{"title":"Universal Lower Bounds for Laplacians on Weighted Graphs","authors":"D. Lenz, P. Stollmann","doi":"10.1017/9781108615259.009","DOIUrl":"https://doi.org/10.1017/9781108615259.009","url":null,"abstract":"We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet Laplacian on subsets that satisfy natural geometric conditions.","PeriodicalId":393578,"journal":{"name":"Analysis and Geometry on Graphs and Manifolds","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131334638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Uniform Existence of the IDS on Lattices and Groups","authors":"C. Schumacher, F. Schwarzenberger, I. Veselić","doi":"10.1017/9781108615259.017","DOIUrl":"https://doi.org/10.1017/9781108615259.017","url":null,"abstract":"We present a general framework for thermodynamic limits and its applications to a variety of models. In particular we will identify criteria such that the limits are uniform in a parameter. All results are illustrated with the example of eigenvalue counting functions converging to the integrated density of states. In this case, the convergence is uniform in the energy.","PeriodicalId":393578,"journal":{"name":"Analysis and Geometry on Graphs and Manifolds","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128692035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lior Alon, R. Band, Michael Bersudsky, Sebastian K. Egger
{"title":"Neumann Domains on Graphs and Manifolds","authors":"Lior Alon, R. Band, Michael Bersudsky, Sebastian K. Egger","doi":"10.1017/9781108615259.011","DOIUrl":"https://doi.org/10.1017/9781108615259.011","url":null,"abstract":"The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. This paper reviews the subject, as appears in a few recent works and points out some open questions and conjectures. The paper concerns both manifolds and metric graphs and the exposition allows for a comparison between the results obtained for each of them.","PeriodicalId":393578,"journal":{"name":"Analysis and Geometry on Graphs and Manifolds","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125412324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Manifolds with Ricci Curvature in the Kato Class: Heat Kernel Bounds and Applications","authors":"Christian Rose, P. Stollmann","doi":"10.1017/9781108615259.006","DOIUrl":"https://doi.org/10.1017/9781108615259.006","url":null,"abstract":"We review recent results about heat kernel estimates based on Kato conditions on the negative part of the Ricci curvature.","PeriodicalId":393578,"journal":{"name":"Analysis and Geometry on Graphs and Manifolds","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116574280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Gromov–Lawson Tunnels with Estimates","authors":"J. Dodziuk","doi":"10.1017/9781108615259.004","DOIUrl":"https://doi.org/10.1017/9781108615259.004","url":null,"abstract":"In an appendix to an earlier paper (cf. arXiv:1703.00984) we showed we showed how to construct tunnels of positive scalar curvature and of arbitrarily small length and volume connecting points in a emph{three dimensional} manifold of emph{constant sectional curvature}. Here we generalize the construction to arbitrary dimensions and require only positivity of the scalar curvature. This version contains an added section on existence of arbitrarily narrow tunnels of prescribed length.","PeriodicalId":393578,"journal":{"name":"Analysis and Geometry on Graphs and Manifolds","volume":"145 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131983082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spectral Properties of Limit-Periodic Operators","authors":"D. Damanik, J. Fillman","doi":"10.1017/9781108615259.016","DOIUrl":"https://doi.org/10.1017/9781108615259.016","url":null,"abstract":"We survey results concerning the spectral properties of limit-periodic operators. The main focus is on discrete one-dimensional Schr\"odinger operators, but other classes of operators, such as Jacobi and CMV matrices, continuum Schr\"odinger operators and multi-dimensional Schr\"odinger operators, are discussed as well. \u0000We explain that each basic spectral type occurs, and it does so for a dense set of limit-periodic potentials. The spectrum has a strong tendency to be a Cantor set, but there are also cases where the spectrum has no gaps at all. The possible regularity properties of the integrated density of states range from extremely irregular to extremely regular. Additionally, we present background about periodic Schr\"odinger operators and almost-periodic sequences. \u0000In many cases we outline the proofs of the results we present.","PeriodicalId":393578,"journal":{"name":"Analysis and Geometry on Graphs and Manifolds","volume":"281 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134486247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Cushing, Shiping Liu, Florentin Münch, N. Peyerimhoff
{"title":"Curvature Calculations for Antitrees","authors":"D. Cushing, Shiping Liu, Florentin Münch, N. Peyerimhoff","doi":"10.1017/9781108615259.003","DOIUrl":"https://doi.org/10.1017/9781108615259.003","url":null,"abstract":"In this article we prove that antitrees with suitable growth properties are examples of infinite graphs exhibiting strictly positive curvature in various contexts: in the normalized and non-normalized Bakry-Emery setting as well in the Ollivier-Ricci curvature case. We also show that these graphs do not have global positive lower curvature bounds, which one would expect in view of discrete analogues of the Bonnet-Myers theorem. The proofs in the different settings require different techniques.","PeriodicalId":393578,"journal":{"name":"Analysis and Geometry on Graphs and Manifolds","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114068073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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