Abstract Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping f : X → Y between oriented cohomology manifolds X and Y induces a pull-back operator f* : Mk,loc(Y) → Mk,loc(X) between the spaces of metric k-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward f* : Mk,loc(X) → Mk,loc(Y). As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology n-manifolds X admitting a BLD-mapping ℝn → X.
{"title":"Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces","authors":"Pekka Pankka, Elefterios Soultanis","doi":"10.1515/agms-2019-0011","DOIUrl":"https://doi.org/10.1515/agms-2019-0011","url":null,"abstract":"Abstract Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping f : X → Y between oriented cohomology manifolds X and Y induces a pull-back operator f* : Mk,loc(Y) → Mk,loc(X) between the spaces of metric k-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward f* : Mk,loc(X) → Mk,loc(Y). As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology n-manifolds X admitting a BLD-mapping ℝn → X.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2019-0011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41328867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We obtain sharp lower bounds on the radii of inscribed balls for strictly convex isoperimetric domains lying in a 2-dimensional Alexandrov metric space of curvature bounded below. We also characterize the case when such bounds are attained.
{"title":"Inradius Estimates for Convex Domains in 2-Dimensional Alexandrov Spaces","authors":"K. Drach","doi":"10.1515/agms-2018-0009","DOIUrl":"https://doi.org/10.1515/agms-2018-0009","url":null,"abstract":"Abstract We obtain sharp lower bounds on the radii of inscribed balls for strictly convex isoperimetric domains lying in a 2-dimensional Alexandrov metric space of curvature bounded below. We also characterize the case when such bounds are attained.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2018-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48214672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let X be a set. Let K(x, y) > 0 be a measure of the affinity between the data points x and y. We prove that K has the structure of a Newtonian potential K(x, y) = φ(d(x, y)) with φ decreasing and d a quasi-metric on X under two mild conditions on K. The first is that the affinity of each x to itself is infinite and that for x ≠ y the affinity is positive and finite. The second is a quantitative transitivity; if the affinity between x and y is larger than λ > 0 and the affinity of y and z is also larger than λ, then the affinity between x and z is larger than ν(λ). The function ν is concave, increasing, continuous from R+ onto R+ with ν(λ) < λ for every λ > 0
{"title":"Affinity and Distance. On the Newtonian Structure of Some Data Kernels","authors":"H. Aimar, I. Gómez","doi":"10.1515/agms-2018-0005","DOIUrl":"https://doi.org/10.1515/agms-2018-0005","url":null,"abstract":"Abstract Let X be a set. Let K(x, y) > 0 be a measure of the affinity between the data points x and y. We prove that K has the structure of a Newtonian potential K(x, y) = φ(d(x, y)) with φ decreasing and d a quasi-metric on X under two mild conditions on K. The first is that the affinity of each x to itself is infinite and that for x ≠ y the affinity is positive and finite. The second is a quantitative transitivity; if the affinity between x and y is larger than λ > 0 and the affinity of y and z is also larger than λ, then the affinity between x and z is larger than ν(λ). The function ν is concave, increasing, continuous from R+ onto R+ with ν(λ) < λ for every λ > 0","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2018-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44838089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of HƩ. These are canonical conformal deformations of HƩ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.
{"title":"Hyperbolic Unfoldings of Minimal Hypersurfaces","authors":"J. Lohkamp","doi":"10.1515/agms-2018-0006","DOIUrl":"https://doi.org/10.1515/agms-2018-0006","url":null,"abstract":"Abstract We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of HƩ. These are canonical conformal deformations of HƩ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2018-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48857381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give conditions under which a ber-preserving quasisymmetric map between open subsets of bered metric spaces is locally biLipschitz. We also show that a quasiconformal map between open subsets of 2-step Carnot groups with reducible rst layer is locally biLipschitz.
{"title":"Rigidity of Local Quasisymmetric Maps on Fibered Spaces","authors":"M. Medwid, Xiangdong Xie","doi":"10.1515/agms-2018-0003","DOIUrl":"https://doi.org/10.1515/agms-2018-0003","url":null,"abstract":"We give conditions under which a ber-preserving quasisymmetric map between open subsets of bered metric spaces is locally biLipschitz. We also show that a quasiconformal map between open subsets of 2-step Carnot groups with reducible rst layer is locally biLipschitz.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2018-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46510248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The objective of this series is to study metric geometric properties of disjoint unions of Cayley graphs of amenable groups by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union admits a fibred coarse embedding into a Hilbert space (as a disjoint union) if and only if the Cayley boundary of the sequence in the space of marked groups is uniformly a-T-menable. We furthermore extend this result to ones with other target spaces. By combining our main results with constructions of Osajda and Arzhantseva–Osajda, we construct two systems of markings of a certain sequence of finite groups with two opposite extreme behaviors of the resulting two disjoint unions: With respect to one marking, the space has property A. On the other hand, with respect to the other, the space does not admit fibred coarse embeddings into Banach spaces with non-trivial type (for instance, uniformly convex Banach spaces) or Hadamard manifolds; the Cayley limit group is, furthermore, non-exact.
