Isotopies of high genus Lagrangian surfaces
R. Hind, A. Ivrii
证明了黎曼曲面余切丛的所有与零截面同伦的拉格朗日子流形必光滑地与零截面同素。
We show that all Lagrangian submanifolds of the cotangent bundle of a Riemann surface which are homotopic to the zero section must be smoothly isotopic to the zero section.
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We study two related invariants of Lagrangian submanifolds in symplectic manifolds. For a Lagrangian torus these invariants are functions on the first cohomology of the torus. The first invariant is of topological nature and is related to the study of Lagrangian isotopies with a given Lagrangian flux. More specifically, it measures the length of straight paths in the first cohomology that can be realized as the Lagrangian flux of a Lagrangian isotopy. The second invariant is of analytical nature and comes from symplectic function theory. It is defined for Lagrangian submanifolds admitting fibr…
Lagrangian isotopies in Stein manifolds
我们建立了某些四维Stein流形中Lagrange球面的唯一性直到Hamilton合痕。
Hamiltonian isotopies of relatively exact Lagrangians are orientation-preserving
本文证明相对正合拉格朗日子流形上的哈密顿微分同胚必保持定向，且无需自旋假设，证明仅依赖模2系数的奇异与Floer上同调环。
Viterbo has conjectured that any Lagrangian in the unit co-disc bundle of a torus which is Hamiltonian isotopic to the zero-section satisfies a uniform bound on its spectral norm; a recent result by Shelukhin showed that this is indeed the case. The modest goal of this note is to explore two natural generalisations of the above geometric setting in which the bound of the spectral norm fails: first, passing to Legendrian isotopies in the contactisation of the unit co-disc bundle (Hamiltonian isotopies lift to Legendrian isotopies) and, second, considering Hamiltonian isotopies but after modifyi…
We exhibit monotone Lagrangian tori inside the standard symplectic four-dimensional unit ball that become Hamiltonian isotopic to the Clifford torus, i.e.~the standard product torus, only when considered inside a strictly larger ball (they are not even symplectomorphic to a standard torus inside the unit ball). These tori are then used to construct new examples of symplectic embeddings of toric domains into the unit ball which are symplectically knotted in the sense of J.~Gutt and M.~Usher. We also give a characterisation of the Clifford torus inside the ball as well as the projective plane in…
Let $P$ be a Delzant polytope in $\mathbb{R}^k$ with $n+k$ facets. We associate a closed Lagrangian submanifold $L$ of $\mathbb{C}^n$ to each Delzant polytope. We prove that $L$ is monotone if and only if and only if the polytope $P$ is Fano. We pose the "Lagrangian version of Delzant Theorem". Then for even $p$ and $n$ we construct $\frac{p}{2}$ monotone Lagrangian embeddings of $S^{p-1} \times S^{n-p-1} \times T^2$ into $\mathbb{C}^n$, no two of which are related by Hamiltonian isotopies. Some of these embeddings are smoothly isotopic and have equal minimal Maslov numbers, but they are not H…
Hofer distance on Lagrangian links inside the disc
本文证明了圆盘内某些圆周并集的 Hamiltonian 同痕集合对于 Hofer 距离是无界的，其证明结合了 Francesco Morabito 的结果与 Michael Khanevsky 的标准论证。
This paper is devoted to deformations of Lagrangian submanifolds contained in the singular locus of a log-symplectic manifold. We prove a normal form result for the log-symplectic structure around such a Lagrangian, which we use to extract algebraic and geometric information about the Lagrangian deformations. We show that the deformation problem is governed by a DGLA, we discuss whether the Lagrangian admits deformations not contained in the singular locus, and we give precise criteria for unobstructedness of first order deformations. We also address equivalences of deformations, showing that …
Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems
设I是一个开区间，M是一个实流形，T*M是它的余切丛，Φ={φ_t}，t在I中，是T*M的一个定义在零截面外的齐次Hamilton合痕.设Λ是与Φ相关联的圆锥拉格朗日子流形（Λ是T*M × T*M × T*I的子集）.本文证明了M × M × I上的层K的存在性和唯一性，K的微支撑包含在Λ和零截面的并中，K对t=0的限制是M × M对角线上的常数层.我们将这个结果应用于接触和辛拓扑中的非线性问题。特别地，我们证明了某些强Morse不等式在Hamilton合痕下是稳定的，并给出了在接触情形下正合痕的不可约性结果。在这个新的版本中，我们取消了主要定理中的一个假设，并推广了正同位素的不可约性结果。
We study Lagrangian cobordism groups of oriented surfaces of genus greater than two. We compute the immersed oriented Lagrangian cobordism group of these surfaces. We show that a variant of this group, with relations given by unobstructed immersed Lagrangian cobordisms computes the Grothendieck group of the derived Fukaya category. The proof relies on an argument of Abouzaid.
