Pub Date : 2021-09-21DOI: 10.1515/crelle-2023-0014
O. Savin, Hui Yu
Abstract We study the regularity of free boundaries in the multiple elastic membrane problem in the plane. We prove the uniqueness of blow-ups, and that the free boundaries are C 1 , log {C^{1,log}} -curves near a regular intersection point.
{"title":"Free boundary regularity in the multiple membrane problem in the plane","authors":"O. Savin, Hui Yu","doi":"10.1515/crelle-2023-0014","DOIUrl":"https://doi.org/10.1515/crelle-2023-0014","url":null,"abstract":"Abstract We study the regularity of free boundaries in the multiple elastic membrane problem in the plane. We prove the uniqueness of blow-ups, and that the free boundaries are C 1 , log {C^{1,log}} -curves near a regular intersection point.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89724697","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 : 2021-09-17DOI: 10.1515/crelle-2023-0016
Gao Chen, Jeff A. Viaclovsky, Ruobing Zhang
Abstract We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG∗ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG∗ manifolds with non-negative Ricci curvature having group Γ = { e } Gamma={e} at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG∗ manifold. A corollary of this is vanishing of the first Betti number for any ALG∗ manifold with non-negative Ricci curvature. Another application of our analysis is to determine the optimal order of ALG∗ gravitational instantons.
{"title":"Hodge theory on ALG∗ manifolds","authors":"Gao Chen, Jeff A. Viaclovsky, Ruobing Zhang","doi":"10.1515/crelle-2023-0016","DOIUrl":"https://doi.org/10.1515/crelle-2023-0016","url":null,"abstract":"Abstract We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG∗ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG∗ manifolds with non-negative Ricci curvature having group Γ = { e } Gamma={e} at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG∗ manifold. A corollary of this is vanishing of the first Betti number for any ALG∗ manifold with non-negative Ricci curvature. Another application of our analysis is to determine the optimal order of ALG∗ gravitational instantons.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77698724","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 : 2021-09-13DOI: 10.1515/crelle-2022-0036
Matthias Paulsen
Abstract Let X ⊂ ℙ 4 {Xsubsetmathbb{P}^{4}} be a very general hypersurface of degree d ≥ 6 {dgeq 6} . Griffiths and Harris conjectured in 1985 that the degree of every curve C ⊂ X {Csubset X} is divisible by d. Despite substantial progress by Kollár in 1991, this conjecture is not known for a single value of d. Building on Kollár’s method, we prove this conjecture for infinitely many d, the smallest one being d = 5005 {d=5005} . The set of these degrees d has positive density. We also prove a higher-dimensional analogue of this result and construct smooth hypersurfaces defined over ℚ {mathbb{Q}} that satisfy the conjecture.
摘要设X∧²4 {Xsubsetmathbb{P} ^{4}}是一个阶数为d≥6d的非常一般的超曲面{geq 6}。Griffiths和Harris在1985年推测，每条曲线C∧X C {subset X的度数}都{可以被d整除。尽管在1991年Kollár取得了实质性进展，但这个猜想并没有一个单一的d值。在Kollár的方法的基础上，我们证明了这个猜想有无限多个d，最小的一个是d=5005} d=5005。这些d度集合的密度是正的。我们也证明了这一结果的高维类比，并构造了定义在π (0) {mathbb{Q}}上的光滑超曲面来满足这个猜想。
{"title":"On the degree of algebraic cycles on hypersurfaces","authors":"Matthias Paulsen","doi":"10.1515/crelle-2022-0036","DOIUrl":"https://doi.org/10.1515/crelle-2022-0036","url":null,"abstract":"Abstract Let X ⊂ ℙ 4 {Xsubsetmathbb{P}^{4}} be a very general hypersurface of degree d ≥ 6 {dgeq 6} . Griffiths and Harris conjectured in 1985 that the degree of every curve C ⊂ X {Csubset X} is divisible by d. Despite substantial progress by Kollár in 1991, this conjecture is not known for a single value of d. Building on Kollár’s method, we prove this conjecture for infinitely many d, the smallest one being d = 5005 {d=5005} . The set of these degrees d has positive density. We also prove a higher-dimensional analogue of this result and construct smooth hypersurfaces defined over ℚ {mathbb{Q}} that satisfy the conjecture.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81154867","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 : 2021-09-08DOI: 10.1515/crelle-2022-0042
Yonatan Harpaz, Dasheng Wei, Olivier Wittenberg
Abstract We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the method unconditional when the degree of the non-split locus is ≤ 2 {leq 2} , as well as in various instances where it is 3. We are also able to obtain improved results in the regime that is conditionally accessible under Schinzel’s hypothesis, by incorporating into it, for the first time, a technique due to Harari for controlling the Brauer–Manin obstruction in families.
