Mei-Lin Yau
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.
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We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon.
We determine the Lagrangian monodromy group L(T) and the smooth monodromy group S(T) of a Clifford torus T in the symplectic 4-space. We show that L(T) is isomorphic to the infinite dihedral group, and S(T) is generated by three reflections. We give explicit formulas for both groups. We also show that if a Lagrangian torus is smoothly isotopic to a Clifford torus then the smooth isotopy can be chosen to be Lagrangian outside of a disc.
We develop a new algebraic theory of positive braids and conjugacy classes in the braid group $B_3$. We use our theory to establish a complete classification of isotopy classes of degree three symplectic curves in $\mathbb{CP}^2$ with only $A_n$-singularities for $n\geq 1$ (an $A_n$-singularity is locally modelled by the equation $z^2 = w^n$) independent of Gromov's theory of pseudoholomorphic curves. We show that if $C$ and $C'$ are degree three symplectic curves in $\mathbb{CP}^2$ with the same numbers of $A_n$-singularities for each $n\geq 1$, then $C$ is isotopic to $C'$. Furthermore, our …
We introduce and develop a language of semigroups over the braid groups for a study of braid monodromy factorizations (bmf's) of plane algebraic curves and other related objects. As an application we give a new proof of Orevkov's theorem on realization of a bmf over a disc by algebraic curves and show that the complexity of such a realization can not be bounded in terms of the types of the factors of the bmf. Besides, we prove that the type of a bmf is distinguishing Hurwitz curves with singularities of inseparable types up to $H$-isotopy and $J$-holomorphic cuspidal curves in $\C P^2$ up to s…
We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space $\mathbb{R}^4$, the projective plane $\mathbb{C}P^2$, and the monotone $S^2 \times S^2$. The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for $T^*\mathbb{T}^2$, i.e.~it is shown that every closed exact Lagrangia…
Homological Lagrangian monodromy and Wang exact sequence
该研究利用能量阈值条件证明 Lagrangian 子流形的同调单值性在无低能 J-全纯球面或圆盘时为平凡。
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.
In this article, using the idea of toric degeneration and the computation of the full potential function of Hirzebruch surface $F_2$, which is \emph{not} Fano, we produce a continuum of Lagrangian tori in $S^2 \times S^2$ which are non-displaceable under the Hamiltonian isotopy.
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。
The presence of focus-focus singularities in semi-toric integrables Hamiltonian systems is one of the reasons why there cannot exist global Action-Angle coordinates on such systems. At focus-focus critical points, the Liouville-Arnold-Mineur theorem does not apply. In particular, the affine structure of the image of the moment map around has non-trivial monodromy. In this article, we establish that the singular behaviour and the multi-valuedness of the Action integrals is given by a complex logarithm. This extends a previous result by Vu Ngoc to any dimension. We also calculate the monodromy m…
A natural way of generalising Hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map. For a class of such ``almost-toric 4-manifolds'' which admits a Hamiltonian $S^1$-action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of Atiyah, Guillemin-Sternberg. As an application, we derive a Duistermaat-Heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system.
Contact homology for Legendrian submanifolds in standard contact $(2n+1)$-space is rigorously defined using moduli spaces of holomorphic disks with Lagrangian boundary conditions in complex $n$-space. It provides new invariants of Legendrian isotopy. Using these invariants the theory of Legendrian isotopy is shown to be very rich. For example, infinite families of pairwise non-isotopic Legendrian $n$-spheres and $n$-tori, which are indistinguishable by means of previously known invariants, are constructed. In a sense, the definition of contact homology presented in this paper is a high dimensi…
We construct a Legendrian 2-torus in the 1-jet space of $S^1\times\R$ (or of $\R^2$) from a loop of Legendrian knots in the 1-jet space of $\R$. The differential graded algebra (DGA) for the Legendrian contact homology of the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit…
We encode the variation structure of a quasihomogeneous polynomial with an isolated singularity as introduced by Nemethi in a set of spectral flows of the signature operator on the Milnor bundle by varying global elliptic boundary conditions in a specific way using the quasihomogeneous circle action on the Brieskorn lattice. For this, we use adiabatic techniques and well-known results on spectral flow and Maslov index. Furthermore we interpret the inequality of a certain member of this family of spectral flows with a spectral flow induced by a Reeb flow on the boundary of the Milnor fibre as g…
