Pub Date : 2018-05-24DOI: 10.4310/jsg.2020.v18.n4.a6
R. Maccheroni
In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not admit fillings by holomorphic discs. The proof relies on a mix of holomorphic curve techniques and on certain convexity results.
{"title":"Complex analytic properties of minimal Lagrangian submanifolds","authors":"R. Maccheroni","doi":"10.4310/jsg.2020.v18.n4.a6","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n4.a6","url":null,"abstract":"In this article we study complex properties of minimal Lagrangian submanifolds in Kaehler ambient spaces, and how they depend on the ambient curvature. In particular, we prove that, in the negative curvature case, minimal Lagrangians do not admit fillings by holomorphic discs. The proof relies on a mix of holomorphic curve techniques and on certain convexity results.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73158464","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 : 2018-05-03DOI: 10.4310/jsg.2022.v20.n5.a5
S. Muller
An embedding $varphi colon (M_1, omega_1) to (M_2, omega_2)$ (of symplectic manifolds of the same dimension) is called $epsilon$-symplectic if the difference $varphi^* omega_2 - omega_1$ is $epsilon$-small with respect to a fixed Riemannian metric on $M_1$. We prove that if a sequence of $epsilon$-symplectic embeddings converges uniformly (on compact subsets) to another embedding, then the limit is $E$-symplectic, where the number $E$ depends only on $epsilon$ and $E (epsilon) to 0$ as $epsilon to 0$. This generalizes $C^0$-rigidity of symplectic embeddings, and answers a question in topological quantum computing by Michael Freedman. As in the symplectic case, this rigidity theorem can be deduced from the existence and properties of symplectic capacities. An $epsilon$-symplectic embedding preserves capacity up to an $epsilon$-small error, and linear $epsilon$-symplectic maps can be characterized by the property that they preserve the symplectic spectrum of ellipsoids (centered at the origin) up to an error that is $epsilon$-small. We sketch an alternative proof using the shape invariant, which gives rise to an analogous characterization and rigidity theorem for $epsilon$-contact embeddings.
如果与$M_1$上的固定黎曼度规的差$varphi^* omega_2 - omega_1$为$epsilon$ -小，则嵌入$varphi colon (M_1, omega_1) to (M_2, omega_2)$(相同维数的辛流形)称为$epsilon$ -辛。我们证明了如果一个$epsilon$ -辛嵌入序列(在紧子集上)一致收敛到另一个嵌入，那么极限是$E$ -辛的，其中$E$只依赖于$epsilon$和$E (epsilon) to 0$作为$epsilon to 0$。这概括了辛嵌入的$C^0$ -刚性，并回答了Michael Freedman在拓扑量子计算中的一个问题。在辛的情况下，刚性定理可以从辛容量的存在性和性质中推导出来。$epsilon$ -辛嵌入将容量保留到$epsilon$ -小误差，线性$epsilon$ -辛映射的特征是它们保留椭球(以原点为中心)的辛谱，误差为$epsilon$ -小。我们用形状不变量勾画了另一种证明，它产生了$epsilon$ -接触嵌入的类似表征和刚性定理。
{"title":"Epsilon-non-squeezing and $C^0$-rigidity of epsilon-symplectic embeddings","authors":"S. Muller","doi":"10.4310/jsg.2022.v20.n5.a5","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n5.a5","url":null,"abstract":"An embedding $varphi colon (M_1, omega_1) to (M_2, omega_2)$ (of symplectic manifolds of the same dimension) is called $epsilon$-symplectic if the difference $varphi^* omega_2 - omega_1$ is $epsilon$-small with respect to a fixed Riemannian metric on $M_1$. We prove that if a sequence of $epsilon$-symplectic embeddings converges uniformly (on compact subsets) to another embedding, then the limit is $E$-symplectic, where the number $E$ depends only on $epsilon$ and $E (epsilon) to 0$ as $epsilon to 0$. This generalizes $C^0$-rigidity of symplectic embeddings, and answers a question in topological quantum computing by Michael Freedman. As in the symplectic case, this rigidity theorem can be deduced from the existence and properties of symplectic capacities. An $epsilon$-symplectic embedding preserves capacity up to an $epsilon$-small error, and linear $epsilon$-symplectic maps can be characterized by the property that they preserve the symplectic spectrum of ellipsoids (centered at the origin) up to an error that is $epsilon$-small. We sketch an alternative proof using the shape invariant, which gives rise to an analogous characterization and rigidity theorem for $epsilon$-contact embeddings.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77865947","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 : 2018-05-02DOI: 10.4310/jsg.2020.v18.n4.a3
Honghao Gao
We classify the simple sheaves microsupported along the conormal bundle of a knot. We also establish a correspondence between simple sheaves up to local systems and augmentations, explaining the underlying reason why knot contact homology representations detect augmentations.
