Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group

TL;DR

This paper studies Semmes surfaces in the Heisenberg group, showing they are Ahlfors-regular and contain large parts of intrinsic Lipschitz graphs, with implications for geometric measure theory and analysis in sub-Riemannian spaces.

Contribution

It establishes that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular and contain big pieces of intrinsic Lipschitz graphs, extending Euclidean results to a sub-Riemannian setting.

Findings

Semmes surfaces are lower Ahlfors-regular in the Heisenberg group.

Semmes surfaces contain big pieces of intrinsic Lipschitz graphs.

Results apply to boundaries of chord-arc domains and reduced isoperimetric sets.

Abstract

A Semmes surface in the Heisenberg group is a closed set $S$ that is upper Ahlfors-regular with codimension one and satisfies the following condition, referred to as Condition B. Every ball $B(x,r)$ with $x \in S$ and $0 < r < \operatorname{diam} S$ contains two balls with radii comparable to $r$ which are contained in different connected components of the complement of $S$. Analogous sets in Euclidean spaces were introduced by Semmes in the late $80$'s. We prove that Semmes surfaces in the Heisenberg group are lower Ahlfors-regular with codimension one and have big pieces of intrinsic Lipschitz graphs. In particular, our result applies to the boundary of chord-arc domains and of reduced isoperimetric sets.