Pub Date : 2018-10-30DOI: 10.4310/JSG.2020.V18.N3.A4
F. Forstnerič
In this paper we introduce the notion of a formal complex contact structure on an odd dimensional complex manifold. Our main result is that every formal complex contact structure on a Stein manifold $X$ is homotopic to a holomorphic contact structure on a Stein domain $Omegasubset X$ which is diffeotopic to $X$. We also prove a parametric h-principle in this setting, analogous to Gromov's h-principle for contact structures on smooth open manifolds. On Stein threefolds we obtain a complete homotopy classification of formal complex contact structures. Our methods also furnish a parametric h-principle for germs of holomorphic contact structures along totally real submanifolds of class $mathscr C^2$ in arbitrary complex manifolds.
{"title":"H-principle for complex contact structures on Stein manifolds","authors":"F. Forstnerič","doi":"10.4310/JSG.2020.V18.N3.A4","DOIUrl":"https://doi.org/10.4310/JSG.2020.V18.N3.A4","url":null,"abstract":"In this paper we introduce the notion of a formal complex contact structure on an odd dimensional complex manifold. Our main result is that every formal complex contact structure on a Stein manifold $X$ is homotopic to a holomorphic contact structure on a Stein domain $Omegasubset X$ which is diffeotopic to $X$. We also prove a parametric h-principle in this setting, analogous to Gromov's h-principle for contact structures on smooth open manifolds. On Stein threefolds we obtain a complete homotopy classification of formal complex contact structures. Our methods also furnish a parametric h-principle for germs of holomorphic contact structures along totally real submanifolds of class $mathscr C^2$ in arbitrary complex manifolds.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86163781","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 : 2018-09-27DOI: 10.4310/jsg.2021.v19.n5.a3
Duvsan Joksimovi'c, I. Marcut
We remark that, as in the symplectic case, the Hofer norm on the Hamiltonian group of a Poisson manifold is non-degenerate. The proof is a straightforward application of tools from symplectic topology.
{"title":"Non-degeneracy of the Hofer norm for Poisson structures","authors":"Duvsan Joksimovi'c, I. Marcut","doi":"10.4310/jsg.2021.v19.n5.a3","DOIUrl":"https://doi.org/10.4310/jsg.2021.v19.n5.a3","url":null,"abstract":"We remark that, as in the symplectic case, the Hofer norm on the Hamiltonian group of a Poisson manifold is non-degenerate. The proof is a straightforward application of tools from symplectic topology.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90174477","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 : 2018-09-11DOI: 10.4310/jsg.2020.v18.n4.a8
M. Munteanu
Given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result. Given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible. We show how the constructed loops become contractible if the target domain becomes large enough. The proof involves studying certain moduli spaces of holomorphic cylinders in families of symplectic cobordisms arising from families of symplectic embeddings.
{"title":"Noncontractible loops of symplectic embeddings between convex toric domains","authors":"M. Munteanu","doi":"10.4310/jsg.2020.v18.n4.a8","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n4.a8","url":null,"abstract":"Given two 4-dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result. Given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible. We show how the constructed loops become contractible if the target domain becomes large enough. The proof involves studying certain moduli spaces of holomorphic cylinders in families of symplectic cobordisms arising from families of symplectic embeddings.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77295833","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 : 2018-08-21DOI: 10.4310/JSG.2020.v18.n5.a1
A. Alekseev, Benjamin Hoffman, J. Lane, Yanpeng Li
For a compact Poisson-Lie group $K$, the homogeneous space $K/T$ carries a family of symplectic forms $omega_xi^s$, where $xi in mathfrak{t}^*_+$ is in the positive Weyl chamber and $s in mathbb{R}$. The symplectic form $omega_xi^0$ is identified with the natural $K$-invariant symplectic form on the $K$ coadjoint orbit corresponding to $xi$. The cohomology class of $omega_xi^s$ is independent of $s$ for a fixed value of $xi$. In this paper, we show that as $sto -infty$, the symplectic volume of $omega_xi^s$ concentrates in arbitrarily small neighbourhoods of the smallest Schubert cell in $K/T cong G/B$. This strengthens earlier results [9,10] and is a step towards a conjectured construction of global action-angle coordinates on $Lie(K)^*$ [4, Conjecture 1.1].
