關於二維旋轉對稱球面之等周長問題的討論

Abstract

文中討論球面上（旋轉對稱同時對赤道對稱在北半球與南半球 分別有單調高斯曲率）的等周長問題的解，使用Sturm’s 比較 定理來分析得出以最短周長包圍面積的區域形狀。

In the thesis we follow the demonstration of Prof. M. Ritoré to solve the isoperimetric problem on roatationally and equatorially symmetric spheres with monotone Gauss curvature from the poles. We first classify all the curves with constant geodesic curvature on a sphere with the above properties. Then we apply Sturm's comparison theorem successively to get the final only possible curve enclosing an isoperimetric domain. On regions with constant Gauss curvature we also invoke the Bol-Fiala inequality to conclude that inside such regions a geodesic circle has the minimal length encircling a domain with a given area.