The Nonlinear Maslov index and the Calabi homomorphism

The Calabi homomorphism, Lagrangian paths and special Lagrangians

Let $\OO$ be an orbit of the group of Hamiltonian symplectomorphisms acting on the space of Lagrangian submanifolds of a symplectic manifold $(X,ω).$ We define a functional $\CC:\OO \to \R$ for each differential form $β$ of middle degree satisfying $β\wedge ω= 0$ and an exactness condition. If the exactness condition does not hold, $\CC$ is defined on the universal cover of $\OO.$ A particular instance of $\CC$ recovers the Calabi homomorphism. If $β$ is the imaginary part of a holomorphic volume form, the critical points of $\CC$ are special Lagrangian submanifolds. We present evidence that $…

纽结谱不变量的次领头渐近行为与曲面的同胚群

Subleading asymptotics of link spectral invariants and homeomorphism groups of surfaces

本文研究亏格为零的曲面上纽结谱不变量的次领头渐近行为，证明其对光滑含时哈密顿量有界、对有限临界值自治圆盘映射可恢复Ruelle不变量，并推导了Calabi同态的延拓性质与hameomorphism群上Calabi同态核的非单性。

The Calabi invariant for some groups of homeomorphisms

We show that the Calabi homomorphism extends to some groups of homeomorphisms on exact symplectic manifolds. The construction is based on the uniqueness of the generating Hamiltonian (proved by Viterbo) of continuous Hamiltonian isotopies (introduced by Mueller and Oh).

Relative Flux Homomorphism in Symplectic Geometry

In this work we define a relative version of the flux homomorphism, introduced by Calabi in 1969, for a symplectic manifold. We use it to study (the universal cover of) the group of symplectomorphisms of a symplectic manifold leaving a Lagrangian submanifold invariant. We show that some quotients of this group are stable under symplectic reduction.

Real Lagrangians in Calabi-Yau Threefolds

We compute the mod $2$ cohomology groups of real Lagrangians in Calabi-Yau threefolds using well-behaved torus fibrations constructed by Gross. To do this we study a long exact sequence introduced by Castaño-Bernard and Matessi, which relates the cohomology of the Lagrangians to the cohomology of the Calabi-Yau. We show that the connecting homomorphism in this sequence is given by the map squaring divisor classes in the mirror Calabi-Yau.

Calabi quasi-morphisms for some non-monotone symplectic manifolds

In this work we construct Calabi quasi-morphisms on the universal cover of the group Ham(M) of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasi-morphisms descend to non-trivial homomorphisms on the fundamental group of Ham(M).

The flux homomorphism and central extensions of diffeomorphism groups

Let $D$ be a 2-dimensional closed unit disk and $\rm{Symp}(D,0)_{\rm{rel}}$ the group of symplectomorphisms preserving the origin and the boundary $\partial D$ pointwise. We consider the $\mathbb{R}$-valued flux homomorphism on $\rm{Symp}(D,0)_{\rm{rel}}$ and define the central $\mathbb{R}$-extension called the $\mathbb{R}$-valued flux extension. We determine the Euler class of this extension and investigate the relation between the extension, the group $2$-cocycle defined by Ismagilov, Losik, and Michor, and the Calabi invariant of $D$.

The Action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures

We extend the definition of Weinstein's Action homomorphism to Hamiltonian actions with equivariant moment maps of (possibly infinite-dimensional) Lie groups on symplectic manifolds, and show that under conditions including a uniform bound on the symplectic areas of geodesic triangles the resulting homomorphism extends to a quasimorphism on the universal cover of the group. We apply these principles to finite dimensional Hermitian Lie groups like Sp(2n,R), reinterpreting the Guichardet-Wigner quasimorphisms, and to the infinite dimensional groups of Hamiltonian diffeomorphisms Ham(M,\om) of cl…

Quasi-morphismes et invariant de Calabi

In this paper, we give two elementary constructions of homogeneous quasi-morphisms defined on the group of Hamiltonian diffeomorphisms of certain closed connected symplectic manifolds (or on its universal cover). The first quasi-morphism, denoted by $\calabi\_{S}$, is defined on the group of Hamiltonian diffeomorphisms of a closed oriented surface $S$ of genus greater than 1. This construction is motivated by a question of M. Entov and L. Polterovich. If $U\subset S$ is a disk or an annulus, the restriction of $\calabi\_{S}$ to the subgroup of diffeomorphisms which are the time one map of a Ha…

定量 Heegaard Floer 上同调与 Calabi 不变量

Quantitative Heegaard Floer cohomology and the Calabi invariant

本文在任意亏格紧致曲面上定义与 Lagrangian links 相关的新谱不变量，证明其极限恢复 Calabi 不变量，并借此解决 Hamiltonian 同胚群非单性、Calabi 同态延拓及二维球面 quasimorphisms 构造等问题。

$\hat{G}$-invariant quasimorphisms and symplectic geometry of surfaces

Let $\hat{G}$ be a group and $G$ its normal subgroup. In this paper, we study $\hat{G}$-invariant quasimorphisms on $G$ which appear in symplectic geometry and low dimensional topology. As its application, we prove the non-existence of a section of the flux homomorphism on closed surfaces of higher genus. We also prove that Py's Calabi quasimorphism and Entov-Polterovich's partial Calabi quasimorphism are non-extendable to the group of symplectomorphisms. We show that Py's Calabi quasimorphism is the unique non-extendable quasimorphism to some group.

Continuous Hamiltonian dynamics and area-preserving homeomorphism group of $D^2$

The main purpose of this paper is to propose a scheme of a proof of the nonsimpleness of the group $Homeo^Ω(D^2,\partial D^2)$ of area preserving homeomorphisms of the 2-disc $D^2$. We first establish the existence of Alexander isotopy in the category of Hamiltonian homeomorphisms. This reduces the question of extendability of the well-known Calabi homomorphism ${\rm Cal}:Diff^Ω(D^1,\partial D^2) \to \mathbb R$ to a homomorphism $\overline{\rm Cal}:Hameo(D^2,\partial D^2) \to \mathbb R$ to that of the vanishing of the basic phase function $f_{\underline{\mathbb F}}$, a Floer theoretic graph se…

特殊Lagrangian网络

Special Lagrangian webbing

论文构造了接近正Lagrangian子流形相交点的虚特殊Lagrangian圆柱体家族，获得了开放正Lagrangian子流形的测地线，并首次尝试构建闭正Lagrangian子流形的测地线。同时，提出了一种证明正Lagrangian测地线在非光滑处C^{1,1}正则性的方法。

由Shelukhin拟同态构造的通量同态与双线性形式

Flux homomorphism and bilinear form constructed from Shelukhin's quasimorphism

本文从Shelukhin拟同态出发，在闭连通辛流形的实一阶上同调上构造了辛映射类群不变的交替双线性形式，并由此给出可交换辛同胚通量的约束，推广了Rousseau关于开辛流形的工作。