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Conformal Geometry and Dynamics

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Thirty-three yes or no questions about mappings, measures, and metrics


Authors: Juha Heinonen and Stephen Semmes
Journal: Conform. Geom. Dyn. 1 (1997), 1-12
MSC (1991): Primary 28A75, 30C65; Secondary 53C23, 57M12
DOI: https://doi.org/10.1090/S1088-4173-97-00012-X
Published electronically: May 22, 1997
MathSciNet review: 1452413
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Abstract: We display thirty-three questions of geometric nature. Most of the problems are of fairly recent origin, and the formulation of each problem is such that it can be answered by one word only.


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  • [AM] S. Akbulut and J. D. McCarthy, Casson's invariant for oriented homology 3-spheres: An exposition, vol. 36, Princeton University Press, Mathematical Notes, Princeton, New Jersey, 1990. MR 90k:57017
  • [A] P. Assouad, Plongements Lipschitziens dans $\mathbf {R}^{n}$, Bull. Soc. Math. France 111 (1983), 429-448. MR 86f:54050
  • [BE] I. Berstein and A. L. Edmonds, On the construction of branched coverings of low-dimensional manifolds, Trans. Amer. Math. Soc. 247 (1979), 87-124. MR 80b:57003
  • [C] J. Cannon, The recognition problem: What is a topological manifold?, Bull. Amer. Math. Soc. 84 (1978), 832-866. MR 58:13043
  • [C1] A. V. Chernavskii, Finite-to-one open mappings of manifolds (in Russian), Mat. Sb. 65 (1964), 357-369. MR 30:2476
  • [C2] A. V. Chernavskii, Remarks on the paper ``Finite-to-one open mappings of manifolds'' (in Russian), Mat. Sb. 66 (1965), 471-472. MR 36:3320
  • [Dn] R. Daverman, Decompositions of manifolds, Academic Press, 1986. MR 88a:57001
  • [Dd] G. David, Morceaux de graphes lipschitziennes et intégrales singulières sur un surface, Rev. Mat. Iberoamericana 4 (1988), 73-114. MR 90h:42026
  • [DS1] G. David and S. Semmes, Strong $A_{\infty }$-weights, Sobolev inequalities, and quasiconformal mappings, Analysis and partial differential equations, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker 122 (1990). MR 91c:30037
  • [DS2] G. David and S. Semmes, Quantitative rectifiability and Lipschitz mappings, Trans. Amer. Math. Soc. 337 (1993), 855-889. MR 93h:42015
  • [DS3] G. David and S. Semmes, Analysis of and on uniformly rectifiable sets, vol. 38, Amer. Math. Soc., Mathematical Surveys and Monographs, 1993. MR 94i:28003
  • [DS4] G. David and S. Semmes, Fractured fractals and broken dreams: Self-similarity through metric and measure, Oxford University Press, to appear.
  • [DK] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford University Press, 1990. MR 92a:57036
  • [DS] S. K. Donaldson and D. P. Sullivan, Quasiconformal $4$-manifolds, Acta Math. 163 (1989), 181-252. MR 91d:57012
  • [E] E. Edwards, The topology of manifolds and cell-like maps, Proceedings ICM, Helsinki (1978), 111-127. MR 81g:57010
  • [F] K. J. Falconer, Fractal Geometry, John Wiley & Sons, 1990. MR 92j:28008
  • [FQ] M. H. Freedman and F. Quinn, Topology of 4-manifolds, vol. 39, Princeton University Press, Princeton Mathematical Series, Princeton, New Jersey, 1990. MR 94b:57021
  • [GS] D. E. Galewski and R. J. Stern, Classification of simplicial triangulations of topological manifolds, Ann. of Math. 111 (1980), 1-34. MR 81f:57012
  • [G1] F. W. Gehring, The Hausdorff measure of sets which link in Euclidean space, Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers, Academic Press, New York, 1974. MR 50:13455
  • [G2] F. W. Gehring, Lower dimensional absolute continuity properties of quasiconformal mappings, Math. Proc. Camb. Phil. Soc. 78 (1975), 81-93. MR 52:3520
  • [H1] J. Heinonen, The boundary absolute continuity of quasiconformal mappings, Amer. J. Math. 116 (1994), 1545-1567. MR 96m:30033
  • [H2] J. Heinonen, A theorem of Semmes and the boundary absolute continuity in all dimensions, Rev. Mat. Iberoamericana 12 (1996), 783-789.
  • [HK1] J. Heinonen and P. Koskela, The boundary distortion of a quasiconformal mapping, Pacific J. Math. 165 (1994), 93-114. MR 95f:30031
  • [HK2] J. Heinonen and P. Koskela, Definitions of quasiconformality, Invent. Math. 120 (1995), 61-79. MR 96e:30051
  • [HK3] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Preprint, University of Jyväskylä (1996).
  • [HR1] J. Heinonen and S. Rickman, Quasiregular maps $\mathbf {S}^{3}\to \mathbf {S}^{3}$ with wild branch sets, Topology (to appear).
  • [HR2] J. Heinonen and S. Rickman, Geometric branched covers between generalized manifolds, in preparation.
  • [HY] J. Heinonen and S. Yang, Strongly uniform domains and periodic quasiconformal maps, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 20 (1995), 123-148. MR 96d:30023
  • [HW] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton, New Jersey, 1948. MR 3:312b