Open Gromov-Witten invariants in genus zero and Lagrangian cobordisms
针对任意闭辛流形和嵌入的相对自旋Lagrangian子流形，利用边界上链构造了亏格零的开放Gromov-Witten不变量，证明了开放WDVV关系及其对近复结构和Hamilton同痕的无关性，并建立了配边Lagrangian之间的不变量关系。
Simultaneous Deformations of Symplectic Forms and Lagrangian Submanifolds
本文研究了辛流形与拉格朗日子流形的同时变形，证明其模空间与第二相对de Rham上同调群的邻域同构，并表明该空间的光滑性和有限维性。
Special Lagrangian cones with higher genus links
对每一个奇自然数g=2d+1，利用几何PDE胶合方法，证明了亏格为g的闭黎曼曲面上C^3中存在可数无限族特殊拉格朗日锥.
Log Floer cohomology for oriented log symplectic surfaces
本文针对在低维子流形上退化的对数辛 Poisson 结构，在可定向曲面上定义对数 Floer 上同调，证明其同痕不变性及单拉格朗日情形下与对数德拉姆上同调的同构。
Intersection theory on the moduli space of holomorphic curves with Lagrangian boundary conditions
在拉格朗日子流形中具有边界的任意亏格的伪全纯曲线的模空间上，基于相交理论定义了一族新的开Gromov-Witten型不变量.我们假设拉格朗日子流形作为反辛对合的不动点而产生，并且具有2或3维。在强半正亏格为0的情况下，新的不变量与实拟全纯曲线的Welschinger不变量计数一致。此外，我们还计算了亏格为0，次数为1的实五次三重数的新不变量为30。
An open-closed Deligne-Mumford field theory associated to a Lagrangian submanifold
本文构造任意亏格、边界在 Lagrangian 子流形上的全纯映射全局 Kuranishi 图，并据此建立统一 Fukaya 代数与 Gromov–Witten 理论的开闭 Deligne–Mumford 场论。
Hofer Energy and Link Preserving Diffeomorphisms in Higher Genus
本文针对预单调拉格朗日链环，建立了保持该链环的哈密顿微分同胚的Hofer能量估计，该估计仅依赖辫型，并利用带边曲面辫群上的范数族，将结果推广至高亏格带边曲面。
In previous papers we introduced the notion of special Bohr - Sommerfeld lagrangian cycles on a compact simply connected symplectic manifold with integer symplectic form, and presented the main interesting case: compact simply connected algebraic variety with an ample line bundle such that the space of Bohr - Sommerfeld lagrangian cycles with respect to a compatible Kahler form of the Hodge type and holomorphic sections of the bundle is finite. The main problem appeared in this way is singular components of the corresponding lagrangian shadows (or sceletons of the corresponding Weinstein domai…
Non-decomposable Lagrangian cobordisms between Legendrian knots
给定\(g>0\)，构建一族标准接触三维球面中（稳定）勒让德纽结间亏格为\(g\)的不可分解拉格朗日配边，用利文斯顿估计法阻碍可分解性。
We prove a version of the Arnol'd conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $β$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $β$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.
Obstruction classes and Open Gromov-Witten invariants
对于Maslov指数为零的Calabi-Yau三重相对自旋拉格朗日子流形，定义亏格为零的开Gromov-Witten不变量.
Unobstructed Lagrangian cobordism groups of surfaces
本文研究了亏格 $g \geq 2$ 的闭辛曲面的无阻碍浸入拉格朗日配边群，基于 Abouzaid 和 Perrier 的工作计算了该群，并证明其同构于曲面的导出 Fukaya 范畴的 Grothendieck 群，证明中利用二维拓扑技术构造了不界定特定全纯多边形的配边。
Global Darboux coordinates for complete Lagrangian fibrations and an application to the deformation space of $\mathbb{R}\mathbb{P}^2$-structures in genus one
本文研究非紧纤维完备 Lagrange 纤维化，证明运动积分可作为作用量坐标，并据此构造环面真凸 $\mathbb{R}\mathbb{P}^2$ 结构形变空间上辛形式的全局 Darboux 坐标及等距表达式。
We show that a quasipositive surface with disconnected boundary induces a map between the knot Floer homology groups of its boundary components preserving the transverse invariant. As an application, we show that this invariant can be used to obstruct decomposable Lagrangian cobordisms of arbitrary genus within Weinstein cobordisms. The construction of our maps rely on the comultiplicativity of the transverse invariant. Along the way, we also recover various naturality statements for the invariant under contact +1 surgery.