对于紧泊松-李群$K$，齐次空间$K/T$携带一族辛形式$omega_xi^s$，其中$xi in mathfrak{t}^*_+$在正Weyl室中，$s in mathbb{R}$。将$omega_xi^0$的辛形式与$xi$对应的$K$共伴随轨道上的自然$K$不变辛形式进行了识别。对于$xi$的固定值，$omega_xi^s$的上同类与$s$无关。在本文中，我们证明了在$sto -infty$中，$omega_xi^s$的辛体积集中在$K/T cong G/B$中最小的Schubert单元的任意小的邻域中。这加强了先前的结果[9,10]，并且朝着在$Lie(K)^*$上推测全局动作角坐标的构造迈出了一步[4，猜想1.1]。
{"title":"Concentration of symplectic volumes on Poisson homogeneous spaces","authors":"A. Alekseev, Benjamin Hoffman, J. Lane, Yanpeng Li","doi":"10.4310/JSG.2020.v18.n5.a1","DOIUrl":"https://doi.org/10.4310/JSG.2020.v18.n5.a1","url":null,"abstract":"For a compact Poisson-Lie group $K$, the homogeneous space $K/T$ carries a family of symplectic forms $omega_xi^s$, where $xi in mathfrak{t}^*_+$ is in the positive Weyl chamber and $s in mathbb{R}$. The symplectic form $omega_xi^0$ is identified with the natural $K$-invariant symplectic form on the $K$ coadjoint orbit corresponding to $xi$. The cohomology class of $omega_xi^s$ is independent of $s$ for a fixed value of $xi$. \u0000In this paper, we show that as $sto -infty$, the symplectic volume of $omega_xi^s$ concentrates in arbitrarily small neighbourhoods of the smallest Schubert cell in $K/T cong G/B$. This strengthens earlier results [9,10] and is a step towards a conjectured construction of global action-angle coordinates on $Lie(K)^*$ [4, Conjecture 1.1].","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80491484","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 : 2018-08-17DOI: 10.4310/jsg.2020.v18.n4.a4
Oleg Lazarev
We prove an existence h-principle for regular Lagrangians with Legendrian boundary in arbitrary Weinstein domains of dimension at least six; this extends a previous result of Eliashberg, Ganatra, and the author for Lagrangians in flexible domains. Furthermore, we show that all regular Lagrangians come from our construction and describe some related decomposition results. We also prove a regular version of Eliashberg and Murphy's h-principle for Lagrangian caps with loose negative end. As an application, we give a new construction of infinitely many regular Lagrangian disks in the standard Weinstein ball.
{"title":"H-principles for regular Lagrangians","authors":"Oleg Lazarev","doi":"10.4310/jsg.2020.v18.n4.a4","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n4.a4","url":null,"abstract":"We prove an existence h-principle for regular Lagrangians with Legendrian boundary in arbitrary Weinstein domains of dimension at least six; this extends a previous result of Eliashberg, Ganatra, and the author for Lagrangians in flexible domains. Furthermore, we show that all regular Lagrangians come from our construction and describe some related decomposition results. We also prove a regular version of Eliashberg and Murphy's h-principle for Lagrangian caps with loose negative end. As an application, we give a new construction of infinitely many regular Lagrangian disks in the standard Weinstein ball.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85364694","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 : 2018-08-13DOI: 10.4310/jsg.2022.v20.n2.a3
Dishant M. Pancholi, Suhas Pandit
The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, xi_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, xi_N),$ provided $M$ contact embeds in $(N, xi_N)$ with a trivial normal bundle and the contact structure induced on $M$ via this embedding is homotopic as an almost-contact structure to $xi_M.$ We apply this result to first establish that a closed contact $3$--manifold having no $2$--torsion in its second integral cohomology iso-contact embeds in the standard contact $5$--sphere if and only if the first Chern class of the contact structure is zero. Finally, we discuss iso-contact embeddings of closed simply connected contact $5$--manifolds.
{"title":"Iso-contact embeddings of manifolds in co-dimension $2$","authors":"Dishant M. Pancholi, Suhas Pandit","doi":"10.4310/jsg.2022.v20.n2.a3","DOIUrl":"https://doi.org/10.4310/jsg.2022.v20.n2.a3","url":null,"abstract":"The purpose of this article is to study co-dimension $2$ iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold $(M^{2n-1}, xi_M)$ iso-contact embeds in a contact manifold $(N^{2n+1}, xi_N),$ provided $M$ contact embeds in $(N, xi_N)$ with a trivial normal bundle and the contact structure induced on $M$ via this embedding is homotopic as an almost-contact structure to $xi_M.$ We apply this result to first establish that a closed contact $3$--manifold having no $2$--torsion in its second integral cohomology iso-contact embeds in the standard contact $5$--sphere if and only if the first Chern class of the contact structure is zero. Finally, we discuss iso-contact embeddings of closed simply connected contact $5$--manifolds.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73371247","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 : 2018-06-30DOI: 10.4310/jsg.2020.v18.n3.a3
Lucas Dahinden
A positive contactomorphism of a contact manifold $M$ is the end point of a contact isotopy on $M$ that is always positively transverse to the contact structure. Assume that $M$ contains a Legendrian sphere $Lambda$, and that $(M,Lambda)$ is fillable by a Liouville domain $(W,omega)$ with exact Lagrangian $L$ such that $omega|_{pi_2(W,L)}=0$. We show that if the exponential growth of the action filtered wrapped Floer homology of $(W,L)$ is positive, then every positive contactomorphism of $M$ has positive topological entropy. This result generalizes the result of Alves and Meiwes from Reeb flows to positive contactomorphisms, and it yields many examples of contact manifolds on which every positive contactomorphism has positive topological entropy, among them the exotic contact spheres found by Alves and Meiwes. A main step in the proof is to show that wrapped Floer homology is isomorphic to the positive part of Lagrangian Rabinowitz-Floer homology.
