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Metric surfaces and conformally removable sets in the plane 平面中的公设曲面和保角可移集
arXiv - MATH - Metric Geometry Pub Date : 2024-08-30 DOI: arxiv-2408.17174
Dimitrios Ntalampekos
{"title":"Metric surfaces and conformally removable sets in the plane","authors":"Dimitrios Ntalampekos","doi":"arxiv-2408.17174","DOIUrl":"https://doi.org/arxiv-2408.17174","url":null,"abstract":"We characterize conformally removable sets in the plane with the aid of the\u0000recent developments in the theory of metric surfaces. We prove that a compact\u0000set in the plane is $S$-removable if and only if there exists a quasiconformal\u0000map from the plane onto a metric surface that maps the given set to a set of\u0000linear measure zero. The statement fails if we consider maps into the plane\u0000rather than metric surfaces. Moreover, we prove that a set is $S$-removable\u0000(resp. $CH$-removable) if and only if every homeomorphism from the plane onto a\u0000metric surface (resp. reciprocal metric surface) that is quasiconformal in the\u0000complement of the given set is quasiconformal everywhere.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187862","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}
引用次数: 0
Persistent equivariant cohomology 持久等变同调
arXiv - MATH - Metric Geometry Pub Date : 2024-08-30 DOI: arxiv-2408.17331
Henry Adams, Evgeniya Lagoda, Michael Moy, Nikola Sadovek, Aditya De Saha
{"title":"Persistent equivariant cohomology","authors":"Henry Adams, Evgeniya Lagoda, Michael Moy, Nikola Sadovek, Aditya De Saha","doi":"arxiv-2408.17331","DOIUrl":"https://doi.org/arxiv-2408.17331","url":null,"abstract":"This article has two goals. First, we hope to give an accessible introduction\u0000to persistent equivariant cohomology. Given a topological group $G$ acting on a\u0000filtered space, persistent Borel equivariant cohomology measures not only the\u0000shape of the filtration, but also attributes of the group action on the\u0000filtration, including in particular its fixed points. Second, we give an\u0000explicit description of the persistent equivariant cohomology of the circle\u0000action on the Vietoris-Rips metric thickenings of the circle, using the Serre\u0000spectral sequence and the Gysin homomorphism. Indeed, if $frac{2pi k}{2k+1}\u0000le r < frac{2pi(k+1)}{2k+3}$, then\u0000$H^*_{S^1}(mathrm{VR}^mathrm{m}(S^1;r))cong\u0000mathbb{Z}[u]/(1cdot3cdot5cdotldots cdot (2k+1), u^{k+1})$ where\u0000$mathrm{deg}(u)=2$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187861","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}
引用次数: 0
Nonsmooth d'Alembertian for Lorentz distance functions 洛伦兹距离函数的非光滑达朗贝尔函数
arXiv - MATH - Metric Geometry Pub Date : 2024-08-29 DOI: arxiv-2408.16525
Mathias Braun
{"title":"Nonsmooth d'Alembertian for Lorentz distance functions","authors":"Mathias Braun","doi":"arxiv-2408.16525","DOIUrl":"https://doi.org/arxiv-2408.16525","url":null,"abstract":"We refine a recent distributional notion of d'Alembertian of a signed Lorentz\u0000distance function to an achronal set in a metric measure spacetime obeying the\u0000timelike measure contraction property. We show precise representation formulas\u0000and comparison estimates (both upper and lower bounds). Under a condition we\u0000call \"infinitesimally strict concavity\" (known for infinitesimally Minkowskian\u0000structures and established here for Finsler spacetimes), we prove the\u0000associated distribution is a signed measure certifying the integration by parts\u0000formula. This treatment of the d'Alembertian using techniques from metric\u0000geometry expands upon its recent elliptic interpretation; even in the smooth\u0000case, our formulas seem to pioneer its exact shape across the timelike cut\u0000locus. Two central ingredients our contribution unifies are the localization\u0000paradigm of Cavalletti-Mondino and the Lorentzian Sobolev calculus of Beran et\u0000al. In the second part of our work, we present several applications of these\u0000insights. First, we show the equivalence of the timelike curvature-dimension\u0000condition with a Bochner-type inequality. Second, we set up synthetic mean\u0000curvature (and barriers for CMC sets) exactly. Third, we prove volume and area\u0000estimates of Heintze-Karcher-type, which enable us to show several synthetic\u0000volume singularity theorems.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187630","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}
引用次数: 0
Merge Trees of Periodic Filtrations 周期过滤的合并树
arXiv - MATH - Metric Geometry Pub Date : 2024-08-29 DOI: arxiv-2408.16575
Herbert Edelsbrunner, Teresa Heiss
