Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group
Fässler, K., Orponen, T., & Rigot, S. (2020). Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group. Transactions of the American Mathematical Society, 373(8), 5957-5996. https://doi.org/10.1090/tran/8146
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© 2020 American Mathematical Society
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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The first author was supported by Swiss National Science Foundation via the project Intrinsic rectifiability and mapping theory on the Heisenberg group, grant no. $161299$. The second author was supported by the Academy of Finland via the project Quantitative rectifiability in Euclidean and non-Euclidean spaces, grant no. $309365$, and by the University of Helsinki via the project Quantitative rectifiability of sets and measures in Euclidean spaces and Heisenberg groups, grant no. $75160012$. The third author was partially supported by the French National Research Agency, Sub-Riemannian Geometry and Interactions ANR-15-CE40-0018 project. ...License
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