, Extremal length and -capacity, Michigan Math. J. 16 (1969), 43-51. MR 40:346

Retrieve articles in Conformal Geometry and Dynamics with MSC (2000): 30C65, 28A78, 46E35, 43A85

Jeremy T. Tyson
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1109
Address at time of publication: Department of Mathematics, State University of New York at Stony Brook, Stony Brook, New York 11794-3651
Email: tyson@math.sunysb.edu

DOI: 10.1090/S1088-4173-01-00064-9
PII: S 1088-4173(01)00064-9
Keywords: Quasiconformal/quasisymmetric map, conformal modulus, Loewner condition, Hausdorff/packing measure, Poincar\'{e} inequality
Received by editor(s): May 31, 2000
Received by editor(s) in revised form: June 4, 2001
Posted: August 8, 2001
Additional Notes: The results of this paper form part of the author's Ph.D. thesis completed at the University of Michigan in 1999. Research supported by an NSF Graduate Research Fellowship and a Sloan Doctoral Dissertation Fellowship.
Copyright of article: Copyright 2001, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-4173

Previous volume \| Table of contents \| Next volume Articles in press \| Most recent volume \| All volumes Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS