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Removable sets for intrinsic metric and for holomorphic functions

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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 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.

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Acknowledgment

We thank the referee for carefully reading themanuscript and suggesting several improvements.

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Authors and Affiliations

  1. School of mathematical sciences, Shanghai Jiao Tong University, 800 Dongchuan RD, Shanghai, 200240, China

    Sergei Kalmykov

  2. Mathematics Department, Syracuse University, 215 Carnegie Building, Syracuse, NY, 13244, USA

    Leonid V. Kovalev

  3. Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), FI-40014, Jyvaskyla, Finland

    Tapio Rajala

Authors
  1. Sergei Kalmykov
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  2. Leonid V. Kovalev
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  3. Tapio Rajala
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Corresponding author

Correspondence to Leonid V. Kovalev.

Additional information

First author supported by NSFC grant 11650110426.

Second author supported by the National Science Foundation grant DMS-1362453.

Third author supported by the Academy of Finland project no. 274372.

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Kalmykov, S., Kovalev, L.V. & Rajala, T. Removable sets for intrinsic metric and for holomorphic functions. JAMA 139, 751–772 (2019). https://doi.org/10.1007/s11854-024-0076-2

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  • DOI: https://doi.org/10.1007/s11854-024-0076-2

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