Pub Date : 2024-11-28DOI: 10.1016/j.geomphys.2024.105386
Michael B. Law, Isaac M. Lopez, Daniel Santiago
We show that the weighted positive mass theorem of Baldauf–Ozuch and Chu–Zhu is equivalent to the usual positive mass theorem under suitable regularity, and can be regarded as a positive mass theorem for smooth metric measure spaces. A stronger weighted positive mass theorem is established, unifying and generalizing their results. We also study Dirac operators on certain warped product manifolds associated to smooth metric measure spaces. Applications of this include, among others, an alternative proof for a special case of our positive mass theorem, eigenvalue bounds for the Dirac operator on closed spin manifolds, and a new way to understand the weighted Dirac operator using warped products.
{"title":"Positive mass and Dirac operators on weighted manifolds and smooth metric measure spaces","authors":"Michael B. Law, Isaac M. Lopez, Daniel Santiago","doi":"10.1016/j.geomphys.2024.105386","DOIUrl":"10.1016/j.geomphys.2024.105386","url":null,"abstract":"<div><div>We show that the weighted positive mass theorem of Baldauf–Ozuch and Chu–Zhu is equivalent to the usual positive mass theorem under suitable regularity, and can be regarded as a positive mass theorem for smooth metric measure spaces. A stronger weighted positive mass theorem is established, unifying and generalizing their results. We also study Dirac operators on certain warped product manifolds associated to smooth metric measure spaces. Applications of this include, among others, an alternative proof for a special case of our positive mass theorem, eigenvalue bounds for the Dirac operator on closed spin manifolds, and a new way to understand the weighted Dirac operator using warped products.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105386"},"PeriodicalIF":1.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096598","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-28DOI: 10.1016/j.geomphys.2024.105387
Xianguo Geng , Feiying Yan , Jiao Wei
In this work, the Cauchy problem for the generalized complex short pulse equation with initial conditions in the weighted Sobolev space is studied by using the Riemann-Hilbert method and the -steepest descent method. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem can be expressed as solution of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, the long-time asymptotics and soliton resolution of the generalized complex short pulse equation in the soliton region are obtained by resorting to the -steepest descent method. The results also indicate that the N-soliton solutions of the generalized complex short pulse equation are asymptotically stable.
{"title":"Soliton resolution for the generalized complex short pulse equation with the weighted Sobolev initial data","authors":"Xianguo Geng , Feiying Yan , Jiao Wei","doi":"10.1016/j.geomphys.2024.105387","DOIUrl":"10.1016/j.geomphys.2024.105387","url":null,"abstract":"<div><div>In this work, the Cauchy problem for the generalized complex short pulse equation with initial conditions in the weighted Sobolev space <span><math><mi>H</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> is studied by using the Riemann-Hilbert method and the <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-steepest descent method. Based on the spectral analysis of the Lax pair, the solution of the Cauchy problem can be expressed as solution of a Riemann-Hilbert problem, which is transformed into a solvable model after a series of deformations. Finally, the long-time asymptotics and soliton resolution of the generalized complex short pulse equation in the soliton region are obtained by resorting to the <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover></math></span>-steepest descent method. The results also indicate that the <em>N</em>-soliton solutions of the generalized complex short pulse equation are asymptotically stable.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105387"},"PeriodicalIF":1.6,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142747797","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-26DOI: 10.1016/j.geomphys.2024.105384
Simone Cristofori, Michela Zedda
In this paper we compute the third coefficient arising from the TYCZ-expansion of the ε-function associated to a Kähler-Einstein metric and discuss the consequences of its vanishing.
