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Entropy and Stability of Hyperbolic Manifolds 双曲流形的熵与稳定性
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2025-05-14 DOI: 10.1007/s00039-025-00711-3
Antoine Song
{"title":"Entropy and Stability of Hyperbolic Manifolds","authors":"Antoine Song","doi":"10.1007/s00039-025-00711-3","DOIUrl":"https://doi.org/10.1007/s00039-025-00711-3","url":null,"abstract":"<p>Let (<i>M</i>,<i>g</i><sub>0</sub>) be a closed oriented hyperbolic manifold of dimension at least 3. By the volume entropy inequality of G. Besson, G. Courtois and S. Gallot, for any Riemannian metric <i>g</i> on <i>M</i> with same volume as <i>g</i><sub>0</sub>, its volume entropy <i>h</i>(<i>g</i>) satisfies <i>h</i>(<i>g</i>)≥<i>n</i>−1 with equality only when <i>g</i> is isometric to <i>g</i><sub>0</sub>. We show that the hyperbolic metric <i>g</i><sub>0</sub> is stable in the following sense: if <i>g</i><sub><i>i</i></sub> is a sequence of Riemaniann metrics on <i>M</i> of same volume as <i>g</i><sub>0</sub> and if <i>h</i>(<i>g</i><sub><i>i</i></sub>) converges to <i>n</i>−1, then there are smooth subsets <i>Z</i><sub><i>i</i></sub>⊂<i>M</i> such that both <span>(operatorname{Vol}(Z_{i},g_{i}))</span> and <span>(operatorname{Area}(partial Z_{i},g_{i}))</span> tend to 0, and (<i>M</i>∖<i>Z</i><sub><i>i</i></sub>,<i>g</i><sub><i>i</i></sub>) converges to (<i>M</i>,<i>g</i><sub>0</sub>) in the measured Gromov-Hausdorff topology. The proof relies on showing that any spherical Plateau solution for <i>M</i> is intrinsically isomorphic to <span>((M,frac{(n-1)^{2}}{4n} g_{0}))</span>.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"29 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143945583","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
Rate Distortion Dimension of Random Brody Curves 随机Brody曲线的速率畸变维数
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2025-05-07 DOI: 10.1007/s00039-025-00709-x
Masaki Tsukamoto
{"title":"Rate Distortion Dimension of Random Brody Curves","authors":"Masaki Tsukamoto","doi":"10.1007/s00039-025-00709-x","DOIUrl":"https://doi.org/10.1007/s00039-025-00709-x","url":null,"abstract":"<p>The main purpose of this paper is to propose an ergodic theoretic approach to the study of entire holomorphic curves. Brody curves are one-Lipschitz holomorphic maps from the complex plane to the complex projective space. They admit a natural group action, and “random Brody curves” in the title refers to invariant probability measures for it. We study their geometric and dynamical properties. Given an invariant probability measure <i>μ</i> on the space of Brody curves, our first main theorem claims that its rate distortion dimension is bounded by the integral of a “geometric potential” over <i>μ</i>. This result is analogous to the Ruelle inequality of smooth ergodic theory. Our second main theorem claims that there exists a rich variety of invariant probability measures attaining equality in this “Ruelle inequality for Brody curves”. The main tools of the proofs are the deformation theory of Brody curves and the variational principle for mean dimension with potential. This approach is motivated by the theory of thermodynamic formalism for Axiom A diffeomorphisms.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"17 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143915961","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
Universal Localizations, Atiyah Conjectures and Graphs of Groups 普适定域、阿蒂亚猜想与群图
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2025-05-05 DOI: 10.1007/s00039-025-00710-4
Pablo Sánchez-Peralta
{"title":"Universal Localizations, Atiyah Conjectures and Graphs of Groups","authors":"Pablo Sánchez-Peralta","doi":"10.1007/s00039-025-00710-4","DOIUrl":"https://doi.org/10.1007/s00039-025-00710-4","url":null,"abstract":"<p>Let <i>G</i> be a countable group that is the fundamental group of a graph of groups with finite edge groups and vertex groups satisfying the strong Atiyah conjecture over <span>(K subseteq mathbb{C})</span> a field closed under complex conjugation. Assume that the orders of finite subgroups of <i>G</i> are bounded above. We show that <i>G</i> satisfies the strong Atiyah conjecture over <i>K</i>. In particular, this implies that the strong Atiyah conjecture is closed under free products. Moreover, we prove that the ∗-regular closure of <i>K</i>[<i>G</i>] in <span>(mathcal{U}(G))</span>, <span>(mathcal{R}_{K[G]})</span>, is a universal localization of the graph of rings associated to the graph of groups, where the rings are the corresponding ∗-regular closures. As a result, we obtain that the