Pub Date : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a12
Andrei Moroianu, Mihaela Pilca
We show that any conformal vector field on a compact lcK manifold is Killing with respect to the Gauduchon metric. Furthermore, we prove that any conformal vector field on a compact lcK manifold whose Kähler cover is neither flat, nor hyperkähler, is holomorphic.
{"title":"Conformal vector fields on lcK manifolds","authors":"Andrei Moroianu, Mihaela Pilca","doi":"10.4310/mrl.2023.v30.n5.a12","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a12","url":null,"abstract":"We show that any conformal vector field on a compact lcK manifold is Killing with respect to the Gauduchon metric. Furthermore, we prove that any conformal vector field on a compact lcK manifold whose Kähler cover is neither flat, nor hyperkähler, is holomorphic.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"25 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941456","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a2
Gabriele Bogo
Let $X = mathbb{H}/Gamma$ be an $n$-punctured sphere, $n gt 3$. We introduce and study $n-3$ deformation operators on the space of modular forms $M_ast (Gamma)$ based on the classical theory of uniformizing differential equations and accessory parameters. When restricting to modular functions, we recover a construction in Teichmüller theory related to the deformation of the complex structure of $X$. We describe the deformation operators in terms of derivations with respect to Eichler integrals of weight-four cusp forms, and in terms of vector-valued modular forms attached to extensions of symmetric tensor representations.
{"title":"Modular forms, deformation of punctured spheres, and extensions of symmetric tensor representations","authors":"Gabriele Bogo","doi":"10.4310/mrl.2023.v30.n5.a2","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a2","url":null,"abstract":"Let $X = mathbb{H}/Gamma$ be an $n$-punctured sphere, $n gt 3$. We introduce and study $n-3$ deformation operators on the space of modular forms $M_ast (Gamma)$ based on the classical theory of uniformizing differential equations and accessory parameters. When restricting to modular functions, we recover a construction in Teichmüller theory related to the deformation of the complex structure of $X$. We describe the deformation operators in terms of derivations with respect to Eichler integrals of weight-four cusp forms, and in terms of vector-valued modular forms attached to extensions of symmetric tensor representations.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"17 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941477","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a5
Estanislao Herscovich
In this article we address Problem 5.12 in $href{https://projecteuclid.org/ebook/Download?urlId=10.2969/aspm/03110051&isFullBook=false}{[9]}$. More precisely, we prove that the singular tensor product introduced by R. Borcherds in the previous reference is part of a 2-monoidal category structure in a certain category of functors. We also complete some missing points in the previously mentioned article, most notably in the definitions of singular tensor products and of vertex algebras themselves, which are however verified in all the examples appearing in that reference. To prove our results it will be extremely useful, if not essential, to frame our objects within the language of bicategories. We also introduce a slightly more general notion of (quantum) vertex algebra than the one in $href{https://projecteuclid.org/ebook/Download?urlId=10.2969/aspm/03110051&isFullBook=false}{[9]}$, that we call categorical (quantum) vertex algebra, enjoying all the properties mentioned by Borcherds in that article and having as particular example the definition presented by that author.