The Hirzebruch--Riemann--Roch theorem in true genus-0 quantum K-theory
本文利用紧致复代数流形的上同调Gromov-Witten不变量，刻画了紧致复代数流形的亏格0 K-理论Gromov-Witten不变量.这是通过应用Kawasaki-Hirzebruch-Riemann-Roch公式的一个虚拟版本，用于表示亏格为0的稳定映射的模空间上的轨道丛的全纯欧拉示性数，分析了这种模空间的惯性栈的复杂组合结构，并采用了来自“fake”的各种量子Riemann-Roch公式。（即orbifold-ignorant）流形和orbifolds的量子K理论（公式，以前从Coates-Givental，Tseng和Coates-Corti-Iritani-Tseng的作品中已知，或Tonita在单独的论文中为此目的新开发）。最终制定结合属性的推翻拉格朗日锥在辛闭路空间（语言，已成为传统的描述生成函数的genus-0 Gromov-Witten理论）与一个新的框架“adelic特征”的这种锥。作为应用，我们证明了量子K-理论的被支配拉格朗日锥的切空间在Novikov变量的有限差分算子代数上具有模的自然结构。作为另一个应用，我们为维度为$\geq l_1^2+...+l_k^2-1$的复射影空间中的度方程$l_1,...,l_k$给出的每个完整交叉计算一个这样的切空间。
Hamiltonian stability and index of minimal Lagrangian surfaces of the complex projective plane
证明了复射影平面CP ^2中的Clifford环面和全测地实射影平面RP ^2是CP ^2中唯一的Hamilton稳定极小Lagrange紧曲面，当曲面可定向时亏格小于等于4，当曲面不可定向时欧拉示性数大于等于-1。我们还刻画了CP^2中的RP^2为CP^2的最小可能指标极小拉格朗日紧致不可定向曲面。
We study symplectic rigidity phenomena for fibers in cotangent bundles of Riemann surfaces. Our main result can be seen as a generalization to open Riemann surfaces of arbitrary genus of work of Eliashberg and Polterovich on the Nearby Lagrangian Conjecture for $T^* \mathbb{R}^2$. As a corollary, we answer a strong version in dimension $2n=4$ of a question of Eliashberg about linking of Lagrangian disks in $T^* \mathbb{R}^n$, which was previously answered by Ekholm and Smith in dimensions $2n \geq 8$.
If a Legendrian knot $Λ$ in the standard contact 3-sphere bounds an orientable exact Lagrangian surface $Σ$ in the standard symplectic 4-ball, then the genus of $Σ$ is equal to the slice genus of (the smooth knot underlying) $Λ$, the sum of the Thurston-Bennequin number of L and the Euler characteristic of $Σ$ is zero as well as the rotation number of $Λ$, and moreover, the linearized contact homology of $Λ$ with respect to the augmentation induced by $Σ$ is isomorphic to the (singular) homology of $Σ$. It was asked in arXiv:1212.1519 whether the converse of this statement is true. We give a n…
Homological mirror symmetry for the four-torus
利用Mau-Wehrheim-Woodward的quilt方法，给出了有限个拉格朗日子流形的集合分裂生成类的一个充分条件，并推导出标准4-环面的同调镜像对称性.作为应用，我们研究了两个具有Lagrange亏格的Maslov零类曲面，导出了这类曲面与线性Lagrange 2-环面在4-环面中相交的数值限制。
Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
本文提出热带子簇的 $B$-可实现性概念，证明若拉格朗日子流形提升具有定义良好的 Floer 上同调且可通过界定上链变为无障碍，则该热带子簇是 $B$-可实现的，并应用于证明亏格零热带曲线及热带阿贝尔曲面内热带曲线的可实现性。
Bohr--Sommerfeld Lagrangians of moduli spaces of Higgs bundles
设$X$是亏格至少为2的紧致连通黎曼曲面.设$M_H(r,d)$表示秩为$r$、度为$d$的$X$上半稳定Higgs丛的模空间.证明了$M_H(r,d)$的紧致复Bohr-Sommerfeld拉格朗日量恰好是$M_H(r,d)$中幂零锥的不可约分量.这推广到Higgs $G$ -束和抛物Higgs束。
Examples of relative quantum cohomology
本文计算了完全交中相对于拉格朗日子流形的量子上同调，并完整计算了二次超曲面的亏格零开Gromov-Witten不变量。
Wavefunctions are proposed for a class of Lagrangian branes in three complex-dimensional space. The branes are asymptotic to Legendrian surfaces of genus g. The expansion of these wavefunctions in appropriate coordinates conjecturally encodes all-genus open Gromov-Witten invariants, i.e. the free energy of the topological open string. This paper is written in physics language, but tries to welcome mathematicians. Most results stem from joint mathematical works with Linhui Shen and David Treumann.
