It is well-known that the planar and spatial circular restricted three-body problem (CR3BP) is of contact type for all energy values below the first critical value. Burgos-Garc'ia and Gidea extended Hill's approach in the CR3BP to the spatial equilateral CR4BP, which can be used to approximate the dynamics of a small body near a Trojan asteroid of a Sun--planet system. Our main result in this paper is that this Hill four-body system also has the contact property. In other words, we can "contact" the Trojan. Such a result enables to use holomorphic curve techniques and Floer theoretical tools in this dynamical system in the energy range where the contact property holds.
{"title":"Contact geometry of Hill's approximation in a spatial restricted four-body problem","authors":"Cengiz Aydin","doi":"arxiv-2407.06927","DOIUrl":"https://doi.org/arxiv-2407.06927","url":null,"abstract":"It is well-known that the planar and spatial circular restricted three-body\u0000problem (CR3BP) is of contact type for all energy values below the first\u0000critical value. Burgos-Garc'ia and Gidea extended Hill's approach in the CR3BP\u0000to the spatial equilateral CR4BP, which can be used to approximate the dynamics\u0000of a small body near a Trojan asteroid of a Sun--planet system. Our main result\u0000in this paper is that this Hill four-body system also has the contact property.\u0000In other words, we can \"contact\" the Trojan. Such a result enables to use\u0000holomorphic curve techniques and Floer theoretical tools in this dynamical\u0000system in the energy range where the contact property holds.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this article is to study operators whose kernel share some key features of Bergman kernels from complex analysis, and are approximate projectors. It turns out that they must be associated with a rich set of geometric data, on the one hand, and that on the other hand, all such operators can be locally conjugated in some sense.
{"title":"Microlocal Projectors","authors":"Yannick Guedes Bonthonneau","doi":"arxiv-2407.06644","DOIUrl":"https://doi.org/arxiv-2407.06644","url":null,"abstract":"The purpose of this article is to study operators whose kernel share some key\u0000features of Bergman kernels from complex analysis, and are approximate\u0000projectors. It turns out that they must be associated with a rich set of\u0000geometric data, on the one hand, and that on the other hand, all such operators\u0000can be locally conjugated in some sense.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the universal covering space of a connected component of a regular level set of a smooth complex valued function on ${mathbb{C}}^2$, which is a smooth affine Riemann surface, is ${mathbb{R}}^2$. This implies that the orbit space of the action of the covering group on ${mathbb{R}}^2$ is the original affine Riemann surface.
{"title":"On affine Riemann surfaces","authors":"Richard Cushman","doi":"arxiv-2407.06332","DOIUrl":"https://doi.org/arxiv-2407.06332","url":null,"abstract":"We show that the universal covering space of a connected component of a\u0000regular level set of a smooth complex valued function on ${mathbb{C}}^2$,\u0000which is a smooth affine Riemann surface, is ${mathbb{R}}^2$. This implies\u0000that the orbit space of the action of the covering group on ${mathbb{R}}^2$ is\u0000the original affine Riemann surface.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove, over any base ring, that the infinity-category of strictly unital A-infinity-categories (and strictly unital functors) is equivalent to the infinity-category of unital A-infinity-categories (and unital functors). We also identify various models for internal homs and mapping spaces in the infinity-categories of dg-categories and of A-infinity--categories, generalizing results of To"en and Faonte.
