We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff measure vanishes. We show that every countably $m$-rectifiable subset of $mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a Hamiltonian diffeomorphism that is arbitrarily $C^infty$-close to the identity. As a consequence, every countably $n$-rectifiable and $n$-negligible subset of $mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both results are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption. The proof of our squeezing result uses folding. Potentially, our folding method can be modified to show that the Gromov width of $B^{2n}_1setminus A$ equals $pi$ for every countably $(n-1)$-rectifiable closed subset $A$ of the open unit ball $B^{2n}_1$. This means that $A$ is not a barrier.
{"title":"Instantaneous Hamiltonian displaceability and arbitrary symplectic squeezability for critically negligible sets","authors":"Yann Guggisberg, Fabian Ziltener","doi":"arxiv-2408.17444","DOIUrl":"https://doi.org/arxiv-2408.17444","url":null,"abstract":"We call a metric space $s$-negligible iff its $s$-dimensional Hausdorff\u0000measure vanishes. We show that every countably $m$-rectifiable subset of\u0000$mathbb{R}^{2n}$ can be displaced from every $(2n-m)$-negligible subset by a\u0000Hamiltonian diffeomorphism that is arbitrarily $C^infty$-close to the\u0000identity. As a consequence, every countably $n$-rectifiable and $n$-negligible\u0000subset of $mathbb{R}^{2n}$ is arbitrarily symplectically squeezable. Both\u0000results are sharp w.r.t. the parameter $s$ in the $s$-negligibility assumption. The proof of our squeezing result uses folding. Potentially, our folding\u0000method can be modified to show that the Gromov width of $B^{2n}_1setminus A$\u0000equals $pi$ for every countably $(n-1)$-rectifiable closed subset $A$ of the\u0000open unit ball $B^{2n}_1$. This means that $A$ is not a barrier.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225932","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}
Generalizing deformation quantizations with separation of variables of a K"ahler manifold $M$, we adopt Fedosov's gluing argument to construct a category $mathsf{DQ}$, enriched over sheaves of $mathbb{C}[[hbar]]$-modules on $M$, as a quantization of the category of Hermitian holomorphic vector bundles over $M$ with morphisms being smooth sections of hom-bundles. We then define quantizable morphisms among objects in $mathsf{DQ}$, generalizing Chan-Leung-Li's notion [4] of quantizable functions. Upon evaluation of quantizable morphisms at $hbar = tfrac{sqrt{-1}}{k}$, we obtain an enriched category $mathsf{DQ}_{operatorname{qu}, k}$. We show that, when $M$ is prequantizable, $mathsf{DQ}_{operatorname{qu}, k}$ is equivalent to the category $mathsf{GQ}$ of holomorphic vector bundles over $M$ with morphisms being holomorphic differential operators, via a functor obtained from Bargmann-Fock actions.
{"title":"Categorical quantization on Kähler manifolds","authors":"YuTung Yau","doi":"arxiv-2408.17201","DOIUrl":"https://doi.org/arxiv-2408.17201","url":null,"abstract":"Generalizing deformation quantizations with separation of variables of a\u0000K\"ahler manifold $M$, we adopt Fedosov's gluing argument to construct a\u0000category $mathsf{DQ}$, enriched over sheaves of $mathbb{C}[[hbar]]$-modules\u0000on $M$, as a quantization of the category of Hermitian holomorphic vector\u0000bundles over $M$ with morphisms being smooth sections of hom-bundles. We then define quantizable morphisms among objects in $mathsf{DQ}$,\u0000generalizing Chan-Leung-Li's notion [4] of quantizable functions. Upon\u0000evaluation of quantizable morphisms at $hbar = tfrac{sqrt{-1}}{k}$, we\u0000obtain an enriched category $mathsf{DQ}_{operatorname{qu}, k}$. We show that,\u0000when $M$ is prequantizable, $mathsf{DQ}_{operatorname{qu}, k}$ is equivalent\u0000to the category $mathsf{GQ}$ of holomorphic vector bundles over $M$ with\u0000morphisms being holomorphic differential operators, via a functor obtained from\u0000Bargmann-Fock actions.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202090","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 construct a mixed Hodge structure on the topological K-theory of smooth Poisson varieties, depending weakly on a choice of compactification. We establish a package of tools for calculations with these structures, such as functoriality statements, projective bundle formulae, Gysin sequences and Torelli properties. We show that for varieties with trivial A-hat class, the corresponding period maps for families can be written as exponential maps for bundles of tori, which we call the "quantum parameters". As justification for the terminology, we show that in many interesting examples, the quantum parameters of a Poisson variety coincide with the parameters appearing in its known deformation quantizations. In particular, we give a detailed implementation of an argument of Kontsevich, to prove that his canonical quantization formula, when applied to Poisson tori, yields noncommutative tori with parameter "$q = e^hbar$".
