Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 36, 2011, 3-20

COMPACTNESS OF A CONFORMAL BOUNDARY OF THE EUCLIDEAN UNIT BALL

Päivi Lammi

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; paivi.e.lammi 'at' jyu.fi

Abstract. We study conformal metrics d_\rho on the Euclidean unit ball Bⁿ. We assume that either the density \rho associated with the metric d_\rho satisfies a logarithmic volume growth condition for small balls or that \rho satisfies a Harnack inequality and a suitable sub-Euclidean volume growth condition. We prove that the \rho-boundary \partial_\rhoBⁿ is homeomorphic to S^n-1 if and only if \partial_\rhoBⁿ is compact. In the planar case, the compactness of \partial_\rhoB² is further equivalent to local connectivity of the \rho-boundary together with the boundedness of (B²,d_\rho).

2000 Mathematics Subject Classification: Primary 30C65.

Key words: Boundary, compactness, conformal metrics.

Reference to this article: P. Lammi: Compactness of a conformal boundary of the Euclidean unit ball. Ann. Acad. Sci. Fenn. Math. 36 (2011), 3-20.

Full document as PDF file

doi:10.5186/aasfm.2011.3601

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