Geometriae Dedicata最新文献

{"title":"Quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces","authors":"Kingshook Biswas","doi":"10.1007/s10711-024-00903-5","DOIUrl":"https://doi.org/10.1007/s10711-024-00903-5","url":null,"abstract":"<p>Hyperbolic fillings of metric spaces are a well-known tool for proving results on extending quasi-Moebius maps between boundaries of Gromov hyperbolic spaces to quasi-isometries between the spaces. For a hyperbolic filling <i>Y</i> of the boundary of a Gromov hyperbolic space <i>X</i>, one has a quasi-Moebius identification between the boundaries <span>(partial Y)</span> and <span>(partial X)</span>. For CAT(-1) spaces, and more generally boundary continuous Gromov hyperbolic spaces, one can refine the quasi-Moebius structure on the boundary to a Moebius structure. It is then natural to ask whether there exists a functorial hyperbolic filling of the boundary by a boundary continuous Gromov hyperbolic space with an identification between boundaries which is not just quasi-Moebius, but in fact Moebius. The filling should be functorial in the sense that a Moebius homeomorphism between boundaries should induce an isometry between there fillings. We give a positive answer to this question for a large class of boundaries satisfying one crucial hypothesis, the <i>antipodal property</i>. This gives a class of compact spaces called <i>quasi-metric antipodal spaces</i>. For any such space <i>Z</i>, we give a functorial construction of a boundary continuous Gromov hyperbolic space <span>(mathcal {M}(Z))</span> together with a Moebius identification of its boundary with <i>Z</i>. The space <span>(mathcal {M}(Z))</span> is maximal amongst all fillings of <i>Z</i>. These spaces <span>(mathcal {M}(Z))</span> give in fact all examples of a natural class of spaces called <i>maximal Gromov hyperbolic spaces</i>. We prove an equivalence of categories between quasi-metric antipodal spaces and maximal Gromov hyperbolic spaces. This is part of a more general equivalence we prove between the larger categories of certain spaces called <i>antipodal spaces</i> and <i>maximal Gromov product spaces</i>. We prove that the injective hull of a Gromov product space <i>X</i> is isometric to the maximal Gromov product space <span>(mathcal {M}(Z))</span>, where <i>Z</i> is the boundary of <i>X</i>. We also show that a Gromov product space is injective if and only if it is maximal.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"33 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Homogeneous semisymmetric neutral 4-manifolds","authors":"","doi":"10.1007/s10711-024-00898-z","DOIUrl":"https://doi.org/10.1007/s10711-024-00898-z","url":null,"abstract":"<h3>Abstract</h3> <p>We determine homogeneous semi-symmetric neutral manifolds of dimension 4. We also describe all the possible semi-symmetric curvature tensors on four-dimensional neutral vector spaces.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"112 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140044809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The residual set dimension of a generalized apollonian packing","authors":"Daniel Lautzenheiser","doi":"10.1007/s10711-024-00899-y","DOIUrl":"https://doi.org/10.1007/s10711-024-00899-y","url":null,"abstract":"<p>We view space-filling circle packings as subsets of the boundary of hyperbolic space subject to symmetry conditions based on a discrete group of isometries. This allows for the application of counting methods which admit rigorous upper and lower bounds on the Hausdorff dimension of the residual set of a generalized Apollonian circle packing. This dimension (which also coincides with a critical exponent) is strictly greater than that of the Apollonian packing.