Duke Mathematical Journal

Conformal dimension does not assume values between zero and one

Leonid V. Kovalev

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We prove that the conformal dimension of any metric space is at least one unless it is zero. This confirms a conjecture of J. T. Tyson [23, Conj. 1.2]

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Duke Math. J., Volume 134, Number 1 (2006), 1-13.

First available in Project Euclid: 4 July 2006

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Zentralblatt MATH identifier

Primary: 51F99: None of the above, but in this section
Secondary: 47H06: Accretive operators, dissipative operators, etc. 46B20: Geometry and structure of normed linear spaces


Kovalev, Leonid V. Conformal dimension does not assume values between zero and one. Duke Math. J. 134 (2006), no. 1, 1--13. doi:10.1215/S0012-7094-06-13411-7. https://projecteuclid.org/euclid.dmj/1152018503

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  • S. Banach, Théorie des óperations linéaires, Chelsea, New York, 1955.
  • C. J. Bishop, A quasi-symmetric surface with no rectifiable curves, Proc. Amer. Math. Soc. 127 (1999), 2035--2040.
  • C. J. Bishop and J. T. Tyson, Conformal dimension of the antenna set, Proc. Amer. Math. Soc. 129 (2001), 3631--3636.
  • —, Locally minimal sets for conformal dimension, Ann. Acad. Sci. Fenn. Math. 26 (2001), 361--373.
  • M. Bonk and B. Kleiner, Conformal dimension and Gromov hyperbolic groups with $2$-sphere boundary, Geom. Topol. 9 (2005), 219--246.
  • M. Bourdon, Au bord de certains polyèdres hyperboliques, Ann. Inst. Fourier (Grenoble) 45 (1995), 119--141.
  • —, Immeubles hyperboliques, dimension conforme et rigidité de Mostow, Geom. Funct. Anal. 7 (1997), 245--268.
  • M. Bourdon and H. Pajot, Cohomologie $l_p$ et espaces de Besov, J. Reine Angew. Math. 558 (2003), 85--108.
  • G. David and T. Toro, Reifenberg flat metric spaces, snowballs, and embeddings, Math. Ann. 315 (1999), 641--710.
  • K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
  • J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer, New York, 2001.
  • 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.
  • W. Hurewicz and H. Wallman, Dimension Theory, Princeton Math. Ser. 4, Princeton Univ. Press, Princeton, 1941.
  • T. Iwaniec, private communication, October 2003.
  • S. Keith and T. Laakso, Conformal Assouad dimension and modulus, Geom. Funct. Anal. 14 (2004), 1278--1321.
  • L. V. Kovalev and D. Maldonado, Mappings with convex potentials and the quasiconformal Jacobian problem, Illinois J. Math. 49 (2005), 1039--1060.
  • P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Stud. Adv. Math. 44, Cambridge Univ. Press, Cambridge, 1995.
  • 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.
  • R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), 395--431.
  • H. L. Royden, Real Analysis, 3rd ed., Macmillan, New York, 1988.
  • J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4) 146 (1987), 65--96.
  • P. Tukia and J. VäIsäLä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 97--114.
  • J. T. Tyson, Sets of minimal Hausdorff dimension for quasiconformal maps, Proc. Amer. Math. Soc. 128 (2000), 3361--3367.
  • —, Lowering the Assouad dimension by quasisymmetric mappings, Illinois J. Math. 45 (2001), 641--656.
  • J. VäIsäLä, Free quasiconformality in Banach spaces, I, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), 355--379.; II, 16 (1991), 255--310.; III, 17 (1992), 393--408.
  • E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators, Springer, New York, 1990.