Removable sets for intrinsic metric and for holomorphic functions
Complex Variables
2021-08-10 v2 Metric Geometry
Abstract
We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of codimension 1 is metrically removable, which answers a question raised by Hakobyan and Herron. The metrically removable sets are shown to be related to other classes of "thin" sets that appeared in the literature. They are also related to the removability problems for classes of holomorphic functions with restrictions on the derivative.
Cite
@article{arxiv.1707.07363,
title = {Removable sets for intrinsic metric and for holomorphic functions},
author = {Sergei Kalmykov and Leonid V. Kovalev and Tapio Rajala},
journal= {arXiv preprint arXiv:1707.07363},
year = {2021}
}
Comments
23 pages; the conjecture is now proved in all dimensions