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Rigidity Theorems for Higher Rank Lattice Actions 高阶晶格作用的刚性定理
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-05-29 DOI: 10.1007/s00039-024-00683-w
Homin Lee
{"title":"Rigidity Theorems for Higher Rank Lattice Actions","authors":"Homin Lee","doi":"10.1007/s00039-024-00683-w","DOIUrl":"https://doi.org/10.1007/s00039-024-00683-w","url":null,"abstract":"<p>Let Γ be a weakly irreducible lattice in a higher rank semisimple algebraic Lie group <i>G</i> without property (T) such as <span>(mathrm {SL}_{2}({mathbb{Z}}[sqrt{2}]))</span> in <span>(mathrm {SL}_{2}({mathbb{R}})times mathrm {SL}_{2}({mathbb{R}}))</span>.</p><p>In this paper, for such Γ, we prove a cocycle superrigidity, <i>Dynamical cocycle superrigidity</i>, for Γ-actions under <i>L</i><sup>2</sup> integrability and irreducibility assumptions. It gives a similar result to Zimmer’s cocycle superrigidity, so we can apply it instead of Zimmer’s cocycle superrigidity in many situations. For instance, in this paper, we obtain global rigidity of Anosov Γ-actions on nilmanifolds under the irreducibility assumption on a fully supported invariant measure.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"32 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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 Parabolic U(1)-Higgs Equations and Codimension-Two Mean Curvature Flows 抛物线 U(1)-Higgs 方程与二维平均曲率流
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-05-29 DOI: 10.1007/s00039-024-00684-9
Davide Parise, Alessandro Pigati, Daniel Stern
{"title":"The Parabolic U(1)-Higgs Equations and Codimension-Two Mean Curvature Flows","authors":"Davide Parise, Alessandro Pigati, Daniel Stern","doi":"10.1007/s00039-024-00684-9","DOIUrl":"https://doi.org/10.1007/s00039-024-00684-9","url":null,"abstract":"<p>We develop the asymptotic analysis as <i>ε</i>→0 for the natural gradient flow of the self-dual <i>U</i>(1)-Higgs energies </p><span>$$ E_{varepsilon }(u,nabla )=int _{M}left (|nabla u|^{2}+ varepsilon ^{2}|F_{nabla }|^{2}+ frac{(1-|u|^{2})^{2}}{4varepsilon ^{2}}right ) $$</span><p> on Hermitian line bundles over closed manifolds (<i>M</i><sup><i>n</i></sup>,<i>g</i>) of dimension <i>n</i>≥3, showing that solutions converge in a measure-theoretic sense to codimension-two mean curvature flows—i.e., integral (<i>n</i>−2)-Brakke flows—generalizing results of (Pigati and Stern in Invent. Math. 223:1027–1095, 2021) from the stationary case. Given any integral (<i>n</i>−2)-cycle Γ<sub>0</sub> in <i>M</i>, these results can be used together with the convergence theory developed in (Parise et al. in Convergence of the self-dual <i>U</i>(1)-Yang–Mills–Higgs energies to the (<i>n</i>−2)-area functional, 2021, arXiv:2103.14615) to produce nontrivial integral Brakke flows starting at Γ<sub>0</sub> with additional structure, similar to those produced via Ilmanen’s elliptic regularization.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"53 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Equilibrium States of Endomorphisms of $mathbb{P}^{k}$ : Spectral Stability and Limit Theorems $mathbb{P}^{k}$的均衡状态：谱稳定性与极限定理
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-05-13 DOI: 10.1007/s00039-024-00678-7
Fabrizio Bianchi, Tien-Cuong Dinh
{"title":"Equilibrium States of Endomorphisms of $mathbb{P}^{k}$ : Spectral Stability and Limit Theorems","authors":"Fabrizio Bianchi, Tien-Cuong Dinh","doi":"10.1007/s00039-024-00678-7","DOIUrl":"https://doi.org/10.1007/s00039-024-00678-7","url":null,"abstract":"<p>We establish the existence of a spectral gap for the transfer operator induced on <span>(mathbb{P}^{k} = mathbb{P}^{k} (mathbb{C}))</span> by a generic holomorphic endomorphism and a suitable continuous weight and its perturbations on various functional spaces, which is new even in dimension one. Thanks to the spectral gap, we establish an exponential speed of convergence for the equidistribution of the backward orbits of points towards the conformal measure and the exponential mixing. Moreover, as an immediate consequence, we obtain a full list of statistical properties for the equilibrium states: CLT, Berry-Esseen Theorem, local CLT, ASIP, LIL, LDP, almost sure CLT. Many of these properties are new even in dimension one, some even in the case of zero weight function (i.e., for the measure of maximal entropy).</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"63 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140914941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Lagrangians, SO(3)-Instantons and Mixed Equation 拉格朗日、SO(3)-等式和混合方程
