Lagrangian isotopies in Stein manifolds

Almost all standard Lagrangian tori in C^n are not Hamiltonian volume minimizing

In 1993, Y.-G. Oh proposed a problem whether standard Lagrangian tori in C^n are volume minimizing under Hamiltonian isotopies of C^n. In this article, we prove that most of them do not have such property if the dimension n is greater than two. We also discuss the existence of Hamiltonian non-volume minimizing Lagrangian torus orbits of compact toric Kahler manifolds.

Non-isotopic monotone Lagrangian submanifolds of $\mathbb{C}^n$

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…

Notes on monotone Lagrangian twist tori

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.

Positive Legendrian isotopies and Floer Theory

Positive loops of Legendrian embeddings are examined from the point of view of Floer homology of Lagrangian cobordisms. This leads to new obstructions to the existence of a positive loop containing a given Legendrian, expressed in terms of the Legendrian contact homology of the Legendrian submanifold. As applications, old and new examples of orderable contact manifolds are obtained and discussed. We also show that contact manifolds admitting a filling of a Liouville domain with non-zero symplectic homology is strongly orderable in the sense of Liu.

Deformations of Lagrangian submanifolds in log-symplectic manifolds

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 …

辛形式与拉格朗日子流形的同时变形

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。

Sheaf quantization of Hamiltonian isotopies and applications to non displaceability problems

Let I be an open interval, M be a real manifold, T*M its cotangent bundle and Φ={ϕ_t}, t in I, a homogeneous Hamiltonian isotopy of T*M defined outside the zero-section. Let Λbe the conic Lagrangian submanifold associated with Φ(Λis a subset of T*M x T*M x T*I). We prove the existence and unicity of a sheaf K on MxMxI whose microsupport is contained in the union of Λand the zero-section and whose restriction to t=0 is the constant sheaf on the diagonal of MxM. We give applications of this result to problems of non displaceability in contact and symplectic topology. In particular we prove that …

Disjoinable Lagrangian spheres and dilations

We consider open symplectic manifolds which admit dilations (in the sense previously introduced by Solomon and the author). We obtain restrictions on collections of Lagrangian submanifolds which are pairwise disjoint (or pairwise disjoinable by Hamiltonian isotopies) inside such manifolds. This includes the Milnor fibres of isolated hypersurface singularities which have been stabilized (by adding quadratic terms) sufficiently often.

圆盘内拉格朗日链上的 Hofer 距离

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。

Symplectic Structures on the cotangent bundles of open 4-manifolds

We show that, for any two orientable smooth open 4-manifolds $X_0,X_1$ which are homeomorphic, their cotangent bundles $T^*X_0,T^*X_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to $\mathbb{R}^4$, the standard Stein structure on $T^*R$ is Stein homotopic to the standard Stein structure on $T^*\mathbb{R}^4 = \mathbb{R}^8$. We use this to show that any exotic $\mathbb{R}^4$ embeds in the standard symplectic $\mathbb{R}^8$ as a Lagrangian submanifold. As a corollary, we show that $\mathbb{R}^8$ has uncountably many smoot…

近 Lefschetz 纤维化、Stein 结构与正则 Lagrangian 子流形

Nearly Lefschetz fibrations, Stein structures, and regular Lagrangians

研究近Lefschetz纤维化上的Stein结构，证明四维Weinstein域中具Legendrian边界的Lagrangian圆盘均为正则，解决邻近猜想与Slice-Ribbon猜想的Lagrangian类比，并给出构造Weinstein Kirby图的算法。

辛通量几何与拉格朗日环面纤维化

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。

Exact Lagrangians in plumbings

Consider a Stein manifold M obtained by plumbing cotangent bundles of manifolds of dimension greater than or equal to 3 at points. We prove that the Fukaya category of closed exact Lagrangians with vanishing Maslov class in M is generated by the compact cores of the plumbing. As applications, we classify exact Lagrangian spheres in A2-Milnor fibres of arbitrary dimension, derive constraints on exact Lagrangian fillings of Legendrian unknots in disk cotangent bundles, and prove that the categorical equivalence given by the spherical twist in a homology sphere is typically not realised by any co…

Classification of multiplicity free Hamiltonian actions of complex tori on Stein manifolds

A Hamiltonian action of a complex torus on a symplectic complex manifold is said to be {\it multiplicity free} if a general orbit is a lagrangian submanifold. To any multiplicity free Hamiltonian action of a complex torus $T\cong (\C^\times)^n$ on a Stein manifold $X$ we assign a certain 5-tuple consisting of a Stein manifold $Y$, an étale map $Y\to \t^*$, a set of divisors on $Y$ and elements of $H^2(Y,\Z)^{\oplus n}, H^2(Y,\C)$. We show that $X$ is uniquely determined by this invariants. Furthermore, we describe all 5-tuples arising in this way.

