Symplectic triangle inequality

Exact triangle for fibered Dehn twists

We use quilted Floer theory to generalize Seidel's long exact sequence in symplectic Floer theory to fibered Dehn twists. We then apply it to construct versions of the Floer and Khovanov-Rozansky exact triangles in Lagrangian Floer theory of moduli spaces of bundles.

Applications of Grothendieck's inequality to linear symplectic geometry

Recently in symplectic geometry there arose an interest in bounding various functionals on spaces of matrices. It appears that Grothendieck's theorems about factorization are a useful tool for proving such bounds. In this note we present two such applications.

Noncontractible loops of symplectic embeddings between convex toric domains

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 i…

辛三分法与附加不等式

Symplectic trisections and the adjunction inequality

该研究为闭辛4-流形建立附加不等式，运用接触几何与4-流形三分法归约问题，通过霍万诺夫同调避免规范理论，可用于给出4-流形拓扑里程碑结果的无规范理论证明。

Symplectic displacement energy for exact Lagrangian immersions

We give an inequality of the displacement energy for exact Lagrangian immersions and the symplectic area of punctured holomorphic discs. Our approach is based on Floer homology for Lagrangian immersions and Chekanov's homotopy technique of continuations.

On a local systolic inequality for odd-symplectic forms

The aim of this paper is to formulate a local systolic inequality for odd-symplectic forms (also known as Hamiltonian structures) and to establish it in some basic cases. Let $Ω$ be an odd-symplectic form on an oriented closed manifold $Σ$ of odd dimension. We say that $Ω$ is Zoll if the trajectories of the flow given by $Ω$ are the orbits of a free $S^1$-action. After defining the volume of $Ω$ and the action of its periodic orbits, we prove that the volume and the action satisfy a polynomial equation, provided $Ω$ is Zoll. This builds the equality case of a conjectural systolic inequality fo…

Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality

We prove upper bounds for the number of critical points in semistable symplectic Lefschetz fibrations. We also obtain a new lower bound for the number of nonseparting vanishing cycles in Lefschetz pencils, and reprove the known lower bounds for the commutator lengths of Dehn twists.

A Brunn-Minkowski type inequality for extended symplectic capacities of convex domains and length estimate for a class of billiard trajectories

In this paper, we firstly generalize the Brunn-Minkowski type inequality for Ekeland-Hofer-Zehnder symplectic capacity of bounded convex domains established by Artstein-Avidan-Ostrover in 2008 to extended symplectic capacities of bounded convex domains constructed by authors based on a class of Hamiltonian non-periodic boundary value problems recently. Then we introduce a class of non-periodic billiards in convex domains, and for them we prove some corresponding results to those for periodic billiards in convex domains obtained by Artstein-Avidan-Ostrover in 2012.

Lagrangian non-squeezing and a geometric inequality

We prove that if the unit codisc bundle of a closed Riemannian manifold embeds symplectically into a symplectic cylinder of radius one then the length of the shortest nontrivial closed geodesic is at most half the area of the unit disc.

An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology

Our main result gives an adjunction inequality for embedded surfaces in certain $4$-manifolds with contact boundary under a non-vanishing assumption on the Bauer--Furuta type invariants. Using this, we give infinitely many knots in $S^3$ that are not smoothly H-slice (that is, bounding a null-homologous disk) in many $4$-manifolds but they are topologically H-slice. In particular, we give such knots in the boundaries of the punctured elliptic surfaces $E(2n)$. In addition, we give obstructions to codimension-0 orientation-reversing embedding of weak symplectic fillings with $b_3=0$ into closed…

A Symplectic Instanton Homology via Traceless Character Varieties

Since its inception, Floer homology has been an important tool in low-dimensional topology. Floer theoretic invariants of $3$-manifolds tend to be either gauge theoretic or symplecto-geometric in nature, and there is a general philosophy that each gauge theoretic Floer homology should have a corresponding symplectic Floer homology and vice-versa. In this article, we construct a Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ (called the symplectic instanton homology of $Y$ and denoted $\mathrm{SI}(Y)$) which is conjecturally equivalent to a Floer homology defined using a c…

Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold

Given an SO(3)-bundle with connection, the associated two-sphere bundle carries a natural closed 2-form. Asking that this be symplectic gives a curvature inequality first considered by Reznikov. We study this inequality in the case when the base has dimension four, with three main aims. Firstly, we use this approach to construct symplectic six-manifolds with c_1=0 which are never Kahler; e.g., we produce such manifolds with b_1=0=b_3 and also with c_2.omega <0, answering questions posed by Smith-Thomas-Yau. Examples come from Riemannian geometry, via the Levi-Civita connection on Lambda^+. The…

