We study compact three-manifolds with boundary obtained by randomly gluing together truncated tetrahedra along their faces. We prove that, asymptotically almost surely as the number of tetrahedra tends to infinity, these manifolds are connected and have a single boundary component. We prove a law of large numbers for the genus of this boundary component, we show that the Heegaard genus of these manifolds is linear in the number of tetrahedra and we bound their first Betti number. We also show that, asymptotically almost surely as the number of tetrahedra tends to infinity, our manifolds admit a unique hyperbolic metric with totally geodesic boundary. We prove a law of large numbers for the volume of this metric, prove that the associated Laplacian has a uniform spectral gap and show that the diameter of our manifolds is logarithmic as a function of their volume. Finally, we determine the Benjamini--Schramm limit of our sequence of random manifolds.
{"title":"A model for random three-manifolds","authors":"Bram Petri, Jean Raimbault","doi":"10.4171/cmh/539","DOIUrl":"https://doi.org/10.4171/cmh/539","url":null,"abstract":"We study compact three-manifolds with boundary obtained by randomly gluing together truncated tetrahedra along their faces. We prove that, asymptotically almost surely as the number of tetrahedra tends to infinity, these manifolds are connected and have a single boundary component. We prove a law of large numbers for the genus of this boundary component, we show that the Heegaard genus of these manifolds is linear in the number of tetrahedra and we bound their first Betti number. \u0000We also show that, asymptotically almost surely as the number of tetrahedra tends to infinity, our manifolds admit a unique hyperbolic metric with totally geodesic boundary. We prove a law of large numbers for the volume of this metric, prove that the associated Laplacian has a uniform spectral gap and show that the diameter of our manifolds is logarithmic as a function of their volume. Finally, we determine the Benjamini--Schramm limit of our sequence of random manifolds.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42076509","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 compute the asymptotic behavior of the average-case filling volume for certain models of random Lipschitz cycles in the unit cube and sphere. For example, we estimate the minimal area of a Seifert surface for a model of random knots first studied by Millett. This is a generalization of the classical Ajtai--Komlos--Tusnady optimal matching theorem from combinatorial probability. The author hopes for applications to the topology of random links, random maps between spheres, and other models of random geometric objects.
{"title":"Filling random cycles","authors":"Fedor Manin","doi":"10.4171/CMH/520","DOIUrl":"https://doi.org/10.4171/CMH/520","url":null,"abstract":"We compute the asymptotic behavior of the average-case filling volume for certain models of random Lipschitz cycles in the unit cube and sphere. For example, we estimate the minimal area of a Seifert surface for a model of random knots first studied by Millett. This is a generalization of the classical Ajtai--Komlos--Tusnady optimal matching theorem from combinatorial probability. The author hopes for applications to the topology of random links, random maps between spheres, and other models of random geometric objects.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43149941","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 extend classical finite-dimensional Fenichel theory in two directions to infinite dimensions. Under comparably weak assumptions we show that the solution of an infinite-dimensional fast-slow system is approximated well by the corresponding slow flow. After that we construct a two-parameter family of slow manifolds $S_{epsilon,zeta}$ under more restrictive assumptions on the linear part of the slow equation. The second parameter $zeta$ does not appear in the finite-dimensional setting and describes a certain splitting of the slow variable space in a fast decaying part and its complement. The finite-dimensional setting is contained as a special case in which $S_{epsilon,zeta}$ does not depend on $zeta$. Finally, we apply our new techniques to three examples of fast-slow systems of partial differential equations.
