论文标题
Carnot组中周长的单调套件和本地最小化器
Monotone sets and local minimizers for the perimeter in Carnot groups
论文作者
论文摘要
Cheeger和Kleiner大约在十年前引入了单调集，他们将Heisenberg Group的非Bilipschitz嵌入性的证据减少为$ L^1 $，以分类其单调子集。后来，单调集合在海森堡环境中与几何测量理论问题有关的几项作品中发挥了重要作用。在本文中，我们在一个任意的Carnot组中工作，并表明其单调子集设置为局部有限的周长，这些周长是周长的局部最小化器。在环境Carnot组的额外条件下，我们证明了它们的度量理论内部和支撑是完全单调的。我们还证明了局部最小化器的拓扑和测量特性，其兴趣与单调集的研究无关。作为我们结果的结合，我们特别是一个足够的条件，在该条件下，任何单调集都可以允许精确单调的理论代表。
Monotone sets have been introduced about ten years ago by Cheeger and Kleiner who reduced the proof of the non biLipschitz embeddability of the Heisenberg group into $L^1$ to the classification of its monotone subsets. Later on, monotone sets played an important role in several works related to geometric measure theory issues in the Heisenberg setting. In this paper, we work in an arbitrary Carnot group and show that its monotone subsets are sets with locally finite perimeter that are local minimizers for the perimeter. Under an additional condition on the ambient Carnot group, we prove that their measure-theoretic interior and support are precisely monotone. We also prove topological and measure-theoretic properties of local minimizers for the perimeter whose interest is independent from the study of monotone sets. As a combination of our results, we get in particular a sufficient condition under which any monotone set admits measure-theoretic representatives that are precisely monotone.
