Additivity of support norm of tight contact structures

Examples of planar tight contact structures with support norm one

We exhibit an infinite family of tight contact structures with the property that none of the supporting open books minimizes the genus and maximizes the Euler characteristic of the page simultaneously, answering a question of Baldwin and Etnyre in arXiv:0910.5021 .

Thurston 范数与紧接触结构的 Euler 类

Thurston norm and Euler classes of tight contact structures

本文针对紧接触结构考察 Thurston 的 Euler 类范数猜想，证明先前紧叶状结构反例中的上同调类可由紧接触结构实现，从而支持相关猜想。

Milnor fillable contact structures are universally tight

We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tight contact structures that are not deformations of taut (or Reebless) foliations. This answers two questions of Etnyre.

使用辛几何采样有限单位范数紧框架

Sampling Finite Unit Norm Tight Frames Using Symplectic Geometry

本文提出Eigenlift算法，利用FUNTF空间的辛几何结构，通过辛约化、Hamilton环面作用及纤维丛提升机制，实现随机有限单位范数紧框架的采样，并在Python中验证了其低维可行性。

Tight neighborhoods of contact submanifolds

We prove that any small enough neighborhood of a closed contact submanifold is always tight under a mild assumption on its normal bundle. The non-existence of $C^0$--small positive loops of contactomorphisms in general overtwisted manifolds is shown as a corollary.

融合框架同伦与通过梯度下降紧致融合框架

Fusion Frame Homotopy and Tightening Fusion Frames by Gradient Descent

本文将单位范数紧致框架的路径连通性和梯度下降构造结果推广至融合框架，利用辛几何与几何不变量理论证明了紧致融合框架集合的路径连通性及融合框架势无虚假局部最小化器。