Yuri Chekanov, Felix Schlenk
We construct monotone Lagrangian tori in the standard symplectic vector space, in the complex projective space and in products of spheres. We explain how to classify these Lagrangian tori up to symplectomorphism and Hamiltonian isotopy, and how to show that they are not displaceable by Hamiltonian isotopies.
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In this note, we prove that two constructions of exotic monotone Lagrangian tori, namely the one by Chekanov and Schlenk and the one obtained by the circle bundle construction of Biran are Hamiltonian isotopic in $\mathbb{C}P^2$ and $S^2 \times S^2$.
We construct infinitely many families of monotone Lagrangian tori in $\mathbb{R}^6$, no two of which are related by Hamiltonian isotopies (or symplectomorphisms). These families are distinguished by the (arbitrarily large) numbers of families of Maslov index 2 pseudo-holomorphic discs that they bound.
We extract from a toric model of the Chekanov-Schlenk exotic torus in $\mathbb{CP}^2$ methods of construction of Lagrangian submanifolds in toric symplectic manifolds. These constructions allow for some control of the monotonicity. We recover this way some known monotone Lagrangians in the toric symplectic manifolds $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$ as well as new examples.
We prove that a monotone Lagrangian torus in $S^2\times S^2$ which suitably sits in a symplectic fibration with two sections in its complement is Hamiltonian isotopic to the Clifford torus.
Infinitely many monotone Lagrangian tori in higher projective spaces
本文将 Vianna 在复射影平面中构造的无穷多个奇异单调 Lagrangian 环面提升到高维射影空间，并借助 Pascaleff–Tonkonog 穿墙公式证明它们互不同痕等价。
We prove a version of Sandon's conjecture on the number of translated points of contactomorphisms for the case of prequantization bundles over certain closed monotone symplectic toric manifolds. Namely we show that any contactomorphism of such a prequantization bundle lying in the identity component of the contactomorphism group possesses at least $N$ translated points, where $N$ is the minimal Chern number of the symplectic toric manifold. The proof relies on the theory of generating functions coupled with equivariant cohomology, whereby we adapt Givental's approach to the Arnold conjecture f…
Related to each degeneration from CP^2 to CP(a^2,b^2,c^2), for (a,b,c) a Markov triple - positive integers satisfying a^2 + b^2 + c^2 = 3abc - there is a monotone Lagrangian torus, which we call T(a^2,b^2,c^2). We employ techniques from symplectic field theory to prove that no two of them are Hamiltonian isotopic to each other.
We construct almost toric fibrations (ATFs) on all del Pezzo surfaces, endowed with a monotone symplectic form. Except for $\mathbb{C}P^2 \# 1 \overline{\mathbb{C}P^2}$ and $\mathbb{C}P^2 \# 2 \overline{\mathbb{C}P^2}$ , we are able to get almost toric base diagrams (ATBDs) of triangular shape and prove the existence of infinitely many symplectomorphism (in particular Hamiltonian isotopy) classes of monotone Lagrangian tori in $\mathbb{C}P^2 \# k \overline{\mathbb{C}P^2}$, for k=0,3,4,5,6,7,8. We name these tori $Θ^{n_1,n_2,n_3}_{p,q,r}$. Using the work of Karpov-Nogin, we are able to classify…
We prove some results about linking of Lagrangian tori in the symplectic vector space $(\mathbb{R}^4, ω)$. 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 $\mathbb{R}^4$.
Homological Lagrangian monodromy for some monotone tori
本文系统研究单调 Lagrangian $n$-环面的同调 Lagrangian 单值群，完整刻画单调环面纤维情形，推进二维分类并提出一般猜想。
We define new Hamiltonian isotopy invariants for a monotone Lagrangian torus embedded in a symplectic 4-manifold. We show that, in the standard symplectic 4-space, these invariants distinguish a monotone Clifford torus from a Chekanov torus.