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-05-02 DOI: 10.1007/s00039-024-00677-8
Aliakbar Daemi, Kenji Fukaya, Maksim Lipyanskiy
{"title":"Lagrangians, SO(3)-Instantons and Mixed Equation","authors":"Aliakbar Daemi, Kenji Fukaya, Maksim Lipyanskiy","doi":"10.1007/s00039-024-00677-8","DOIUrl":"https://doi.org/10.1007/s00039-024-00677-8","url":null,"abstract":"<p>The <i>mixed equation</i>, defined as a combination of the anti-self-duality equation in gauge theory and Cauchy–Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah–Floer conjecture.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"58 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140819341","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Commensurations of Aut(FN) and Its Torelli Subgroup Aut(FN) 及其 Torelli 子群的共形
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-04-22 DOI: 10.1007/s00039-024-00681-y
Martin R. Bridson, Richard D. Wade
{"title":"Commensurations of Aut(FN) and Its Torelli Subgroup","authors":"Martin R. Bridson, Richard D. Wade","doi":"10.1007/s00039-024-00681-y","DOIUrl":"https://doi.org/10.1007/s00039-024-00681-y","url":null,"abstract":"<p>For <i>N</i>≥3, the abstract commensurators of both Aut(<i>F</i><sub><i>N</i></sub>) and its Torelli subgroup IA<sub><i>N</i></sub> are isomorphic to Aut(<i>F</i><sub><i>N</i></sub>) itself.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"100 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140632140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Sections and Unirulings of Families over $mathbb{P}^{1}$ $mathbb{P}^{1}$上族的分段和单圈
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-04-18 DOI: 10.1007/s00039-024-00679-6
Alex Pieloch
{"title":"Sections and Unirulings of Families over $mathbb{P}^{1}$","authors":"Alex Pieloch","doi":"10.1007/s00039-024-00679-6","DOIUrl":"https://doi.org/10.1007/s00039-024-00679-6","url":null,"abstract":"<p>We consider morphisms <span>(pi : X to mathbb{P}^{1})</span> of smooth projective varieties over <span>(mathbb{C})</span>. We show that if <i>π</i> has at most one singular fibre, then <i>X</i> is uniruled and <i>π</i> admits sections. We reach the same conclusions, but with genus zero multisections instead of sections, if <i>π</i> has at most two singular fibres, and the first Chern class of <i>X</i> is supported in a single fibre of <i>π</i>.</p><p>To achieve these result, we use action completed symplectic cohomology groups associated to compact subsets of convex symplectic domains. These groups are defined using Pardon’s virtual fundamental chains package for Hamiltonian Floer cohomology. In the above setting, we show that the vanishing of these groups implies the existence of unirulings and (multi)sections.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"100 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140608063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Dominated Splitting from Constant Periodic Data and Global Rigidity of Anosov Automorphisms 恒定周期数据的支配分裂与阿诺索夫自动形的全局刚性
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-04-15 DOI: 10.1007/s00039-024-00680-z
Jonathan DeWitt, Andrey Gogolev
{"title":"Dominated Splitting from Constant Periodic Data and Global Rigidity of Anosov Automorphisms","authors":"Jonathan DeWitt, Andrey Gogolev","doi":"10.1007/s00039-024-00680-z","DOIUrl":"https://doi.org/10.1007/s00039-024-00680-z","url":null,"abstract":"<p>We show that a <span>(operatorname{GL}(d,mathbb{R}))</span> cocycle over a hyperbolic system with constant periodic data has a dominated splitting whenever the periodic data indicates it should. This implies global periodic data rigidity of generic Anosov automorphisms of <span>(mathbb{T}^{d})</span>. Further, our approach also works when the periodic data is narrow, that is, sufficiently close to constant. We can show global periodic data rigidity for certain non-linear Anosov diffeomorphisms in a neighborhood of an irreducible Anosov automorphism with simple spectrum.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"37 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140553531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Large Genus Bounds for the Distribution of Triangulated Surfaces in Moduli Space 模空间三角曲面分布的大属界
