On the capacity of Lagrangians in the cotangent disc bundle of the torus

平坦环面圆盘余切丛的辛容量

Symplectic capacities of disc cotangent bundles of flat tori

本文证明了平坦Riemannian环面及Finsler环面单位圆盘余切丛上的归一化容量等于2倍systole。

Peterson conjecture via Lagrangian correspondences and wonderful compactifications

For a simply-connected compact semisimple Lie group $G$ and its maximal torus $T$, we study the $A_{\infty}$-functor associated to the moment Lagrangian correspondence from the cotangent bundle $T^*G$ to the square $G/T^{-} \times G/T$. In particular, we compute the leading term of the $A_{\infty}$-homomorphism from the wrapped Floer cohomology $HW^*(T^*_e G, T^*_e G)$ of the cotangent fiber $T_e^*G$ to the Floer cohomology $HF^*(Δ, Δ)$ of the diagonal $Δ$ in the square $G/T^{-} \times G/T$ by determining the count of certain pseudo-holomorphic quilts. As a consequence, we prove that the Floer…

Symplectic homology of disc cotangent bundles of domains in Euclidean space

Let $V$ be a bounded domain with smooth boundary in $\R^n$, and $D^*V$ denote its disc cotangent bundle. We compute symplectic homology of $D^*V$, in terms of relative homology of loop spaces on the closure of $V$. We use this result to show that Floer-Hofer capacity of $D^*V$ is between $2r(V)$ and $2(n+1)r(V)$, where $r(V)$ denotes inradius of $V$. As an application, we study periodic billiard trajectories on $V$.

Fillings and fittings of unit cotangent bundles of odd-dimensional spheres

We introduce the concept of fittings to symplectic fillings of the unit cotangent bundle of odd-dimensional spheres. Assuming symplectic asphericity we show that all fittings are diffeomorphic to the respective unit co-disc bundle.

圆盘余切丛中关于谱距离的广义Viterbo猜想的一个反例

A counterexample to the generalized Viterbo conjecture on the spectral distance in a disc cotangent bundle

该研究构造了圆盘余切丛中广义Viterbo猜想的反例，证明特定拉格朗日量空间上的谱距离无界，还通过Floer理论分类了ℤ₂上该猜想成立的所有曲面。

Floer homology in the cotangent bundle of a closed Finsler manifold and noncontractible periodic orbits

We show that the existence of noncontractible periodic orbits for compactly supported time-dependent Hamiltonian on the disk cotangent bundle of a Finsler manifold provided that the Hamiltonian is sufficiently large over the zero section. We generalize the BPS capacities and earlier constructions of Weber (2006 Duke Math. J. 133, 527-568) and other authors Biran et al (2003 Duke Math. J. 119, 65-118) to the Finsler setting. We then obtain a number of applications including: (1) generalizing the main theorem of Xue (2017 J. Symplectic Geom. 15, 905-936) to the Lie group setting, (2) preservatio…

Lagrangian isotopy of tori in $S^2 \times S^2$ and $\mathbb{C}P^2$

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…

Exact Lagrangian tori in $T^*\mathbb{T}^n$

We show that for $n\geq 2$ there exists an exact Lagrangian submanifold $L$ in the cotangent bundle $T^*\mathbb{T}^n$ of the $n$-dimensional torus $\mathbb{T}^n$ such that $L$ is symplectically but not Hamiltonian isotopic to the zero section of $T^*\mathbb{T}^n$.

Invariance of symplectic cohomology and twisted cotangent bundles over surfaces

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be non-exact and non-compactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of t…

余切丛中的黎曼距离与辛嵌入

Riemannian distance and symplectic embeddings in cotangent bundle

本文利用标准球到余切丛邻域的辛嵌入定义类距离函数，证明其在单位圆盘余切丛情形恢复黎曼度量，并证明强 Viterbo 猜想及两球辛填充数界。

The symplectic geometry of cotangent bundles from a categorical viewpoint

We describe various approaches to understanding Fukaya categories of cotangent bundles. All of the approaches rely on introducing a suitable class of noncompact Lagrangian submanifolds. We review the work of Nadler-Zaslow (math/0604379, math/0612399) and the authors (math/0701783), before discussing a new approach using family Floer cohomology and the ``wrapped Fukaya category''. The latter, inspired by Viterbo's symplectic homology, emphasises the connection to loop spaces, hence seems particularly suitable when trying to extend the existing theory beyond the simply-connected case.