{"title":"Rational points on fibrations with few non-split fibres","authors":"Yonatan Harpaz, Dasheng Wei, Olivier Wittenberg","doi":"10.1515/crelle-2022-0042","DOIUrl":"https://doi.org/10.1515/crelle-2022-0042","url":null,"abstract":"Abstract We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the method unconditional when the degree of the non-split locus is ≤ 2 {leq 2} , as well as in various instances where it is 3. We are also able to obtain improved results in the regime that is conditionally accessible under Schinzel’s hypothesis, by incorporating into it, for the first time, a technique due to Harari for controlling the Brauer–Manin obstruction in families.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76136472","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 : 2021-09-06DOI: 10.1515/crelle-2022-0050
Q. Ding
Abstract In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding’s method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C Y × ℝ n - k {CYtimesmathbb{R}^{n-k}} ( 2 ≤ k ≤ n {2leq kleq n} ) in a non-collapsing metric cone C X × ℝ n - k {CXtimesmathbb{R}^{n-k}} obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY. Namely, Y has only one connected component in X.
{"title":"Minimal hypersurfaces in manifolds of Ricci curvature bounded below","authors":"Q. Ding","doi":"10.1515/crelle-2022-0050","DOIUrl":"https://doi.org/10.1515/crelle-2022-0050","url":null,"abstract":"Abstract In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding’s method in [T. H. Colding, Ricci curvature and volume convergence, Ann. of Math. (2) 145 1997, 3, 477–501]. With Cheeger–Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric cone C Y × ℝ n - k {CYtimesmathbb{R}^{n-k}} ( 2 ≤ k ≤ n {2leq kleq n} ) in a non-collapsing metric cone C X × ℝ n - k {CXtimesmathbb{R}^{n-k}} obtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY. Namely, Y has only one connected component in X.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78437381","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 : 2021-09-03DOI: 10.1515/crelle-2021-0045
M. A. D. de Cataldo, J. Heinloth, L. Migliorini
Abstract We compute the supports of the perverse cohomology sheaves of the Hitchin fibration for GLn{mathrm{GL}_{n}} over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every n≥2{ngeq{2}}. A key ingredient is a restriction result for intersection cohomology sheaves that allows us to compare the fibration to the one defined over versal deformations of spectral curves.
{"title":"A support theorem forbreak the Hitchin fibration: The case of GLn and KC","authors":"M. A. D. de Cataldo, J. Heinloth, L. Migliorini","doi":"10.1515/crelle-2021-0045","DOIUrl":"https://doi.org/10.1515/crelle-2021-0045","url":null,"abstract":"Abstract We compute the supports of the perverse cohomology sheaves of the Hitchin fibration for GLn{mathrm{GL}_{n}} over the locus of reduced spectral curves. In contrast to the case of meromorphic Higgs fields we find additional supports at the loci of reducible spectral curves. Their contribution to the global cohomology is governed by a finite twist of Hitchin fibrations for Levi subgroups. The corresponding summands give non-trivial contributions to the cohomology of the moduli spaces for every n≥2{ngeq{2}}. A key ingredient is a restriction result for intersection cohomology sheaves that allows us to compare the fibration to the one defined over versal deformations of spectral curves.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75295125","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 : 2021-09-03DOI: 10.1515/crelle-2021-0035
C. Escher, C. Searle
Abstract Let ℳ0n{mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M∈ℳ0n{Minmathcal{M}_{0}^{n}}, then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M∈ℳ0n{Minmathcal{M}_{0}^{n}}. Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.
{"title":"Torus actions, maximality, and non-negative curvature","authors":"C. Escher, C. Searle","doi":"10.1515/crelle-2021-0035","DOIUrl":"https://doi.org/10.1515/crelle-2021-0035","url":null,"abstract":"Abstract Let ℳ0n{mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M∈ℳ0n{Minmathcal{M}_{0}^{n}}, then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M∈ℳ0n{Minmathcal{M}_{0}^{n}}. Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83027536","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 : 2021-09-02DOI: 10.1515/crelle-2023-0032
Yichao Tian
Abstract Let ( A , I ) {(A,I)} be a bounded prism, and let X be a smooth p-adic formal scheme over Spf ( A / I ) {operatorname{Spf}(A/I)} . We consider the notion of crystals on Bhatt–Scholze’s prismatic site ( X / A ) Δ Δ {(X/A)_{{kern-0.284528pt{Delta}kern-5.975079pt{Delta}}}} of X relative to A. We prove that if X is proper over Spf ( A / I ) {operatorname{Spf}(A/I)} of relative dimension n, then the cohomology of a prismatic crystal is a perfect complex of A-modules with tor-amplitude in degrees [ 0 , 2 n ] {[0,2n]} . We also establish a Poincaré duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of ( X / A ) Δ Δ {(X/A)_{{kern-0.284528pt{Delta}kern-5.975079pt{Delta}}}} . The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.
