Let $\Sigma$ be a closed subset of $\mathbb{R}^{n+1}$ which is parabolic Ahlfors-David regular and assume that $\Sigma$ satisfies a 2-sided corkscrew condition. Assume, in addition, that $\Sigma$ is either time-forwards Ahlfors-David regular, time-backwards Ahlfors-David regular, or parabolic uniform rectifiable. We then first prove that $\Sigma$ satisfies a weak synchronized two cube condition. Based on this we are able to revisit the argument of Nyström and Strömqvist (2009) and prove that $\Sigma$ contain suniform big pieces of Lip(1,1/2) graphs. When $\Sigma$ is parabolic uniformly rectifiable the construction can be refined and in this case we prove that $\Sigma$ contains uniform big pieces of regular parabolic Lip(1,1/2) graphs. Similar results hold if $\Omega\subset\mathbb{R}^{n+1}$ is a connected component of $\mathbb{R}^{n+1}\setminus\Sigma$ and in this context we also give a parabolic counterpart of the main result of Azzam et al. (2017) by proving that if $\Omega$ is a one-sided parabolic chord arc domain, and if $\Sigma$ is parabolic uniformly rectifiable, then $\Omega$ is in fact a parabolic chord arc domain. Our results give a flexible parabolic version of the classical (elliptic) result of David and Jerison (1990) concerning the existence of uniform big pieces of Lipschitz graphs for sets satisfying a two disc condition.
Simon Bortz
1
;
John Hoffman
2
;
Steve Hofmann
2
;
Jose Luis Luna-Garcia
2
;
Kaj Nyström
3
1
University of Alabama, Department of Mathematics
2
University of Missouri, Department of Mathematics
3
Uppsala University, Department of Mathematics
Simon Bortz; John Hoffman; Steve Hofmann; Jose Luis Luna-Garcia; Kaj Nyström. On big pieces approximations of parabolic hypersurfaces. Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 533-571. doi: 10.54330/afm.115417
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author = {Simon Bortz and John Hoffman and Steve Hofmann and Jose Luis Luna-Garcia and Kaj Nystr\"om},
title = {On big pieces approximations of parabolic hypersurfaces},
journal = {Annales Fennici Mathematici},
pages = {533--571},
year = {2022},
volume = {47},
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doi = {10.54330/afm.115417},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.54330/afm.115417/}
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