{"title":"Positive topological entropy of positive contactomorphisms","authors":"Lucas Dahinden","doi":"10.4310/jsg.2020.v18.n3.a3","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n3.a3","url":null,"abstract":"A positive contactomorphism of a contact manifold $M$ is the end point of a contact isotopy on $M$ that is always positively transverse to the contact structure. Assume that $M$ contains a Legendrian sphere $Lambda$, and that $(M,Lambda)$ is fillable by a Liouville domain $(W,omega)$ with exact Lagrangian $L$ such that $omega|_{pi_2(W,L)}=0$. We show that if the exponential growth of the action filtered wrapped Floer homology of $(W,L)$ is positive, then every positive contactomorphism of $M$ has positive topological entropy. This result generalizes the result of Alves and Meiwes from Reeb flows to positive contactomorphisms, and it yields many examples of contact manifolds on which every positive contactomorphism has positive topological entropy, among them the exotic contact spheres found by Alves and Meiwes. \u0000A main step in the proof is to show that wrapped Floer homology is isomorphic to the positive part of Lagrangian Rabinowitz-Floer homology.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84795003","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 : 2018-06-20DOI: 10.4310/jsg.2020.v18.n2.a3
Laurent Cot'e
We prove some results about linking of Lagrangian tori in the symplectic vector space $(mathbb{R}^4, omega)$. We show that certain enumerative counts of holomophic disks give useful information about linking. This enables us to prove, for example, that any two Clifford tori are unlinked in a strong sense. We extend work of Dimitroglou Rizell and Evans on linking of monotone Lagrangian tori to a class of non-monotone tori in $mathbb{R}^4$ and also strengthen their conclusions in the monotone case in $mathbb{R}^4$.
{"title":"On linking of Lagrangian tori in $mathbb{R}^4$","authors":"Laurent Cot'e","doi":"10.4310/jsg.2020.v18.n2.a3","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n2.a3","url":null,"abstract":"We prove some results about linking of Lagrangian tori in the symplectic vector space $(mathbb{R}^4, omega)$. We show that certain enumerative counts of holomophic disks give useful information about linking. This enables us to prove, for example, that any two Clifford tori are unlinked in a strong sense. We extend work of Dimitroglou Rizell and Evans on linking of monotone Lagrangian tori to a class of non-monotone tori in $mathbb{R}^4$ and also strengthen their conclusions in the monotone case in $mathbb{R}^4$.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88060285","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 : 2018-06-20DOI: 10.4310/jsg.2020.v18.n5.a5
Caroline Vernier
This paper is concerned with the existence of metrics of constant Hermitian scalar curvature on almost-Kahler manifolds obtained as smoothings of a constant scalar curvature Kahler orbifold, with A1 singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphic vector fields, we show that an almost-Kahler smoothing (Me, ωe) admits an almost-Kahler structure (Je, ge) of constant Hermitian curvature. Moreover, we show that for e > 0 small enough, the (Me, ωe) are all symplectically equivalent to a fixed symplectic manifold (M , ω) in which there is a surface S homologous to a 2-sphere, such that [S] is a vanishing cycle that admits a representant that is Hamiltonian stationary for ge.
{"title":"Almost-Kähler smoothings of compact complex surfaces with $A_1$ singularities","authors":"Caroline Vernier","doi":"10.4310/jsg.2020.v18.n5.a5","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n5.a5","url":null,"abstract":"This paper is concerned with the existence of metrics of constant Hermitian scalar curvature on almost-Kahler manifolds obtained as smoothings of a constant scalar curvature Kahler orbifold, with A1 singularities. More precisely, given such an orbifold that does not admit nontrivial holomorphic vector fields, we show that an almost-Kahler smoothing (Me, ωe) admits an almost-Kahler structure (Je, ge) of constant Hermitian curvature. Moreover, we show that for e > 0 small enough, the (Me, ωe) are all symplectically equivalent to a fixed symplectic manifold (M , ω) in which there is a surface S homologous to a 2-sphere, such that [S] is a vanishing cycle that admits a representant that is Hamiltonian stationary for ge.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77696488","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 : 2018-06-17DOI: 10.4310/jsg.2020.v18.n4.a2
Y. Ganor
We suggest a homotopical description of the Poisson bracket invariants for tuples of closed sets in symplectic manifolds. It implies that these invariants depend only on the union of the sets along with topological data.
{"title":"A homotopical viewpoint at the Poisson bracket invariants for tuples of sets","authors":"Y. Ganor","doi":"10.4310/jsg.2020.v18.n4.a2","DOIUrl":"https://doi.org/10.4310/jsg.2020.v18.n4.a2","url":null,"abstract":"We suggest a homotopical description of the Poisson bracket invariants for tuples of closed sets in symplectic manifolds. It implies that these invariants depend only on the union of the sets along with topological data.","PeriodicalId":50029,"journal":{"name":"Journal of Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2018-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76753983","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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