\u0000</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045123","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The sub-Riemannian length spectrum for screw motions of constant pitch on flat and hyperbolic 3-manifolds","authors":"Marcos Salvai","doi":"10.1007/s10711-024-00896-1","DOIUrl":"https://doi.org/10.1007/s10711-024-00896-1","url":null,"abstract":"<p>Let <i>M</i> be an oriented three-dimensional Riemannian manifold of constant sectional curvature <span>(k=0,1,-1)</span> and let <span>(SOleft( Mright) )</span> be its direct orthonormal frame bundle (direct refers to positive orientation), which may be thought of as the set of all positions of a small body in <i>M</i>. Given <span>( lambda in {mathbb {R}})</span>, there is a three-dimensional distribution <span>(mathcal { D}^{lambda })</span> on <span>(SOleft( Mright) )</span> accounting for infinitesimal rototranslations of constant pitch <span>(lambda )</span>. When <span>(lambda ne k^{2})</span>, there is a canonical sub-Riemannian structure on <span>({mathcal {D}}^{lambda })</span>. We present a geometric characterization of its geodesics, using a previous Lie theoretical description. For <span>(k=0,-1)</span>, we compute the sub-Riemannian length spectrum of <span>(left( SOleft( Mright) ,{mathcal {D}} ^{lambda }right) )</span> in terms of the complex length spectrum of <i>M</i> (given by the lengths and the holonomies of the periodic geodesics) when <i>M</i> has positive injectivity radius. In particular, for two complex length isospectral closed hyperbolic 3-manifolds (even if they are not isometric), the associated sub-Riemannian metrics on their direct orthonormal bundles are length isospectral.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"4 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Infinitesimal symmetries of bundle gerbes and Courant algebroids","authors":"","doi":"10.1007/s10711-024-00897-0","DOIUrl":"https://doi.org/10.1007/s10711-024-00897-0","url":null,"abstract":"<h3>Abstract</h3> <p>Let <em>M</em> be a smooth manifold and let <span> <span>(chi in Omega ^3(M))</span> </span> be closed differential form with integral periods. We show the Lie 2-algebra <span> <span>(mathbb {L}(C_chi ))</span> </span> of sections of the <span> <span>(chi )</span> </span>-twisted Courant algebroid <span> <span>(C_chi )</span> </span> on <em>M</em> is quasi-isomorphic to the Lie 2-algebra of connection-preserving multiplicative vector fields on an <span> <span>(S^1)</span> </span>-bundle gerbe with connection (over <em>M</em>) whose 3-curvature is <span> <span>(chi )</span> </span>. </p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"2015 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A family of Andrews–Curtis trivializations via 4-manifold trisections","authors":"Ethan Romary, Alexander Zupan","doi":"10.1007/s10711-024-00891-6","DOIUrl":"https://doi.org/10.1007/s10711-024-00891-6","url":null,"abstract":"<p>An R-link is an <i>n</i>-component link <i>L</i> in <span>(S^3)</span> such that Dehn surgery on <i>L</i> yields <span>(#^n(S^1 times S^2))</span>. Every R-link <i>L</i> gives rise to a geometrically simply-connected homotopy 4-sphere <span>(X_L)</span>, which in turn can be used to produce a balanced presentation of the trivial group. Adapting work of Gompf, Scharlemann, and Thompson, Meier and Zupan produced a family of R-links <i>L</i>(<i>p</i>, <i>q</i>; <i>c</i>/<i>d</i>), where the pairs (<i>p</i>, <i>q</i>) and (<i>c</i>, <i>d</i>) are relatively prime and <i>c</i> is even. Within this family, <span>(L(3,2;2n/(2n+1)))</span> induces the infamous trivial group presentation <span>(langle x,y , | , xyx=yxy, x^{n+1}=y^n rangle )</span>, a popular collection of potential counterexamples to the Andrews–Curtis conjecture for <span>(n ge 3)</span>. In this paper, we use 4-manifold trisections to show that the group presentations corresponding to a different subfamily, <i>L</i>(3, 2; 4/<i>d</i>), are Andrews–Curtis trivial for all <i>d</i>.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"31 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139921301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A cyclotomic family of thin hypergeometric monodromy groups in $${text {Sp}}_4({mathbb {R}})$$","authors":"Simion Filip, Charles Fougeron","doi":"10.1007/s10711-024-00893-4","DOIUrl":"https://doi.org/10.1007/s10711-024-00893-4","url":null,"abstract":"<p>We exhibit an infinite family of discrete subgroups of <span>({{,mathrm{textbf{Sp}},}}_4(mathbb {R}))</span> which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters <span>(left( tfrac{N-3}{2N},tfrac{N-1}{2N}, tfrac{N+1}{2N}, tfrac{N+3}{2N}right) )</span> at infinity and maximal unipotent monodromy at zero, for any integer <span>(Nge 4)</span>. Additionally, we relate the cones used for ping-pong in <span>(mathbb {R}^4)</span> with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian. These domains of discontinuity lead to uniformizations of variations of Hodge structure with Hodge numbers (1, 1, 1, 1).