Illinois Journal of Mathematics

Uniformity from Gromov hyperbolicity

David Herron, Nageswari Shanmugalingam, and Xiangdong Xie

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Abstract

We show that in a metric space $X$ with annular convexity, uniform domains are precisely those Gromov hyperbolic domains whose quasiconformal structure on the Gromov boundary agrees with that on the boundary in $X$. As an application, we show that quasimöbius maps between geodesic spaces with annular convexity preserve uniform domains. These results are quantitative.

Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1065-1109.

Dates
First available in Project Euclid: 18 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258554351

Digital Object Identifier
doi:10.1215/ijm/1258554351

Mathematical Reviews number (MathSciNet)
MR2595756

Zentralblatt MATH identifier
1189.30055

Subjects
Primary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Citation

Herron, David; Shanmugalingam, Nageswari; Xie, Xiangdong. Uniformity from Gromov hyperbolicity. Illinois J. Math. 52 (2008), no. 4, 1065--1109. doi:10.1215/ijm/1258554351. https://projecteuclid.org/euclid.ijm/1258554351


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References

  • H. Aikawa, Potential-theoretic characterizations of nonsmooth domains, Bull. London Math. Soc. 36 (2004), 469–482.
  • H. Aikawa, Characterization of a uniform domain by the boundary Harnack principle, Harmonic analysis and its applications, Yokohama Publ., Yokohama, 2006, pp. 1–17.
  • M. Bonk, J. Heinonen and P. Koskela, Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001).
  • S. Buckley, D. Herron and X. Xie, Metric space inversions, quasihyperbolic distance, and uniform spaces, Indiana Univ. Math. J. 57 (2008), 837–890.
  • M. Bonk and B. Kleiner, Rigidity for quasi-Möbius group actions, J. Differential Geom. 61 (2002), 81–106.
  • M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), 266–306.
  • J. Björn and N. Shanmugalingam, Poincaré inequalities, uniform domains and extension properties for Newton–Sobolev functions in metric spaces, J. Math. Anal. Appl. 332 (2007), 190–208.
  • M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes, Lecture Notes in Math., vol. 1441, Springer, Berlin, 1990.
  • L. Capogna, N. Garofalo and D.-M. Nhieu, Examples of uniform and NTA domains in Carnot groups, Proceedings on Analysis and Geometry (Novosibirsk), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., 2000 (Novosibirsk Akad., 1999), pp. 103–121.
  • L. Capogna and P. Tang, Uniform domains and quasiconformal mappings on the Heisenberg group, Manuscripta Math. 86 (1995), 267–281.
  • E. Ghys and P. de la Harpe (editors), Sur les groupes hyperboliques d'aprés Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser, Boston, MA, 1990.
  • F. W. Gehring, Uniform domains and the ubiquitous quasidisk, Jahresber. Deutsch. Math.-Verein 89 (1987), 88–103.
  • A. V. Greshnov, On uniform and NTA-domains on Carnot groups, Sibirsk. Mat. Zh. 42 (2001), 1018–1035.
  • F. Gehring and B. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199.
  • P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147 (1981), 71–88.
  • R. Korte, Geometric implications of the Poincaré inequality, Results Math. 50 (2007), 93–107.
  • F. Paulin, Un groupe hyperbolique est déterminé par son bord, J. London Math. Soc. (2) 54 (1996), 50–74.
  • J. Väisälä, Hyperbolic and uniform domains in Banach spaces, Ann. Acad. Sci. Fenn. Math. 30 (2005), 261–302.
  • J. Väisälä, The free quasiworld. Freely quasiconformal and related maps in Banach spaces, Quasiconformal geometry and dynamics, Banach Center Publ. (Lublin), vol. 48, Polish Acad. Sci., Warsaw, 1996, pp. 55–118.
  • J. Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), 187–231.
  • J. Väisälä, Uniform domains, Tôhoku Math. J. 40 (1988), 101–118.
  • X. Xie, Quasimöbius maps preserve uniform domains, Ann. Acad. Sci. Fenn. Math. 32 (2007), 481–495.