algebraic and center-valued Atiyah conjecture over <i>K</i> are also closed under the graph of groups construction as long as the edge groups are finite. We also infer some consequences on the structure of the <i>K</i><sub>0</sub> and <i>K</i><sub>1</sub>-groups of <span>(mathcal{R}_{K[G]})</span>. The techniques developed enable us to prove that <i>K</i>[<i>G</i>] fulfills the strong, algebraic and center-valued Atiyah conjectures, and that <span>(mathcal{R}_{K[G]})</span> is the universal localization of <i>K</i>[<i>G</i>] over the set of all matrices that become invertible in <span>(mathcal{U}(G))</span>, provided that <i>G</i> belongs to a certain class of groups <span>(mathcal{T}_{mathcal{VLI}})</span>, which contains in particular virtually-{locally indicable} groups that are the fundamental group of a graph of virtually free groups.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"15 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143910696","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 Continuous Cusp Closing Process for Negative Kähler-Einstein Metrics 负Kähler-Einstein指标的连续尖峰关闭过程
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2025-03-04 DOI: 10.1007/s00039-025-00708-y
Xin Fu, Hans-Joachim Hein, Xumin Jiang
{"title":"A Continuous Cusp Closing Process for Negative Kähler-Einstein Metrics","authors":"Xin Fu, Hans-Joachim Hein, Xumin Jiang","doi":"10.1007/s00039-025-00708-y","DOIUrl":"https://doi.org/10.1007/s00039-025-00708-y","url":null,"abstract":"<p>We give an example of a family of smooth complex algebraic surfaces of degree 6 in <span>(mathbb{CP}^{3})</span> developing an isolated elliptic singularity. We show via a gluing construction that the unique Kähler-Einstein metrics of Ricci curvature −1 on these sextics develop a complex hyperbolic cusp in the limit, and that near the tip of the forming cusp a Tian-Yau gravitational instanton bubbles off.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"2 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143539114","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 Shapes of Rational Lemniscates 论理性lemnisces的形状
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2025-02-18 DOI: 10.1007/s00039-025-00704-2
Christopher J. Bishop, Alexandre Eremenko, Kirill Lazebnik
{"title":"On the Shapes of Rational Lemniscates","authors":"Christopher J. Bishop, Alexandre Eremenko, Kirill Lazebnik","doi":"10.1007/s00039-025-00704-2","DOIUrl":"https://doi.org/10.1007/s00039-025-00704-2","url":null,"abstract":"<p>A rational lemniscate is a level set of |<i>r</i>| where <span>(r: widehat {mathbb{C}}rightarrow widehat {mathbb{C}})</span> is rational. We prove that any planar Euler graph can be approximated, in a strong sense, by a homeomorphic rational lemniscate. This generalizes Hilbert’s lemniscate theorem; he proved that any Jordan curve can be approximated (in the same strong sense) by a polynomial lemniscate that is also a Jordan curve. As consequences, we obtain a sharp quantitative version of the classical Runge’s theorem on rational approximation, and we give a new result on the approximation of planar continua by Julia sets of rational maps.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"49 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143435144","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
Suppression of Chemotactic Singularity by Buoyancy 浮力对趋化奇异性的抑制
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2025-02-13 DOI: 10.1007/s00039-025-00706-0
Zhongtian Hu, Alexander Kiselev, Yao Yao
{"title":"Suppression of Chemotactic Singularity by Buoyancy","authors":"Zhongtian Hu, Alexander Kiselev, Yao Yao","doi":"10.1007/s00039-025-00706-0","DOIUrl":"https://doi.org/10.1007/s00039-025-00706-0","url":null,"abstract":"<p>Chemotactic singularity formation in the context of the Patlak-Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of fluid advection can arrest the singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong - this effect is conjectured to hold for more general classes of nonlinear PDEs. In this paper, we consider the Patlak-Keller-Segel equation coupled with a fluid flow that obeys Darcy’s law for incompressible porous media via buoyancy force. We prove that in contrast with passive advection, this active fluid coupling is capable of suppressing singularity formation at arbitrary small coupling strength: namely, the system always has globally regular solutions.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"78 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143401930","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
