Calculus of Variations and Geometric Measure Theory

C. De Lellis - I. Fleschler

An elementary rectifiability lemma and some applications

created by delellis on 06 Jul 2023
modified by fleschler on 04 Sep 2024

[BibTeX]

Accepted Paper: Mathematical Research Letters

Inserted: 6 jul 2023
Last Updated: 4 sep 2024

Year: 2023

ArXiv: 2307.02866 PDF
Notes:

The second version contains minor corrections, an additional general statement in a class of metric spaces, and an outline of the argument for its validity


Abstract:

We generalize a classical theorem of Besicovitch, showing that, for any positive integers $k<n$, if $E\subset \mathbb R^n$ is a Souslin set which is not $\mathcal{H}^k$-$\sigma$-finite, then $E$ contains a purely unrectifiable closed set $F$ with $0< \mathcal{H}^k (F) < \infty$. Therefore, if $E\subset \mathbb R^n$ is a Souslin set with the property that every closed subset with finite $\mathcal{H}^k$ measure is $k$-rectifiable, then $E$ is $k$-rectifiable. Our interest is motivated by recent studies of the structure of the singular sets of several objects in geometric analysis and we explain the usefulness of our lemma with some examples.

Keywords: Rectifiable sets, Hausdorff measure, Non-sigma finite sets, Quantitative rectifiability, Structure of singular sets


Download: