Thirty-three yes or no questions about mappings, measures, and metrics
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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.References
- Selman Akbulut and John D. McCarthy, Casson’s invariant for oriented homology $3$-spheres, Mathematical Notes, vol. 36, Princeton University Press, Princeton, NJ, 1990. An exposition. MR 1030042, DOI 10.1515/9781400860623
- Patrice Assouad, Plongements lipschitziens dans $\textbf {R}^{n}$, Bull. Soc. Math. France 111 (1983), no. 4, 429–448 (French, with English summary). MR 763553, DOI 10.24033/bsmf.1997
- Israel Berstein and Allan L. Edmonds, On the construction of branched coverings of low-dimensional manifolds, Trans. Amer. Math. Soc. 247 (1979), 87–124. MR 517687, DOI 10.1090/S0002-9947-1979-0517687-9
- J. W. Cannon, The recognition problem: what is a topological manifold?, Bull. Amer. Math. Soc. 84 (1978), no. 5, 832–866. MR 494113, DOI 10.1090/S0002-9904-1978-14527-3
- A. V. Černavskiĭ, Finite-to-one open mappings of manifolds, Mat. Sb. (N.S.) 65 (107) (1964), 357–369 (Russian). MR 0172256
- A. V. Černavskiĭ, Addendum to the paper “Finite-to-one open mappings of manifolds”, Mat. Sb. (N.S.) 66 (108) (1965), 471–472 (Russian). MR 0220254
- Robert J. Daverman, Decompositions of manifolds, Pure and Applied Mathematics, vol. 124, Academic Press, Inc., Orlando, FL, 1986. MR 872468
- Guy David, Morceaux de graphes lipschitziens et intégrales singulières sur une surface, Rev. Mat. Iberoamericana 4 (1988), no. 1, 73–114 (French). MR 1009120, DOI 10.4171/RMI/64
- 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, Quantitative rectifiability and Lipschitz mappings, Trans. Amer. Math. Soc. 337 (1993), no. 2, 855–889. MR 1132876, DOI 10.1090/S0002-9947-1993-1132876-8
- Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061, DOI 10.1090/surv/038
- G. David and S. Semmes, Fractured fractals and broken dreams: Self-similarity through metric and measure, Oxford University Press, to appear.
- S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
- S. K. Donaldson and D. P. Sullivan, Quasiconformal $4$-manifolds, Acta Math. 163 (1989), no. 3-4, 181–252. MR 1032074, DOI 10.1007/BF02392736
- Robert D. Edwards, The topology of manifolds and cell-like maps, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp. 111–127. MR 562601
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
- Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton Mathematical Series, vol. 39, Princeton University Press, Princeton, NJ, 1990. MR 1201584
- David E. Galewski and Ronald J. Stern, Classification of simplicial triangulations of topological manifolds, Ann. of Math. (2) 111 (1980), no. 1, 1–34. MR 558395, DOI 10.2307/1971215
- 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, pp. 159–167. MR 0361008
- F. W. Gehring, Lower dimensional absolute continuity properties of quasiconformal mappings, Math. Proc. Cambridge Philos. Soc. 78 (1975), 81–93. MR 382638, DOI 10.1017/S0305004100051513
- Juha Heinonen, The boundary absolute continuity of quasiconformal mappings, Amer. J. Math. 116 (1994), no. 6, 1545–1567. MR 1305877, DOI 10.2307/2375058
- J. Heinonen, A theorem of Semmes and the boundary absolute continuity in all dimensions, Rev. Mat. Iberoamericana 12 (1996), 783–789.
- Juha Heinonen and Pekka Koskela, The boundary distortion of a quasiconformal mapping, Pacific J. Math. 165 (1994), no. 1, 93–114. MR 1285566, DOI 10.2140/pjm.1994.165.93
- Juha Heinonen and Pekka Koskela, Definitions of quasiconformality, Invent. Math. 120 (1995), no. 1, 61–79. MR 1323982, DOI 10.1007/BF01241122
- J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Preprint, University of Jyväskylä (1996).
- J. Heinonen and S. Rickman, Quasiregular maps $\mathbf {S}^{3}\to \mathbf {S}^{3}$ with wild branch sets, Topology (to appear).
