Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group

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 1980s. 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. The proof of the main result uses the concept of quantitative non-monotonicity developed by Cheeger, Kleiner, Naor, and Young. The approach also yields a new proof for the big pieces of Lipschitz graphs property of Semmes surfaces in Euclidean spaces.