Mironov, Panov and Kotelskiy studied Hamiltonian-minimal Lagrangians inside $\mathbb{C}^n$. They associated a closed embedded Lagrangian $L$ to each Delzant polytope $P$. In this paper we develop their ideas and prove that $L$ is monotone if and only if the polytope $P$ is Fano. In some examples, we further compute the minimal Maslov numbers. Namely, let $\mathcal{N}\to T^k$ be some fibration over the $k$-dimensional torus with a fiber equal to either $S^k \times S^l$, or $S^k \times S^l \times S^m$, or $\#_5(S^{2p-1} \times S^{n-2p-2})$. We construct monotone Lagrangian embeddings $\mathcal{N…
Twist tori are examples of exotic monotone lagrangian tori, presented in [1]. This tree of examples grew up over the first one --- the torus $Θ\in \R^4$, constructured in [2] and [3]. On the other hand, in [4] and [5] we proposed a new structure which generalizes the notion of toric structure. One calls this generalization pseudo toric structure, and several examples were given which show that certain toric symplectic manifolds can carry the structre and that certain non toric symplectic manifolds do the same. Below we show that any twist torus $Θ^k \subset \R^{2k+2}$, defined in [1], can be c…
By a result due to Ziltener, there exist no closed embedded Bohr-Sommerfeld Lagrangians inside $\mathbb CP^n$ for the prequantisation bundle whose total space is the standard contact sphere. On the other hand, any embedded monotone Lagrangian torus has a canonical nontrivial cover which is a Bohr-Sommerfeld immersion. We draw the front projections for the corresponding Legendrian lifts inside a contact Darboux ball of the threefold covers of both the two-dimensional Clifford and Chekanov tori (the former is the Legendrian link of the Harvey-Lawson special Lagrangian cone), and compute the asso…
We use Floer cohomology to prove the monotone version of a conjecture of Audin: the minimal Maslov number of a monotone Lagrangian torus in C^n is 2. Our approach is based on the study of the quantum cup product on Floer cohomology and in particular the behaviour of Oh's spectral sequence with respect to this product. As further applications we prove existence of holomorphic disks with boundaries on Lagrangians as well as new results on Lagrangian intersections.
Tropical disk potential for almost toric manifolds
本文基于先前工作，为近toric四流形中的Lagrangian tori推导出热带公式，扩展了Mikhalkin关于射影平面holomorphic sphere的结果，并通过del Pezzo表面的例子验证了该公式。
Cluster algebras and monotone Lagrangian tori
本文研究簇代数与单调Lagrangian环面的联系，给出充分条件构造无穷多个互不等价的单调Lagrangian环面，并应用于旗流形。
We investigate leaf-wise intersection points on hypersurfaces of contact type in monotone symplectic manifolds. We show that monotone Floer-essential Lagrangians detect periodic leaf-wise intersection points in hypersurfaces of contact type whose Reeb flow is Zoll. Examples include the prequantization bundles appearing in monotone toric negative line bundles. Generalizing, we prove the existence of leaf-wise intersection points for certain annulus subbundles in weak+-monotone negative line bundles, not necessarily toric. The proofs combine reduced symplectic cohomology with the original method…
This small note contains two simple remarks about Bohr - Sommerfeld with respect to the anticanonical class lagrangian tori in the projective spaces, which are the most popular examples of monotone symplectic manifolds.
Fixed point Floer cohomology of disjoint Dehn twists on a w+-monotone manifold with rational symplectic form
本文显式描述了 w+-单调有理辛流形中不相交 Lagrangian 球面 Dehn 扭曲的 Floer 上同调，通过计数半带条在单调辛流形中定义并证明了一类不动点 Floer 上同调的消失性，计划借此为 Seidel 长正合序列提供新的几何证明。
We consider various constructions of monotone Lagrangian submanifolds of $C P^n, S^2\times S^2$, and quadric hypersurfaces of $C P^n$. In $S^2\times S^2$ and $C P^2$ we show that several different known constructions of exotic monotone tori yield results that are Hamiltonian isotopic to each other, in particular answering a question of Wu by showing that the monotone fiber of a toric degeneration model of $C P^2$ is Hamiltonian isotopic to the Chekanov torus. Generalizing our constructions to higher dimensions leads us to consider monotone Lagrangian submanifolds (typically not tori) of quadri…
Can a given Lagrangian submanifold be realized as the fixed point set of an anti-symplectic involution? If so, it is called \emph{real}. We give an obstruction for a closed Lagrangian submanifold to be real in terms of the displacement energy of nearby Lagrangians. Applying this obstruction to toric fibres, we obtain that the central fibre of many (and probably all) toric monotone symplectic manifolds is real only if the corresponding moment polytope is centrally symmetric. Furthermore, we embed the Chekanov torus in all toric monotone symplectic manifolds and show that it is exotic and not re…
Quantum cohomology and Fukaya summands from monotone Lagrangian tori
针对紧辛流形内的单调拉格朗日环面，证明几何闭开映射诱导量子上同调分解为含雅可比环局部化的乘积，通过描述福卡亚范畴直和项验证镜像对称预期并建立生成准则，为理解量子上同调结构和约束超势提供支撑。
We construct an exotic monotone Lagrangian torus in CP^2 using techniques motivated by mirror symmetry. We show that it bounds 10 families of Maslov index 2 holomorphic discs, and it follows that this exotic torus is not Hamiltonian isotopic to the known Clifford and Chekanov tori.