本文将讨论 $href{https://projecteuclid.org/ebook/Download?urlId=10.2969/aspm/03110051&isFullBook=false}{[9]}$ 中的问题 5.12。更确切地说，我们证明了 R. Borcherds 在前一篇参考文献中引入的奇异张量积是某个函子范畴中的 2 单调范畴结构的一部分。我们还完成了前面提到的文章中的一些缺失点，尤其是奇异张量积和顶点代数本身的定义，不过这些定义在该参考文献中出现的所有例子中都得到了验证。为了证明我们的结果，将我们的研究对象置于二范畴的语言中将是非常有用的，甚至是必不可少的。我们还引入了一个比$href{https://projecteuclid.org/ebook/Download?urlId=10.2969/aspm/03110051&isFullBook=false}{[9]}$中的（量子）顶点代数稍微更一般的概念，我们称之为分类（量子）顶点代数，它享有鲍彻尔斯在那篇文章中提到的所有性质，并以该作者提出的定义为例。
{"title":"Vertex algebras and $2$-monoidal categories","authors":"Estanislao Herscovich","doi":"10.4310/mrl.2023.v30.n5.a5","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a5","url":null,"abstract":"In this article we address Problem 5.12 in $href{https://projecteuclid.org/ebook/Download?urlId=10.2969/aspm/03110051&isFullBook=false}{[9]}$. More precisely, we prove that the singular tensor product introduced by R. Borcherds in the previous reference is part of a 2-monoidal category structure in a certain category of functors. We also complete some missing points in the previously mentioned article, most notably in the definitions of singular tensor products and of vertex algebras themselves, which are however verified in all the examples appearing in that reference. To prove our results it will be extremely useful, if not essential, to frame our objects within the language of bicategories. We also introduce a slightly more general notion of (quantum) vertex algebra than the one in $href{https://projecteuclid.org/ebook/Download?urlId=10.2969/aspm/03110051&isFullBook=false}{[9]}$, that we call categorical (quantum) vertex algebra, enjoying all the properties mentioned by Borcherds in that article and having as particular example the definition presented by that author.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"29 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941416","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 : 2024-05-14DOI: 10.4310/mrl.2023.v30.n5.a11
Erik Lindgren, Peter Lindqvist
We study $infty$-Ground states in convex domains in the plane. In a polygon, the points where an $infty$-Ground state does not satisfy the $infty$-Laplace Equation are characterized: they are restricted to lie on specific curves, which are acting as attracting (fictitious) streamlines. The gradient is continuous outside these curves and no streamlines can meet there.
{"title":"On $infty$-ground states in the plane","authors":"Erik Lindgren, Peter Lindqvist","doi":"10.4310/mrl.2023.v30.n5.a11","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n5.a11","url":null,"abstract":"We study $infty$-Ground states in convex domains in the plane. In a polygon, the points where an $infty$-Ground state does not satisfy the $infty$-Laplace Equation are characterized: they are restricted to lie on specific curves, which are acting as attracting (fictitious) streamlines. The gradient is continuous outside these curves and no streamlines can meet there.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"21 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941457","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 : 2024-04-03DOI: 10.4310/mrl.2023.v30.n4.a7
Yijie Lin
We generalize the construction of a moduli space of semistable pairs parametrizing isomorphism classes of morphisms from a fixed coherent sheaf to any sheaf with fixed Hilbert polynomial under a notion of stability to the case of projective Deligne–Mumford stacks.We study the deformation and obstruction theories of stable pairs, and then prove the existence of virtual fundamental classes for some cases of dimension two and three. This leads to a definition of Pandharipande–Thomas invariants on three-dimensional smooth projective Deligne–Mumford stacks.
{"title":"Moduli spaces of semistable pairs on projective Deligne–Mumford stacks","authors":"Yijie Lin","doi":"10.4310/mrl.2023.v30.n4.a7","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n4.a7","url":null,"abstract":"We generalize the construction of a moduli space of semistable pairs parametrizing isomorphism classes of morphisms from a fixed coherent sheaf to any sheaf with fixed Hilbert polynomial under a notion of stability to the case of projective Deligne–Mumford stacks.We study the deformation and obstruction theories of stable pairs, and then prove the existence of virtual fundamental classes for some cases of dimension two and three. This leads to a definition of Pandharipande–Thomas invariants on three-dimensional smooth projective Deligne–Mumford stacks.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"30 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565776","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 : 2024-04-03DOI: 10.4310/mrl.2023.v30.n4.a6
Naoki Koseki
We investigate the stronger form of the Bogomolov–Gieseker inequality on smooth hypersurfaces in the projective space of any degree and dimension. The main technical tool is the theory of tilt-stability conditions in the derived category.
{"title":"On the Bogomolov–Gieseker inequality for hypersurfaces in the projective spaces","authors":"Naoki Koseki","doi":"10.4310/mrl.2023.v30.n4.a6","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n4.a6","url":null,"abstract":"We investigate the stronger form of the Bogomolov–Gieseker inequality on smooth hypersurfaces in the projective space of any degree and dimension. The main technical tool is the theory of tilt-stability conditions in the derived category.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"14 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565785","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 : 2024-04-03DOI: 10.4310/mrl.2023.v30.n4.a4
Marco Golla, Kyle Larson
We produce a rational homology 3‑sphere that does not smoothly bound either a positive or negative definite 4‑manifold. Such a 3‑manifold necessarily cannot be rational homology cobordant to a Seifert fibered space or any 3‑manifold obtained by Dehn surgery on a knot. The proof requires an analysis of short characteristic covectors in bimodular lattices.
