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A Metric Fixed Point Theorem and Some of Its Applications 公设定点定理及其一些应用
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-07 DOI: 10.1007/s00039-024-00658-x
Anders Karlsson
{"title":"A Metric Fixed Point Theorem and Some of Its Applications","authors":"Anders Karlsson","doi":"10.1007/s00039-024-00658-x","DOIUrl":"https://doi.org/10.1007/s00039-024-00658-x","url":null,"abstract":"<p>A general fixed point theorem for isometries in terms of metric functionals is proved under the assumption of the existence of a conical bicombing. It is new for isometries of convex sets of Banach spaces as well as for non-locally compact CAT(0)-spaces and injective spaces. Examples of actions on non-proper CAT(0)-spaces come from the study of diffeomorphism groups, birational transformations, and compact Kähler manifolds. A special case of the fixed point theorem provides a novel mean ergodic theorem that in the Hilbert space case implies von Neumann’s theorem. The theorem accommodates classically fixed-point-free isometric maps such as those of Kakutani, Edelstein, Alspach and Prus. Moreover, from the main theorem together with some geometric arguments of independent interest, one can deduce that every bounded invertible operator of a Hilbert space admits a nontrivial invariant metric functional on the space of positive operators. This is a result in the direction of the invariant subspace problem although its full meaning is dependent on a future determination of such metric functionals.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139705033","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
On the Almost Reducibility Conjecture 关于几乎可重复性猜想
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-05 DOI: 10.1007/s00039-024-00671-0
{"title":"On the Almost Reducibility Conjecture","authors":"","doi":"10.1007/s00039-024-00671-0","DOIUrl":"https://doi.org/10.1007/s00039-024-00671-0","url":null,"abstract":"<h3>Abstract</h3> <p>Avila’s Almost Reducibility Conjecture (ARC) is a powerful statement linking purely analytic and dynamical properties of analytic one-frequency <span> <span>(SL(2,{mathbb{R}}))</span> </span> cocycles. It is also a fundamental tool in the study of spectral theory of analytic one-frequency Schrödinger operators, with many striking consequences, allowing to give a detailed characterization of the subcritical region. Here we give a proof, completely different from Avila’s, for the important case of Schrödinger cocycles with trigonometric polynomial potentials and non-exponentially approximated frequencies, allowing, in particular, to obtain all the desired spectral consequences in this case.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695624","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
Kaufman and Falconer Estimates for Radial Projections and a Continuum Version of Beck’s Theorem 径向投影的考夫曼和法尔科纳估计以及贝克定理的连续版本
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-05 DOI: 10.1007/s00039-024-00660-3
Tuomas Orponen, Pablo Shmerkin, Hong Wang
{"title":"Kaufman and Falconer Estimates for Radial Projections and a Continuum Version of Beck’s Theorem","authors":"Tuomas Orponen, Pablo Shmerkin, Hong Wang","doi":"10.1007/s00039-024-00660-3","DOIUrl":"https://doi.org/10.1007/s00039-024-00660-3","url":null,"abstract":"<p>We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let <span>(X,Y subset mathbb{R}^{2})</span> be non-empty Borel sets. If <i>X</i> is not contained in any line, we prove that </p><span>$$ sup _{x in X} dim _{mathrm {H}}pi _{x}(Y , setminus , {x}) geq min { dim _{mathrm {H}}X,dim _{mathrm {H}}Y,1}. $$</span><p> If dim<sub>H</sub><i>Y</i>&gt;1, we have the following improved lower bound: </p><span>$$ sup _{x in X} dim _{mathrm {H}}pi _{x}(Y , setminus , {x}) geq min { dim _{mathrm {H}}X + dim _{mathrm {H}}Y - 1,1}. $$</span><p> Our results solve conjectures of Lund-Thang-Huong, Liu, and the first author. Another corollary is the following continuum version of Beck’s theorem in combinatorial geometry: if <span>(X subset mathbb{R}^{2})</span> is a Borel set with the property that dim<sub>H</sub>(<i>X</i> ∖ <i>ℓ</i>)=dim<sub>H</sub><i>X</i> for all lines <span>(ell subset mathbb{R}^{2})</span>, then the line set spanned by <i>X</i> has Hausdorff dimension at least min{2dim<sub>H</sub><i>X</i>,2}.</p><p>While the results above concern <span>(mathbb{R}^{2})</span>, we also derive some counterparts in <span>(mathbb{R}^{d})</span> by means of integralgeometric considerations. The proofs are based on an <i>ϵ</i>-improvement in the Furstenberg set problem, due to the two first authors, a bootstrapping scheme introduced by the second and third author, and a new planar incidence estimate due to Fu and Ren.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695631","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
