Stability problem of equilibrium discrete planar curves

空间形式中的离散曲线理论：平面弹性和面积约束弹性曲线

Discrete curve theory in space forms: planar elastic and area-constrained elastic curves

本文提出空间形式中离散弹性与面积约束弹性曲线的概念，推广离散欧几里得曲率方程，利用Bäcklund变换构建离散流，并证明此类曲线可刻画为流不变量的层级元素，同时给出离散Frenet公式与基本定理。

Infinitesimal Darboux transformation and semi-discrete mKdV equation

We connect certain continuous motions of discrete planar curves resulting in semi-discrete potential Korteweg-de Vries (mKdV) equation with Darboux transformations of smooth planar curves. In doing so, we define infinitesimal Darboux transformations that include the aforementioned motions, and also give an alternate geometric interpretation for establishing the semi-discrete potential mKdV equation.

关于若干离散的Bonnesen型等周不等式

on some discrete Bonnesen-style isoperimetric inequalities

本文研究平面凸多边形的精确离散等周不等式，建立了基于Schur凸函数的解析不等式，并由此推导出Bonnesen型及其逆向等周不等式。

从离散全纯映射得到具有球面参数线的等温网

Isothermic nets with spherical parameter lines from discrete holomorphic maps

本文证明了具有平面或球面曲率线的离散等温网可通过提升折叠从特殊离散全纯映射获得，该新方法推广并离散化了经典光滑曲面理论，并为构造离散等温拓扑环面提供了有效途径。

Affine subspaces of curvature functions from closed planar curves

Given a pair of real functions $(k,f)$, we study the conditions they must satisfy for $k+λf$ to be the curvature in the arc-length of a closed planar curve for all real $λ$. Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied. Finally, the characterization obtained is used to show that a sufficient analogue of the 4-vertex theorem cannot be developed.

某些模空间的辛结构和Poisson结构。II。上紧离散平面群的射影表示

Symplectic and Poisson structures of certain moduli spaces. II. Projective representations of cocompact discrete planar groups

设$G$是具有双不变度量的李群，不一定正定。本文证明了在早期文献中对闭曲面的基本群所作的某种构造可以推广到欧氏或非欧氏运动的任意无限保向余紧平面离散群$π$上，并得到（i）包含同态空间$\roman{Hom}(π,G)$的光滑流形$\Cal M$上的辛结构，此外，（ii）一个哈密顿$G$ -作用在$\Cal M$上，保持辛结构以及动量映射，使得约化空间等于表示空间$\roman{Rep}(π,G)$。更一般地说，这种构造也适用于某些射影表示空间。对于$G$紧的，所得到的表示空间继承了{ \it分层辛空间\/}的结构，使得分层具有有限的辛体积。例如，具有有理权的半稳定全纯抛物丛的{ \smc Mehta-Seshadri}模空间或与它们密切相关的空间通过{ \it在有限维中的辛约化以这种方式产生。