{"title":"On the third coefficient in the TYCZ–expansion of the epsilon function of Kähler–Einstein manifolds","authors":"Simone Cristofori, Michela Zedda","doi":"10.1016/j.geomphys.2024.105384","DOIUrl":"10.1016/j.geomphys.2024.105384","url":null,"abstract":"<div><div>In this paper we compute the third coefficient arising from the TYCZ-expansion of the <em>ε</em>-function associated to a Kähler-Einstein metric and discuss the consequences of its vanishing.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105384"},"PeriodicalIF":1.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722576","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.geomphys.2024.105375
Miguel Manzano, Marc Mars
In this work we provide a definition of the constraint tensor of a null hypersurface data which is completely explicit in the extrinsic geometry of the hypersurface. The definition is fully covariant and applies for any topology of the hypersurface. For data embedded in a spacetime, the constraint tensor coincides with the pull-back of the ambient Ricci tensor. As applications of the results, we find three geometric quantities on any transverse submanifold S of the data with remarkably simple gauge behaviour, and prove that the restriction of the constraint tensor to S takes a very simple form in terms of them. We also obtain an identity that generalizes the standard near horizon equation of isolated horizons to totally geodesic null hypersurfaces with any topology. Finally, we prove that when a null hypersurface has product topology, its extrinsic curvature can be uniquely reconstructed from the constraint tensor plus suitable initial data on a cross-section.
在这项工作中，我们给出了空超曲面数据的约束张量定义，该定义在超曲面的外几何学中是完全明确的。该定义是完全协变的，适用于超曲面的任何拓扑结构。对于嵌入时空的数据，约束张量与环境里奇张量的回拉重合。作为结果的应用，我们在数据的任意横向子满面 S 上发现了三个几何量，它们具有非常简单的轨距行为，并证明了约束张量对 S 的限制采用了非常简单的形式。我们还得到了一个特性，它将孤立地平线的标准近地平线方程推广到具有任意拓扑结构的完全大地空超曲面。最后，我们证明了当空超曲面具有积拓扑结构时，其外曲率可以通过约束张量加上横截面上合适的初始数据唯一地重建。
{"title":"The constraint tensor for null hypersurfaces","authors":"Miguel Manzano, Marc Mars","doi":"10.1016/j.geomphys.2024.105375","DOIUrl":"10.1016/j.geomphys.2024.105375","url":null,"abstract":"<div><div>In this work we provide a definition of the constraint tensor of a null hypersurface data which is completely explicit in the extrinsic geometry of the hypersurface. The definition is fully covariant and applies for any topology of the hypersurface. For data embedded in a spacetime, the constraint tensor coincides with the pull-back of the ambient Ricci tensor. As applications of the results, we find three geometric quantities on any transverse submanifold <em>S</em> of the data with remarkably simple gauge behaviour, and prove that the restriction of the constraint tensor to <em>S</em> takes a very simple form in terms of them. We also obtain an identity that generalizes the standard near horizon equation of isolated horizons to totally geodesic null hypersurfaces with any topology. Finally, we prove that when a null hypersurface has product topology, its extrinsic curvature can be uniquely reconstructed from the constraint tensor plus suitable initial data on a cross-section.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"208 ","pages":"Article 105375"},"PeriodicalIF":1.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142721953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.geomphys.2024.105373
Liqiang Cai , Zhuo Chen , Honglei Lang , Maosong Xiang
Given a vector bundle A over a smooth manifold M such that the square root of the line bundle exists, the Clifford bundle associated to the standard split pseudo-Euclidean vector bundle admits a spinor bundle , whose section space consists of Berezinian half-densities of the graded manifold . Inspired by Kosmann-Schwarzbach's formula of deriving operator of split Courant algebroid (or proto-bialgebroid) structures on , we give an explicit construction of the associate Dirac generating operator introduced by Alekseev and Xu. We prove that the square of the Dirac generating operator is an invariant of the corresponding split Courant algebroid, and also give an explicit expression of this invariant in terms of modular elements.
