Volume (2001) no. 270

Uniformizing Gromov hyperbolic spaces

Bonk, Mario; Heinonen, Juha; Koskela, Pekka

Astérisque, no. 270 (2001) , 107 p.

MR 1829896 | Zbl 0970.30010 | 1 citation in Numdam

@book{AST_2001__270__R1_0,
     author = {Bonk, Mario and Heinonen, Juha and Koskela, Pekka},
     title = {Uniformizing Gromov hyperbolic spaces},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {270},
     year = {2001},
     zbl = {0970.30010},
     mrnumber = {1829896},
     language = {en},
     url = {http://www.numdam.org/item/AST_2001__270__R1_0}
}

Bonk, Mario; Heinonen, Juha; Koskela, Pekka. Uniformizing Gromov hyperbolic spaces. Astérisque, no. 270 (2001), 107 p. http://www.numdam.org/item/AST_2001__270__R1_0/

References

[A1] A. Ancona, "On strong barriers and an inequality of Hardy for domains in $ℝ^{n}$ ", J. London Math. Soc. 34 (1986), 274-290. | Article | MR 856511 | Zbl 0629.31002

[A2] A. Ancona, "Negatively curved manifolds, elliptic operators, and the Martin boundary", Ann. Math. 125 (1987), 495-536. | Article | MR 890161 | Zbl 0652.31008

[A3] A. Ancona, "Positive harmonic functions and hyperbolicity", In "Potential Theory Surveys and Problems", Lecture Notes in Mathematics 1344, Springer, Berlin, 1988, pp. 1-23. | Article | MR 973878 | Zbl 0677.31006

[Ar] H. Arai, "Degenerate elliptic operators, Hardy spaces and diffusions on strongly pseudoconvex domains", Tôhoku Math. J. (2) 46 (1994), 469-498. | Article | MR 1301285 | Zbl 0823.32002

[BB1] Z. Balogh and M. Bonk, "Pseudoconvexity and Gromov hyperbolicity", C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), 597-602. | Article | MR 1679982 | Zbl 0940.32007

[BB2] Z. Balogh and M. Bonk, "Gromov hyperbolicity and the Kobayashi metric in strictly pseudoconvex domains", Comment. Math. Helv. 75 (2000), 504-533. | Article | MR 1793800 | Zbl 0986.32012

[BV1] Z. Balogh and A. Volberg, "Geometric localization, uniformly John property and separated semihyperbolic dynamics", Ark. Math. 34 (1996), 21-49. | Article | MR 1396621 | Zbl 0855.30022

[BV2] Z. Balogh and A. Volberg, "Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications", Rev. Mat. Iberoamericana 12, (1996), 299-336. | Article | MR 1402670 | Zbl 0857.31006

[BePo] A. Beardon and Ch. Pommerenke, "The Poincaré metric of plane domains", J. London Math. Soc. 18 (1978), 475-483. | Article | MR 518232 | Zbl 0399.30008

[Bo] M. Bonk, "Quasi-geodesic segments and Gromov hyperbolic spaces", Geom. Dedicata 62 (1996), 281-298. | Article | MR 1406442 | Zbl 0864.53031

[BKR] M. Bonk, P. Koskela, and S. Rohde, "Conformal metrics on the unit ball in Euclidean space", Proc. London Math. Soc. 77 (1998), 635-664. | Article | MR 1643421 | Zbl 0916.30017

[BoPa] M. Bourdon and H. Pajot, "Poincaré inequalities and quasiconformal structure on the boundary of some hyperbolic buildings", Proc. Amer. Math. Soc. 127 (1999), 2315-2324. | Article | MR 1610912 | Zbl 0924.30030

[BoSc] M. Bonk and O. Schramm, "Embeddings of Gromov hyperbolic spaces", Geom. Funct. Anal. 10 (2000), 266-306. | Article | MR 1771428 | Zbl 0972.53021

[BuSt] S. M. Buckley and A. Stanoyevitch, "Weak slice conditions and product domains", Preprint, 1999. | MR 1900897

[Bu] P. Buser, "A note on the isoperimetric constant", Ann. Sci. École Norm. Sup. 15 (1982), 213-230. | Article | Numdam | MR 683635 | Zbl 0501.53030

[C] C. B. Croke, "Some isoperimetric inequalities and eigenvalue estimates", Ann. Sci. École Norm. Sup. 13 (1980), 419-435. | Article | Numdam | MR 608287 | Zbl 0465.53032

[CG] L. Capogna and N. Garofalo, "Boundary behavior of nonnegative solutions of subelliptic equations in NTA domains for Carnot-Carathéodory metrics", J. Fourier Anal. Appl. 4 (1998), 403-432. | Article | MR 1658616 | Zbl 0926.35043

[CDP] M. Coornaert, M. Delzant and T. Papadopoulos, "Géométrie et théorie des groupes. Les groupes hyperboliques de Gromov", Lecture Notes in Mathematics, 1441, Springer, Berlin, 1990. | MR 1075994 | Zbl 0727.20018