{"title":"3-manifolds that bound no definite 4-manifolds","authors":"Marco Golla, Kyle Larson","doi":"10.4310/mrl.2023.v30.n4.a4","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n4.a4","url":null,"abstract":"We produce a rational homology 3‑sphere that does not smoothly bound either a positive <i>or</i> negative definite 4‑manifold. Such a 3‑manifold necessarily cannot be rational homology cobordant to a Seifert fibered space or any 3‑manifold obtained by Dehn surgery on a knot. The proof requires an analysis of short characteristic covectors in bimodular lattices.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"6 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565786","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 : 2024-04-03DOI: 10.4310/mrl.2023.v30.n4.a1
Aynur Bulut, Manh Huynh Khang
In this paper, we use frequency decomposition techniques to give a direct proof of global existence and regularity for the Navier–Stokes equations on two-dimensional Riemannian manifolds without boundary. Our techniques are inspired by an approach of Mattingly and Sinai $href{https://doi.org/10.1142/S0219199799000183}{[15]}$ which was developed in the context of periodic boundary conditions on a flat background, and which is based on a maximum principle for Fourier coefficients. The extension to general manifolds requires several new ideas, connected to the less favorable spectral localization properties in our setting. Our arguments make use of frequency projection operators, multilinear estimates that originated in the study of the non-linear Schr¨odinger equation, and ideas from microlocal analysis.
{"title":"A geometric trapping approach to global regularity for 2D Navier–Stokes on manifolds","authors":"Aynur Bulut, Manh Huynh Khang","doi":"10.4310/mrl.2023.v30.n4.a1","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n4.a1","url":null,"abstract":"In this paper, we use frequency decomposition techniques to give a direct proof of global existence and regularity for the Navier–Stokes equations on two-dimensional Riemannian manifolds without boundary. Our techniques are inspired by an approach of Mattingly and Sinai $href{https://doi.org/10.1142/S0219199799000183}{[15]}$ which was developed in the context of periodic boundary conditions on a flat background, and which is based on a maximum principle for Fourier coefficients. The extension to general manifolds requires several new ideas, connected to the less favorable spectral localization properties in our setting. Our arguments make use of frequency projection operators, multilinear estimates that originated in the study of the non-linear Schr¨odinger equation, and ideas from microlocal analysis.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"17 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565908","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 : 2024-04-03DOI: 10.4310/mrl.2023.v30.n4.a9
Fanjun Meng
We prove some vanishing and torsion-freeness results for higher direct images of adjoint pairs satisfying relative abundance and nefness conditions. These are applied to generic vanishing and weak positivity.
{"title":"On vanishing and torsion-freeness results for adjoint pairs","authors":"Fanjun Meng","doi":"10.4310/mrl.2023.v30.n4.a9","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n4.a9","url":null,"abstract":"We prove some vanishing and torsion-freeness results for higher direct images of adjoint pairs satisfying relative abundance and nefness conditions. These are applied to generic vanishing and weak positivity.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"6 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565990","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 : 2024-04-03DOI: 10.4310/mrl.2023.v30.n4.a10
Ruoyu P. T. Wang
We study the decay of global energy for the wave equation with Hölder continuous damping placed on the $C^{1,1}$-boundary of compact and non-compact waveguides with star-shaped cross-sections. We show there is sharp $t^{-1/2}$-decay when the damping is uniformly bounded from below on the cylindrical wall of product cylinders where the Geometric Control Condition is violated. On non-product cylinders, we also show that there is $t^{-1/3}$-decay when the damping is uniformly bounded from below on the cylindrical wall.
{"title":"Sharp polynomial decay for waves damped from the boundary in cylindrical waveguides","authors":"Ruoyu P. T. Wang","doi":"10.4310/mrl.2023.v30.n4.a10","DOIUrl":"https://doi.org/10.4310/mrl.2023.v30.n4.a10","url":null,"abstract":"We study the decay of global energy for the wave equation with Hölder continuous damping placed on the $C^{1,1}$-boundary of compact and non-compact waveguides with star-shaped cross-sections. We show there is sharp $t^{-1/2}$-decay when the damping is uniformly bounded from below on the cylindrical wall of product cylinders where the Geometric Control Condition is violated. On non-product cylinders, we also show that there is $t^{-1/3}$-decay when the damping is uniformly bounded from below on the cylindrical wall.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"23 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140565773","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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