On Big Pieces approximations of parabolic hypersurfaces

Let $Σ$ be a closed subset of $\mathbb{R}^ {n+1}$ which is parabolic Ahlfors-David regular and assume that $Σ$ satisfies a 2-sided corkscrew condition. Assume, in addition, that $Σ$ is either time-forwards Ahlfors-David regular, time-backwards Ahlfors-David regular, or parabolic uniform rectifiable. We then first prove that $Σ$ satisfies a {\it weak synchronized two cube condition}. Based on this we are able to revisit the argument in \cite{NS} and prove that $Σ$ contains {\it uniform big pieces of Lip(1,1/2) graphs}. When $Σ$ is parabolic uniformly rectifiable the construction can be refined and in this case we prove that $Σ$ contains {\it uniform big pieces of regular parabolic Lip(1,1/2) graphs}. Similar results hold if $Ω\subset\mathbb R^{n+1}$ is a connected component of $\mathbb R^{n+1}\setminusΣ$ and in this context we also give a parabolic counterpart of the main result in \cite{AHMNT} by proving that if $Ω$ is a one-sided parabolic chord arc domain, and if $Σ$ is parabolic uniformly rectifiable, then $Ω$ is in fact a parabolic chord arc domain. Our results give a flexible parabolic version of the classical (elliptic) result of G. David and D. Jerison concerning the existence of uniform big pieces of Lipschitz graphs for sets satisfying a two disc condition.

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