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Extremal Affine Subspaces and Khintchine-Jarník Type Theorems 极值仿射子空间和Khintchine-Jarník类型定理
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-01 DOI: 10.1007/s00039-024-00665-y
{"title":"Extremal Affine Subspaces and Khintchine-Jarník Type Theorems","authors":"","doi":"10.1007/s00039-024-00665-y","DOIUrl":"https://doi.org/10.1007/s00039-024-00665-y","url":null,"abstract":"<h3>Abstract</h3> <p>We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of <span> <span>(mathbb{R}^{n})</span> </span>. We also give an essentially complete classification of all Khintchine type affine subspaces, except for some boundary cases within two logarithmic scales. More general Jarník type theorems are proved as well, sometimes without the monotonicity of the approximation function. These results follow as consequences of our novel estimates for the number of rational points close to an affine subspace in terms of diophantine properties of its defining matrix. Our main tool is the multidimensional large sieve inequality and its dual form.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"177 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660183","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
Gromov’s Tori Are Optimal 格罗莫夫的托里是最优的
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-01 DOI: 10.1007/s00039-024-00663-0
{"title":"Gromov’s Tori Are Optimal","authors":"","doi":"10.1007/s00039-024-00663-0","DOIUrl":"https://doi.org/10.1007/s00039-024-00663-0","url":null,"abstract":"<h3>Abstract</h3> <p>We give an optimal bound on normal curvatures of immersed <em>n</em>-torus in a Euclidean ball of large dimension.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"51 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139659972","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
Mean Convex Smoothing of Mean Convex Cones 均值凸锥的均值凸平滑化
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-01 DOI: 10.1007/s00039-024-00666-x
Zhihan Wang
{"title":"Mean Convex Smoothing of Mean Convex Cones","authors":"Zhihan Wang","doi":"10.1007/s00039-024-00666-x","DOIUrl":"https://doi.org/10.1007/s00039-024-00666-x","url":null,"abstract":"<p>We show that any minimizing hypercone can be perturbed into one side to a properly embedded smooth minimizing hypersurface in the Euclidean space, and every viscosity mean convex cone admits a properly embedded smooth mean convex self-expander asymptotic to it near infinity. These two together confirm a conjecture of Lawson (Geom. Meas. Theor. Calcu. Var. 44:441, 1986, Problem 5.7).</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"16 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660218","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
Two Rigidity Results for Stable Minimal Hypersurfaces 稳定最小超曲面的两个刚性结果
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-01 DOI: 10.1007/s00039-024-00662-1
Giovanni Catino, Paolo Mastrolia, Alberto Roncoroni
{"title":"Two Rigidity Results for Stable Minimal Hypersurfaces","authors":"Giovanni Catino, Paolo Mastrolia, Alberto Roncoroni","doi":"10.1007/s00039-024-00662-1","DOIUrl":"https://doi.org/10.1007/s00039-024-00662-1","url":null,"abstract":"<p>The aim of this paper is to prove two results concerning the rigidity of complete, immersed, orientable, stable minimal hypersurfaces: we show that they are hyperplane in <i>R</i><sup>4</sup>, while they do not exist in positively curved closed Riemannian (<i>n</i>+1)-manifold when <i>n</i>≤5; in particular, there are no stable minimal hypersurfaces in <i>S</i><sup><i>n</i>+1</sup> when <i>n</i>≤5. The first result was recently proved also by Chodosh and Li, and the second is a consequence of a more general result concerning minimal surfaces with finite index. Both theorems rely on a conformal method, inspired by a classical work of Fischer-Colbrie.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"37 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139659970","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
Odd Distances in Colourings of the Plane 平面着色中的奇异距离
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-01-30 DOI: 10.1007/s00039-024-00659-w
James Davies
{"title":"Odd Distances in Colourings of the Plane","authors":"James Davies","doi":"10.1007/s00039-024-00659-w","DOIUrl":"https://doi.org/10.1007/s00039-024-00659-w","url":null,"abstract":"<p>We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd distance from each other.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644131","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
A New Regularized Siegel-Weil Type Formula. Part I 一种新的正规化西格尔-韦尔公式。第一部分
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-01-22 DOI: 10.1007/s00039-024-00657-y
David Ginzburg, David Soudry
{"title":"A New Regularized Siegel-Weil Type Formula. Part I","authors":"David Ginzburg, David Soudry","doi":"10.1007/s00039-024-00657-y","DOIUrl":"https://doi.org/10.1007/s00039-024-00657-y","url":null,"abstract":"<p>In this paper, we prove a formula, realizing certain residual Eisenstein series on symplectic groups as regularized kernel integrals. These Eisenstein series, as well as the kernel integrals, are attached to Speh representations. This forms an initial step to a new type of a regularized Siegel-Weil formula that we propose. This new formula bears the same relation to the generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan, as does the regularized Siegel-Weil formula to the doubling integrals of Piatetski-Shapiro and Rallis.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"2 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139510830","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