给定光滑流形 M 上的向量束 A，使得线束 ∧topA⁎⊗∧topT⁎M 的平方根 L 存在，那么与标准分裂伪欧几里得向量束相关联的克利福德束（E=A⊕A⁎、⋅,〉）允许一个旋量束∧-A⊗L，其截面空间由分级流形 A⁎[1] 的贝雷津半密度组成。受科斯曼-施瓦茨巴赫（Kosmann-Schwarzbach）关于 A⊕A⁎上的分裂库朗网格（或原边网格）结构的导出算子公式的启发，我们给出了阿列克谢耶夫（Alekseev）和徐（Xu）引入的关联狄拉克生成算子的明确构造。我们证明了狄拉克生成算子的平方是相应的分裂库朗拟合结构的不变量，并给出了这个不变量在模元方面的明确表达式。
{"title":"Dirac generating operators of split Courant algebroids","authors":"Liqiang Cai , Zhuo Chen , Honglei Lang , Maosong Xiang","doi":"10.1016/j.geomphys.2024.105373","DOIUrl":"10.1016/j.geomphys.2024.105373","url":null,"abstract":"<div><div>Given a vector bundle <em>A</em> over a smooth manifold <em>M</em> such that the square root <span><math><mi>L</mi></math></span> of the line bundle <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mi>top</mi></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>⊗</mo><msup><mrow><mo>∧</mo></mrow><mrow><mi>top</mi></mrow></msup><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mi>M</mi></math></span> exists, the Clifford bundle associated to the standard split pseudo-Euclidean vector bundle <span><math><mo>(</mo><mi>E</mi><mo>=</mo><mi>A</mi><mo>⊕</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>,</mo><mo>〈</mo><mo>⋅</mo><mo>,</mo><mo>⋅</mo><mo>〉</mo><mo>)</mo></math></span> admits a spinor bundle <span><math><msup><mrow><mo>∧</mo></mrow><mrow><mo>•</mo></mrow></msup><mi>A</mi><mo>⊗</mo><mi>L</mi></math></span>, whose section space consists of Berezinian half-densities of the graded manifold <span><math><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>[</mo><mn>1</mn><mo>]</mo></math></span>. Inspired by Kosmann-Schwarzbach's formula of deriving operator of split Courant algebroid (or proto-bialgebroid) structures on <span><math><mi>A</mi><mo>⊕</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span>, we give an explicit construction of the associate Dirac generating operator introduced by Alekseev and Xu. We prove that the square of the Dirac generating operator is an invariant of the corresponding split Courant algebroid, and also give an explicit expression of this invariant in terms of modular elements.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"208 ","pages":"Article 105373"},"PeriodicalIF":1.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142721793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.geomphys.2024.105374
Huynh M. Hien
It is well-known that hyperbolic flows admit Markov partitions of arbitrarily small size. However, the constructions of Markov partitions for general hyperbolic flows are quite abstract and not easy to understand. To establish a more detailed understanding of Markov partitions, in this paper we consider the geodesic flow on Riemann surfaces of constant negative curvature. We provide a more complete construction of Markov partitions for this hyperbolic flow with explicit forms of rectangles and local cross sections. The local product structure is also calculated in detail.
{"title":"Construction of Markov partitions for the geodesic flow on compact Riemann surfaces of constant negative curvature","authors":"Huynh M. Hien","doi":"10.1016/j.geomphys.2024.105374","DOIUrl":"10.1016/j.geomphys.2024.105374","url":null,"abstract":"<div><div>It is well-known that hyperbolic flows admit Markov partitions of arbitrarily small size. However, the constructions of Markov partitions for general hyperbolic flows are quite abstract and not easy to understand. To establish a more detailed understanding of Markov partitions, in this paper we consider the geodesic flow on Riemann surfaces of constant negative curvature. We provide a more complete construction of Markov partitions for this hyperbolic flow with explicit forms of rectangles and local cross sections. The local product structure is also calculated in detail.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"208 ","pages":"Article 105374"},"PeriodicalIF":1.6,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142721954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-17DOI: 10.1016/j.geomphys.2024.105372
Shuyu Xiao
We investigate the abelianization of a Lie algebroid and provide a necessary and sufficient condition for its existence. We also study the abelianization of groupoids and provide sufficient conditions for its existence in the smooth category and a necessary and sufficient condition for its existence in the diffeological category.
{"title":"Abelianization of Lie algebroids and Lie groupoids","authors":"Shuyu Xiao","doi":"10.1016/j.geomphys.2024.105372","DOIUrl":"10.1016/j.geomphys.2024.105372","url":null,"abstract":"<div><div>We investigate the abelianization of a Lie algebroid and provide a necessary and sufficient condition for its existence. We also study the abelianization of groupoids and provide sufficient conditions for its existence in the smooth category and a necessary and sufficient condition for its existence in the diffeological category.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"207 ","pages":"Article 105372"},"PeriodicalIF":1.6,"publicationDate":"2024-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-17DOI: 10.1016/j.geomphys.2024.105370
Albert Much
We introduce a non-commutative product for curved spacetimes, that can be regarded as a generalization of the Rieffel (or Moyal-Weyl) product. This product employs the exponential map and a Poisson tensor, and the deformed product maintains associativity under the condition that the Poisson tensor Θ satisfies , in relation to a Levi-Cevita connection. We proceed to solve the associativity condition for various physical spacetimes, uncovering non-commutative structures with compelling properties.