关于奥克利 - 厄舍型拉格朗日量

On Lagrangians of Oakley-Usher type

研究非环面拉格朗日子流形辛（不）打结性，构造受奥克利 - 厄舍拉格朗日量启发，比较其与相关对象关系；还研究\(Q^3\)中单调拉格朗日环面，计算相关不变量，证明不可位移性并联系镜像对称。

关于余切丛中投影哈密顿轨道交集的研究

On the intersections of projected Hamiltonian orbits in cotangent bundles

本文研究了余切丛中哈密顿轨道投影后的交集行为，证明了在特定条件下交点可被扰动消除，并应用于Reeb流和Finsler流的研究。

余切丛与plumbing上的Hodge微层

Hodge microsheaves on cotangent bundles and plumbings

本文引入Hodge微层范畴，研究缠绕层的Hodge理论版本，并应用于代数簇环路空间上同调的Hain Hodge结构及Ginzburg代数的Koszul对偶。

关于奇异拉格朗日纤维化及其在辛嵌入中的应用 I

On singular Lagrangian fibrations and applications to symplectic embeddings I

本文在 4 维和 6 维圆盘余切丛上构造奇异拉格朗日纤维化，可用于获取环面域，恢复了相关结果，还能用于研究辛嵌入问题。

无需 fittings 的填充

Fitting without fittings

本文证明奇维球面单位余切丛的辛非球面填充必微分同胚于单位余圆盘丛，从而无需使用此前提出的 fittings 概念。

对称空间作为伴随轨道及其几何

Symmetric spaces as adjoint orbits and their geometries

本文通过将经典对称空间实现为对称旗流形的余切丛，研究其测地线与拉格朗日子流形，并首次构造出单连通流形上不能承载非负截面曲率度量的向量丛例子。

Holomorphic disc, spin structures and Floer cohomology of the Clifford torus

We compute the Bott-Morse Floer cohomology of the Clifford torus in $\CP^n$ with all possible spin-structures. Each spin structure is known to determine an orientation of the moduli space of holomorphic discs, and we analyze the change of orientation according to the change of spin structure of the Clifford torus. Also, we classify all holomorphic discs with boundary lying on the Clifford torus by establishing a Maslov index formula for such discs. As a result, we show that in odd dimensions there exist two spin structures which give non-vanishing Floer cohomology of the Clifford torus, and in…

A nonpolynomially convex isotropic two-torus with no attached discs

We show --- with the means of a real-analytic example in $\mathbb{C}^3$ --- that Gromov's theorem on the presence of attached holomorphic discs for compact Lagrangian manifolds is not true in the isotropic (subcritical) case, even in the absence of an obvious obstruction, i.e, polynomial convexity.

Constructing local models for Lagrangian torus fibrations

We give a construction of Lagrangian torus fibrations with controlled discriminant locus on certain affine varieties. In particular, we apply our construction in the following ways. We find a Lagrangian torus fibration on the 3-fold negative vertex whose discriminant locus has codimension 2; this provides a local model for finding torus fibrations on compact Calabi-Yau 3-folds with codimension 2 discriminant locus. Then, we find a Lagrangian torus fibration on a neighbourhood of the one-dimensional stratum of a simple normal crossing divisor (satisfying certain conditions) such that the base o…

Lagrangian torus fibration of quintic Calabi-Yau hypersurfaces I: Fermat quintic case

In this paper we give a construction of Lagrangian torus fibration for Fermat type quintic \cy hypersurfaces via the method of gradient flow. We also compute the monodromy of the expected special Lagrangian torus fibration and discuss structures of singular fibers.