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Proceedings of the American Mathematical Society
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An $n$-dimensional space that admits a Poincaré inequality but has no manifold points

Author(s): Bruce Hanson; Juha Heinonen
Journal: Proc. Amer. Math. Soc. 128 (2000), 3379-3390.
MSC (1991): Primary 43A85; Secondary 28A75
Posted: May 18, 2000
MathSciNet review: 1690990
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Abstract | References | Similar articles | Additional information

Abstract:

For each integer $n\ge 2$ we construct a compact, geodesic metric space $X$ which has topological dimension $n$, is Ahlfors $n$-regular, satisfies the Poincaré inequality, possesses $\mathbb R^n$ as a unique tangent cone at $\mathcal{H}_n$ almost every point, but has no manifold points.

References:

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J. Cheeger, ``Differentiability of Lipschitz functions on metric measure spaces'', GAFA, Geom. Funct. Anal. 9 (1999), 428-517.

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G. David and S. Semmes, ``Fractured fractals and broken dreams; self-similar geometry through metric and measure", Oxford Lecture Series in Mathematics and its Applications 7, Clarendon Press, Oxford, 1997. MR 99h:28018

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M. Gromov, ``Structures Métriques pour les Variétés Riemanniennes", (J. Lafontaine and P. Pansu, eds.), Cedic/Fernand Nathan, Paris, 1981. MR 85e:53051

[HaK 1]
P. Hajlasz and P. Koskela, ``Sobolev meets Poincaré", C.R. Acad. Sci. Paris 320 (1995), 1211-1215. MR 96f:46062

[HaK 2]
P. Hajlasz and P. Koskela, ``Sobolev met Poincaré", Mem. Amer. Math. Soc. (to appear).

[HeK]
J. Heinonen and P. Koskela, ``Quasiconformal maps in metric spaces with controlled geometry", Acta Math. 181 (1998), 1-61. MR 99j:30025

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T.J. Laakso, ``Ahlfors $Q$-regular spaces with arbitrary $Q$admitting weak Poincaré inequality", GAFA, Geom. Funct. Anal. (to appear).

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Additional Information:

Bruce Hanson
Affiliation: Department of Mathematics, St. Olaf College, Northfield, Minnesota 55057
Email: hansonb@stolaf.edu

Juha Heinonen
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

DOI: 10.1090/S0002-9939-00-05453-8
PII: S 0002-9939(00)05453-8
Keywords: Poincaré inequality, Ahlfors $n$-regular, manifold point.
Received by editor(s): August 14, 1998
Received by editor(s) in revised form: January 18, 1999
Posted: May 18, 2000
Additional Notes: The second author was supported by NSF grant DMS 96-22844
Communicated by: Albert Baernstein II
Copyright of article: Copyright 2000, American Mathematical Society




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