On big pieces approximations of parabolic hypersurfaces

Abstract

Let $\mathrm{\Sigma }$ be a closed subset of ${\mathbb{R}}^{n+1}$ which is parabolic Ahlfors-David regular and assume that $\mathrm{\Sigma }$ satisfies a 2-sided corkscrew condition. Assume, in addition, that $\mathrm{\Sigma }$ is either time-forwards Ahlfors-David regular, time-backwards Ahlfors-David regular, or parabolic uniform rectifiable. We then first prove that $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$ contain suniform big pieces of Lip(1,1/2) graphs. When $\mathrm{\Sigma }$ is parabolic uniformly rectifiable the construction can be refined and in this case we prove that $\mathrm{\Sigma }$ contains uniform big pieces of regular parabolic Lip(1,1/2) graphs. Similar results hold if $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$ is a connected component of ${\mathbb{R}}^{n+1}\setminus \mathrm{\Sigma }$ and in this context we also give a parabolic counterpart of the main result of Azzam et al. (2017) by proving that if $\mathrm{\Omega }$ is a one-sided parabolic chord arc domain, and if $\mathrm{\Sigma }$ is parabolic uniformly rectifiable, then $\mathrm{\Omega }$ 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.