Lefschetz Fibrations on Compact Stein Manifolds

Here we prove that up to diffeomorphism every compact Stein manifold W of dimension 2n+2>4 admits a Lefschetz fibration over the two-disk with Stein regular fibers, such that the monodromy of the fibration is a symplectomorphism induced by compositions of right-handed Dehn twists along embedded Lagrangian n-spheres on the generic fiber. This generalizes the Stein surface case of n=1, previously proven by Loi-Piergallini and Akbulut-Ozbagci. More precisely, we show that up to Liouville isomorphism any Weinstein domain W admits a compatible compact convex Lefschetz fibration with Weinstein regul…

Infinitely many Lagrangian fillings

We prove that all maximal-tb Legendrian torus links (n,m) in the standard contact 3-sphere, except for (2,m),(3,3),(3,4) and (3,5), admit infinitely many Lagrangian fillings in the standard symplectic 4-ball. This is proven by constructing infinite order Lagrangian concordances which induce faithful actions of the modular group PSL(2,Z) and the mapping class group M(0,4) into the coordinate rings of algebraic varieties associated to Legendrian links. Our results imply that there exist Lagrangian concordance monoids with subgroups of exponential-growth, and yield Stein surfaces homotopic to a 2…

四维空间中的半局域奇异拉格朗日环面

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。

干净的拉格朗日相交的无结性保持

Persistence of unknottedness of clean Lagrangian intersections

该研究证明了在特定条件下，两个拉格朗日球面的无结相交性质在附近哈密顿同伦下保持，回答了 Smith 的问题。

The Kunneth formula in Floer homology for manifolds with restricted contact type boundary

We prove the Kunneth formula in Floer (co)homology for manifolds with restricted contact type boundary. We use Viterbo's definition of Floer homology, involving the symplectic completion by adding a positive cone over the boundary. The Kunneth formula implies the vanishing of Floer (co)homology for subcritical Stein manifolds. Other applications include the Weinstein conjecture in certain product manifolds, obstructions to exact Lagrangian embeddings, existence of holomorphic curves with Lagrangian boundary condition, as well as symplectic capacities.

部分全纯关系的局部 h-原理

Local H-Principles for Partial Holomorphic Relations

本文通过引入部分全纯关系的实化概念，证明了当开部分全纯关系的实化满足相对 h-原理时，Stein 流形上的形式解可在 Lagrangian 骨架邻域内形变为全纯解，从而将局部 h-原理推广到复偶接触、复 Engel 及复共形辛结构等情形。

Propagation in Hamiltonian dynamics and relative symplectic homology

The main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof is based on Floer homology and on the notion of a relative symplectic capacity. Applications include results about propagation properties of sequential Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle actions, and smooth Lagrangian skeletons in Stein manifolds.

Double bubble plumbings and two-curve flops

We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to local threefolds which contain two floppable $(-1,-1)$-curves meeting at a point. Using contraction algebras we classify spherical objects on the B-side, and derive topological consequences including a complete description of the homology classes realised by graded exact Lagrangians.

Hamiltonian dynamics on convex symplectic manifolds

We study the dynamics of Hamiltonian diffeomorphisms on convex symplectic manifolds. To this end we first establish the Piunikhin-Salamon-Schwarz isomorphism between the Floer homology and the Morse homology of such a manifold, and then use this isomorphism to construct a biinvariant metric on the group of compactly supported Hamiltonian diffeomorphisms analogous to the metrics constructed by Viterbo and Schwarz. These tools are then applied to prove and reprove results in Hamiltonian dynamics. Our applications comprise a uniform lower estimate for the slow entropy of a compactly supported Ham…