{"title":"Semiclassical Hodge theory for log Poisson manifolds","authors":"Aidan Lindberg, Brent Pym","doi":"arxiv-2408.16685","DOIUrl":"https://doi.org/arxiv-2408.16685","url":null,"abstract":"We construct a mixed Hodge structure on the topological K-theory of smooth\u0000Poisson varieties, depending weakly on a choice of compactification. We\u0000establish a package of tools for calculations with these structures, such as\u0000functoriality statements, projective bundle formulae, Gysin sequences and\u0000Torelli properties. We show that for varieties with trivial A-hat class, the\u0000corresponding period maps for families can be written as exponential maps for\u0000bundles of tori, which we call the \"quantum parameters\". As justification for\u0000the terminology, we show that in many interesting examples, the quantum\u0000parameters of a Poisson variety coincide with the parameters appearing in its\u0000known deformation quantizations. In particular, we give a detailed\u0000implementation of an argument of Kontsevich, to prove that his canonical\u0000quantization formula, when applied to Poisson tori, yields noncommutative tori\u0000with parameter \"$q = e^hbar$\".","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202092","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 two-dimensional totally real concordances can be approximated by Lagrangian concordances after sufficiently many positive and negative stabilisations of the Legendrian boundaries. The applications of this result are the construction of knotted Lagrangian concordances, and knotted Lagrangian tori in symplectisations of overtwisted contact manifolds.
{"title":"Lagrangian Approximation of Totally Real Concordances","authors":"Georgios Dimitroglou Rizell","doi":"arxiv-2408.16614","DOIUrl":"https://doi.org/arxiv-2408.16614","url":null,"abstract":"We show that two-dimensional totally real concordances can be approximated by\u0000Lagrangian concordances after sufficiently many positive and negative\u0000stabilisations of the Legendrian boundaries. The applications of this result\u0000are the construction of knotted Lagrangian concordances, and knotted Lagrangian\u0000tori in symplectisations of overtwisted contact manifolds.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202093","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 paper, we explore scaling symmetries within the framework of symplectic geometry. We focus on the action $Phi$ of the multiplicative group $G = mathbb{R}^+$ on exact symplectic manifolds $(M, omega,theta)$, with $omega = -dtheta$, where $ theta $ is a given primitive one-form. Extending established results in symplectic geometry and Hamiltonian dynamics, we introduce conformally symplectic maps, conformally Hamiltonian systems, conformally symplectic group actions, and the notion of conformal invariance. This framework allows us to generalize the momentum map to the conformal momentum map, which is crucial for understanding scaling symmetries. Additionally, we provide a generalized Hamiltonian Noether's theorem for these symmetries. We introduce the (conformal) augmented Hamiltonian $H_{xi}$ and prove that the relative equilibria of scaling symmetries are solutions to equations involving $ H _{ xi } $ and the primitive one-form $theta$. We derive their main properties, emphasizing the differences from relative equilibria in traditional symplectic actions. For cotangent bundles, we define a scaled cotangent lifted action and derive explicit formulas for the conformal momentum map. We also provide a general definition of central configurations for Hamiltonian systems on cotangent bundles that admit scaling symmetries. Applying these results to simple mechanical systems, we introduce the augmented potential $U_{xi}$ and show that the relative equilibria of scaling symmetries are solutions to an equation involving $ U _{ xi } $ and the Lagrangian one-form $theta_L$. Finally, we apply our general theory to the Newtonian $n$-body problem, recovering the classical equations for central configurations.
