David Bate

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math.MG2025

Structure of Metric $1$ -currents: approximation by normal currents and representation results

David Bate, Emanuele Caputo, Jakub Takáč +2

We prove the $1$ -dimensional flat chain conjecture in any complete and quasiconvex metric space, namely that metric $1$ -currents can be approximated in mass by normal $1$ -currents.…

math.MG2025

Characterising 1-rectifiable metric spaces via connected tangent spaces

David Bate, Phoebe Valentine

We prove that in a complete metric space $X$ , $1$ -rectifiability of a set $E \subset X$ with $H^{1} (E) < \infty$ and positive lower density $H^{1}$ -a.e. is implied by…

math.MG2025

On 1-regular and 1-uniform metric measure spaces

David Bate

A metric measure space $(X, μ)$ is 1-regular if \[0< \lim_{r\to 0} \frac{μ(B(x,r))}{r}<\infty\] for $μ$ -a.e $x \in X$ . We give a complete geometric characterisation of the rectifiabl…

math.MG2025

Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE

David Bate, Julian Weigt

We give a sufficient condition for a Borel subset $E \subset X$ of a complete metric space with $H^{n} (E) < \infty$ to be $n$ -rectifiable. This condition involves a decomposit…

Structure of Metric 1-currents: approximation by normal currents and representation results

Characterising 1-rectifiable metric spaces via connected tangent spaces

On 1-regular and 1-uniform metric measure spaces

Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE

Structure of Metric $1$ -currents: approximation by normal currents and representation results