In this paper, we give a formula for the Maslov index of a gradient holomorphic disc, which is a relative version of the Chern number formula of a gradient holomorphic sphere for a Hamiltonian $S^1$-action. Using the formula, we classify all monotone Lagrangian non-toric fibers of Gelfand-Cetlin systems on partial flag manifolds.
Compact monotone tall complexity one $T$-spaces
本文研究紧单调高复杂度一$T$-空间，利用分类与单调性条件证明其同构由Duistermaat-Heckman测度唯一决定，刻画矩多胞体性质与测度形式，得到有限性结果，并证明其可延拓作用且等变辛同胚于Fano流形。
On Hamiltonian stationarity of twisted Lagrangian tori in ${\mathbb{C}}^2$
本文研究 ${\u000bC}^2$ 中由平面简单闭曲线扭曲生成的 Lagrangian 环面，证明在 Hamilton 形变下只有乘积环面面积极小，从而说明 Chekanov 奇异环面在其同痕类中不是面积极小。
Monotone Lagrangians in cotangent bundles of spheres
本文研究了 $T^*S^n$ 的紧致单调 Fukaya 范畴，证明其由零截面 $S^n$ 和一族单参数单调拉格朗日环面分裂生成，并得出具有非平凡 Floer 上同调的闭定向 spin 单调拉格朗日子流形不可分离的推论。
We show that a monotone Lagrangian $L$ in $\mathbb{CP}^n$ of minimal Maslov number $n + 1$ is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to $\mathbb{RP}^n$. To prove this we use Zapolsky's canonical pearl complex for $L$ with coefficients in $\mathbb{Z}$, and various twisted versions thereof, where the twisting is determined by connected covers of $L$. The main tool is the action of the quantum cohomology of $\mathbb{CP}^n$ on the resulting Floer homologies.
Unlinking and unknottedness of monotone Lagrangian submanifolds
该研究在特定拓扑假设下证明了同单调性常数的单调拉格朗日子流形的无环绕性与纽结类型由同伦论数据决定，并推导出复维数≥5的奇数维辛向量空间中所有单调拉格朗日环面光滑同痕。
We construct a family of Lagrangian tori $Θ^n_s$ $\subset$ $(\mathbb{C}P^1)^n$, $s \in (0,1)$, where $Θ^n_{1/2} = Θ^n$, is the monotone twist Lagrangian torus described by Chekanov-Schlenk. We show that for $n = 2m$ and $s \ge 1/2$ these tori are non-displaceable. Then by considering $Θ^{k_1}_{s_1}$ $ \times$ $\cdots$ $\times$ $ Θ^{k_l}_{s_l}$ $ \times$ $ (S^2_{\mathrm{eq}})^{n - \sum_i k_i}$ $ \subset $ $(\mathbb{C}P^1)^n$, with $s_i \in [1/2,1)$ and $k_i \in 2\mathbb{Z}_{>0}$, $\sum_i k_i \le n$ we get several $l$-dimensional families of non-displaceable Lagrangian tori. We also show that th…
We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states), some non-monotone Lagrangian torus fibers. We also extend the results by Oh and the author about the computations of Floer cohomology of Lagrangian torus fibers to the case of all toric Fano manifolds, removing the convexity assumption in the prev…
Two degenerate intersections of Clifford tori in \(S^2\times S^2\) and \(\CP^2\)
本文在单调流形上构造时间周期哈密顿量，使 Clifford 环面与其像仅有两个孤立退化交点。
Semi-Local Exotic Lagrangian Tori in Dimension Four
该研究针对四维奇异拉格朗日环面展开，在Stein域$B_{dpq}$中找到$d+1$族辛同胚互异的单调拉格朗日环面，证明其嵌入几何有界四维辛流形后仍互异，且紧致近环面流形中存在任意多此类互异环面，突破了此前仅del Pezzo曲面有相关结论的限制。
This note studies the geometric structure of monotone moment polytopes (the duals of smooth Fano polytopes) using probes. The latter are line segments that enter the polytope at an interior point of a facet and whose direction is integrally transverse to this facet. A point inside the polytope is displaceable by a probe if it lies less than half way along it. Using a construction due to Fukaya-Oh-Ohta-Ono, we show that every rational polytope has a central point that is not displaceable by probes. In the monotone (or more generally, the reflexive) case, this central point is its unique interio…
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…
Local exotic tori
该研究针对维数≥6的几何有界辛流形，证明其每个开子集内存在无穷多辛微分同胚下不同但拉格朗日同痕、经典不变量相同的局部exotic拉格朗日环面，借助位移能量芽局部性在达布图中计算，揭示非单调环面的Maslov指标2的J-全纯圆盘计数不构成不变量。
We prove the Hamiltonian unknottedness of real Lagrangian tori in the monotone $S^2\times S^2$, namely any real Lagrangian torus in $S^2\times S^2$ is Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$. The proof is based on a neck-stretching argument, Gromov's foliation theorem, and the Cieliebak-Schwingenheuer criterion.