{"title":"Quantum spacetimes from general relativity?","authors":"Albert Much","doi":"10.1016/j.geomphys.2024.105370","DOIUrl":"10.1016/j.geomphys.2024.105370","url":null,"abstract":"<div><div>We introduce a non-commutative product for curved spacetimes, that can be regarded as a generalization of the Rieffel (or Moyal-Weyl) product. This product employs the exponential map and a Poisson tensor, and the deformed product maintains associativity under the condition that the Poisson tensor Θ satisfies <span><math><msup><mrow><mi>Θ</mi></mrow><mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mrow><mi>∇</mi></mrow><mrow><mi>ν</mi></mrow></msub><msup><mrow><mi>Θ</mi></mrow><mrow><mi>ρ</mi><mi>σ</mi></mrow></msup><mo>=</mo><mn>0</mn></math></span>, in relation to a Levi-Cevita connection. We proceed to solve the associativity condition for various physical spacetimes, uncovering non-commutative structures with compelling properties.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105370"},"PeriodicalIF":1.6,"publicationDate":"2024-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-14DOI: 10.1016/j.geomphys.2024.105366
W.K. Schief , U. Hertrich-Jeromin , B.G. Konopelchenko
It is shown that a canonical geometric setting of the integrable TED equation is a Kählerian tangent bundle of an affine manifold. The remarkable multi-dimensional consistency of this 4+4-dimensional dispersionless partial differential equation arises naturally in this context. In a particular 4-dimensional reduction, the affine manifolds turn out to be self-dual Einstein spaces of neutral signature governed by Plebański's first heavenly equation. In another reduction, the affine manifolds are Hessian, governed by compatible general heavenly equations. The Legendre invariance of the latter gives rise to a (dual) Hessian structure. Foliations of affine manifolds in terms of self-dual Einstein spaces are also shown to arise in connection with a natural 5-dimensional reduction.
{"title":"Affine manifolds: The differential geometry of the multi-dimensionally consistent TED equation","authors":"W.K. Schief , U. Hertrich-Jeromin , B.G. Konopelchenko","doi":"10.1016/j.geomphys.2024.105366","DOIUrl":"10.1016/j.geomphys.2024.105366","url":null,"abstract":"<div><div>It is shown that a canonical geometric setting of the integrable TED equation is a Kählerian tangent bundle of an affine manifold. The remarkable multi-dimensional consistency of this 4+4-dimensional dispersionless partial differential equation arises naturally in this context. In a particular 4-dimensional reduction, the affine manifolds turn out to be self-dual Einstein spaces of neutral signature governed by Plebański's first heavenly equation. In another reduction, the affine manifolds are Hessian, governed by compatible general heavenly equations. The Legendre invariance of the latter gives rise to a (dual) Hessian structure. Foliations of affine manifolds in terms of self-dual Einstein spaces are also shown to arise in connection with a natural 5-dimensional reduction.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"207 ","pages":"Article 105366"},"PeriodicalIF":1.6,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-14DOI: 10.1016/j.geomphys.2024.105364
Jinghong Lin , Xiaomeng Xu
In this paper, we compute the Stokes matrices of a special quantum confluent hypergeometric system with Poincaré rank one. The interest in the Stokes phenomenon of such a system arises from representation theory and the theory of isomonodromy deformation.
{"title":"Explicit evaluation of the Stokes matrices for certain quantum confluent hypergeometric equations","authors":"Jinghong Lin , Xiaomeng Xu","doi":"10.1016/j.geomphys.2024.105364","DOIUrl":"10.1016/j.geomphys.2024.105364","url":null,"abstract":"<div><div>In this paper, we compute the Stokes matrices of a special quantum confluent hypergeometric system with Poincaré rank one. The interest in the Stokes phenomenon of such a system arises from representation theory and the theory of isomonodromy deformation.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105364"},"PeriodicalIF":1.6,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143092483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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