{"title":"Simple sheaves for knot conormals","authors":"Honghao Gao","doi":"10.4310/jsg.2020.v18.n4.a3","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n4.a3","url":null,"abstract":"We classify the simple sheaves microsupported along the conormal bundle of a knot. We also establish a correspondence between simple sheaves up to local systems and augmentations, explaining the underlying reason why knot contact homology representations detect augmentations.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86785102","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 : 2018-04-17DOI: 10.4310/JSG.2020.V18.N2.A2
P. Bieliavsky, C. Esposito, R. Nest
In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfel'd twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups compatible with the 2-cocycle stucture and we discuss a concrete example. This allows us to construct, out of the classical momentum map, a quantum momentum map in the setting of Hopf coactions and to quantize it by using Drinfel'd approach.
{"title":"Quantization of Hamiltonian coactions via twist","authors":"P. Bieliavsky, C. Esposito, R. Nest","doi":"10.4310/JSG.2020.V18.N2.A2","DOIUrl":"https://doi.org/10.4310/JSG.2020.V18.N2.A2","url":null,"abstract":"In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfel'd twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups compatible with the 2-cocycle stucture and we discuss a concrete example. This allows us to construct, out of the classical momentum map, a quantum momentum map in the setting of Hopf coactions and to quantize it by using Drinfel'd approach.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87624886","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 : 2018-04-11DOI: 10.4310/JSG.2021.v19.n1.a4
Douglas Schultz
We consider a fibered Lagrangian $L$ in a compact symplectic fibration with small monotone fibers, and develop a strategy for lifting $J$-holomorphic disks with Lagrangian boundary from the base to the total space. In case $L$ is a product, we use this machinery to give a formula for the leading order potential and formulate an unobstructedness criteria for the $A_infty$ algebra. We provide some explicit computations, one of which involves finding an embedded 2n+k dimensional submanifold of Floer-non-trivial tori in an 2n+2k dimensional fiber bundle.
{"title":"Holomorphic disks and the disk potential for a fibered Lagrangian","authors":"Douglas Schultz","doi":"10.4310/JSG.2021.v19.n1.a4","DOIUrl":"https://doi.org/10.4310/JSG.2021.v19.n1.a4","url":null,"abstract":"We consider a fibered Lagrangian $L$ in a compact symplectic fibration with small monotone fibers, and develop a strategy for lifting $J$-holomorphic disks with Lagrangian boundary from the base to the total space. In case $L$ is a product, we use this machinery to give a formula for the leading order potential and formulate an unobstructedness criteria for the $A_infty$ algebra. We provide some explicit computations, one of which involves finding an embedded 2n+k dimensional submanifold of Floer-non-trivial tori in an 2n+2k dimensional fiber bundle.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78581615","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 : 2018-03-19DOI: 10.4310/jsg.2020.v18.n1.a9
Alexandre Vérine
We prove that every closed Bohr-Sommerfeld Lagrangian submanifold $Q$ of a symplectic/K"ahler manifold $X$ can be realised as a Morse-Bott minimum for some 'convex' exhausting function defined in the complement of a symplectic/complex hyperplane section $Y$. In the K"ahler case, 'convex' means strictly plurisubharmonic while, in the symplectic case, it refers to the existence of a Liouville pseudogradient. In particular, $Qsubset Xsetminus Y$ is a regular Lagrangian submanifold in the sense of Eliashberg-Ganatra-Lazarev.