  • [J] D. Jerison, The Poincaré inequality for vector fields satisfying Hörmander's condition, Duke Math. J. 53 (1986), 503-523. MR 87i:35027
  • [K] B. Kirchheim, Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), 113-123. MR 94g:28013
  • [KM] P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Preprint (1997).
  • [KR1] A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), 309-338. MR 86m:32035
  • [KR2] A. Korányi and H. M. Reimann, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. 111 (1995), 1-87. MR 96c:30021
  • [MRV] O. Martio, S. Rickman, and J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 488 (1971), 1-31. MR 45:8830
  • [MV] O. Martio and J. Väisälä, Elliptic equations and maps of bounded length distortion, Math. Ann. 282 (1988), 423-443. MR 89m:35062
  • [Pa] P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. Math. 129 (1989), 1-60. MR 90e:53058
  • [Pi] R. Piergallini, Four-manifolds as $4$-fold branched covers of $\mathbf {S}^{4}$, Topology 34 (1995), 497-508. MR 96g:57003
  • [Re] Yu. G. Reshetnyak, Space mappings with bounded distortion, Translation of Mathematical Monographs 73, American Mathematical Society, Providence, RI, 1989. MR 90d:30067
  • [Ri1] S. Rickman, Existence of quasiregular mappings, Proceedings of the Workshop on Holomorphic Functions and Moduli I. Math. Sci. Res. Inst. Publ. Berkeley, Springer-Verlag 10 (1988), 179-185. MR 90g:30026
  • [Ri2] S. Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete 26, Springer-Verlag, Berlin Heidelberg New York, 1993. MR 95g:30026
  • [Ri3] S. Rickman, Construction of quasiregular mappings, In the volume in honor of F. Gehring, Springer-Verlag (to appear).
  • [S1] S. Semmes, Chord-arc surfaces with small constant. II. Good parameterizations, Adv. Math. 88 (1991), 170-199. MR 93d:42019b
  • [S2] S. Semmes, Bi-Lipschitz mappings and strong $A_{\infty }$ weights, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), 211-248. MR 95g:30032
  • [S3] S. Semmes, Good metric spaces without good parameterizations, Rev. Mat. Iberoamericana 12 (1996), 187-275. CMP 96:12
  • [S4] S. Semmes, On the nonexistence of bilipschitz parameterizations and geometric problems about $A_{\infty }$ weights, Rev. Mat. Iberoamericana 12 (1996), 337-410. CMP 96:16
  • [S5] S. Semmes, Finding Curves on General Spaces through Quantitative Topology with Applications for Sobolev and Poincaré inequalities, Selecta Math. (N.S.) 2 (1996), 155-295. CMP 97:03
  • [S6] S. Semmes, Metric structures in Riemannian spaces, M. Gromov et. al., Birkhäuser (to appear).
  • [S7] S. Semmes, Some remarks about metric spaces, spherical mappings, functions and their derivatives, Publicacions Matemàtiques 40 (1996), no. 2, 411-430. CMP 97:06
  • [S8] S. Semmes, Quasisymmetry, measure, and a question of Heinonen, Rev. Mat. Iberoamericana 12 (1996), 727-781.
  • [SS] L. Siebenmann and D. Sullivan, On complexes that are Lipschitz manifolds, Geometric topology (Proceedings Georgia Topology Conf., Athens, Ga. 1977) edited by J. C. Cantrell, Academic Press, New York, N.Y. - London (1979), 503-525. MR 80h:57027
  • [Sp] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. MR 35:1007
  • [Sr] U. Srebro, Topological properties of quasiregular maps, In Quasiconformal space mappings; a collection of surveys 1960-1990, M. Vuorinen Ed. Springer Verlag, Lecture Notes 1508 (1992), 104-118. CMP 93:02
  • [Su] D. P. Sullivan, Hyperbolic geometry and homeomorphisms, Geometric topology (Proceedings Georgia Topology Conf., Athens, Ga. 1977) edited by J. C. Cantrell, Academic Press, New York, N.Y. - London (1979), 543-555. MR 81m:57012
  • [TV1] 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
  • [TV2] P. Tukia and J. Väisälä, Bi-Lipschitz extensions of maps having quasiconformal extensions, Math. Ann. 269 (1984), 561-572. MR 86c:30041
  • [V1] J. Väisälä, Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I 392 (1966), 1-10. MR 34:814
  • [V2] J. Väisälä, Quasi-symmetric embeddings in euclidean spaces, Trans. Amer. Math. Soc. 264 (1981), 191-204. MR 82i:30031
  • [V3] J. Väisälä, The wall conjecture on domains in Euclidean spaces, Manuscripta Math. (to appear).

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Additional Information

Juha Heinonen
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: juha.heinonen@umich.edu

Stephen Semmes
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77251
Email: semmes@math.rice.edu

DOI: https://doi.org/10.1090/S1088-4173-97-00012-X
Received by editor(s): February 17, 1997
Published electronically: May 22, 1997
Additional Notes: The first author is supported by NSF grant DMS 96-22844. The second author is supported by NSF grant DMS 94-00230. Both authors wish to thank the Institut des Hautes Études Scientifiques for its support.
Article copyright: © Copyright 1997 American Mathematical Society

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