In this article, we consider the Floer cohomology (with $\Z_2$ coefficients) between torus fibers and the real Lagrangian in Fano toric manifolds. We first investigate the conditions under which the Floer cohomology is defined, and then develop a combinatorial description of the Floer complex based on the polytope of the toric manifold. We show that if the Floer cohomology is defined, and the Floer cohomology of the torus fiber is non-zero, then the Floer cohomology of the pair is non-zero. We use this result to develop some applications to non-displaceability and the minimum number of interse…
We show the smooth version of the nearby Lagrangian conjecture for the 2-dimensional pair of pants and the Hamiltonian version for the cylinder. In other words, for any closed exact Lagrangian submanifold of $T^{*}M$, there is a smooth or Hamiltonian isotopy, when $M$ is a pair of pants or a cylinder respectively, from it to the 0-section. For the cylinder we modify a result of G. Dimitroglou Rizell for certain Lagrangian tori to show that it gives the Hamiltonian isotopy for a Lagrangian cylinder. For the pair of pants, we first study some results from pseudo-holomorphic curve theory and the …
The Lagrangian flux-monodromy morphism
研究在拉格朗日子流形空间基本群上定义的新同态，考虑拉格朗日通量和单值性，通过研究其像的离散性与哈密顿轨道拓扑的关系，对\(\mathbb{R}^4\)、\(\mathbb{C}P^2\)及\(\mathbb{C}P^n\)中的拉格朗日环面进行描述并提升相关结果拓扑。
A near-symplectic structure on a 4-manifold is a closed 2-form that is symplectic away from the 1-dimensional submanifold along which it vanishes and that satisfies a certain transversality condition along this vanishing locus. We investigate near-symplectic 4-manifolds equipped with singular Lagrangian torus fibrations which are locally induced by effective Hamiltonian torus actions. We show how such a structure is completely characterized by a singular integral affine structure on the base of the fibration whenever the vanishing locus is nonempty. The base equipped with this geometric struct…
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。
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。
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…
We construct distinguished elements in the embedded contact homology (and monopole Floer homology) of a 3-torus, associated with Lagrangian tori in symplectic 4-manifolds and their isotopy classes. They turn out not to be new invariants, instead they repackage the Gromov (and Seiberg-Witten) invariants of various torus surgeries. We then recover a result of Morgan-Mrowka-Szabó on product formulas for the Seiberg-Witten invariants along 3-tori.
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…
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 classify weakly exact, rational Lagrangian tori in $T^* \mathbb{T}^2- 0_{\mathbb{T}^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on $\mathbb{T}^n$ and also has applications to symplectic topology. As a first corollary, we strengthen a result due independently to Eliashberg-Polterovich and to Giroux describing Lagrangian tori in $T^* \mathbb{T}^2-0_{\mathbb{T}^2}$ which are homologous to the zero section. As a second corollary, we exhibit pairs of disjoint totally real tori $K_1, K_2 \subset T^*\mathbb{T}^2$, each of which is isotopic…
Geometry of symplectic flux and Lagrangian torus fibrations
本文通过Fukaya代数定义数值不变量研究辛通量，揭示其在线性通量同伦上的凹性，并推导Fano簇Gelfand-Cetlin纤维的约束及复射影平面纤维空间描述。
M. M. Nekhoroshev put forward the problem of to find the Complex Germ on a isotropic invariant torus with respect to Hamiltonian phases flows which come from k-functions in involution. This statement was partially solved in [9] establishing that if certain simplectic operator has a simple spectrum then the complex germ exist. In this work we solve this problem, providing a full solution, i.e. we present conditions for the existence and uniqueness of complex germ through the monodromy operator constructed in [9], but without the simple spectrum condition. We study also the Hamiltonian system wi…
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 $…
In this paper, we first provide an explicit description of {\it all} holomorphic discs (``disc instantons'') attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute Fukaya-Oh-Ohta-Ono's (FOOO's) obstruction (co)chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter $T^{2π} = e^{-1}$, our computation verifies the folklore that FOOO's obstruction (co)chains correspond to the Landau-Ginzburg superpotentials under the mirror symmetry correspo…
Exact Lagrangians in four-dimensional symplectisations
本文研究过度扭转接触三维流形辛化中的精确 Lagrangian 嵌入，显式构造环面与 Klein 瓶，并利用正回路展示拓扑环绕却动态非互链的 Legendrian 例子。
Lagrangian Spheres and Cyclic Quotient T-singularities
本文通过构造Lefschetz纤维化与J-全纯叶状结构，将循环商T-奇点Milnor纤维中拉格朗日球面的同位素分类归结为映射类群计算，并证明其辛映射类群由Dehn扭转及特定单值化辛同胚生成。
In this paper we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, S^2 or RP^2, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction this is a natural extension of McDuff's connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoreti…
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…
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