A local systolic-diastolic inequality in contact and symplectic geometry

Let $Σ$ be a connected closed three-manifold, and let $t_Σ$ be the order of the torsion subgroup of $H_1(Σ;\mathbb Z)$. For a contact form $α$ on $Σ$, we denote by $\mathrm{Volume}(α)$ the contact volume of $α$, and by $T_{\min}(α)$ and $T_{\max}(α)$ the minimal period and the maximal period of prime periodic orbits of the Reeb flow of $α$ respectively. We say that $α$ is Zoll if its Reeb flow generates a free $S^1$-action on $Σ$. We prove that every Zoll contact form $α_*$ on $Σ$ admits a $C^3$-neighbourhood $\mathcal U$ in the space of contact forms such that \[ t_ΣT_{\min}(α)^2\leq \mathrm{…

Symplectic embeddings and continued fractions: a survey

As has been known since the time of Gromov's Nonsqueezing Theorem, symplectic embedding questions lie at the heart of symplectic geometry. After surveying some of the most important ways of measuring the size of a symplectic set, these notes discuss some recent developments concerning the question of when a 4-dimensional ellipsoid can be symplectically embedded in a ball. This problem turns out to have unexpected relations to the properties of continued fractions and of exceptional curves in blow ups of the complex projective plane. It is also related to questions of lattice packing of planar …

圆盘状截面与辛容量

Disk-like surfaces of section and symplectic capacities

本文证明R^4中动态凸区域的柱容量等于其边界Reeb流圆盘状全局截面的最小辛面积，并对C^3接近圆球的凸区域证明强Viterbo猜想，推广了已有收缩不等式。

The equivariant pair-of-pants product in fixed point Floer cohomology

The Floer cohomology of a symplectic automorphism and that of its square are related by the pair-of-pants product. For exact symplectic automorphisms, we introduce an equivariant version of that product, and use it to prove a Smith-type inequality of ranks between Floer cohomology groups. Under additional topological assumptions, the same inequality was previously proved by Hendricks, using a different strategy.

Local middle dimensional symplectic non-squeezing in the analytic setting

We prove the following middle-dimensional non-squeezing result for analytic symplectic embeddings of domains in $\mathbb{R}^{2n}$. Let $φ: D \hookrightarrow \mathbb{R}^{2n}$ be an analytic symplectic embedding of a domain $D \subset \mathbb{R}^{2n}$ and $P$ be a symplectic projector onto a linear $2k$-dimensional symplectic subspace $V\subset \mathbb{R}^{2n}$. Then there exists a positive function $r_0:D\rightarrow (0,+ \infty)$, bounded away from $0$ on compact subsets $K \subset D$, such that the inequality $Vol_{2k}(Pφ(B_r(x)),ω^k _{0|V})\geq π^{k} r^{2k}$ holds for every $x \in D$ and for …

A Frölicher-type inequality for generalized complex manifolds

We prove a Frölicher-type inequality for a compact generalized complex manifold $M$, and show that the equality holds if and only if $M$ satisfies the generalized $\partial\bar{\partial}$-Lemma. In particular, this gives a unified proof of analogous results in the complex and symplectic cases.

The codisc radius capacity

We define a capacity which measures the size of Weinstein tubular neighbourhoods of Lagrangian submanifolds. In symplectic vector spaces this leads to bounds on the codisc radius for any closed Lagrangian submanifold in terms of Viterbo's isoperimetric inequality. Moreover, we prove a generalization of Gromov's packing inequality concerning symplectic embeddings of the boundaries of two balls of equal radius into the open unit ball. If the interior components of the image spheres are disjoint, then the radii are less than the square root of one half. Furthermore, we introduce the spherical var…

Limits of Riemannian 4-manifolds and the symplectic geometry of their twistor spaces

The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $ω$. This article studies limits of manifolds for which $ω$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperkähler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorph…

Localization for involutions in Floer cohomology

We consider Lagrangian Floer cohomology for a pair of Lagrangian submanifolds in a symplectic manifold M. Suppose that M carries a symplectic involution, which preserves both submanifolds. Under various topological hypotheses, we prove a localization theorem for Floer cohomology, which implies a Smith-type inequality for the Floer cohomology groups in M and its fixed point set. Two applications to symplectic Khovanov cohomology are included.

A linear isoperimetric inequality for the punctured Euclidean plane

It follows from a general theorem of Bonk and Eremenko that closed plane curves which are contractible in the complement to the integral lattice satisfy a linear isoperimetric inequality. We give an alternative proof of this fact. Our approach is based on a non-standard combinatorial isoperimetric inequality which requires a refinement of the small cancellation theory. We present an application of the isoperimetric inequality for the punctured plane to Hamiltonian dynamics. Combining it with methods of symplectic topology we show that every non-identical Hamiltonian diffeomorphism of the 2-tor…