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Analysis and Geometry in Metric Spaces最新文献

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On the Regularity of Alexandrov Surfaces with Curvature Bounded Below 曲率有界的Alexandrov曲面的正则性
IF 1 3区 数学
Analysis and Geometry in Metric Spaces Pub Date : 2016-11-10 DOI: 10.1515/agms-2016-0012
L. Ambrosio, J. Bertrand
{"title":"On the Regularity of Alexandrov Surfaces with Curvature Bounded Below","authors":"L. Ambrosio, J. Bertrand","doi":"10.1515/agms-2016-0012","DOIUrl":"https://doi.org/10.1515/agms-2016-0012","url":null,"abstract":"Abstract In this note, we prove that on a surface with Alexandrov’s curvature bounded below, the distance derives from a Riemannian metric whose components, for any p ∈ [1, 2), locally belong to W1,p out of a discrete singular set. This result is based on Reshetnyak’s work on the more general class of surfaces with bounded integral curvature.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"4 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2016-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2016-0012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166588","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}
引用次数: 12
Convex Hull Property and Exclosure Theorems for H-Minimal Hypersurfaces in Carnot Groups 卡诺群中h -极小超曲面的凸包性质及包合定理
IF 1 3区 数学
Analysis and Geometry in Metric Spaces Pub Date : 2016-09-20 DOI: 10.1515/agms-2016-0008
F. Montefalcone
{"title":"Convex Hull Property and Exclosure Theorems for H-Minimal Hypersurfaces in Carnot Groups","authors":"F. Montefalcone","doi":"10.1515/agms-2016-0008","DOIUrl":"https://doi.org/10.1515/agms-2016-0008","url":null,"abstract":"Abstract In this paper, we generalize to sub-Riemannian Carnot groups some classical results in the theory of minimal submanifolds. Our main results are for step 2 Carnot groups. In this case, we will prove the convex hull property and some “exclosure theorems” for H-minimal hypersurfaces of class C2 satisfying a Hörmander-type condition.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"4 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2016-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2016-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166333","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}
引用次数: 0
The kinematic formula in the 3D-Heisenberg group 三维海森堡群的运动公式
IF 1 3区 数学
Analysis and Geometry in Metric Spaces Pub Date : 2016-09-10 DOI: 10.1515/agms-2016-0020
Yen-Chang Huang
{"title":"The kinematic formula in the 3D-Heisenberg group","authors":"Yen-Chang Huang","doi":"10.1515/agms-2016-0020","DOIUrl":"https://doi.org/10.1515/agms-2016-0020","url":null,"abstract":"By studying the group of rigid motions, $PSH(1)$, in the 3D-Heisenberg group $H_1$, we define the density and the measure for the sets of horizontal lines. We show that the volume of a convex domain $Dsubset H_1$ is equal to the integral of length of chord over all horizontal lines intersecting $D$. As the classical result in integral geometry, we also define the kinematic density for $PSH(1)$ and show the probability of randomly throwing a vector $v$ interesting the convex domain $Dsubset D_0$ under the condition that $v$ is contained in $D_0$. Both results show the relationship connecting the geometric probability and the natural geometric quantity in Cheng-Hwang-Malchiodi-Yang's work approached by the variational method.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2016-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166917","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}
引用次数: 3
Quasi-Isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology 拟等距不需要在Gromov积拓扑下导出收缩边界的同胚
IF 1 3区 数学
Analysis and Geometry in Metric Spaces Pub Date : 2016-05-05 DOI: 10.1515/agms-2016-0011
Christopher H. Cashen
{"title":"Quasi-Isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology","authors":"Christopher H. Cashen","doi":"10.1515/agms-2016-0011","DOIUrl":"https://doi.org/10.1515/agms-2016-0011","url":null,"abstract":"Abstract We consider a ‘contracting boundary’ of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space.We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by constructing an explicit example.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"4 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2016-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166525","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}
引用次数: 12
Some Invariant Properties of Quasi-Möbius Maps Quasi-Möbius映射的一些不变性
IF 1 3区 数学
Analysis and Geometry in Metric Spaces Pub Date : 2016-03-24 DOI: 10.1515/agms-2017-0004
Loreno Heer
{"title":"Some Invariant Properties of Quasi-Möbius Maps","authors":"Loreno Heer","doi":"10.1515/agms-2017-0004","DOIUrl":"https://doi.org/10.1515/agms-2017-0004","url":null,"abstract":"Abstract We investigate properties which remain invariant under the action of quasi-Möbius maps of quasimetric spaces. A metric space is called doubling with constant D if every ball of finite radius can be covered by at most D balls of half the radius. It is shown that the doubling property is an invariant property for (quasi-)Möbius maps. Additionally it is shown that the property of uniform disconnectedness is an invariant for (quasi-)Möbius maps as well.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"5 1","pages":"69 - 77"},"PeriodicalIF":1.0,"publicationDate":"2016-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2017-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166993","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}