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-03-04 DOI: 10.1007/s00039-023-00656-5
Sahana Vasudevan
{"title":"Large Genus Bounds for the Distribution of Triangulated Surfaces in Moduli Space","authors":"Sahana Vasudevan","doi":"10.1007/s00039-023-00656-5","DOIUrl":"https://doi.org/10.1007/s00039-023-00656-5","url":null,"abstract":"<p>Triangulated surfaces are compact Riemann surfaces equipped with a conformal triangulation by equilateral triangles. In 2004, Brooks and Makover asked how triangulated surfaces are distributed in the moduli space of Riemann surfaces as the genus tends to infinity. Mirzakhani raised this question in her 2010 ICM address. We show that in the large genus case, triangulated surfaces are well distributed in moduli space in a fairly strong sense. We do this by proving upper and lower bounds for the number of triangulated surfaces lying in a Teichmüller ball in moduli space. In particular, we show that the number of triangulated surfaces lying in a Teichmüller unit ball is at most exponential in the number of triangles, independent of the genus.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"106 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140032103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Coalescence of Geodesics and the BKS Midpoint Problem in Planar First-Passage Percolation 平面第一通道渗流中的大地线凝聚和 BKS 中点问题
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-21 DOI: 10.1007/s00039-024-00672-z
Barbara Dembin, Dor Elboim, Ron Peled
{"title":"Coalescence of Geodesics and the BKS Midpoint Problem in Planar First-Passage Percolation","authors":"Barbara Dembin, Dor Elboim, Ron Peled","doi":"10.1007/s00039-024-00672-z","DOIUrl":"https://doi.org/10.1007/s00039-024-00672-z","url":null,"abstract":"<p>We consider first-passage percolation on <span>(mathbb{Z}^{2})</span> with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics.</p><p>The result leads to a quantitative resolution of the Benjamini–Kalai–Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge.</p><p>We further prove that the limit shape assumption is satisfied for a specific family of distributions.</p><p>Lastly, related to the 1965 Hammersley–Welsh highways and byways problem, we prove that the expected fraction of the square {−<i>n</i>,…,<i>n</i>}<sup>2</sup> which is covered by infinite geodesics starting at the origin is at most an inverse power of <i>n</i>. This result is obtained without explicit limit shape assumptions.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"35 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139915821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
Augmentations, Fillings, and Clusters 增量、填充和集群
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-21 DOI: 10.1007/s00039-024-00673-y
Honghao Gao, Linhui Shen, Daping Weng
{"title":"Augmentations, Fillings, and Clusters","authors":"Honghao Gao, Linhui Shen, Daping Weng","doi":"10.1007/s00039-024-00673-y","DOIUrl":"https://doi.org/10.1007/s00039-024-00673-y","url":null,"abstract":"<p>We investigate positive braid Legendrian links via a Floer-theoretic approach and prove that their augmentation varieties are cluster K<sub>2</sub> (aka. <span>(mathcal{A})</span>-) varieties. Using the exact Lagrangian cobordisms of Legendrian links in Ekholm et al. (J. Eur. Math. Soc. 18(11):2627–2689, 2016), we prove that a large family of exact Lagrangian fillings of positive braid Legendrian links correspond to cluster seeds of their augmentation varieties. We solve the infinite-filling problem for positive braid Legendrian links; i.e., whenever a positive braid Legendrian link is not of type ADE, it admits infinitely many exact Lagrangian fillings up to Hamiltonian isotopy.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"12 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139915861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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
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