Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group

Fässler, Katrin; Orponen, Tuomas; Rigot, Séverine

doi:10.1090/tran/8146

Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group
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by Katrin Fässler, Tuomas Orponen and Séverine Rigot PDF

Trans. Amer. Math. Soc. 373 (2020), 5957-5996 Request permission

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.

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