Parabolic rectifiability, tangent planes and tangent measures

TL;DR

This paper introduces a new notion of rectifiability in a parabolic metric space, characterizes it via tangent planes and measures, and explores its relation to classical rectifiability concepts.

Contribution

It defines parabolic rectifiability using $C^1$ and Lipschitz graphs and characterizes it through tangent measures and planes, linking it to existing rectifiability notions.

Findings

Parabolic rectifiability characterized by tangent measures.

Relation established between parabolic and classical rectifiability.

Framework for analyzing rectifiability in parabolic metric spaces.

Abstract

We define rectifiability in $\mathbb{R}^{n}\times\mathbb{R}$ with a parabolic metric in terms of $C^1$ graphs and Lipschitz graphs with small Lipschitz constants and we characterize it in terms of approximate tangent planes and tangent measures. We also discuss relations between the parabolic rectifiability and other notions of rectifiability.