{"title":"Slow manifolds for infinite-dimensional evolution equations","authors":"Felix Hummel, C. Kuehn","doi":"10.4171/cmh/527","DOIUrl":"https://doi.org/10.4171/cmh/527","url":null,"abstract":"We extend classical finite-dimensional Fenichel theory in two directions to infinite dimensions. Under comparably weak assumptions we show that the solution of an infinite-dimensional fast-slow system is approximated well by the corresponding slow flow. After that we construct a two-parameter family of slow manifolds $S_{epsilon,zeta}$ under more restrictive assumptions on the linear part of the slow equation. The second parameter $zeta$ does not appear in the finite-dimensional setting and describes a certain splitting of the slow variable space in a fast decaying part and its complement. The finite-dimensional setting is contained as a special case in which $S_{epsilon,zeta}$ does not depend on $zeta$. Finally, we apply our new techniques to three examples of fast-slow systems of partial differential equations.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42703909","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}
When $G$ is a Polish group, metrizability of the universal minimal flow has been shown to be a robust dividing line in the complexity of the topological dynamics of $G$. We introduce a class of groups, the CAP groups, which provides a neat generalization of this dividing line to all topological groups. We prove a number of characterizations of this class, having very different flavors, and use these to prove that the class of CAP groups enjoys a number of nice closure properties. As a concrete application, we compute the universal minimal flow of the homeomorphism groups of several scattered topological spaces, building on recent work of Gheysens.
{"title":"Topological dynamics beyond Polish groups","authors":"Gianluca Basso, Andy Zucker","doi":"10.4171/CMH/521","DOIUrl":"https://doi.org/10.4171/CMH/521","url":null,"abstract":"When $G$ is a Polish group, metrizability of the universal minimal flow has been shown to be a robust dividing line in the complexity of the topological dynamics of $G$. We introduce a class of groups, the CAP groups, which provides a neat generalization of this dividing line to all topological groups. We prove a number of characterizations of this class, having very different flavors, and use these to prove that the class of CAP groups enjoys a number of nice closure properties. As a concrete application, we compute the universal minimal flow of the homeomorphism groups of several scattered topological spaces, building on recent work of Gheysens.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49641591","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 combine the DPW method and opening nodes to construct embedded surfaces of positive constant mean curvature with Delaunay ends in euclidean space, with no limitation to the genus or number of ends.
{"title":"Opening nodes in the DPW method: Co-planar case","authors":"M. Traizet","doi":"10.4171/cmh/524","DOIUrl":"https://doi.org/10.4171/cmh/524","url":null,"abstract":"We combine the DPW method and opening nodes to construct embedded surfaces of positive constant mean curvature with Delaunay ends in euclidean space, with no limitation to the genus or number of ends.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45961161","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}
Let $N$ be a closed manifold and $U subset T^*(N)$ a bounded domain in the cotangent bundle of $N$, containing the zero-section. A conjecture due to Viterbo asserts that the spectral metric for Lagrangian submanifolds that are exact-isotopic to the zero-section is bounded. In this paper we establish an upper bound on the spectral distance between two such Lagrangians $L_0, L_1$, which depends linearly on the boundary depth of the Floer complexes of $(L_0, F)$ and $(L_1, F)$, where $F$ is a fiber of the cotangent bundle.
{"title":"Bounds on the Lagrangian spectral metric in cotangent bundles","authors":"P. Biran, O. Cornea","doi":"10.4171/cmh/522","DOIUrl":"https://doi.org/10.4171/cmh/522","url":null,"abstract":"Let $N$ be a closed manifold and $U subset T^*(N)$ a bounded domain in the cotangent bundle of $N$, containing the zero-section. A conjecture due to Viterbo asserts that the spectral metric for Lagrangian submanifolds that are exact-isotopic to the zero-section is bounded. In this paper we establish an upper bound on the spectral distance between two such Lagrangians $L_0, L_1$, which depends linearly on the boundary depth of the Floer complexes of $(L_0, F)$ and $(L_1, F)$, where $F$ is a fiber of the cotangent bundle.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44801235","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 show how self-referential discs in 4-manifolds lead to the construction of pairs of discs with a common geometrically dual sphere which are properly homotopic rel $partial$ and coincide near their boundaries, yet are not properly isotopic. This occurs in manifolds without 2-torsion in their fundamental group, thereby exhibiting phenomena not seen with spheres, e.g. the boundary connect sum of $S^2times D^2$ and $S^1times B^3$. On the other hand we show that two such discs are isotopic rel $partial$ if the manifold is simply connected. We construct in $S^2times D^2natural S^1times B^3$ a properly embedded 3-ball properly homotopic to a $z_0times B^3$ but not properly isotopic to $z_0times B^3$.
