Intrinsic Lipschitz graphs and vertical $\beta$-numbers in the Heisenberg group

TL;DR

This paper explores the concept of intrinsic Lipschitz graphs in the Heisenberg group, demonstrating their geometric properties and establishing a connection to quantitative rectifiability, which differs from Euclidean cases and relates to nonlinear PDEs.

Contribution

It introduces the notion of quantitative rectifiability in the Heisenberg group and proves that intrinsic Lipschitz graphs satisfy a weak geometric lemma, extending Euclidean results.

Findings

Intrinsic Lipschitz graphs satisfy a weak geometric lemma.

3-Ahlfors-David regular sets with certain properties contain big pieces of intrinsic Lipschitz graphs.

New phenomena in geometric measure theory arise in the Heisenberg group compared to Euclidean space.

Abstract

The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 90's. The theory in $\mathbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in $\mathbb{R}^{3}$. Our main object of study are the intrinsic Lipschitz graphs in $\mathbb{H}$, introduced by B. Franchi, R. Serapioni and F. Serra Cassano in 2006. We claim that these $3$-dimensional sets in $\mathbb{H}$, if any, deserve to be called quantitatively $3$-rectifiable. Our main result is that the intrinsic Lipschitz graphs satisfy a weak geometric lemma with respect to vertical $\beta$-numbers. Conversely, extending a result of David and Semmes from $\mathbb{R}^{n}$, we prove that a $3$-Ahlfors-David regular subset in $\mathbb{H}$, which satisfies the weak geometric lemma and has big vertical projections, necessarily has big pieces of intrinsic Lipschitz graphs.