Mathematische Annalen最新文献

{"title":"A Mattila–Sjölin theorem for simplices in low dimensions","authors":"Eyvindur Ari Palsson, Francisco Romero Acosta","doi":"10.1007/s00208-024-02948-z","DOIUrl":"https://doi.org/10.1007/s00208-024-02948-z","url":null,"abstract":"<p>In this paper we show that if a compact set <span>(E subset {mathbb {R}}^d)</span>, <span>(d ge 3)</span>, has Hausdorff dimension greater than <span>(frac{(4k-1)}{4k}d+frac{1}{4})</span> when <span>(3 le d&lt;frac{k(k+3)}{(k-1)})</span> or <span>(d- frac{1}{k-1})</span> when <span>(frac{k(k+3)}{(k-1)} le d)</span>, then the set of congruence class of simplices with vertices in <i>E</i> has nonempty interior. By set of congruence class of simplices with vertices in <i>E</i> we mean </p><span>$$begin{aligned} Delta _{k}(E) = left{ textbf{t} = left( t_{ij} right) : |x_i-x_j|=t_{ij}; x_i,x_j in E; 0le i &lt; j le k right} subset {mathbb {R}}^{frac{k(k+1)}{2}} end{aligned}$$</span><p>where <span>(2 le k &lt;d)</span>. This result improves the previous best results in the sense that we now can obtain a Hausdorff dimension threshold which allow us to guarantee that the set of congruence class of triangles formed by triples of points of <i>E</i> has nonempty interior when <span>(d=3)</span> as well as extending to all simplices. The present work can be thought of as an extension of the Mattila–Sjölin theorem which establishes a non-empty interior for the distance set instead of the set of congruence classes of simplices.\u0000</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"231 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141740678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A new gauge-theoretic construction of the 4-dimensional hyperkähler ALE spaces","authors":"Jiajun Yan","doi":"10.1007/s00208-024-02944-3","DOIUrl":"https://doi.org/10.1007/s00208-024-02944-3","url":null,"abstract":"<p>Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of complete non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay Correspondence. The 4-dimensional hyperkähler ALE spaces are first classified by Peter Kronheimer via a finite-dimensional hyperkähler reduction. In this paper, we give a new gauge-theoretic construction of these spaces. More specifically, we realize each 4-dimensional hyperkähler ALE space as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a gauge group action. The construction given in this paper parallels Kronheimer’s original construction and hence can also be thought of as a gauge-theoretic interpretation of Kronheimer’s construction of these spaces.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"11 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141745946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the volumes of linear subvarieties in moduli spaces of projectivized Abelian differentials","authors":"Duc-Manh Nguyen","doi":"10.1007/s00208-024-02945-2","DOIUrl":"https://doi.org/10.1007/s00208-024-02945-2","url":null,"abstract":"<p>Let <span>(overline{{mathcal {H}}}_{g,n})</span> denote the Hodge bundle over <span>(overline{{mathfrak {M}}}_{g,n})</span>, and <span>({mathbb {P}}overline{{mathcal {H}}}_{g,n})</span> its associated projective bundle. Let <span>({mathcal {H}}_{g,n})</span> and <span>({mathbb {P}}{mathcal {H}}_{g,n})</span> be respectively the restriction of <span>(overline{{mathcal {H}}}_{g,n})</span> and <span>({mathbb {P}}overline{{mathcal {H}}}_{g,n})</span> to the smooth part <span>({mathfrak {M}}_{g,n})</span> of <span>(overline{{mathfrak {M}}}_{g,n})</span>. The Hodge norm provides us with a Hermtian metric on <span>({mathscr {O}}(-1)_{{mathbb {P}}{mathcal {H}}_{g,n}})</span>. Let <span>(Theta )</span> denote the curvature form of this metric. In this paper, we show that if <span>(overline{{mathcal {N}}})</span> is a subvariety of <span>({mathbb {P}}overline{{mathcal {H}}}_{g,n})</span> that intersects <span>({mathcal {H}}_{g,n})</span>, then the integral of the top power of <span>(Theta )</span> over the smooth part of <span>(overline{{mathcal {N}}}cap {mathbb {P}}{mathcal {H}}_{g,n})</span> equals the self-intersection number of the tautological divisor <span>(c_1({mathscr {O}}(-1)_{{mathbb {P}}overline{{mathcal {H}}}_{g,n}})cap overline{{mathcal {N}}})</span> in <span>(overline{{mathcal {N}}})</span>. This implies that the volume of a linear subvariety of <span>({mathbb {P}}{mathcal {H}}_{g,n})</span> whose local coordinates do not involve relative periods can be computed by the intersection number of its closure in <span>({mathbb {P}}overline{{mathcal {H}}}_{g,n})</span> with some power of any divisor representing the tautological line bundle. We also genralize this statement to the bundles <span>({mathbb {P}}overline{{mathcal {H}}}^{(k)}_{g,n})</span>, <span>(k in {mathbb {Z}}_{ge 2})</span>, of <i>k</i>-differentials with poles of order at most <span>((k-1))</span> over <span>(overline{{mathfrak {M}}}_{g,n})</span>. To obtain these results, we use the existence of an appropriate desingularization of <span>(overline{{mathcal {N}}})</span> and a deep result of Kollár (Subadditivity of the Kodaira Dimension: Fiber of General Type, Algebraic Geometry, Sendai, 1985, Advanced studies in Pure Math. (1987)) on variation of Hodge structure.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"33 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Tube formulas for valuations in complex space forms","authors":"Gil Solanes, Juan Andrés Trillo","doi":"10.1007/s00208-024-02929-2","DOIUrl":"https://doi.org/10.1007/s00208-024-02929-2","url":null,"abstract":"<p>Given an isometry invariant valuation on a complex space form we compute its value on the tubes of sufficiently small radii around a set of positive reach. This generalizes classical formulas of Weyl, Gray and others about the volume of tubes. We also develop a general framework on tube formulas for valuations in Riemannian manifolds.