Pub Date : 2025-10-31DOI: 10.1007/s00039-025-00723-z
Aditya Kumar, Balarka Sen
We construct infinitely many examples of macroscopically large manifolds of dimension $m geq 4$m≥4 equipped with circle bundles whose total spaces admit metrics of positive scalar curvature and have macroscopic dimension at most $lceil m/2 rceil + 1$⌈m/2⌉+1 . In particular, we answer a question of Gromov on the existence of circle bundles over enlargeable manifolds whose total spaces admit metrics of positive scalar curvature, in all dimensions. Our constructions are based on techniques from symplectic geometry.
{"title":"Circle Bundles with PSC over Large Manifolds","authors":"Aditya Kumar, Balarka Sen","doi":"10.1007/s00039-025-00723-z","DOIUrl":"https://doi.org/10.1007/s00039-025-00723-z","url":null,"abstract":"We construct infinitely many examples of macroscopically large manifolds of dimension <jats:inline-formula> <jats:alternatives> <jats:tex-math>$m geq 4$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:math> </jats:alternatives> </jats:inline-formula> equipped with circle bundles whose total spaces admit metrics of positive scalar curvature and have macroscopic dimension at most <jats:inline-formula> <jats:alternatives> <jats:tex-math>$lceil m/2 rceil + 1$</jats:tex-math> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>⌈</mml:mo> <mml:mi>m</mml:mi> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>⌉</mml:mo> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> </jats:alternatives> </jats:inline-formula> . In particular, we answer a question of Gromov on the existence of circle bundles over enlargeable manifolds whose total spaces admit metrics of positive scalar curvature, in all dimensions. Our constructions are based on techniques from symplectic geometry.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145404392","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}
Pub Date : 2025-10-10DOI: 10.1007/s00039-025-00722-0
Sergei Kuksin, Armen Shirikyan
The paper deals with the problem of long-time asymptotic behaviour of solutions for classes of ODEs and PDEs, perturbed by stationary noises. The latter are not assumed to be δdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$delta $end{document}-correlated in time, therefore the evolution in question is not necessarily Markovian. We first prove an abstract result which implies the mixing for random dynamical systems satisfying appropriate dissipativity and controllability conditions. It is applicable to a large class of evolution equations, and we illustrate this on the examples of a chain of anharmonic oscillators coupled to heat reservoirs, the 2d Navier–Stokes system, and a complex Ginzburg–Landau equation. Our results also apply to the general theory of random processes on the 1d lattice and allow one to get for them results related to Dobrushin’s theorems on reconstructing processes via their conditional distributions. The proof is based on an iterative construction with Newton’s quadratic approximation. It uses the method of Kantorovich functional, introduced earlier by the authors in the context of randomly forced PDEs, and some ideas used by them in the Markovian case to prove mixing with the help of controllability properties of an associated system.
The paper deals with the problem of long-time asymptotic behaviour of solutions for classes of ODEs and PDEs, perturbed by stationary noises. The latter are not assumed to be δdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$delta $end{document}-correlated in time, therefore the evolution in question is not necessarily Markovian. We first prove an abstract result which implies the mixing for random dynamical systems satisfying appropriate dissipativity and controllability conditions. It is applicable to a large class of evolution equations, and we illustrate this on the examples of a chain of anharmonic oscillators coupled to heat reservoirs, the 2d Navier–Stokes system, and a complex Ginzburg–Landau equation. Our results also apply to the general theory of random processes on the 1d lattice and allow one to get for them results related to Dobrushin’s theorems on reconstructing processes via their conditional distributions. The proof is based on an iterative construction with Newton’s quadratic approximation. It uses the method of Kantorovich functional, introduced earlier by the authors in the context of randomly forced PDEs, and some ideas used by them in the Markovian case to prove mixing with the help of controllability properties of an associated system.
{"title":"Mixing for Dynamical Systems Driven by Stationary Noises","authors":"Sergei Kuksin, Armen Shirikyan","doi":"10.1007/s00039-025-00722-0","DOIUrl":"https://doi.org/10.1007/s00039-025-00722-0","url":null,"abstract":"The paper deals with the problem of long-time asymptotic behaviour of solutions for classes of ODEs and PDEs, perturbed by stationary noises. The latter are not assumed to be <inline-formula><alternatives><mml:math><mml:mi>δ</mml:mi></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$delta $end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"39_2025_722_Article_IEq1.gif\"></inline-graphic></alternatives></inline-formula>-correlated in time, therefore the evolution in question is not necessarily Markovian. We first prove an abstract result which implies the mixing for random dynamical systems satisfying appropriate dissipativity and controllability conditions. It is applicable to a large class of evolution equations, and we illustrate this on the examples of a chain of anharmonic oscillators coupled to heat reservoirs, the 2d Navier–Stokes system, and a complex Ginzburg–Landau equation. Our results also apply to the general theory of random processes on the 1d lattice and allow one to get for them results related to Dobrushin’s theorems on reconstructing processes via their conditional distributions. The proof is based on an iterative construction with Newton’s quadratic approximation. It uses the method of Kantorovich functional, introduced earlier by the authors in the context of randomly forced PDEs, and some ideas used by them in the Markovian case to prove mixing with the help of controllability properties of an associated system.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"186 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914890","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}
Pub Date : 2025-10-10DOI: 10.1007/s00039-025-00721-1
Dorin Bucur, Richard S. Laugesen, Eloi Martinet, Mickaël Nahon
We prove the existence of an open set Ω⊂S2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$Omega subset mathbb{S}^{2}$end{document} for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving the maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.
We prove the existence of an open set Ω⊂S2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$Omega subset mathbb{S}^{2}$end{document} for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving the maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.
{"title":"Spherical Caps do Not Always Maximize Neumann Eigenvalues on the Sphere","authors":"Dorin Bucur, Richard S. Laugesen, Eloi Martinet, Mickaël Nahon","doi":"10.1007/s00039-025-00721-1","DOIUrl":"https://doi.org/10.1007/s00039-025-00721-1","url":null,"abstract":"We prove the existence of an open set <inline-formula><alternatives><mml:math><mml:mi mathvariant=\"normal\">Ω</mml:mi><mml:mo>⊂</mml:mo><mml:msup><mml:mi mathvariant=\"double-struck\">S</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math><tex-math>documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$Omega subset mathbb{S}^{2}$end{document}</tex-math><inline-graphic mime-subtype=\"GIF\" specific-use=\"web\" xlink:href=\"39_2025_721_Article_IEq1.gif\"></inline-graphic></alternatives></inline-formula> for which the first positive eigenvalue of the Laplacian with Neumann boundary condition exceeds that of the geodesic disk having the same area. This example holds for large areas and contrasts with results by Bandle and later authors proving the maximality of the disk under additional topological or geometric conditions, thereby revealing such conditions to be necessary.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"43 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145914889","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}
Pub Date : 2025-09-01DOI: 10.1007/s00039-025-00718-w
Boaz Klartag, Joseph Lehec
We provide the final step in the resolution of Bourgain’s slicing problem in the affirmative. Thus we establish the following theorem: for any convex body K subseteq mathbb{R}^{n} of volume one, there exists a hyperplane H subseteq mathbb{R}^{n} such that
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