</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"76 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139921249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Complete Calabi–Yau metrics from smoothing Calabi–Yau complete intersections","authors":"Benjy J. Firester","doi":"10.1007/s10711-024-00886-3","DOIUrl":"https://doi.org/10.1007/s10711-024-00886-3","url":null,"abstract":"<p>We construct complete Calabi–Yau metrics on non-compact manifolds that are smoothings of an initial complete intersection <span>(V_0)</span> that is a Calabi–Yau cone, extending the work of Székelyhidi (Duke Math J 168(14):2651–2700, 2019). The constructed Calabi–Yau manifold has tangent cone at infinity given by <span>({mathbb {C}}times V_0)</span>. This construction produces Calabi–Yau metrics with fibers having varying complex structures and possibly isolated singularities.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"11 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139928208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On branched coverings of singular (G, X)-manifolds","authors":"Léo Brunswic","doi":"10.1007/s10711-023-00873-0","DOIUrl":"https://doi.org/10.1007/s10711-023-00873-0","url":null,"abstract":"<p>Branched coverings boast a rich history, ranging from the ramification of Riemann surfaces to the realization of 3-manifolds as coverings branched over knots and spanning both geometric topology and algebraic geometry. This work delves into branched coverings “à la Fox” of (<i>G</i>, <i>X</i>)-manifolds, encompassing three main avenues: Firstly, we introduce a comprehensive class of singular (<i>G</i>, <i>X</i>)-manifolds, elucidating elementary theory paired with illustrative examples to showcase its efficacy and universality. Secondly, building on Montesinos’ work, we revisit and augment the prevailing knowledge, formulating a Galois theory tailored for such branched coverings. This includes a detailed portrayal of the fiber above branching points. Lastly, we identify local attributes that guarantee the existence of developing maps for singular (<i>G</i>, <i>X</i>)-manifolds within the branched coverings framework. Notably, we pinpoint conditions that ensure the existence of developing maps for these singular manifolds. This research proves especially pertinent for non-metric singular (<i>G</i>, <i>X</i>)-manifolds like those of Lorentzian or projective nature, as discussed by Barbot, Bonsante, Suhyoung Choi, Danciger, Seppi, Schlenker, and the author, among others. While examples are sprinkled throughout, a standout application presented is a uniformization theorem “à la Mess” for singular locally Minkowski manifolds exhibiting BTZ-like singularities.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"17 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Unique continuation problem on RCD Spaces. I","authors":"Qin Deng, Xinrui Zhao","doi":"10.1007/s10711-024-00890-7","DOIUrl":"https://doi.org/10.1007/s10711-024-00890-7","url":null,"abstract":"<p>In this note we establish the weak unique continuation theorem for caloric functions on compact <i>RCD</i>(<i>K</i>, 2) spaces and show that there exists an <i>RCD</i>(<i>K</i>, 4) space on which there exist non-trivial eigenfunctions of the Laplacian and non-stationary solutions of the heat equation which vanish up to infinite order at one point . We also establish frequency estimates for eigenfunctions and caloric functions on the metric horn. In particular, this gives a strong unique continuation type result on the metric horn for harmonic functions with a high rate of decay at the horn tip, where it is known that the standard strong unique continuation property fails.</p>","PeriodicalId":55103,"journal":{"name":"Geometriae Dedicata","volume":"178 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139752288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}