[ET] V. A. Efremovich and E. S. Tihomirova, "Equimorphisms of hyperbolic spaces", Izv. Akad. Nauk SSSR 28 (1964), 1139-1144 (Russian). | MR 169121 | Zbl 0134.16002

[Fe] H. Federer, "Geometric measure theory", Springer, Berlin, 1969. | MR 257325 | Zbl 0874.49001 | Zbl 0176.00801

[Fl] W. J. Floyd, "Group completions and limit sets of Kleinian groups", Invent. Math. 57 (1980), 205-218. | Article | MR 568933 | Zbl 0428.20022

[GN] N. Garofalo and D.-N. Nhieu, "Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces", J. Analyse Math. 74 (1998), 67-97. | Article | MR 1631642 | Zbl 0906.46026

[Ge] F. W. Gehring, "Uniform domains and the ubiquitous quasidisk", Jahresber. Deutsch. Math.-Verein. 89 (1987), 88-103. | MR 880189 | Zbl 0615.30016

[GeHa] F. W. Gehring and W. K. Hayman, "An inequality in the theory of conformal mapping", J. Math. Pures Appl. 41 (1962), 353-361. | MR 148884 | Zbl 0105.28002

[GM] F. W. Gehring and O. Martio, "Quasiextremal distance domains and extension of quasiconformal mappings", J. Analyse Math. 45 (1985), 181-206. | Article | MR 833411 | Zbl 0596.30031

[GO] F. W. Gehring and B. G. Osgood, "Uniform domains and the quasi-hyperbolic metric", J. Analyse Math. 36 (1979), 50-74. | Article | MR 581801 | Zbl 0449.30012

[GePa] F. W. Gehring and B. P. Palka, "Quasiconformally homogeneous domains", J. Analyse Math. 30 (1976), 172-199. | Article | MR 437753 | Zbl 0349.30019

[GhHa] E. Ghys and P. De La Harpe, "Sur les Groupes Hyperboliques d'après Mikhael Gromov", Birkhäuser, Progress in Mathematics, Boston-Basel-Berlin, 1990. | MR 1086650 | Zbl 0731.20025

[Gr1] M. Gromov, "Metric Structures for Riemannian and Non-Riemannian Spaces", Birkhäuser, Boston, 1999. | MR 1699320 | Zbl 0953.53002

[Gr2] M. Gromov, "Hyperbolic Groups", Essays in Group Theory, S. Gersten, Editor, MSRI Publication, Springer, 1987, pp. 75-265. | Article | MR 919829 | Zbl 0634.20015

[Gr3] M. Gromov, "Asymptotic invariants of infinite groups", in: Geometric Group Theory, London Math. Soc., Lecture Notes Series 182 (1993). | MR 1253544

[GrPa] M. Gromov and P. Pansu, "Rigidity of Lattices: An Introduction." In Geometric Topology: Recent Developments, Lecture Notes in Mathematics 1504, Springer, Berlin, 1991. | MR 1168043 | Zbl 0786.22015

[HS1] Z-X. He and O. Schramm, "Fixed points, Koebe uniformization and circle packings", Ann. Math. 137 (1993), 369-406. | Article | MR 1207210 | Zbl 0777.30002

[HS2] Z-X. He and O. Schramm, "Koebe uniformization for "almost circle domains"", Amer. J. Math. 117 (1995), 653-667. | Article | MR 1333941 | Zbl 0830.30006

[Hei] J. Heinonen, "Quasiconformal mappings onto John domains", Rev. Mat. Iberoamericana 5 (1989), 97-123. | Article | MR 1077206 | Zbl 0712.30017

[HeiK] J. Heinonen and P. Koskela, "Quasiconformal maps in metric spaces with controlled geometry", Acta Math. 181 (1998), 1-61. | Article | MR 1654771 | Zbl 0915.30018

[HN] J. Heinonen and R. Näkki, "Quasiconformal distortion on arcs", J. Analyse Math. 63 (1994), 19-53. | Article | MR 1269214 | Zbl 0804.30016

[HR] J. Heinonen and S. Rohde, "The Gehring-Hayman theorem for quasihyperbolic geodesics", Math. Proc. Camb. Phil. Soc. 114 (1993), 393-405 | Article | MR 1235987 | Zbl 0791.30015

[He1] L. L. Helms, "Introduction to potential theory", Wiley-Interscience, New York, 1969. | MR 261018 | Zbl 0188.17203

[Her] D. A. Herron, "The geometry of uniform, quasicircle, and circle domains", Ann. Acad. Sci. Fenn. Ser. A.I. Math. 12 (1987), 217-228. | Article | MR 951971 | Zbl 0596.30028

[HerK] D. A. Herron and P. Koskela, "Quasiextremal distance domains and conformal mappings onto circle domains", Complex Variables Theory Appl. 15 (1990), 167-179. | Article | MR 1074059 | Zbl 0682.30011

[HY] J. G. Hocking and G. S. Young, "Topology", Academic Press, 1959. | MR 125557 | Zbl 0135.22701

