Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

Details Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

2006-09-12

arxiv.org/abs/math/0609327v1

Mathematics

Complex Variables

31 pages, 1 figure

Scientific paper

The classical Painlev\'e theorem tells that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains the corresponding critical dimension is $\frac{2}{K+1}$. We show that when $K>1$, unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $\sigma$-finite Hausdorff $\frac{2}{K+1}$-measure are removable for bounded $K$-quasiregular mappings. On the other hand, $\dim(E) = \frac{2}{K+1}$ is not enough to guarantee this property. We also study absolute continuity properties of pull-backs of Hausdorff measures under $K$-quasiconformal mappings, in particular at the relevant dimensions 1 and $\frac{2}{K+1}$. For general Hausdorff measures ${\cal H}^t$, $0 < t < 2$, we reduce the absolute continuity properties to an open question on conformal mappings.

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