{"title":"Finiteness and duality for the cohomology of prismatic crystals","authors":"Yichao Tian","doi":"10.1515/crelle-2023-0032","DOIUrl":"https://doi.org/10.1515/crelle-2023-0032","url":null,"abstract":"Abstract Let ( A , I ) {(A,I)} be a bounded prism, and let X be a smooth p-adic formal scheme over Spf ( A / I ) {operatorname{Spf}(A/I)} . We consider the notion of crystals on Bhatt–Scholze’s prismatic site ( X / A ) Δ Δ {(X/A)_{{kern-0.284528pt{Delta}kern-5.975079pt{Delta}}}} of X relative to A. We prove that if X is proper over Spf ( A / I ) {operatorname{Spf}(A/I)} of relative dimension n, then the cohomology of a prismatic crystal is a perfect complex of A-modules with tor-amplitude in degrees [ 0 , 2 n ] {[0,2n]} . We also establish a Poincaré duality for the reduced prismatic crystals, i.e. the crystals over the reduced structural sheaf of ( X / A ) Δ Δ {(X/A)_{{kern-0.284528pt{Delta}kern-5.975079pt{Delta}}}} . The key ingredient is an explicit local description of reduced prismatic crystals in terms of Higgs modules.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91224867","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 : 2021-08-20DOI: 10.1515/crelle-2021-0084
E. Carneiro, Vorrapan Chandee, Andrés Chirre, M. Milinovich
Abstract We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery’s function F ( α , T ) {F(alpha,T)} in bounded intervals, the second is an integral introduced by Selberg related to estimating the variance of primes in short intervals, and the last is the second moment of the logarithmic derivative of the Riemann zeta-function near the critical line. The conjectured asymptotic for any of these three integrals is equivalent to Montgomery’s pair correlation conjecture. Assuming the Riemann hypothesis, we substantially improve the known upper and lower bounds for these integrals by introducing new connections to certain extremal problems in Fourier analysis. In an appendix, we study the intriguing problem of establishing the sharp form of an embedding between two Hilbert spaces of entire functions naturally connected to Montgomery’s pair correlation conjecture.
{"title":"On Montgomery’s pair correlation conjecture: A tale of three integrals","authors":"E. Carneiro, Vorrapan Chandee, Andrés Chirre, M. Milinovich","doi":"10.1515/crelle-2021-0084","DOIUrl":"https://doi.org/10.1515/crelle-2021-0084","url":null,"abstract":"Abstract We study three integrals related to the celebrated pair correlation conjecture of H. L. Montgomery. The first is the integral of Montgomery’s function F ( α , T ) {F(alpha,T)} in bounded intervals, the second is an integral introduced by Selberg related to estimating the variance of primes in short intervals, and the last is the second moment of the logarithmic derivative of the Riemann zeta-function near the critical line. The conjectured asymptotic for any of these three integrals is equivalent to Montgomery’s pair correlation conjecture. Assuming the Riemann hypothesis, we substantially improve the known upper and lower bounds for these integrals by introducing new connections to certain extremal problems in Fourier analysis. In an appendix, we study the intriguing problem of establishing the sharp form of an embedding between two Hilbert spaces of entire functions naturally connected to Montgomery’s pair correlation conjecture.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76297546","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 : 2021-08-16DOI: 10.1515/crelle-2023-0053
C. Frances, K. Melnick
Abstract We prove that, for a compact, 3-dimensional, real-analytic, Lorentzian manifold, if the group of conformal transformations does not preserve any metric in the conformal class, then the metric is conformally flat.
{"title":"The Lorentzian Lichnerowicz conjecture for real-analytic, three-dimensional manifolds","authors":"C. Frances, K. Melnick","doi":"10.1515/crelle-2023-0053","DOIUrl":"https://doi.org/10.1515/crelle-2023-0053","url":null,"abstract":"Abstract We prove that, for a compact, 3-dimensional, real-analytic, Lorentzian manifold, if the group of conformal transformations does not preserve any metric in the conformal class, then the metric is conformally flat.","PeriodicalId":54896,"journal":{"name":"Journal fur die Reine und Angewandte Mathematik","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78441804","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}
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