Kähler流形超曲面中单调Lagrange算子的一个拓扑约束

Families of monotone Lagrangians in Brieskorn--Pham hypersurfaces

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 $…

Periodic leaf-wise intersection points from Lagrangians

We investigate leaf-wise intersection points on hypersurfaces of contact type in monotone symplectic manifolds. We show that monotone Floer-essential Lagrangians detect periodic leaf-wise intersection points in hypersurfaces of contact type whose Reeb flow is Zoll. Examples include the prequantization bundles appearing in monotone toric negative line bundles. Generalizing, we prove the existence of leaf-wise intersection points for certain annulus subbundles in weak+-monotone negative line bundles, not necessarily toric. The proofs combine reduced symplectic cohomology with the original method…

On the Fukaya category of a Fano hypersurface in projective space

This paper is about the Fukaya category of a Fano hypersurface $X \subset \mathbb{CP}^n$. Because these symplectic manifolds are monotone, both the analysis and the algebra involved in the definition of the Fukaya category simplify considerably. The first part of the paper is devoted to establishing the main structures of the Fukaya category in the monotone case: the closed-open string maps, weak proper Calabi-Yau structure, Abouzaid's split-generation criterion, and their analogues when weak bounding cochains are included. We then turn to computations of the Fukaya category of the hypersurfac…

负单调流形的谱直径

Spectral diameter of negatively monotone manifolds

本研究针对闭负单调辛流形，在含不可压缩重拉格朗日子流形的条件下构造欧氏空间到其哈密顿微分同胚群的拟等距嵌入，并证明其骨架相对Donaldson超曲面的超重性，推进辛拓扑谱直径相关研究。

关于$S^2\times S^2$和$C P^n$的某些拉格朗日子流形

On Certain Lagrangian Submanifolds of $S^2\times S^2$ and $C P^n$

我们考虑$C P^n, S^2\times S^2$的单调拉格朗日子流形和$C P^n$的二次超曲面的各种构造。在$S^2\times S^2$和$C P^2$，我们表明，几种不同的已知结构的外来单调环面产生的结果是哈密顿同位素彼此，特别是回答吴的问题，表明单调纤维的复曲面退化模型的$C P^2$是哈密顿同位素的切卡诺夫环面。将我们的构造推广到更高的维度，使我们考虑二次曲面和$C P^n$的单调拉格朗日子流形（通常不是环面），这可以用$T^*S^n$上的测地线流或Biran圆丛构造来理解。与以前已知的单连通闭辛流形的单调拉格朗日子流形不同，我们的许多高维拉格朗日子流形是可移位的。

齐次等参超曲面高斯象的Hamilton稳定性

Hamiltonian stability of the Gauss images of homogeneous isoparametric hypersurfaces

单位标准球面$S^{n+1}(1)$的任何定向等参超曲面的高斯映射的像是复超二次曲面$Q_n({\mathbf C})$中的极小拉格朗日子流形。本文证明了$S^{n+1}(1)$中具有$g$不同常主曲率的紧致定向等参超曲面的Gauss象是具有最小Maslov数$2n/g$的紧致单调循环嵌入拉格朗日子流形.本文的主要结果是利用齐次空间上的调和分析和齐次等参超曲面上的纤维化，完全确定了单位球面上的齐次等参超曲面的高斯映射的象所构成的复超二次曲面中所有紧致极小拉格朗日子流形的Hamilton稳定性.另外，对例外黎曼对称空间$(E_6, U(1)\cdot Spin(10))$和相应的高斯象的讨论也有自己的兴趣。