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arXiv · 2608.23250

Big Pieces of Regular Parabolic (bi-)Lipschitz Images is Equivalent to Parabolic Uniform Rectifiability

Simon Bortz·Matthew Hyde·Mason Sharp
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Abstract

We define the notion of regular parabolic (bi-)Lipschitz images as the parabolic (bi-)Lipschitz maps from $n$-dimensional space time which, up to translation, fix the $t$ variable and whose spatial components are each regular parabolic Lipschitz functions. We show that any parabolic Ahlfors-David regular is parabolic uniformly rectifiable if and only if it has big pieces of parabolic Lipschitz images of $n$-dimensional space time if and only if it has big pieces of parabolic bi-Lipschitz images of $n$-dimensional space time. This further extends the David-Semmes theory to the parabolic setting. Our proof combines the ideas of the first authors previous work [BH12,BHH+22] and some ideas of Azzam and Schul [AS12]. The proof easily adapts (and is far less complicated) to the Euclidean case to give an alternative proof of the analogous fact.

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Simon Bortz, Matthew Hyde, Mason Sharp. 2026-08-24. Big Pieces of Regular Parabolic (bi-)Lipschitz Images is Equivalent to Parabolic Uniform Rectifiability. https://arxiv.org/abs/2608.23250

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