Relations between scaling exponents in unimodular random graphs 单模随机图中标度指数之间的关系
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-11-09 DOI: 10.1007/s00039-023-00654-7
James R. Lee
{"title":"Relations between scaling exponents in unimodular random graphs","authors":"James R. Lee","doi":"10.1007/s00039-023-00654-7","DOIUrl":"https://doi.org/10.1007/s00039-023-00654-7","url":null,"abstract":"<p>We investigate the validity of the “Einstein relations” in the general setting of unimodular random networks. These are equalities relating scaling exponents: </p><span> $$begin{aligned} d_{w} &amp;= d_{f} + tilde{zeta }, d_{s} &amp;= 2 d_{f}/d_{w}, end{aligned}$$ </span><p> where <i>d</i><sub><i>w</i></sub> is the walk dimension, <i>d</i><sub><i>f</i></sub> is the fractal dimension, <i>d</i><sub><i>s</i></sub> is the spectral dimension, and <span>(tilde{zeta })</span> is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if <i>d</i><sub><i>f</i></sub> and <span>(tilde{zeta } geqslant 0)</span> exist, then <i>d</i><sub><i>w</i></sub> and <i>d</i><sub><i>s</i></sub> exist, and the aforementioned equalities hold. Moreover, our primary new estimate <span>(d_{w} geqslant d_{f} + tilde{zeta })</span> is established for all <span>(tilde{zeta } in mathbb{R})</span>.</p><p>For the uniform infinite planar triangulation (UIPT), this yields the consequence <i>d</i><sub><i>w</i></sub>=4 using <i>d</i><sub><i>f</i></sub>=4 (Angel in Geom. Funct. Anal. 13(5):935–974, 2003) and <span>(tilde{zeta }=0)</span> (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2020 and (Ding and Gwynne in Commun. Math. Phys. 374(3):1877–1934, 2020)). The conclusion <i>d</i><sub><i>w</i></sub>=4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is that <i>d</i><sub><i>w</i></sub>=<i>d</i><sub><i>f</i></sub> for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since <i>d</i><sub><i>f</i></sub>&gt;2.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"56 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72364946","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: 4
GOE statistics on the moduli space of surfaces of large genus 大亏格曲面模空间的GOE统计
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-11-02 DOI: 10.1007/s00039-023-00655-6
Zeév Rudnick
{"title":"GOE statistics on the moduli space of surfaces of large genus","authors":"Zeév Rudnick","doi":"10.1007/s00039-023-00655-6","DOIUrl":"https://doi.org/10.1007/s00039-023-00655-6","url":null,"abstract":"<p>For a compact hyperbolic surface, we define a smooth linear statistic, mimicking the number of Laplace eigenvalues in a short energy window. We study the variance of this statistic, when averaged over the moduli space <span>(mathcal{M}_{g})</span> of all genus <i>g</i> surfaces with respect to the Weil-Petersson measure. We show that in the double limit, first taking the large genus limit and then the short window limit, we recover GOE statistics for the variance. The proof makes essential use of Mirzakhani’s integration formula.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"25 11","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509201","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: 11
Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case 经典波动方法和现代规范变换：一维情况下的谱渐近性
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-10-31 DOI: 10.1007/s00039-023-00650-x
Jeffrey Galkowski, Leonid Parnovski, Roman Shterenberg
{"title":"Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case","authors":"Jeffrey Galkowski, Leonid Parnovski, Roman Shterenberg","doi":"10.1007/s00039-023-00650-x","DOIUrl":"https://doi.org/10.1007/s00039-023-00650-x","url":null,"abstract":"<p>In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let <span>(H: L^{2}(mathbb{R})to L^{2}(mathbb{R}))</span> have the form </p><span>$$ H:=-frac{d^{2}}{dx^{2}}+Q, $$</span><p> where <i>Q</i> is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spectral projector, <span>({1}_{(-infty ,rho ^{2}]}(H))</span>, has a complete asymptotic expansion in powers of <i>ρ</i>. This settles the 1-dimensional case of a conjecture made by the last two authors.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509199","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
Bers’ simultaneous uniformization and the intersection of Poincaré holonomy varieties Bers的同时一致化与Poincaréholonomy变种的交集
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-10-31 DOI: 10.1007/s00039-023-00653-8
Shinpei Baba
{"title":"Bers’ simultaneous uniformization and the intersection of Poincaré holonomy varieties","authors":"Shinpei Baba","doi":"10.1007/s00039-023-00653-8","DOIUrl":"https://doi.org/10.1007/s00039-023-00653-8","url":null,"abstract":"<p>We consider the space of ordered pairs of distinct <span>({mathbb{C}{mathrm{P}}}^{1})</span>-structures on Riemann surfaces (of any orientations) which have identical holonomy, so that the quasi-Fuchsian space is identified with a connected component of this space. This space holomorphically maps to the product of the Teichmüller spaces minus its diagonal.</p><p>In this paper, we prove that this mapping is a complete local branched covering map. As a corollary, we reprove Bers’ simultaneous uniformization theorem without any quasi-conformal deformation theory. Our main theorem is that the intersection of arbitrary two Poincaré holonomy varieties (<span>(operatorname{SL}_{2}mathbb{C})</span>-opers) is a non-empty discrete set, which is closely related to the mapping.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"25 12","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509200","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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