{"title":"Merge Trees of Periodic Filtrations","authors":"Herbert Edelsbrunner, Teresa Heiss","doi":"arxiv-2408.16575","DOIUrl":"https://doi.org/arxiv-2408.16575","url":null,"abstract":"Motivated by applications to crystalline materials, we generalize the merge\u0000tree and the related barcode of a filtered complex to the periodic setting in\u0000Euclidean space. They are invariant under isometries, changing bases, and\u0000indeed changing lattices. In addition, we prove stability under perturbations\u0000and provide an algorithm that under mild geometric conditions typically\u0000satisfied by crystalline materials takes $mathcal{O}({(n+m) log n})$ time, in\u0000which $n$ and $m$ are the numbers of vertices and edges in the quotient\u0000complex, respectively.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187629","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}
引用次数: 0
Tubes in sub-Riemannian geometry and a Weyl's invariance result for curves in the Heisenberg groups 亚黎曼几何中的管子和海森堡群曲线的韦尔不变性结果
arXiv - MATH - Metric Geometry Pub Date : 2024-08-29 DOI: arxiv-2408.16838
Tania Bossio, Luca Rizzi, Tommaso Rossi
{"title":"Tubes in sub-Riemannian geometry and a Weyl's invariance result for curves in the Heisenberg groups","authors":"Tania Bossio, Luca Rizzi, Tommaso Rossi","doi":"arxiv-2408.16838","DOIUrl":"https://doi.org/arxiv-2408.16838","url":null,"abstract":"The purpose of the paper is threefold: first, we prove optimal regularity\u0000results for the distance from $C^k$ submanifolds of general rank-varying\u0000sub-Riemannian structures. Then, we study the asymptotics of the volume of\u0000tubular neighbourhoods around such submanifolds. Finally, for the case of\u0000curves in the Heisenberg groups, we prove a Weyl's invariance result: the\u0000volume of small tubes around a curve does not depend on the way the curve is\u0000isometrically embedded, but only on its Reeb angle. The proof does not need the\u0000computation of the actual volume of the tube, and it is new even for the\u0000three-dimensional Heisenberg group.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187628","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}
引用次数: 0
Convexity of sums of eigenvalues of a segment of unitaries 单元段特征值之和的凸性
arXiv - MATH - Metric Geometry Pub Date : 2024-08-29 DOI: arxiv-2408.16906
Gabriel Larotonda, Martin Miglioli
{"title":"Convexity of sums of eigenvalues of a segment of unitaries","authors":"Gabriel Larotonda, Martin Miglioli","doi":"arxiv-2408.16906","DOIUrl":"https://doi.org/arxiv-2408.16906","url":null,"abstract":"For a $ntimes n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles\u0000of $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1le\u0000mle n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path\u0000$tmapsto e^{tx}e^y$ of unitary matrices, is a convex function of $t$ (provided\u0000the path stays in a vecinity of the identity matrix). This vecinity is\u0000described in terms of the opertor norm of matrices, and it is optimal. We show\u0000that the when all the maps $tmapsto s_m(t)$ are linear, then $x$ commutes with\u0000$y$. Several application to unitarily invariant norms in the unitary group are\u0000given. Then we extend these applications to $Ad$-invariant Finsler norms in the\u0000special unitary group of matrices. This last result is obtained by proving that\u0000any $Ad$-invariant Finsler norm in a compact semi-simple Lie group $K$ is the\u0000supremum of a family of what we call orbit norms, induced by the Killing form\u0000of $K$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187627","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}
引用次数: 0
The Riesz $α$-energy of log-concave functions and related Minkowski problem 对数凹函数的里兹α元能及相关闵科夫斯基问题
arXiv - MATH - Metric Geometry Pub Date : 2024-08-28 DOI: arxiv-2408.16141
Niufa Fang, Deping Ye, Zengle Zhang
{"title":"The Riesz $α$-energy of log-concave functions and related Minkowski problem","authors":"Niufa Fang, Deping Ye, Zengle Zhang","doi":"arxiv-2408.16141","DOIUrl":"https://doi.org/arxiv-2408.16141","url":null,"abstract":"We calculate the first order variation of the Riesz $alpha$-energy of a\u0000log-concave function $f$ with respect to the Asplund sum. Such a variational\u0000formula induces the Riesz $alpha$-energy measure of log-concave function $f$,\u0000which will be denoted by $mathfrak{R}_{alpha}(f, cdot)$. We pose the related\u0000Riesz $alpha$-energy Minkowski problem aiming to find necessary and/or\u0000sufficient conditions on a pregiven Borel measure $mu$ defined on $Rn$ so\u0000that $mu=mathfrak{R}_{alpha}(f,cdot)$ for some log-concave function $f$.\u0000Assuming enough smoothness, the Riesz $alpha$-energy Minkowski problem reduces\u0000to a new Monge-Amp`{e}re type equation involving the Riesz $alpha$-potential.\u0000Moreover, this new Minkowski problem can be viewed as a functional counterpart\u0000of the recent Minkowski problem for the chord measures in integral geometry\u0000posed by Lutwak, Xi, Yang and Zhang (Comm. Pure Appl. Math., 2024). The\u0000Riesz $alpha$-energy Minkowski problem will be solved under certain mild\u0000conditions on $mu$.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187633","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}
