Enrico Le Donne, Daniele Morbidelli, Séverine Rigot
{"title":"Horizontally Affine Functions on Step-2 Carnot Algebras.","authors":"Enrico Le Donne, Daniele Morbidelli, Séverine Rigot","doi":"10.1007/s12220-023-01360-4","DOIUrl":null,"url":null,"abstract":"<p><p>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-<i>n</i> Carnot algebra is isomorphic to the exterior algebra of <math><msup><mrow><mi>R</mi></mrow><mi>n</mi></msup></math>. 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.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10492776/pdf/","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Geometric Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s12220-023-01360-4","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2023/9/9 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
Abstract
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-n 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.
期刊介绍:
JGA publishes both research and high-level expository papers in geometric analysis and its applications. There are no restrictions on page length.