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"12 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720352","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Nonlinear stability of non-rotating gaseous stars","authors":"Zhiwu Lin, Yucong Wang, Hao Zhu","doi":"10.1007/s00208-024-02940-7","DOIUrl":"https://doi.org/10.1007/s00208-024-02940-7","url":null,"abstract":"<p>For the non-rotating gaseous stars modeled by the compressible Euler–Poisson system with general pressure law, Lin and Zeng (Comm Pure Appl Math 75: 2511–2572, 2022) proved a turning point principle, which gives the sharp linear stability/instability criteria for the non-rotating gaseous stars. In this paper, we prove that the sharp linear stability criterion for the non-rotating stars also implies nonlinear orbital stability against general perturbations provided the global weak solutions exist. If the perturbations are further restricted to be spherically symmetric, then nonlinear stability holds true unconditionally in the sense that the existence of global weak solutions near the non-rotating star can be proved.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"3 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cardinality and IOD-type continuity of pullback attractors for random nonlocal equations on unbounded domains","authors":"Yangrong Li, Tomás Caraballo, Fengling Wang","doi":"10.1007/s00208-024-02938-1","DOIUrl":"https://doi.org/10.1007/s00208-024-02938-1","url":null,"abstract":"<p>We study the continuity set (the set of all continuous points) of pullback random attractors from a parametric space into the space of all compact subsets of the state space with Hausdorff metric. We find a general theorem that the continuity set is an IOD-type (a countable <i>intersection</i> of <i>open dense</i> sets) with the local similarity under appropriate conditions of random dynamical systems, and we further show that any IOD-type set in the parametric space has the continuous cardinality, which affirmatively answers the unsolved question about the cardinality of the continuity set of attractors in the literature. Applying to the random nonautonomous nonlocal parabolic equations on an unbounded domain driven by colored noise, we establish the existence and IOD-type continuity of pullback random attractors in time, sample-translation and noise-size, moreover, we prove that the continuity set of the pullback random attractor on the plane of time and sample-translation is composed of diagonal rays whose number of bars is the continuous cardinality.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"55 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141608971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Harmonic weak Maass forms and periods II","authors":"Claudia Alfes, Jan Hendrik Bruinier, Markus Schwagenscheidt","doi":"10.1007/s00208-024-02927-4","DOIUrl":"https://doi.org/10.1007/s00208-024-02927-4","url":null,"abstract":"<p>In this paper we investigate the Fourier coefficients of harmonic Maass forms of negative half-integral weight. We relate the algebraicity of these coefficients to the algebraicity of the coefficients of certain canonical meromorphic modular forms of positive even weight with poles at Heegner divisors. Moreover, we give an explicit formula for the coefficients of harmonic Maass forms in terms of periods of certain meromorphic modular forms with algebraic coefficients.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"32 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141608972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Differentiable approximation of continuous definable maps that preserves the image","authors":"Antonio Carbone","doi":"10.1007/s00208-024-02921-w","DOIUrl":"https://doi.org/10.1007/s00208-024-02921-w","url":null,"abstract":"<p>Recently Pawłucki showed that compact sets that are definable in some o-minimal structure admit triangulations of class <span>({{mathcal {C}}}^p)</span> for each integer <span>(pge 1)</span>. In this work, we make use of these new techniques of triangulation to show that all continuous definable maps between compact definable sets can be approximated by differentiable maps without changing their image. The argument is an interplay between o-minimal geometry and PL geometry and makes use of a ‘surjective definable version’ of the finite simplicial approximation theorem that we prove here.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"18 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141585141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algebraic threefolds of general type with small volume","authors":"Yong Hu, Tong Zhang","doi":"10.1007/s00208-024-02933-6","DOIUrl":"https://doi.org/10.1007/s00208-024-02933-6","url":null,"abstract":"<p>It is known that the optimal Noether inequality <span>({text {vol} }(X) ge frac{4}{3}p_g(X) - frac{10}{3})</span> holds for every 3-fold <i>X</i> of general type with <span>(p_g(X) ge 11)</span>. In this paper, we give a complete classification of 3-folds <i>X</i> of general type with <span>(p_g(X) ge 11)</span> satisfying the above equality by giving the explicit structure of a relative canonical model of <i>X</i>. This model coincides with the canonical model of <i>X</i> when <span>(p_g(X) ge 23)</span>. We also establish the second and third optimal Noether inequalities for 3-folds <i>X</i> of general type with <span>(p_g(X) ge 11)</span>. A novel phenomenon shows that there is a one-to-one correspondence between the three Noether inequalities and three possible residues of <span>(p_g(X))</span> modulo 3. These results answer two open questions by Chen et al. (Duke Math J 169(9):1603–1164, 2020), and in dimension three an open question by Chen and Lai (Int J Math 31(1):2050005, 2020).</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"18 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Piecewise regularity results for linear elliptic systems with piecewise regular coefficients","authors":"Youchan Kim","doi":"10.1007/s00208-024-02924-7","DOIUrl":"https://doi.org/10.1007/s00208-024-02924-7","url":null,"abstract":"<p>We obtain piecewise regularity results for linear elliptic systems with piecewise regular coefficients arising from composite materials. We find Hölder continuous functions related to the higher-order derivatives by compensating the possible discontinuity due to the geometry and the discontinuity of the coefficients. This leads to the piecewise regularity of the higher-order derivatives and solves an open problem introduced by Li and Vogelius, and Li and Nirenberg for general composite geometry.</p>","PeriodicalId":18304,"journal":{"name":"Mathematische Annalen","volume":"58 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}