{"title":"Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings","authors":"M. Mimura, Hiroki Sako","doi":"10.1515/agms-2019-0005","DOIUrl":"https://doi.org/10.1515/agms-2019-0005","url":null,"abstract":"Abstract The objective of this series is to study metric geometric properties of disjoint unions of Cayley graphs of amenable groups by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union admits a fibred coarse embedding into a Hilbert space (as a disjoint union) if and only if the Cayley boundary of the sequence in the space of marked groups is uniformly a-T-menable. We furthermore extend this result to ones with other target spaces. By combining our main results with constructions of Osajda and Arzhantseva–Osajda, we construct two systems of markings of a certain sequence of finite groups with two opposite extreme behaviors of the resulting two disjoint unions: With respect to one marking, the space has property A. On the other hand, with respect to the other, the space does not admit fibred coarse embeddings into Banach spaces with non-trivial type (for instance, uniformly convex Banach spaces) or Hadamard manifolds; the Cayley limit group is, furthermore, non-exact.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2019-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46722213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.
{"title":"Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces","authors":"Shouhei Honda","doi":"10.1515/agms-2018-0007","DOIUrl":"https://doi.org/10.1515/agms-2018-0007","url":null,"abstract":"Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied for the glued space of any two (not necessary same dimensional) closed pointed Riemannian manifolds at their base points. This tells us that the BE condition is strictly weaker than the RCD condition even in this setting, and that the local dimension is not constant even if the space satisfies the BE condition with the coincidence between the induced distance by the Cheeger energy and the original distance. In particular, the glued space gives a first example with a Ricci bound from below in the Bakry-Émery sense, whose local dimension is not constant. We also give a necessary and sufficient condition for such spaces to be RCD(K, N) spaces.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2018-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42287979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We introduce and study the conical curvature-dimension condition, CCD(K, N), for finite graphs.We show that CCD(K, N) provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincaré inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs. Another application of the conical curvature-dimension analysis is finding a sharp estimate on the curvature of complete graphs
{"title":"A Global Poincaré inequality on Graphs via a Conical Curvature-Dimension Condition","authors":"Sajjad Lakzian, Zachary Mcguirk","doi":"10.1515/agms-2018-0002","DOIUrl":"https://doi.org/10.1515/agms-2018-0002","url":null,"abstract":"Abstract We introduce and study the conical curvature-dimension condition, CCD(K, N), for finite graphs.We show that CCD(K, N) provides necessary and sufficient conditions for the underlying graph to satisfy a sharp global Poincaré inequality which in turn translates to a sharp lower bound for the first eigenvalues of these graphs. Another application of the conical curvature-dimension analysis is finding a sharp estimate on the curvature of complete graphs","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2018-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43718388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study the long-scale Ollivier Ricci curvature of graphs as a function of the chosen idleness. Similarly to the previous work on the short-scale case, we show that this idleness function is concave and piecewise linear with at most 3 linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long-scale curvature for the Cartesian product of two regular graphs.
{"title":"Long-Scale Ollivier Ricci Curvature of Graphs","authors":"D. Cushing, S. Kamtue","doi":"10.1515/agms-2019-0003","DOIUrl":"https://doi.org/10.1515/agms-2019-0003","url":null,"abstract":"Abstract We study the long-scale Ollivier Ricci curvature of graphs as a function of the chosen idleness. Similarly to the previous work on the short-scale case, we show that this idleness function is concave and piecewise linear with at most 3 linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long-scale curvature for the Cartesian product of two regular graphs.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2019-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67167142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between (n + 1) points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.
{"title":"Scalar Curvature via Local Extent","authors":"G. Veronelli","doi":"10.1515/agms-2018-0008","DOIUrl":"https://doi.org/10.1515/agms-2018-0008","url":null,"abstract":"Abstract We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between (n + 1) points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2017-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2018-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44555502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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