Removable sets for intrinsic metric and for holomorphic functions

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 closed totally disconnected set with finite Hausdorff measure of codimension 1 is metrically removable, which answers a...

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Bibliographic Details
Published in:Journal d'analyse mathématique (Jerusalem) 2019-10, Vol.139 (2), p.751-772
Main Authors: Kalmykov, Sergei, Kovalev, Leonid V., Rajala, Tapio
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Analysis
Analytic functions
Dynamical Systems and Ergodic Theory
Functional Analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Metric space
Partial Differential Equations
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Summary: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 closed 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.
ISSN:0021-7670
1565-8538
DOI:10.1007/s11854-024-0076-2