Abstract This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ2n, as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of ℝ2n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these planes have positive Hausdorff m-measure. In addition, if A is a measurable set of Hausdorff dimension greater than m, then there is a set B ⊂ ℝ2n with dim B ⩽ m such that for all x ∈ ℝ2nB there is a positive measure set of isotropic m-planes for which the translate by x of the orthogonal complement of each such plane, intersects A on a set of dimension dim A – m. These results are then applied to obtain analogous results on the nth Heisenberg group.
{"title":"Intersections of Projections and Slicing Theorems for the Isotropic Grassmannian and the Heisenberg group","authors":"Fernando Roman-Garcia","doi":"10.1515/agms-2020-0002","DOIUrl":"https://doi.org/10.1515/agms-2020-0002","url":null,"abstract":"Abstract This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ2n, as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of ℝ2n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these planes have positive Hausdorff m-measure. In addition, if A is a measurable set of Hausdorff dimension greater than m, then there is a set B ⊂ ℝ2n with dim B ⩽ m such that for all x ∈ ℝ2nB there is a positive measure set of isotropic m-planes for which the translate by x of the orthogonal complement of each such plane, intersects A on a set of dimension dim A – m. These results are then applied to obtain analogous results on the nth Heisenberg group.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49500834","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 deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.
{"title":"Higher Dimensional Holonomy Map for Ruled Submanifolds in Graded Manifolds","authors":"Gianmarco Giovannardi","doi":"10.1515/agms-2020-0105","DOIUrl":"https://doi.org/10.1515/agms-2020-0105","url":null,"abstract":"Abstract The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0105","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46994243","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 prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.
{"title":"Duality of Moduli and Quasiconformal Mappings in Metric Spaces","authors":"Rebekah Jones, P. Lahti","doi":"10.1515/agms-2020-0112","DOIUrl":"https://doi.org/10.1515/agms-2020-0112","url":null,"abstract":"Abstract We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2020-0112","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42965023","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 show that if a CD(K, n) space (X, d, f ℋn) with n ≥ 2 has curvature bounded above by κ in the sense of Alexandrov then f is constant.
摘要证明了在Alexandrov意义下，如果一个n≥2的CD(K, n)空间(X, d, f h n)的曲率以K为界，则f是常数。
{"title":"Weakly Noncollapsed RCD Spaces with Upper Curvature Bounds","authors":"V. Kapovitch, C. Ketterer","doi":"10.1515/agms-2019-0010","DOIUrl":"https://doi.org/10.1515/agms-2019-0010","url":null,"abstract":"Abstract We show that if a CD(K, n) space (X, d, f ℋn) with n ≥ 2 has curvature bounded above by κ in the sense of Alexandrov then f is constant.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2019-0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42734816","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 aim of this note is to generalize to the class of non collapsed RCD(K, N) metric measure spaces the volume bound for the effective singular strata obtained by Cheeger and Naber for non collapsed Ricci limits in [13]. The proof, which is based on a quantitative differentiation argument, closely follows the original one. As a simple outcome we provide a volume estimate for the enlargement of Gigli-DePhilippis’ boundary ([20, Remark 3.8]) of ncRCD(K, N) spaces.
{"title":"Volume Bounds for the Quantitative Singular Strata of Non Collapsed RCD Metric Measure Spaces","authors":"Gioacchino Antonelli, Elia Brué, Daniele Semola","doi":"10.1515/agms-2019-0008","DOIUrl":"https://doi.org/10.1515/agms-2019-0008","url":null,"abstract":"Abstract The aim of this note is to generalize to the class of non collapsed RCD(K, N) metric measure spaces the volume bound for the effective singular strata obtained by Cheeger and Naber for non collapsed Ricci limits in [13]. The proof, which is based on a quantitative differentiation argument, closely follows the original one. As a simple outcome we provide a volume estimate for the enlargement of Gigli-DePhilippis’ boundary ([20, Remark 3.8]) of ncRCD(K, N) spaces.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2019-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46594417","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 theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.
{"title":"Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces","authors":"Anders Björn, Daniel Hansevi","doi":"10.1515/agms-2019-0009","DOIUrl":"https://doi.org/10.1515/agms-2019-0009","url":null,"abstract":"Abstract The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2019-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42010068","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 provide a general strategy to construct multilinear inequalities of Brascamp–Lieb type on compact homogeneous spaces of Lie groups. As an application we obtain sharp integral inequalities on the real unit sphere involving functions with some degree of symmetry.
