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# Title: Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group

Abstract: A Semmes surface in the first Heisenberg group is a closed upper $3$-regular set $S$ that satisfies the following condition, referred to as Condition B: every ball $B(x,r)$ with $x \in S$ and $0 < r < \mathrm{diam}(S)$ contains two balls with radii comparable to $r$ which are contained in different connected components of the complement of $S$. Analogous surfaces in Euclidean spaces were introduced by Semmes in the late $80$'s. We prove that Semmes surfaces in the Heisenberg group are lower $3$-regular, and have big pieces of intrinsic Lipschitz graphs. In particular, our result applies to the boundaries of chord-arc domains in the Heisenberg group.
 Comments: 32 pages, 5 figures Subjects: Classical Analysis and ODEs (math.CA); Metric Geometry (math.MG) MSC classes: 28A75 (Primary) 28A78 (Secondary) Cite as: arXiv:1803.04819 [math.CA] (or arXiv:1803.04819v1 [math.CA] for this version)

## Submission history

From: Tuomas Orponen [view email]
[v1] Tue, 13 Mar 2018 14:08:13 GMT (155kb)