Uniqueness of Tangent Flows at Infinity for Finite-Entropy Shortening Curves 有限熵缩短曲线无穷远处切线流动的唯一性
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2025-02-13 DOI: 10.1007/s00039-025-00705-1
Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su, Kai-Wei Zhao
{"title":"Uniqueness of Tangent Flows at Infinity for Finite-Entropy Shortening Curves","authors":"Kyeongsu Choi, Dong-Hwi Seo, Wei-Bo Su, Kai-Wei Zhao","doi":"10.1007/s00039-025-00705-1","DOIUrl":"https://doi.org/10.1007/s00039-025-00705-1","url":null,"abstract":"<p>In this paper, we prove that an ancient smooth curve-shortening flow with finite entropy embedded in <span>(mathbb{R}^{2})</span> has a unique tangent flow at infinity. To this end, we show that its rescaled flows backwardly converge to a line with multiplicity <i>m</i>≥3 exponentially fast in any compact region, unless the flow is a shrinking circle, a static line, a paper clip, or a translating grim reaper. In addition, we figure out the exact numbers of tips, vertices, and inflection points of the curves at negative enough time. Moreover, the exponential growth rate of graphical radius and the convergence of vertex regions to grim reaper curves will be shown.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"62 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143401541","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 Spielman-Teng Conjecture 关于斯皮尔伯格-邓猜想
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2025-02-13 DOI: 10.1007/s00039-025-00707-z
Ashwin Sah, Julian Sahasrabudhe, Mehtaab Sawhney
{"title":"On the Spielman-Teng Conjecture","authors":"Ashwin Sah, Julian Sahasrabudhe, Mehtaab Sawhney","doi":"10.1007/s00039-025-00707-z","DOIUrl":"https://doi.org/10.1007/s00039-025-00707-z","url":null,"abstract":"<p>Let <i>M</i> be an <i>n</i>×<i>n</i> matrix with iid subgaussian entries with mean 0 and variance 1 and let <i>σ</i><sub><i>n</i></sub>(<i>M</i>) denote the least singular value of <i>M</i>. We prove that </p><span>$$ mathbb{P}big( sigma _{n}(M) leqslant varepsilon n^{-1/2} big) = (1+o(1)) varepsilon + e^{- Omega (n)} $$</span><p> for all 0⩽<i>ε</i>≪1. This resolves, up to a 1+<i>o</i>(1) factor, a seminal conjecture of Spielman and Teng.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"62 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143401542","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
Geometric Langlands Duality for Periods 周期的几何朗兰对偶
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2025-02-06 DOI: 10.1007/s00039-025-00702-4
Tony Feng, Jonathan Wang
{"title":"Geometric Langlands Duality for Periods","authors":"Tony Feng, Jonathan Wang","doi":"10.1007/s00039-025-00702-4","DOIUrl":"https://doi.org/10.1007/s00039-025-00702-4","url":null,"abstract":"<p>We study conjectures of Ben-Zvi–Sakellaridis–Venkatesh that categorify the relationship between automorphic periods and <i>L</i>-functions in the context of the Geometric Langlands equivalence. We provide evidence for these conjectures in some low-rank examples, by using derived Fourier analysis and the theory of chiral algebras to categorify the Rankin-Selberg unfolding method.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"26 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143192061","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
Optimal Rigidity and Maximum of the Characteristic Polynomial of Wigner Matrices Wigner矩阵特征多项式的最优刚度和最大值
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2025-02-05 DOI: 10.1007/s00039-025-00701-5
Paul Bourgade, Patrick Lopatto, Ofer Zeitouni
{"title":"Optimal Rigidity and Maximum of the Characteristic Polynomial of Wigner Matrices","authors":"Paul Bourgade, Patrick Lopatto, Ofer Zeitouni","doi":"10.1007/s00039-025-00701-5","DOIUrl":"https://doi.org/10.1007/s00039-025-00701-5","url":null,"abstract":"<p>We determine to leading order the maximum of the characteristic polynomial for Wigner matrices and <i>β</i>-ensembles. In the special case of Gaussian-divisible Wigner matrices, our method provides universality of the maximum up to tightness. These are the first universal results on the Fyodorov–Hiary–Keating conjectures for these models, and in particular answer the question of optimal rigidity for the spectrum of Wigner matrices.</p><p>Our proofs combine dynamical techniques for universality of eigenvalue statistics with ideas surrounding the maxima of log-correlated fields and Gaussian multiplicative chaos.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"13 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143125161","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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