多边形空间中的面积-周长对偶性

Hyperbolic polygons of minimal perimeter with given angles

We prove that, among all convex hyperbolic polygons with given angles, the perimeter is minimized by the unique polygon with an inscribed circle. The proof relies on work of J.-M.\ Schlenker.

Hyperbolic polygons of minimal perimeter in punctured discs

We prove that, among the polygons in a punctured disc with fixed angles, the perimeter is minimized by the polygon with an inscribed horocycle centered at the puncture. We generalize this to a disc with a cone point and to an annulus with a geodesic boundary component and a complete end. Then we apply this result to describe the minimum of the spine systole on the moduli space of punctured surfaces.

固定周长的多边形极值面积

Extremal areas of polygons with fixed perimeter

本文研究固定周长平面n边形的构型空间，证明定向面积函数临界点最少，描述正则星形临界点并计算Morse情形下的指标。

Polygons with prescribed edge slopes: configuration space and extremal points of perimeter

We describe the configuration space $\mathbf{S}$ of polygons with prescribed edge slopes, and study the perimeter $\mathcal{P}$ as a Morse function on $\mathbf{S}$. We characterize critical points of $\mathcal{P}$ (these are \textit{tangential} polygons) and compute their Morse indices. This setup is motivated by a number of results about critical points and Morse indices of the oriented area function defined on the configuration space of polygons with prescribed edge lengths (flexible polygons). As a by-product, we present an independent computation of the Morse index of the area function (ob…

等周不等式的一些类似结论

Some analogues of isoperimetric inequality

本文证明了双曲几何中循环和切向多边形的离散等周不等式类似结论，并建立了多个多边形在面积或周长限制下的等周不等式版本。