在本文中，我们将在折射几何的框架内探索缩放对称性。我们聚焦于乘法组$G = mathbb{R}^+$在精确交映流形$(M, omega,theta)$上的作用$Phi$，其中$omega = -dtheta$是一个给定的原始单形式。通过扩展交映几何学和哈密顿动力学的既定结果，我们引入了共形交映映射、共形哈密顿系统、共形交映群作用以及共形不变性概念。这个框架使我们能够将动量映射推广到共形动量映射，这对于理解缩放对称性至关重要。我们引入了（共形）增强哈密顿方程 $H_{xi}$，并证明了缩放对称性的相对平衡是涉及 $ H _{ xi }$ 和原始单形式 $H _{ xi }$ 的方程的解。和原始单形式 $theta$ 的方程。我们推导了它们的主要性质，强调了它们与传统交映作用中的相对均衡的区别。对于余切束，我们定义了尺度余切提升作用，并推导了共形动量映射的明确公式。我们还为承认缩放对称性的余切束上的哈密顿系统提供了中心构型的一般定义。将这些结果应用于简单机械系统，我们引入了增强势 $U_{xi}$，并证明了缩放对称的相对平衡是涉及 $ U _{ xi }$ 和拉格朗日方程的解。和拉格朗日单形式 $theta_L$ 的方程的解。最后，我们将我们的一般理论应用于牛顿$n$体问题，恢复了中心构型的经典方程。
{"title":"Relative Equilibria for Scaling Symmetries and Central Configurations","authors":"Giovanni Rastelli, Manuele Santoprete","doi":"arxiv-2408.15191","DOIUrl":"https://doi.org/arxiv-2408.15191","url":null,"abstract":"In this paper, we explore scaling symmetries within the framework of\u0000symplectic geometry. We focus on the action $Phi$ of the multiplicative group\u0000$G = mathbb{R}^+$ on exact symplectic manifolds $(M, omega,theta)$, with\u0000$omega = -dtheta$, where $ theta $ is a given primitive one-form. Extending\u0000established results in symplectic geometry and Hamiltonian dynamics, we\u0000introduce conformally symplectic maps, conformally Hamiltonian systems,\u0000conformally symplectic group actions, and the notion of conformal invariance.\u0000This framework allows us to generalize the momentum map to the conformal\u0000momentum map, which is crucial for understanding scaling symmetries.\u0000Additionally, we provide a generalized Hamiltonian Noether's theorem for these\u0000symmetries. We introduce the (conformal) augmented Hamiltonian $H_{xi}$ and prove that\u0000the relative equilibria of scaling symmetries are solutions to equations\u0000involving $ H _{ xi } $ and the primitive one-form $theta$. We derive their\u0000main properties, emphasizing the differences from relative equilibria in\u0000traditional symplectic actions. For cotangent bundles, we define a scaled cotangent lifted action and derive\u0000explicit formulas for the conformal momentum map. We also provide a general\u0000definition of central configurations for Hamiltonian systems on cotangent\u0000bundles that admit scaling symmetries. Applying these results to simple\u0000mechanical systems, we introduce the augmented potential $U_{xi}$ and show\u0000that the relative equilibria of scaling symmetries are solutions to an equation\u0000involving $ U _{ xi } $ and the Lagrangian one-form $theta_L$. Finally, we apply our general theory to the Newtonian $n$-body problem,\u0000recovering the classical equations for central configurations.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202095","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 paper, we introduce a broad class of phenomena which appear when you intersect a given Lagrangian submanifold $K$ with a family of Lagrangian submanifolds $L_t$ (all Hamiltonian isotopic to one another). We establish that this phenomenon occurs in a particular situation, which lets us give a lower bound for the volume of any Lagrangian torus in $mathbb{CP}^2$ which is Hamiltonian isotopic to the Chekanov torus. The rest of the paper is a discussion of why we should expect these phenomena to be very common, motivated by Oh's conjecture on the volume-minimising property of the Clifford torus and the concurrent normals conjecture in convex geometry. We pose many open questions.