我们证明，在任何基环上，严格无素 A 无穷范畴（和严格无素函子）的无穷范畴等价于无素 A 无穷范畴（和无素函子）的无穷范畴。我们还识别了dg-范畴的无穷范畴和A-无穷范畴中的内部原子和映射空间的各种模型，概括了To"en 和 Faonte的结果。
{"title":"Unitalities and mapping spaces in $A_infty$-categories","authors":"Hiro Lee Tanaka","doi":"arxiv-2407.05532","DOIUrl":"https://doi.org/arxiv-2407.05532","url":null,"abstract":"We prove, over any base ring, that the infinity-category of strictly unital\u0000A-infinity-categories (and strictly unital functors) is equivalent to the\u0000infinity-category of unital A-infinity-categories (and unital functors). We\u0000also identify various models for internal homs and mapping spaces in the\u0000infinity-categories of dg-categories and of A-infinity--categories,\u0000generalizing results of To\"en and Faonte.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain topological obstructions to the existence of a complete Riemannian metric with uniformly positive scalar curvature on certain (non-compact) $4$-manifolds. In particular, such a metric on the interior of a compact contractible $4$-manifold uniquely distinguishes the standard $4$-ball up to diffeomorphism among Mazur manifolds and up to homeomorphism in general. We additionally show there exist uncountably many exotic $mathbb{R}^4$'s that do not admit such a metric and that any (non-compact) tame $4$-manifold has a smooth structure that does not admit such a metric.
{"title":"Complete Riemannian 4-manifolds with uniformly positive scalar curvature","authors":"Otis Chodosh, Davi Maximo, Anubhav Mukherjee","doi":"arxiv-2407.05574","DOIUrl":"https://doi.org/arxiv-2407.05574","url":null,"abstract":"We obtain topological obstructions to the existence of a complete Riemannian\u0000metric with uniformly positive scalar curvature on certain (non-compact)\u0000$4$-manifolds. In particular, such a metric on the interior of a compact\u0000contractible $4$-manifold uniquely distinguishes the standard $4$-ball up to\u0000diffeomorphism among Mazur manifolds and up to homeomorphism in general. We additionally show there exist uncountably many exotic $mathbb{R}^4$'s\u0000that do not admit such a metric and that any (non-compact) tame $4$-manifold\u0000has a smooth structure that does not admit such a metric.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"2016 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we explore the possibility of the existence of Kirby move of type 1 for contact surgery diagrams. In particular, we give the necessary conditions on a contact surgery diagram to become a potential candidate for contact Kirby move of type 1. We prove that no other contact integral surgery diagram satisfies those conditions except for contact $(+2)$-surgery on Legendrian unknot with Thruston--Bennequin number $-1$.
{"title":"On A Potential Contact Analogue Of Kirby Move Of Type 1","authors":"Prerak Deep, Dheeraj Kulkarni","doi":"arxiv-2407.04395","DOIUrl":"https://doi.org/arxiv-2407.04395","url":null,"abstract":"In this note, we explore the possibility of the existence of Kirby move of\u0000type 1 for contact surgery diagrams. In particular, we give the necessary\u0000conditions on a contact surgery diagram to become a potential candidate for\u0000contact Kirby move of type 1. We prove that no other contact integral surgery\u0000diagram satisfies those conditions except for contact $(+2)$-surgery on\u0000Legendrian unknot with Thruston--Bennequin number $-1$.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that, on any closed manifold of dimension at least two with non-trivial first Betti number, a $C^infty$ generic Riemannian metric has infinitely many closed geodesics, and indeed closed geodesics of arbitrarily large length. We derive this existence result combining a theorem of Ma~n'e together with the following new theorem of independent interest: the existence of minimal closed geodesics, in the sense of Aubry-Mather theory, implies the existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic flow of a suitable $C^infty$-close Riemannian metric.
{"title":"Closed geodesics and the first Betti number","authors":"Gonzalo Contreras, Marco Mazzucchelli","doi":"arxiv-2407.02995","DOIUrl":"https://doi.org/arxiv-2407.02995","url":null,"abstract":"We prove that, on any closed manifold of dimension at least two with\u0000non-trivial first Betti number, a $C^infty$ generic Riemannian metric has\u0000infinitely many closed geodesics, and indeed closed geodesics of arbitrarily\u0000large length. We derive this existence result combining a theorem of Ma~n'e\u0000together with the following new theorem of independent interest: the existence\u0000of minimal closed geodesics, in the sense of Aubry-Mather theory, implies the\u0000existence of a transverse homoclinic, and thus of a horseshoe, for the geodesic\u0000flow of a suitable $C^infty$-close Riemannian metric.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141551152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe an obstruction to smoothing stable maps in smooth projective varieties, which generalizes some previously known obstructions. Our obstruction comes from the non-existence of certain rational functions on the ghost components, with prescribed simple poles and residues.
