On Big Pieces approximations of parabolic hypersurfaces

Abstract

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 {\it weak synchronized two cube condition}. Based on this we are able to revisit the argument in \cite{NS} and prove that $\Sigma$ contains {\it uniform 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 {\it 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 in \cite{AHMNT} 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 G. David and D. Jerison concerning the existence of uniform big pieces of Lipschitz graphs for sets satisfying a two disc condition.