{"title":"Brascamp–Lieb Inequalities on Compact Homogeneous Spaces","authors":"R. Bramati","doi":"10.1515/agms-2019-0007","DOIUrl":"https://doi.org/10.1515/agms-2019-0007","url":null,"abstract":"Abstract We provide a general strategy to construct multilinear inequalities of Brascamp–Lieb type on compact homogeneous spaces of Lie groups. As an application we obtain sharp integral inequalities on the real unit sphere involving functions with some degree of symmetry.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2019-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49251228","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 use continuous and discrete unification to prove that standard triple bubbles in ℝ2 and 𝕊2 are the minimizers of perimeter, among all clusters (Definition 2.3) enclosing the same triple of areas. Unification defines a unified measurement that allows all configurations, regardless of areas, to compete together. Continuous unification proves that if a unified minimizer were better than expected, it would have to have at least one interior bubble component. Discrete unification proves there can only be one interior bubble and that it must be connected. This leaves only the “daisy” configurations: one interior bubble surrounded by an even number of “petals.” A more careful analysis also eliminates these, leaving only the standard triple bubbles as minimizers. The result on the sphere is new; the result in the plane is due to Wichiramala [11]. The double bubble in the sphere was done by Masters [6].
{"title":"Perimeter-Minimizing Triple Bubbles in the Plane and the 2-Sphere","authors":"G. Lawlor","doi":"10.1515/agms-2019-0004","DOIUrl":"https://doi.org/10.1515/agms-2019-0004","url":null,"abstract":"Abstract We use continuous and discrete unification to prove that standard triple bubbles in ℝ2 and 𝕊2 are the minimizers of perimeter, among all clusters (Definition 2.3) enclosing the same triple of areas. Unification defines a unified measurement that allows all configurations, regardless of areas, to compete together. Continuous unification proves that if a unified minimizer were better than expected, it would have to have at least one interior bubble component. Discrete unification proves there can only be one interior bubble and that it must be connected. This leaves only the “daisy” configurations: one interior bubble surrounded by an even number of “petals.” A more careful analysis also eliminates these, leaving only the standard triple bubbles as minimizers. The result on the sphere is new; the result in the plane is due to Wichiramala [11]. The double bubble in the sphere was done by Masters [6].","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2019-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48062094","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 G be a group and let S be a generating set of G. In this article,we introduce a metric dC on G with respect to S, called the cardinal metric.We then compare geometric structures of (G, dC) and (G, dW), where dW denotes the word metric. In particular, we prove that if S is finite, then (G, dC) and (G, dW) are not quasiisometric in the case when (G, dW) has infinite diameter and they are bi-Lipschitz equivalent otherwise. We also give an alternative description of cardinal metrics by using Cayley color graphs. It turns out that colorpermuting and color-preserving automorphisms of Cayley digraphs are isometries with respect to cardinal metrics.
{"title":"Geometry of Generated Groups with Metrics Induced by Their Cayley Color Graphs","authors":"T. Suksumran","doi":"10.1515/agms-2019-0002","DOIUrl":"https://doi.org/10.1515/agms-2019-0002","url":null,"abstract":"Abstract Let G be a group and let S be a generating set of G. In this article,we introduce a metric dC on G with respect to S, called the cardinal metric.We then compare geometric structures of (G, dC) and (G, dW), where dW denotes the word metric. In particular, we prove that if S is finite, then (G, dC) and (G, dW) are not quasiisometric in the case when (G, dW) has infinite diameter and they are bi-Lipschitz equivalent otherwise. We also give an alternative description of cardinal metrics by using Cayley color graphs. It turns out that colorpermuting and color-preserving automorphisms of Cayley digraphs are isometries with respect to cardinal metrics.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2019-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41421077","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 prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.
{"title":"Antisymmetry of the Stochastical Order on all Ordered Topological Spaces","authors":"T. Fritz","doi":"10.1515/agms-2019-0012","DOIUrl":"https://doi.org/10.1515/agms-2019-0012","url":null,"abstract":"Abstract In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2018-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2019-0012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42608305","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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