{"title":"Bohr–Sommerfeld Lagrangian submanifolds as minima of convex functions","authors":"Alexandre Vérine","doi":"10.4310/jsg.2020.v18.n1.a9","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n1.a9","url":null,"abstract":"We prove that every closed Bohr-Sommerfeld Lagrangian submanifold $Q$ of a symplectic/K\"ahler manifold $X$ can be realised as a Morse-Bott minimum for some 'convex' exhausting function defined in the complement of a symplectic/complex hyperplane section $Y$. In the K\"ahler case, 'convex' means strictly plurisubharmonic while, in the symplectic case, it refers to the existence of a Liouville pseudogradient. In particular, $Qsubset Xsetminus Y$ is a regular Lagrangian submanifold in the sense of Eliashberg-Ganatra-Lazarev.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72633027","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 : 2018-03-10DOI: 10.4310/JSG.2021.v19.n2.a2
U. Frauenfelder, Joa Weber
In this article we give a uniform proof why the shift map on Floer homology trajectory spaces is scale smooth. This proof works for various Floer homologies, periodic, Lagrangian, Hyperk"ahler, elliptic or parabolic, and uses Hilbert space valued Sobolev theory.
{"title":"The shift map on Floer trajectory spaces","authors":"U. Frauenfelder, Joa Weber","doi":"10.4310/JSG.2021.v19.n2.a2","DOIUrl":"https://doi.org/10.4310/JSG.2021.v19.n2.a2","url":null,"abstract":"In this article we give a uniform proof why the shift map on Floer homology trajectory spaces is scale smooth. This proof works for various Floer homologies, periodic, Lagrangian, Hyperk\"ahler, elliptic or parabolic, and uses Hilbert space valued Sobolev theory.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86250147","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 : 2018-02-28DOI: 10.4310/jsg.2020.v18.n1.a3
A. Futaki, Hajime Ono
We show that if a compact Kaehler manifold $M$ admits closed Fedosov's star product then the reduced Lie algebra of holomorphic vector fields on $M$ is reductive. This comes in pair with the obstruction previously found by La Fuente-Gravy. More generally we consider the squared norm of Cahen-Gutt moment map as in the same spirit of Calabi functional for the scalar curvature in cscK problem, and prove a Cahen-Gutt version of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kaehler manifolds. The proof uses a Hessian formula for the squared norm of Cahen-Gutt moment map.
{"title":"Cahen–Gutt moment map, closed Fedosov star product and structure of the automorphism group","authors":"A. Futaki, Hajime Ono","doi":"10.4310/jsg.2020.v18.n1.a3","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n1.a3","url":null,"abstract":"We show that if a compact Kaehler manifold $M$ admits closed Fedosov's star product then the reduced Lie algebra of holomorphic vector fields on $M$ is reductive. This comes in pair with the obstruction previously found by La Fuente-Gravy. More generally we consider the squared norm of Cahen-Gutt moment map as in the same spirit of Calabi functional for the scalar curvature in cscK problem, and prove a Cahen-Gutt version of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kaehler manifolds. The proof uses a Hessian formula for the squared norm of Cahen-Gutt moment map.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73448156","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 : 2018-02-27DOI: 10.4310/jsg.2020.v18.n2.a1
D. Barilari, I. Beschastnyi, A. Lerário
We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure
{"title":"Volume of small balls and sub-Riemannian curvature in 3D contact manifolds","authors":"D. Barilari, I. Beschastnyi, A. Lerário","doi":"10.4310/jsg.2020.v18.n2.a1","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n2.a1","url":null,"abstract":"We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78655718","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 : 2018-02-25DOI: 10.4310/JSG.2021.V19.N1.A2
Alexandru Doicu, Urs Fuchs
$mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. In this paper we compactify the moduli space of $mathcal{H}-$holomorphic curves with a priori bounds on the harmonic $1-$forms.
{"title":"A compactness result for $mathcal{H}$‑holomorphic curves in symplectizations","authors":"Alexandru Doicu, Urs Fuchs","doi":"10.4310/JSG.2021.V19.N1.A2","DOIUrl":"https://doi.org/10.4310/JSG.2021.V19.N1.A2","url":null,"abstract":"$mathcal{H}-$holomorphic curves are solutions of a specific modification of the pseudoholomorphic curve equation in symplectizations involving a harmonic $1-$form as perturbation term. In this paper we compactify the moduli space of $mathcal{H}-$holomorphic curves with a priori bounds on the harmonic $1-$forms.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78013029","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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