{"title":"Lagrangian surplusection phenomena","authors":"Georgios Dimitroglou Rizell, Jonathan David Evans","doi":"arxiv-2408.14883","DOIUrl":"https://doi.org/arxiv-2408.14883","url":null,"abstract":"In this paper, we introduce a broad class of phenomena which appear when you\u0000intersect a given Lagrangian submanifold $K$ with a family of Lagrangian\u0000submanifolds $L_t$ (all Hamiltonian isotopic to one another). We establish that\u0000this phenomenon occurs in a particular situation, which lets us give a lower\u0000bound for the volume of any Lagrangian torus in $mathbb{CP}^2$ which is\u0000Hamiltonian isotopic to the Chekanov torus. The rest of the paper is a\u0000discussion of why we should expect these phenomena to be very common, motivated\u0000by Oh's conjecture on the volume-minimising property of the Clifford torus and\u0000the concurrent normals conjecture in convex geometry. We pose many open\u0000questions.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202096","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}
Given an anticanonical divisor in a projective variety, one naturally obtains a monotone K"ahler manifold. In this paper, for divisors in a certain class (larger than normal crossings), we construct smoothing families of contact hypersurfaces with controlled Reeb dynamics. We use these to adapt arguments of Borman, Sheridan and Varolgunes to obtain analogous results about symplectic cohomology with supports in the divisor complement. In particular, we will show that several examples of Lagrangian skeleta of such divisor complements are superheavy, in cases where applying Lagrangian Floer theory may be intractable.
{"title":"Superheavy Skeleta for non-Normal Crossings Divisors","authors":"Elliot Gathercole","doi":"arxiv-2408.13187","DOIUrl":"https://doi.org/arxiv-2408.13187","url":null,"abstract":"Given an anticanonical divisor in a projective variety, one naturally obtains\u0000a monotone K\"ahler manifold. In this paper, for divisors in a certain class\u0000(larger than normal crossings), we construct smoothing families of contact\u0000hypersurfaces with controlled Reeb dynamics. We use these to adapt arguments of\u0000Borman, Sheridan and Varolgunes to obtain analogous results about symplectic\u0000cohomology with supports in the divisor complement. In particular, we will show\u0000that several examples of Lagrangian skeleta of such divisor complements are\u0000superheavy, in cases where applying Lagrangian Floer theory may be intractable.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202097","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 develop an affine scheme-theoretic version of Hamiltonian reduction by symplectic groupoids. It works over $Bbbk=mathbb{R}$ or $Bbbk=mathbb{C}$, and is formulated for an affine symplectic groupoid $mathcal{G}rightrightarrows X$, an affine Hamiltonian $mathcal{G}$-scheme $mu:Mlongrightarrow X$, a coisotropic subvariety $Ssubseteq X$, and a stabilizer subgroupoid $mathcal{H}rightrightarrows S$. Our first main result is that the Poisson bracket on $Bbbk[M]$ induces a Poisson bracket on the subquotient $Bbbk[mu^{-1}(S)]^{mathcal{H}}$. The Poisson scheme $mathrm{Spec}(Bbbk[mu^{-1}(S)]^{mathcal{H}})$ is then declared to be a Hamiltonian reduction of $M$. Other main results include sufficient conditions for $mathrm{Spec}(Bbbk[mu^{-1}(S)]^{mathcal{H}})$ to inherit a residual Hamiltonian scheme structure. Our main results are best viewed as affine scheme-theoretic counterparts to an earlier paper, where we simultaneously generalize several Hamiltonian reduction processes. In this way, the present work yields scheme-theoretic analogues of Marsden-Ratiu reduction, Mikami-Weinstein reduction, '{S}niatycki-Weinstein reduction, and symplectic reduction along general coisotropic submanifolds. The initial impetus for this work was its utility in formulating and proving generalizations of the Moore-Tachikawa conjecture.