- J. Heinonen and S. Rickman, Geometric branched covers between generalized manifolds, in preparation.
- Juha Heinonen and Shan Shuang Yang, Strongly uniform domains and periodic quasiconformal maps, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 1, 123–148. MR 1304111
- Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
- David Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (1986), no. 2, 503–523. MR 850547, DOI 10.1215/S0012-7094-86-05329-9
- Bernd Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), no. 1, 113–123. MR 1189747, DOI 10.1090/S0002-9939-1994-1189747-7
- P. Koskela and P. MacManus, Quasiconformal mappings and Sobolev spaces, Preprint (1997).
- A. Korányi and H. M. Reimann, Quasiconformal mappings on the Heisenberg group, Invent. Math. 80 (1985), no. 2, 309–338. MR 788413, DOI 10.1007/BF01388609
- A. Korányi and H. M. Reimann, Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. 111 (1995), no. 1, 1–87. MR 1317384, DOI 10.1006/aima.1995.1017
- O. Martio, S. Rickman, and J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A. I. 488 (1971), 31. MR 299782
- O. Martio and J. Väisälä, Elliptic equations and maps of bounded length distortion, Math. Ann. 282 (1988), no. 3, 423–443. MR 967022, DOI 10.1007/BF01460043
- Pierre Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60 (French, with English summary). MR 979599, DOI 10.2307/1971484
- R. Piergallini, Four-manifolds as $4$-fold branched covers of $S^4$, Topology 34 (1995), no. 3, 497–508. MR 1341805, DOI 10.1016/0040-9383(94)00034-I
- Yu. G. Reshetnyak, Space mappings with bounded distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden. MR 994644, DOI 10.1090/mmono/073
- Seppo Rickman, Existence of quasiregular mappings, Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986) Math. Sci. Res. Inst. Publ., vol. 10, Springer, New York, 1988, pp. 179–185. MR 955819, DOI 10.1007/978-1-4613-9602-4_{1}5
- Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
- S. Rickman, Construction of quasiregular mappings, In the volume in honor of F. Gehring, Springer-Verlag (to appear).
- Stephen Semmes, Chord-arc surfaces with small constant. I, Adv. Math. 85 (1991), no. 2, 198–223. MR 1093006, DOI 10.1016/0001-8708(91)90056-D
- Stephen Semmes, Bi-Lipschitz mappings and strong $A_\infty$ weights, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 2, 211–248. MR 1234732
- S. Semmes, Good metric spaces without good parameterizations, Rev. Mat. Iberoamericana 12 (1996), 187–275.
- S. Semmes, On the nonexistence of bilipschitz parameterizations and geometric problems about $A_{\infty }$ weights, Rev. Mat. Iberoamericana 12 (1996), 337–410.
- 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.
- S. Semmes, Metric structures in Riemannian spaces, M. Gromov et. al., Birkhäuser (to appear).
- S. Semmes, Some remarks about metric spaces, spherical mappings, functions and their derivatives, Publicacions Matemàtiques 40 (1996), no. 2, 411–430.
- S. Semmes, Quasisymmetry, measure, and a question of Heinonen, Rev. Mat. Iberoamericana 12 (1996), 727–781.
- L. Siebenmann and D. Sullivan, On complexes that are Lipschitz manifolds, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) Academic Press, New York-London, 1979, pp. 503–525. MR 537747
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- 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.
- Dennis Sullivan, Hyperbolic geometry and homeomorphisms, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) Academic Press, New York-London, 1979, pp. 543–555. MR 537749
- 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
- Pekka Tukia and Jussi Väisälä, Bi-Lipschitz extensions of maps having quasiconformal extensions, Math. Ann. 269 (1984), no. 4, 561–572. MR 766014, DOI 10.1007/BF01450765
- Jussi Väisälä, Discrete open mappings on manifolds, Ann. Acad. Sci. Fenn. Ser. A I No. 392 (1966), 10. MR 0200928
- 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
- J. Väisälä, The wall conjecture on domains in Euclidean spaces, Manuscripta Math. (to appear).
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
- 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.
- © Copyright 1997 American Mathematical Society
- 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
- MathSciNet review: 1452413