2021/02/23 by Simon Bortz, John P. Hoffman, Bortz, Simon +7
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2102.11912
openalex publication_date 2021/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Σ be a closed subset of 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 \citeNS 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⊂\mathbb Rn+1 is a connected component of \mathbb Rn+1∖Σ and in this context we also give a parabolic counterpart of the main result in \citeAHMNT 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.