{"title":"An obstruction to smoothing stable maps","authors":"Fatemeh Rezaee, Mohan Swaminathan","doi":"arxiv-2407.01845","DOIUrl":"https://doi.org/arxiv-2407.01845","url":null,"abstract":"We describe an obstruction to smoothing stable maps in smooth projective\u0000varieties, which generalizes some previously known obstructions. Our\u0000obstruction comes from the non-existence of certain rational functions on the\u0000ghost components, with prescribed simple poles and residues.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529753","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We push forward the study of higher dimensional stable Hamiltonian topology by establishing two non-density results. First, we prove that stable hypersurfaces are not $C^2$-dense in any isotopy class of embedded hypersurfaces on any ambient symplectic manifold of dimension $2ngeq 8$. Our second result is that on any manifold of dimension $2m+1geq 5$, the set of non-degenerate stable Hamiltonian structures is not $C^2$-dense among stable Hamiltonian structures in any given stable homotopy class that satisfies a mild assumption. The latter generalizes a result by Cieliebak and Volkov to arbitrary dimensions.
{"title":"Non-density results in high dimensional stable Hamiltonian topology","authors":"Robert Cardona, Fabio Gironella","doi":"arxiv-2407.01357","DOIUrl":"https://doi.org/arxiv-2407.01357","url":null,"abstract":"We push forward the study of higher dimensional stable Hamiltonian topology\u0000by establishing two non-density results. First, we prove that stable\u0000hypersurfaces are not $C^2$-dense in any isotopy class of embedded\u0000hypersurfaces on any ambient symplectic manifold of dimension $2ngeq 8$. Our\u0000second result is that on any manifold of dimension $2m+1geq 5$, the set of\u0000non-degenerate stable Hamiltonian structures is not $C^2$-dense among stable\u0000Hamiltonian structures in any given stable homotopy class that satisfies a mild\u0000assumption. The latter generalizes a result by Cieliebak and Volkov to\u0000arbitrary dimensions.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For a given closed two-form, we introduce the cone Yang-Mills functional which is a Yang-Mills-type functional for a pair $(A,B)$, a connection one-form $A$ and a scalar $B$ taking value in the adjoint representation of a Lie group. The functional arises naturally from dimensionally reducing the Yang-Mills functional over the fiber of a circle bundle with the two-form being the Euler class. We write down the Euler-Lagrange equations of the functional and present some of the properties of its critical solutions, especially in comparison with Yang-Mills solutions. We show that a special class of three-dimensional solutions satisfy a duality condition which generalizes the Bogomolny monopole equations. Moreover, we analyze the zero solutions of the cone Yang-Mills functional and give an algebraic classification characterizing principal bundles that carry such cone-flat solutions when the two-form is non-degenerate.
{"title":"Mapping Cone Connections and their Yang-Mills Functional","authors":"Li-Sheng Tseng, Jiawei Zhou","doi":"arxiv-2407.01508","DOIUrl":"https://doi.org/arxiv-2407.01508","url":null,"abstract":"For a given closed two-form, we introduce the cone Yang-Mills functional\u0000which is a Yang-Mills-type functional for a pair $(A,B)$, a connection one-form\u0000$A$ and a scalar $B$ taking value in the adjoint representation of a Lie group.\u0000The functional arises naturally from dimensionally reducing the Yang-Mills\u0000functional over the fiber of a circle bundle with the two-form being the Euler\u0000class. We write down the Euler-Lagrange equations of the functional and present\u0000some of the properties of its critical solutions, especially in comparison with\u0000Yang-Mills solutions. We show that a special class of three-dimensional\u0000solutions satisfy a duality condition which generalizes the Bogomolny monopole\u0000equations. Moreover, we analyze the zero solutions of the cone Yang-Mills\u0000functional and give an algebraic classification characterizing principal\u0000bundles that carry such cone-flat solutions when the two-form is\u0000non-degenerate.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141508418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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