我们展示了4-流形中的自指圆盘如何导致具有共同几何对偶球体的圆盘对的构造，这些圆盘对是适当的同位rel$partial$，并且在它们的边界附近重合，但不是适当的同位素。这发生在基本群中没有2-扭转的流形中，从而表现出球面所没有的现象，例如$S^2乘以D^2和$S^1乘以B^3的边界连接和。另一方面，我们证明了如果流形是简单连接的，那么两个这样的圆盘是同位素rel$partial$。我们在$S^2 times D^2 natural S^1 times B^3$中构造了一个与$z_0 times B ^3$适当嵌入的三球适当同位，但与$z_ 0times B ^ 3$不适当同位。
{"title":"Self-referential discs and the light bulb lemma","authors":"David Gabai","doi":"10.4171/cmh/518","DOIUrl":"https://doi.org/10.4171/cmh/518","url":null,"abstract":"We show how self-referential discs in 4-manifolds lead to the construction of pairs of discs with a common geometrically dual sphere which are properly homotopic rel $partial$ and coincide near their boundaries, yet are not properly isotopic. This occurs in manifolds without 2-torsion in their fundamental group, thereby exhibiting phenomena not seen with spheres, e.g. the boundary connect sum of $S^2times D^2$ and $S^1times B^3$. On the other hand we show that two such discs are isotopic rel $partial$ if the manifold is simply connected. We construct in $S^2times D^2natural S^1times B^3$ a properly embedded 3-ball properly homotopic to a $z_0times B^3$ but not properly isotopic to $z_0times B^3$.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48567466","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}
Given an open cover of a closed symplectic manifold, consider all smooth partitions of unity consisting of functions supported in the covering sets. The Poisson bracket invariant of the cover measures how much the functions from such a partition of unity can become close to being Poisson commuting. We introduce a new approach to this invariant, which enables us to prove the lower bound conjectured by L. Polterovich, in dimension 2.
{"title":"Poisson brackets of partitions of unity on surfaces","authors":"Lev Buhovsky, Alexander Logunov, Shira Tanny","doi":"10.4171/cmh/487","DOIUrl":"https://doi.org/10.4171/cmh/487","url":null,"abstract":"Given an open cover of a closed symplectic manifold, consider all smooth partitions of unity consisting of functions supported in the covering sets. The Poisson bracket invariant of the cover measures how much the functions from such a partition of unity can become close to being Poisson commuting. We introduce a new approach to this invariant, which enables us to prove the lower bound conjectured by L. Polterovich, in dimension 2.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138540702","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 prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is a locally bi-Lipschitz embedding map to a sphere, which generalizes a fundamental theorem of Takahashi in submanifold theory to a non-smooth setting. Applications of these results include a topological sphere theorem and topological finiteness theorems, which are new even for closed Riemannian manifolds.
{"title":"Isometric immersions of RCD spaces","authors":"Shouhei Honda","doi":"10.4171/cmh/519","DOIUrl":"https://doi.org/10.4171/cmh/519","url":null,"abstract":"We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is a locally bi-Lipschitz embedding map to a sphere, which generalizes a fundamental theorem of Takahashi in submanifold theory to a non-smooth setting. Applications of these results include a topological sphere theorem and topological finiteness theorems, which are new even for closed Riemannian manifolds.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42422768","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}
On any smooth compact and connected manifold of dimension 2 admitting a smooth nontrivial circle action we construct C∞-diffeomorphisms of topological entropy zero whose differential is ergodic with respect to a smooth measure in the projectivization of the tangent bundle. The proof is based on a version of the “approximation by conjugation”-method.
{"title":"Smooth zero-entropy diffeomorphisms with ergodic derivative extension","authors":"Philipp Kunde","doi":"10.4171/cmh/478","DOIUrl":"https://doi.org/10.4171/cmh/478","url":null,"abstract":"On any smooth compact and connected manifold of dimension 2 admitting a smooth nontrivial circle action we construct C∞-diffeomorphisms of topological entropy zero whose differential is ergodic with respect to a smooth measure in the projectivization of the tangent bundle. The proof is based on a version of the “approximation by conjugation”-method.","PeriodicalId":50664,"journal":{"name":"Commentarii Mathematici Helvetici","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/cmh/478","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45394854","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：