We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite subRiemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every constant-normal set in a Carnot group its subRiemannian-Lebesgue representative is regularly open, contractible, and its topological boundary coincides with the reduced boundary and with the measure-theoretic boundary. We infer these properties from a metric cone property. Such a cone will be a semisubgroup with nonempty interior that is canonically associated with the normal direction. We characterize the constant-normal sets exactly as those that are arbitrary unions of translations of such semisubgroups. Second, making use of such a characterization, we provide some pathological examples in the specific case of the free-Carnot group of step 3 and rank 2. Namely, we construct a constant normal set that, with respect to any Riemannian metric, is not of locally finite perimeter; we also construct an example with non-unique intrinsic blowup at some point, showing that it has different upper and lower subRiemannian density at the origin. Third, we show that in Carnot groups of step 4 or less, every constant-normal set is intrinsically rectifiable, in the sense of Franchi, Serapioni, and Serra Cassano.

中文翻译：

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卡诺组中法线不变的集合：属性和示例

我们分析了具有固有常数法线的卡诺组的子集，这些子集出现在具有有限次黎曼周长的集合的爆破研究中。本文的目的是三方面的。首先，我们证明了任意Carnot组的一些适度的规律性和结构性结果。即，我们表明，对于卡诺组中的每个常态正态集合，其次黎曼－勒贝格代表是有规律的，可收缩的，并且其拓扑边界与缩减边界以及测度理论边界重合。我们从度量圆锥属性推断出这些属性。这样的圆锥将是具有非空内部的半子群，其内部与法线方向规范关联。我们将常态-正规集的特征恰好描述为此类半子群的翻译的任意并集。第二，利用这种特征，我们在步骤3和等级2的自由卡诺组的特定情况下提供了一些病理学示例。即，我们构造了一个恒定的法线集，对于任何黎曼度量而言，该法线集都不是局部的有限周长 我们还构造了一个实例，该实例在某个点具有非唯一的固有爆燃，表明其在原点具有不同的上下亚黎曼密度。第三，我们表明，在第4步或更少的卡诺组中，从Franchi，Serapioni和Serra Cassano的意义上讲，每个恒常态集合在本质上都是可校正的。我们还构造了一个实例，该实例在某个点具有非唯一的固有爆燃，表明其在原点具有不同的上下亚黎曼密度。第三，我们表明，在第4步或更少的卡诺组中，从Franchi，Serapioni和Serra Cassano的意义上讲，每个恒常态集合在本质上都是可校正的。我们还构造了一个实例，该实例在某个点具有非唯一的固有爆燃，表明其在原点具有不同的上下亚黎曼密度。第三，我们表明，在第4步或更少的卡诺组中，从Franchi，Serapioni和Serra Cassano的意义上讲，每个恒常态集合在本质上都是可校正的。