引用次数: 0
Iterated graph systems and the combinatorial Loewner property 迭代图系统和组合洛夫纳特性
arXiv - MATH - Metric Geometry Pub Date : 2024-08-28 DOI: arxiv-2408.15692
Riku Anttila, Sylvester Eriksson-Bique
{"title":"Iterated graph systems and the combinatorial Loewner property","authors":"Riku Anttila, Sylvester Eriksson-Bique","doi":"arxiv-2408.15692","DOIUrl":"https://doi.org/arxiv-2408.15692","url":null,"abstract":"We introduce iterated graph systems which yield fractal spaces through a\u0000projective sequence of self-similar graphs. Our construction yields new\u0000examples of self-similar fractals, such as the pentagonal Sierpi'nski carpet\u0000and a pillow-space, and it yields a new framework to study potential theory on\u0000many standard examples in the literature, including generalized Sierpi'nski\u0000carpets. We demonstrate that a fractal arising from an iterated graph system\u0000satisfying mild geometric conditions and having enough reflection symmetries,\u0000satisfies the combinatorial Loewner property (CLP) of Bourdon and Kleiner. For\u0000certain exponents, one can also obtain the conductive homogeneity condition of\u0000Kigami. This shows, in a precise sense, that self-similarity and sufficient\u0000symmetry yields CLP. Furthermore, we show that our examples satisfy important asymptotic behaviors\u0000of discrete moduli. In particular, we establish the so-called\u0000super-multiplicative inequality for our examples - which when $p=2$ plays a\u0000crucial role in the study of Dirichlet forms and random walks on fractals. This\u0000implies the super-multiplicativity bound also for many fractals for which it\u0000was not known before, such as the Menger sponge. A particular feature here is\u0000using replacement flows and the notion of a flow basis. This yields a simpler\u0000way to establish many known estimates, and allows us to prove several new ones.\u0000Further, super-multiplicativity shows that one can effectively estimate the\u0000conformal dimensions numerically for many self-similar fractals.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187632","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}
引用次数: 0
A Young type integration on self-similar sets in intervals 区间自相似集上的杨式积分
arXiv - MATH - Metric Geometry Pub Date : 2024-08-28 DOI: arxiv-2408.15468
Takashi Maruyama, Tatsuki Seto
{"title":"A Young type integration on self-similar sets in intervals","authors":"Takashi Maruyama, Tatsuki Seto","doi":"arxiv-2408.15468","DOIUrl":"https://doi.org/arxiv-2408.15468","url":null,"abstract":"We introduce a generalization of the Young integration on self-similar sets\u0000defined in a closed interval and give a sufficient condition of its\u0000integrability. We also prove integration by substitution, integration by parts\u0000and term-by-term integration and give examples of the properties.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187638","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}
引用次数: 0
A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes 公度量时空中的非线性达朗贝尔比较定理和因果微分学
arXiv - MATH - Metric Geometry Pub Date : 2024-08-28 DOI: arxiv-2408.15968
Tobias Beran, Mathias Braun, Matteo Calisti, Nicola Gigli, Robert J. McCann, Argam Ohanyan, Felix Rott, Clemens Sämann
{"title":"A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes","authors":"Tobias Beran, Mathias Braun, Matteo Calisti, Nicola Gigli, Robert J. McCann, Argam Ohanyan, Felix Rott, Clemens Sämann","doi":"arxiv-2408.15968","DOIUrl":"https://doi.org/arxiv-2408.15968","url":null,"abstract":"We introduce a variational first-order Sobolev calculus on metric measure\u0000spacetimes. The key object is the maximal weak subslope of an arbitrary causal\u0000function, which plays the role of the (Lorentzian) modulus of its differential.\u0000It is shown to satisfy certain chain and Leibniz rules, certify a locality\u0000property, and be compatible with its smooth analog. In this setup, we propose a\u0000quadraticity condition termed infinitesimal Minkowskianity, which singles out\u0000genuinely Lorentzian structures among Lorentz-Finsler spacetimes. Moreover, we\u0000establish a comparison theorem for a nonlinear yet elliptic $p$-d'Alembertian\u0000in a weak form under the timelike measure contraction property. As a particular\u0000case, this extends Eschenburg's classical estimate past the timelike cut locus.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187631","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}
引用次数: 0
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