Pub Date : 2022-01-01Epub Date: 2022-01-12DOI: 10.1007/s12220-021-00745-7
Zohreh Fathi, Sajjad Lakzian
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.
{"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":"https://doi.org/10.1007/s12220-021-00745-7","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":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8753965/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39687129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01Epub Date: 2022-02-01DOI: 10.1007/s12220-021-00860-5
Gunther Leobacher, Alexander Steinicke
We consider a class of functions defined on metric spaces which generalizes the concept of piecewise Lipschitz continuous functions on an interval or on polyhedral structures. The study of such functions requires the investigation of their exception sets where the Lipschitz property fails. The newly introduced notion of permeability describes sets which are natural exceptions for Lipschitz continuity in a well-defined sense. One of the main results states that continuous functions which are intrinsically Lipschitz continuous outside a permeable set are Lipschitz continuous on the whole domain with respect to the intrinsic metric. We provide examples of permeable sets in , which include Lipschitz submanifolds.
我们考虑的是一类定义在度量空间上的函数，它概括了区间上或多面体结构上的片状 Lipschitz 连续函数的概念。要研究这类函数，就必须研究它们的例外集，在这些例外集中，利普希兹特性失效。新引入的渗透性概念描述了在明确定义的意义上作为利普齐兹连续性自然例外的集合。其中一个主要结果表明，在渗透集外本质上是利普齐兹连续的连续函数，在整个域上相对于内在度量也是利普齐兹连续的。我们举例说明了 R d 中的可渗透集，其中包括 Lipschitz 子线面。
{"title":"Exception Sets of Intrinsic and Piecewise Lipschitz Functions.","authors":"Gunther Leobacher, Alexander Steinicke","doi":"10.1007/s12220-021-00860-5","DOIUrl":"10.1007/s12220-021-00860-5","url":null,"abstract":"<p><p>We consider a class of functions defined on metric spaces which generalizes the concept of piecewise Lipschitz continuous functions on an interval or on polyhedral structures. The study of such functions requires the investigation of their exception sets where the Lipschitz property fails. The newly introduced notion of permeability describes sets which are natural exceptions for Lipschitz continuity in a well-defined sense. One of the main results states that continuous functions which are intrinsically Lipschitz continuous outside a permeable set are Lipschitz continuous on the whole domain with respect to the intrinsic metric. We provide examples of permeable sets in <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> , which include Lipschitz submanifolds.</p>","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"32 4","pages":"118"},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8807473/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"39914097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-05-26DOI: 10.1007/S12220-021-00693-2
A. Boggess, A. Raich
{"title":"The Fundamental Solution to $$Box _b$$ on Quadric Manifolds: Part 3. Asymptotics for a Codimension 2 Case in $${mathbb {C}}^4$$","authors":"A. Boggess, A. Raich","doi":"10.1007/S12220-021-00693-2","DOIUrl":"https://doi.org/10.1007/S12220-021-00693-2","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2021-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S12220-021-00693-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42658630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-03-29DOI: 10.1007/S12220-021-00651-Y
Haremy Zúñiga
{"title":"Discrete Subgroups of $$text{ PSL }(n+1,{mathbb {C}})$$ Acting on the Grassmannians","authors":"Haremy Zúñiga","doi":"10.1007/S12220-021-00651-Y","DOIUrl":"https://doi.org/10.1007/S12220-021-00651-Y","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"1 1","pages":"1-28"},"PeriodicalIF":1.1,"publicationDate":"2021-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S12220-021-00651-Y","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49335966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-27DOI: 10.1007/S12220-021-00623-2
D. Chakrabarti, Phillip S. Harrington
{"title":"A Modified Morrey-Kohn-Hörmander Identity and Applications to the $$overline{partial }$$-Problem","authors":"D. Chakrabarti, Phillip S. Harrington","doi":"10.1007/S12220-021-00623-2","DOIUrl":"https://doi.org/10.1007/S12220-021-00623-2","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"1 1","pages":"1-38"},"PeriodicalIF":1.1,"publicationDate":"2021-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S12220-021-00623-2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41562911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-17DOI: 10.1007/s12220-022-00950-y
A. Clop, L. Hitruhin, B. Sengupta
{"title":"Rotation Bounds for Hölder Continuous Homeomorphisms with Integrable Distortion","authors":"A. Clop, L. Hitruhin, B. Sengupta","doi":"10.1007/s12220-022-00950-y","DOIUrl":"https://doi.org/10.1007/s12220-022-00950-y","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"32 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2021-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44367622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-05DOI: 10.1007/S12220-020-00583-Z
Jianxin Sun, Jie Zhou
{"title":"Compactness of Surfaces in $$pmb {mathbb {R}}^n$$ with Small Total Curvature","authors":"Jianxin Sun, Jie Zhou","doi":"10.1007/S12220-020-00583-Z","DOIUrl":"https://doi.org/10.1007/S12220-020-00583-Z","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"1 1","pages":"1-33"},"PeriodicalIF":1.1,"publicationDate":"2021-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/S12220-020-00583-Z","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47757234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-02-01DOI: 10.1007/S12220-019-00332-X
L. Ha
{"title":"$$C^k$$-Estimates for $$bar{partial }$$-Equation on Certain Convex Domains of Infinite Type in $$mathbb {C}^n$$","authors":"L. Ha","doi":"10.1007/S12220-019-00332-X","DOIUrl":"https://doi.org/10.1007/S12220-019-00332-X","url":null,"abstract":"","PeriodicalId":56121,"journal":{"name":"Journal of Geometric Analysis","volume":"402 1","pages":"2058-2087"},"PeriodicalIF":1.1,"publicationDate":"2021-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86831125","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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