Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

Thursday, July 15, 17:10 ~ 17:40 UTC-3

Conformal removability is hard

Christopher Bishop

Stony Brook University, United States of America   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Suppose E is a compact set in the complex plane and U is its complement. The set E is called removable for a property P, if any holomorphic function on U with this property extends to be holomorphic on the whole plane. This is an important concept with applications in complex analysis, dynamics and probability. Tolsa famously characterized removable sets for bounded holomorphic functions, but such a characterization remains unknown for conformal maps on U that extend homeomorphically to the boundary. We offer an explanation for why the latter problem is actually harder: the collection of removable sets for bounded holomorphic maps is a G-delta set in the space of compact planar sets with the Hausdorff metric, but the collection of conformally removable sets is not even a Borel subset of this space. These results follow from known facts, but they suggest a number of new questions about fractals, removable curves and conformal welding.

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