Removable sets for intrinsic metric and for holomorphic functions

TL;DR

This paper investigates metrically removable sets in metric spaces, establishing conditions under which totally disconnected sets with finite Hausdorff measure are removable, and explores their relation to holomorphic function theory.

Contribution

It proves that totally disconnected sets with finite Hausdorff measure of codimension 1 are metrically removable, answering a previously posed question and linking to holomorphic function removability.

Findings

Totally disconnected sets with finite Hausdorff measure of codimension 1 are metrically removable.

Metrically removable sets relate to other 'thin' sets in the literature.

Removability of holomorphic functions with derivative restrictions is connected to these sets.

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.