Le Donne Enrico, Morbidelli Daniele, Rigot Séverine
Département de Mathématiques, Université de Fribourg, Ch. du musée 23, 1700 Fribourg, Switzerland.
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box (MaD), 40014 Jyvaskyla, Finland.
J Geom Anal. 2023;33(11):359. doi: 10.1007/s12220-023-01360-4. Epub 2023 Sep 9.
In this paper, we introduce the notion of horizontally affine, h-affine in short, function and give a complete description of such functions on step-2 Carnot algebras. We show that the vector space of h-affine functions on the free step-2 rank- Carnot algebra is isomorphic to the exterior algebra of . Using that every Carnot algebra can be written as a quotient of a free Carnot algebra, we shall deduce from the free case a description of h-affine functions on arbitrary step-2 Carnot algebras, together with several characterizations of those step-2 Carnot algebras where h-affine functions are affine in the usual sense of vector spaces. Our interest for h-affine functions stems from their relationship with a class of sets called precisely monotone, recently introduced in the literature, as well as from their relationship with minimal hypersurfaces.
在本文中，我们引入了水平仿射函数（简称为h - 仿射函数）的概念，并给出了此类函数在二阶卡诺代数上的完整描述。我们证明了自由二阶秩 - 卡诺代数上的h - 仿射函数的向量空间同构于 的外代数。利用每个卡诺代数都可以写成自由卡诺代数的商这一事实，我们将从自由情形推导出任意二阶卡诺代数上h - 仿射函数的描述，以及那些h - 仿射函数在向量空间通常意义下是仿射的二阶卡诺代数的几个特征。我们对h - 仿射函数的兴趣源于它们与文献中最近引入的一类称为精确单调的集合的关系，以及它们与极小超曲面的关系。