Givental's Lagrangian Cone and S^1-Equivariant Gromov-Witten Theory
在Givental发展的Gromov-Witten理论的方法中，流形X的亏格为零的Gromov-Witten不变量在某个无限维辛向量空间中被拉格朗日锥编码。我们根据乘积X \times P^1的S^1-等变Gromov-Witten理论，按照S^1-等变Floer理论的精神，给出了这个锥的构造。这给出了一个概念上的理解“的”转移：变量的变化，其中发挥了至关重要的作用，吉文塔尔的理论。
Minimal surfaces in germs of hyperbolic 3--manifolds
本文从某个双曲三维流形中的紧的、定向的、正亏格的极小曲面引入了一个泛模空间，其原型元是由度量和第二基本型组成的一对.这个模空间是一个光滑的、有限维的流形，它具有到Teichmueller空间的余切丛和给定亏格曲面的SO（3，C）表示空间的正则映射。这两个映射将泛模空间作为拉格朗日子流形嵌入到后两个空间的乘积中。
Given two Fuchsian representations $ρ_l$ and $ρ_r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, we study the relation between Lagrangian submanifolds of $M_ρ=(\mathbb{H}^2/ρ_l(π_1(S)))\times (\mathbb{H}^2/ρ_r(π_1(S)))$ and $ρ$-equivariant embeddings $σ$ of $\widetilde S$ into Anti-de Sitter space, where $ρ=(ρ_l,ρ_r)$ is the corresponding representation into $\mathrm{PSL}_2\mathbb R\times \mathrm{PSL}_2\mathbb R$. It is known that, if $σ$ is a maximal embedding, then its Gauss map takes values in the unique minimal Lagrangian submanifold $Λ_{\mathrm{ML}}$ of $M_ρ…
The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems
本文讨论了最近由Franks和Handel \cite{FrHa}证明的Conley猜想（对于正亏格曲面），Hingston \cite{Hi}（对于托里）和金兹伯格\cite{Gi}（对于闭辛非球面流形），对无边界$C^3$ -光滑紧致流形$M$的余切丛上的$C^1$ -哈密顿系统，证明了的时间1-周期$C^2$ -光滑哈密顿量$H:\R\times T^\ast M\to\R$，它是强凸的，在纤维上具有二次增长。也就是说，我们表明，这样的哈密顿系统有一个无限序列的可收缩的积分周期解，使他们中的任何一个不能从其他的迭代。若对任何$(t,q, p)\in\R\times T^\ast M$，$H$也满足$H(-t,q, -p)=H(t,q, p)$，则证明了Hamilton系统的时一映射（若存在）有无穷多个周期点位于$T^\ast M$的零区.若$M$是$C^5$ -光滑的，且$\dim M>1$，$H$是$C^4$类的，且与时间$t$无关，则对任何$τ>0$，相应的系统都有无穷多个周期为$τ$整数倍的可收缩周期解序列，使得其中任何一个都不能通过迭代或旋转从其它解中得到.这些结果是通过证明$H$的Fenchel变换的Lagrangian系统$L:\R\times TM\to\R$的类似结果而得到的，它被证明是强凸的，并且在速度上具有二次增长。
Modularity of Open Gromov-Witten Potentials of Elliptic Orbifolds
本文研究了椭圆型轨道的亏格零开Gromov-Witten势的模性及其生成矩阵分解。这些对象构造的拉格朗日弗洛尔理论是一个先验良好的定义，只有周围的大体积限制。从模性出发，它们可以在整体Kähler模空间上解析延拓。
Open enumerative mirror symmetry for lines in the mirror quintic
本文假设负循环开闭映射存在，从同调镜像对称推导出五次三维流形某些拉格朗日子流形的开放Gromov-Witten不变量的镜像定理，其OGW不变量与Walcher的相对周期积分一致，且涉及Dirichlet L函数特殊值，还解释了相关双曲拉格朗日存在性预测。
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