{"title":"Bakry-Émery Ricci Curvature Bounds for Doubly Warped Products of Weighted Spaces.","authors":"Zohreh Fathi, Sajjad Lakzian","doi":"10.1007/s12220-021-00745-7","DOIUrl":null,"url":null,"abstract":"<p><p>We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry-Émery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate analysis of the quadratic forms involved, prompting the introduction of some crucial notions such as curvature saturation at a vertex. In the spirit of being thorough and to provide a frame of reference, we also introduce the <math> <mfenced><msub><mi>R</mi> <mn>1</mn></msub> <mo>,</mo> <msub><mi>R</mi> <mn>2</mn></msub> </mfenced> </math> -doubly warped products of smooth measure spaces and establish <math><mi>N</mi></math> -Bakry-Émery Ricci curvature (lower) bounds thereof in terms of those of the factors. At the end of these notes, we present examples and demonstrate applications of warped products with some toy models.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"32 3","pages":"79"},"PeriodicalIF":1.2000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8753965/pdf/","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Geometric Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s12220-021-00745-7","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2022/1/12 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
Abstract
We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry-Émery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate analysis of the quadratic forms involved, prompting the introduction of some crucial notions such as curvature saturation at a vertex. In the spirit of being thorough and to provide a frame of reference, we also introduce the -doubly warped products of smooth measure spaces and establish -Bakry-Émery Ricci curvature (lower) bounds thereof in terms of those of the factors. At the end of these notes, we present examples and demonstrate applications of warped products with some toy models.
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.