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.

中文翻译：

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海森堡群中的 Semmes 曲面和内在 Lipschitz 图

第一个海森堡群中的 Semmes 曲面是一个封闭的上 $3$-正则集合 $S$，它满足以下条件，称为条件 B：每个球 $B(x,r)$ 与 $x \in S$ 和$0 < r < \mathrm{diam}(S)$ 包含两个半径与 $r$ 相当的球，它们包含在 $S$ 的补集的不同连通分量中。Semmes 在 80 美元后期引入了欧几里得空间中的类似曲面。我们证明了 Heisenberg 组中的 Semmes 曲面低于 $3$-regular，并且具有大量的内在 Lipschitz 图。特别是，我们的结果适用于海森堡群中弦弧域的边界。