[Ho] I. Holopainen, "Rough isometries and $p$ -harmonic functions with finite Dirichlet integral", Rev. Mat. Iberoamericana 10 (1994), 143-176. | Article | MR 1271760 | Zbl 0797.31008

[JK] D. S. Jerison and C. E. Kenig, "Boundary behavior of harmonic functions in non-tangentially accessible domains", Adv. Math. 46 (1982), 80-147. | Article | MR 676988 | Zbl 0514.31003

[J] P. W. Jones, "Quasiconformal mappings and extendability of functions in Sobolev spaces", Acta Math. 147 (1981), 71-88. | Article | MR 631089 | Zbl 0489.30017

[La] T. J. Laakso, "Ahlfors $𝒬$ -regular spaces with arbitrary $𝒬 > 1$ admitting weak Poincaré inequality", Geom. Funct. Anal. 10 (2000), 111-123. | Article | MR 1748917 | Zbl 0962.30006

[Le] J. L. Lewis, "Uniformly fat sets", Trans. Amer. Math. Soc. 830 (1988), 177-196. | Article | MR 946438 | Zbl 0668.31002

[M] P. Macmanus, "Catching sets with quasicircles", Rev. Mat. Iberoamericana 15 (1999), 267-277. | Article | MR 1715408 | Zbl 0944.30013

[MS] O. Martio and J. Sarvas, "Injectivity theorems in plane and space", Ann. Acad. Sci. Fenn. Ser. A. I. Math. 4 (1978), 383-401. | Article | MR 565886 | Zbl 0406.30013

[NV] R. Näkki and J. Väisälä, "John disks", Expo. Math. 9 (1991), 3-43. | MR 1101948 | Zbl 0757.30028

[N] D.-M. Nhieu, "Extension of Sobolev spaces on the Heisenberg group", C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), 1559-1564. | MR 1367807 | Zbl 0841.43007

[Pa] P. Pansu, "Quasiconformal mappings and manifolds of negative curvature", Curvature and topology of Riemannian manifolds (Katata, 1985), Lecture Notes in Mathematics, 1201, Springer, Berlin, 1986, pp. 212-229. | Article | MR 859587 | Zbl 0592.53031

[Po] Ch. Pommerenke, "On uniformly perfect sets and Fuchsian groups", Analysis 4 (1984), 299-321. | Article | MR 780609 | Zbl 0501.30036

[SC] L. Saloff-Coste, "Uniformly elliptic operators on Riemannian manifolds", J. Differential Geom. 36 (1992), 417-450. | Article | MR 1180389 | Zbl 0735.58032

[Sc] O. Schramm, "Transboundary extremal length", J. Analyse Math. 66 (1995), 307-329. | Article | MR 1370355 | Zbl 0842.30006

[Seg] S. Segawa, "Martin boundaries of Denjoy domains and quasiconformal mappings", J. Math. Kyoto Univ. 30 (1990), 297-316. | Article | MR 1068793 | Zbl 0736.30015

[ST] S. Segawa and T. Tada, "Martin compactifications and quasiconformal mappings", Proc. Amer. Math. Soc. 93 (1985), 242-244. | Article | MR 770529 | Zbl 0575.30017

[Sem] S. Semmes, "Some novel types of fractal geometry", Book Manuscript, 1999. | MR 1815356 | Zbl 0970.28001

[St] E. M. Stein, "Singular integrals and differentiability properties of functions", Princeton University Press, Princeton, New Jersey, 1970. | MR 290095 | Zbl 0207.13501

[Ts] M. Tsuji, "Potential theory in modern function theory", Chelsea, 1959. | MR 414898 | Zbl 0322.30001 | Zbl 0087.28401

[Ty] J. Tyson, "Geometric and analytic applications of a generalized definition of the conformal modulus", Thesis, University of Michigan, Ann Arbor, 1999. | MR 2699421

[TV] P. Tukia and J. Väisälä, "Quasisymmetric embeddings of metric spaces", Ann. Acad. Sci. Fenn. Ser. A. I. Math. 5 (1980), 97-114. | Article | MR 595180 | Zbl 0403.54005

[V1] J. Väisälä, "Quasimöbius maps", J. Analyse Math. 44 (1984/85), 218-234. | Article | MR 801295 | Zbl 0593.30022

[V2] J. Väisälä, "Quasiconformal maps of cylindrical domains", Acta Math. 162 (1989), 201-225. | Article | MR 989396 | Zbl 0674.30017

[V3] J. Väisälä, "Relatively and inner uniform domains", Conformal Geom. Dynam. 2 (1998), 56-88. | Article | MR 1637079 | Zbl 0902.30017

[V4] J. Väisälä, "The free quasiworld. Freely quasiconformal and related maps in Banach spaces", Banach Center Publications 48, Polish Academy of Sciences, Warszawa, 1999. | Article | MR 1709974 | Zbl 0934.30018

[VG] S. K. Vodopyanov and A. V. Greshnov, "On the continuation of functions of bounded mean oscillation on spaces of homogeneous type with intrinsic metric", Siberian Math. J. 36 (1995), 873-901. | Article | MR 1373594 | Zbl 0865.30029