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RESEARCH PRODUCT
Semmes surfaces and intrinsic Lipschitz graphs in the Heisenberg group
Séverine RigotKatrin FässlerKatrin FässlerTuomas Orponensubject
Closed setApplied MathematicsGeneral Mathematics010102 general mathematicsBoundary (topology)Metric Geometry (math.MG)CodimensionLipschitz continuitySurface (topology)01 natural sciencesCombinatorics28A75 (Primary) 28A78 (Secondary)Mathematics - Metric GeometryMathematics - Classical Analysis and ODEsClassical Analysis and ODEs (math.CA)FOS: MathematicsHeisenberg groupMathematics::Metric Geometrymittateoria[MATH]Mathematics [math]0101 mathematicsIsoperimetric inequalityComputingMilieux_MISCELLANEOUSMathematicsComplement (set theory)description
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2018-03-13 | Transactions of the American Mathematical Society |