We show that a closed weakly-monotone symplectic manifold of dimension $2n$ which has minimal Chern number greater than or equal to $n+1$ and admits a Hamiltonian toric pseudo-rotation is necessarily monotone and its quantum homology is isomorphic to that of the complex projective space. As a consequence when $n=2$, the manifold is symplectomorphic to $\mathbb{C}P^2$.
Finiteness and boundedness of positive monotone Hamiltonian GKM$_3$ spaces
针对正单调紧辛流形上的GKM$_3$作用，证明了在固定维数和欧拉示性数下，复配边类有限，矩映射像可嵌入有界盒子，且陈数满足定量界。
On the fragility of periodic tori for families of symplectic twist maps
本文研究辛扭转映射族中Lagrange周期环面在势扰动下的脆弱性，证明非退化条件下指定旋转向量的周期环面仅能在有限参数值存在，并推出可积映射的刚性结果。
We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to certain birational surgeries like blow-downs and flips. As a consequence, there are many non-toric (non-compact) monotone symplectic manifolds whose wrapped Fukaya categories are proper.
Monotone twist maps and Dowker-type theorems
本文证明了关于多边形面积与周长的四个经典 Dowker 型定理均是相应台球系统 Mather $β$ 函数凸性质的体现，并据此为多种台球系统及更高旋转数推导出了新的几何不等式。
Floer potentials, cluster algebras and quiver representations
本文利用cluster代数，将toric del Pezzo曲面上单调Lagrangian环面的Floer势解释为quiver表示的cluster特征。
On Lagrangian Tori in $S^2\times S^2$
本文对 FOOO 构造的 $S^2\times S^2$ 中的环面纤维进行了全面分类，判断其是否与标准环面结构的纤维哈密顿同痕。
Lagrangian split tori in $S^2 \times S^2$ and billiards
本文证明 $S^2 \times S^2$ 中分裂拉格朗日环面的哈密顿同痕分类等价于矩形台球问题，并据此解决拉格朗日填充数、环面空间拓扑及辛球嵌入像等问题。
In this paper we extend the discussion on Homological Mirror Symmetry for Fano toric varieties presented by Hori and Vafa to more general case of monotone symplectic manifolds with real polarizations. We claim that the Hori -- Vafa prediction, proven by Cho and Oh for toric Fano varieties, can be checked in much more wider context. Then the notion of Bohr - Sommerfeld with respect to the canonical class lagrangian submanifold appears and plays an important role. The discussion presents a bridge between Geometric Quantization and Homological Mirror Symmetry programmes both applied to the projec…
We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn-Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian $…
We prove a general form of the wall-crossing formula which relates the disk potentials of monotone Lagrangian submanifolds with their Floer-theoretic behavior away from a Donaldson divisor. We define geometric operations called mutations of Lagrangian tori in del Pezzo surfaces and in toric Fano varieties of higher dimension, and study the corresponding wall-crossing formulas that compute the disk potential of a mutated torus from that of the original one. In the case of del Pezzo surfaces, this justifies the connection between Vianna's tori and the theory of mutations of Landau-Ginzburg seeds…
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