On the Dimension of Exceptional Parameters for Nonlinear Projections, and the Discretized Elekes-Rónyai Theorem 论非线性投影的异常参数维度和离散化的埃莱克斯-罗尼亚伊定理
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-05 DOI: 10.1007/s00039-024-00664-z
Orit E. Raz, Joshua Zahl
{"title":"On the Dimension of Exceptional Parameters for Nonlinear Projections, and the Discretized Elekes-Rónyai Theorem","authors":"Orit E. Raz, Joshua Zahl","doi":"10.1007/s00039-024-00664-z","DOIUrl":"https://doi.org/10.1007/s00039-024-00664-z","url":null,"abstract":"<p>We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and Shmerkin. (3) The parallelizability of planar <i>d</i>-webs. (4) The Elekes-Rónyai theorem on expanding polynomials.</p><p>Given a Borel set <i>A</i> in the plane, we study the set of exceptional vantage points, for which the pinned distance Δ<sub><i>p</i></sub>(<i>A</i>) has small dimension, that is, close to (dim<i>A</i>)/2. We show that if this set has positive dimension, then it must have very special structure. This result follows from a more general single-scale nonlinear projection theorem, which says that if <i>ϕ</i><sub>1</sub>, <i>ϕ</i><sub>2</sub>, <i>ϕ</i><sub>3</sub> are three smooth functions whose associated 3-web has non-vanishing Blaschke curvature, and if <i>A</i> is a (<i>δ</i>,<i>α</i>)<sub>2</sub>-set in the sense of Katz and Tao, then at least one of the images <i>ϕ</i><sub><i>i</i></sub>(<i>A</i>) must have measure much larger than |<i>A</i>|<sup>1/2</sup>, where |<i>A</i>| stands for the measure of <i>A</i>. We prove analogous results for <i>d</i> smooth functions <i>ϕ</i><sub>1</sub>,…,<i>ϕ</i><sub><i>d</i></sub>, whose associated <i>d</i>-web is not parallelizable.</p><p>We use similar tools to characterize when bivariate real analytic functions are “dimension expanding” when applied to a Cartesian product: if <i>P</i> is a bivariate real analytic function, then <i>P</i> is either locally of the form <i>h</i>(<i>a</i>(<i>x</i>)+<i>b</i>(<i>y</i>)), or <i>P</i>(<i>A</i>,<i>B</i>) has dimension at least <i>α</i>+<i>c</i> whenever <i>A</i> and <i>B</i> are Borel sets with Hausdorff dimension <i>α</i>. Again, this follows from a single-scale estimate, which is an analogue of the Elekes-Rónyai theorem in the setting of the Katz-Tao discretized ring conjecture.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695834","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
CAT(0) Spaces of Higher Rank I CAT(0) I 类高级职位空缺
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-02-02 DOI: 10.1007/s00039-024-00661-2
{"title":"CAT(0) Spaces of Higher Rank I","authors":"","doi":"10.1007/s00039-024-00661-2","DOIUrl":"https://doi.org/10.1007/s00039-024-00661-2","url":null,"abstract":"<h3>Abstract</h3> <p>A CAT(0) space has rank at least <em>n</em> if every geodesic lies in an <em>n</em>-flat. Ballmann’s Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least 2 with a geometric group action is <em>rigid</em> – isometric to a Riemannian symmetric space, a Euclidean building, or splits as a metric product. This paper is the first in a series motivated by Ballmann’s conjecture. Here we prove that a CAT(0) space of rank at least <em>n</em>≥2 is rigid if it contains a periodic <em>n</em>-flat and its Tits boundary has dimension (<em>n</em>−1). This does not require a geometric group action. The result relies essentially on the study of flats which do not bound flat half-spaces – so-called <em>Morse flats</em>. We show that the Tits boundary <em>∂</em><sub><em>T</em></sub><em>F</em> of a periodic Morse <em>n</em>-flat <em>F</em> contains a <em>regular point</em> – a point with a Tits-neighborhood entirely contained in <em>∂</em><sub><em>T</em></sub><em>F</em>. More precisely, we show that the set of singular points in <em>∂</em><sub><em>T</em></sub><em>F</em> can be covered by finitely many round spheres of positive codimension.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139676933","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
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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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
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