我们开发了一种仿射方案理论版的交映群体哈密顿还原法。它适用于$Bbbk=mathbb{R}$或$Bbbk=mathbb{C}$，并针对仿交映群元$mathcal{G}rightrightarrows X$、仿哈密顿$mathcal{G}$-scheme$mu：X$, a coisotropic subvariety $Ssubseteq X$, and astabilizer subgroupoid $mathcal{H}rightrightarrows S$.我们的第一个主要结果是，$Bbbk[M]$ 上的泊松括号会在子集$Bbbk[mu^{-1}(S)]^{mathcal{H}}$ 上引起泊松括号。然后宣布泊松方案$mathrm{Spec}(Bbbk[mu^{-1}(S)]^{mathcal{H}})$ 是$M$ 的哈密顿还原。其他主要结果包括$mathrm{Spec}(Bbbk[mu^{-1}(S)]^{mathcal{H}})$继承残余哈密顿方案结构的充分条件。我们的主要结果最好被视为早先论文的仿射方案理论对应物，在这篇论文中，我们同时归纳了几个哈密顿还原过程。通过这种方式，本研究产生了马斯登-拉蒂乌还原、米卡米-韦恩斯坦还原、尼亚茨基-韦恩斯坦还原以及沿着一般各向异性子满的交点还原的方案理论模拟。这项工作的最初推动力是它对摩尔-立川猜想的广义化和证明的实用性。
{"title":"Scheme-theoretic coisotropic reduction","authors":"Peter Crooks, Maxence Mayrand","doi":"arxiv-2408.11932","DOIUrl":"https://doi.org/arxiv-2408.11932","url":null,"abstract":"We develop an affine scheme-theoretic version of Hamiltonian reduction by\u0000symplectic groupoids. It works over $Bbbk=mathbb{R}$ or $Bbbk=mathbb{C}$,\u0000and is formulated for an affine symplectic groupoid\u0000$mathcal{G}rightrightarrows X$, an affine Hamiltonian $mathcal{G}$-scheme\u0000$mu:Mlongrightarrow X$, a coisotropic subvariety $Ssubseteq X$, and a\u0000stabilizer subgroupoid $mathcal{H}rightrightarrows S$. Our first main result\u0000is that the Poisson bracket on $Bbbk[M]$ induces a Poisson bracket on the\u0000subquotient $Bbbk[mu^{-1}(S)]^{mathcal{H}}$. The Poisson scheme\u0000$mathrm{Spec}(Bbbk[mu^{-1}(S)]^{mathcal{H}})$ is then declared to be a\u0000Hamiltonian reduction of $M$. Other main results include sufficient conditions\u0000for $mathrm{Spec}(Bbbk[mu^{-1}(S)]^{mathcal{H}})$ to inherit a residual\u0000Hamiltonian scheme structure. Our main results are best viewed as affine scheme-theoretic counterparts to\u0000an earlier paper, where we simultaneously generalize several Hamiltonian\u0000reduction processes. In this way, the present work yields scheme-theoretic\u0000analogues of Marsden-Ratiu reduction, Mikami-Weinstein reduction,\u0000'{S}niatycki-Weinstein reduction, and symplectic reduction along general\u0000coisotropic submanifolds. The initial impetus for this work was its utility in\u0000formulating and proving generalizations of the Moore-Tachikawa conjecture.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202098","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 employ the curve shortening flow to establish three new theorems on the dynamics of geodesic flows of closed Riemannian surfaces. The first one is the stability, under $C^0$-small perturbations of the Riemannian metric, of certain flat links of closed geodesics. The second one is a forced existence theorem for orientable closed Riemannian surfaces of positive genus, asserting that the existence of a contractible simple closed geodesic $gamma$ forces the existence of infinitely many closed geodesics intersecting $gamma$ in every primitive free homotopy class of loops. The third theorem asserts the existence of Birkhoff sections for the geodesic flow of any closed orientable Riemannian surface of positive genus.
{"title":"From curve shortening to flat link stability and Birkhoff sections of geodesic flows","authors":"Marcelo R. R. Alves, Marco Mazzucchelli","doi":"arxiv-2408.11938","DOIUrl":"https://doi.org/arxiv-2408.11938","url":null,"abstract":"We employ the curve shortening flow to establish three new theorems on the\u0000dynamics of geodesic flows of closed Riemannian surfaces. The first one is the\u0000stability, under $C^0$-small perturbations of the Riemannian metric, of certain\u0000flat links of closed geodesics. The second one is a forced existence theorem\u0000for orientable closed Riemannian surfaces of positive genus, asserting that the\u0000existence of a contractible simple closed geodesic $gamma$ forces the\u0000existence of infinitely many closed geodesics intersecting $gamma$ in every\u0000primitive free homotopy class of loops. The third theorem asserts the existence\u0000of Birkhoff sections for the geodesic flow of any closed orientable Riemannian\u0000surface of positive genus.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202099","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}
{"title":"Unknotting Lagrangian $mathrm{S}^1timesmathrm{S}^{n-1}$ in $mathbb{R}^{2n}$","authors":"Stefan Nemirovski","doi":"arxiv-2408.10916","DOIUrl":"https://doi.org/arxiv-2408.10916","url":null,"abstract":"Lagrangian embeddings\u0000$mathrm{S}^1timesmathrm{S}^{n-1}hookrightarrowmathbb{R}^{2n}$ are\u0000classified up to smooth isotopy for all $nge 3$.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202100","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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