引用次数: 9
On the Hausdorff Dimension of CAT(κ) Surfaces CAT(κ)曲面的Hausdorff维数
IF 1 3区 数学
Analysis and Geometry in Metric Spaces Pub Date : 2016-03-01 DOI: 10.1515/agms-2016-0010
D. Constantine, J.-F. Lafont
{"title":"On the Hausdorff Dimension of CAT(κ) Surfaces","authors":"D. Constantine, J.-F. Lafont","doi":"10.1515/agms-2016-0010","DOIUrl":"https://doi.org/10.1515/agms-2016-0010","url":null,"abstract":"Abstract We prove that a closed surface with a CAT(κ) metric has Hausdorff dimension = 2, and that there are uniform upper and lower bounds on the two-dimensional Hausdorff measure of small metric balls. We also discuss a connection between this uniformity condition and some results on the dynamics of the geodesic flow for such surfaces. Finally,we give a short proof of topological entropy rigidity for geodesic flow on certain CAT(−1) manifolds.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"16 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2016-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166467","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}
引用次数: 3
Multiscale Analysis of 1-rectifiable Measures II: Characterizations 1-可校正措施的多尺度分析II:特征
IF 1 3区 数学
Analysis and Geometry in Metric Spaces Pub Date : 2016-02-11 DOI: 10.1515/agms-2017-0001
Matthew Badger, Raanan Schul
{"title":"Multiscale Analysis of 1-rectifiable Measures II: Characterizations","authors":"Matthew Badger, Raanan Schul","doi":"10.1515/agms-2017-0001","DOIUrl":"https://doi.org/10.1515/agms-2017-0001","url":null,"abstract":"Abstract A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between μ and 1-dimensional Hausdorff measure H1. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L2 variant of P. Jones’ traveling salesman construction, which is of independent interest.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"5 1","pages":"1 - 39"},"PeriodicalIF":1.0,"publicationDate":"2016-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2017-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166932","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}
引用次数: 39
Applications of the ‘Ham Sandwich Theorem’ to Eigenvalues of the Laplacian “火腿三明治定理”在拉普拉斯特征值中的应用
IF 1 3区 数学
Analysis and Geometry in Metric Spaces Pub Date : 2016-01-05 DOI: 10.1515/agms-2016-0015
Kei Funano
{"title":"Applications of the ‘Ham Sandwich Theorem’ to Eigenvalues of the Laplacian","authors":"Kei Funano","doi":"10.1515/agms-2016-0015","DOIUrl":"https://doi.org/10.1515/agms-2016-0015","url":null,"abstract":"Abstract We apply Gromov’s ham sandwich method to get: (1) domain monotonicity (up to a multiplicative constant factor); (2) reverse domain monotonicity (up to a multiplicative constant factor); and (3) universal inequalities for Neumann eigenvalues of the Laplacian on bounded convex domains in Euclidean space.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"4 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2016-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166281","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}
引用次数: 4
Isometric Embeddings of Pro-Euclidean Spaces 前欧几里得空间的等距嵌入
IF 1 3区 数学
Analysis and Geometry in Metric Spaces Pub Date : 2015-10-29 DOI: 10.1515/agms-2015-0019
B. Minemyer
{"title":"Isometric Embeddings of Pro-Euclidean Spaces","authors":"B. Minemyer","doi":"10.1515/agms-2015-0019","DOIUrl":"https://doi.org/10.1515/agms-2015-0019","url":null,"abstract":"Abstract In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E2n+1. Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C0 version of) the famous Nash isometric embedding theorem from [10].","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"3 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2015-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67166009","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}
引用次数: 1
A Universal Separable Diversity 普遍可分的多样性
IF 1 3区 数学
Analysis and Geometry in Metric Spaces Pub Date : 2015-09-23 DOI: 10.1515/agms-2017-0008
David Bryant, A. Nies, P. Tupper
{"title":"A Universal Separable Diversity","authors":"David Bryant, A. Nies, P. Tupper","doi":"10.1515/agms-2017-0008","DOIUrl":"https://doi.org/10.1515/agms-2017-0008","url":null,"abstract":"Abstract The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points.We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities.","PeriodicalId":48637,"journal":{"name":"Analysis and Geometry in Metric Spaces","volume":"8 1","pages":"138 - 151"},"PeriodicalIF":1.0,"publicationDate":"2015-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/agms-2017-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67167112","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}
引用次数: 9
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