论文标题
在简单各向同性空间
Catenaries and singular minimal surfaces in the simply isotropic space
论文作者
论文摘要
本文研究了简单的各向同性平面中的悬挂链问题及其在简单各向同性空间中的二维类似物。简单的各向同性平面和空间是二维和三维的几何形状，配备了一个归化度度量的核的核具有尺寸1。尽管度量是退化的，但如果我们采用相对弧的长度和相对面积来测量重量。在这里，通过将简单的各向同性几何形状视为相对几何形状，相对弧长和相对区域的概念出现。除了表征简单的各向同性链状（即悬挂链问题的解决方案）外，我们还证明了它是简单的各向同性空间中最小革命表面的生成曲线。最后，我们获得了所谓的奇异最小表面的二维类似物，并确定简单各向同性空间中革命悬挂表面的形状。
This paper investigates the hanging chain problem in the simply isotropic plane as well as its 2-dimensional analog in the simply isotropic space. The simply isotropic plane and space are two- and three-dimensional geometries equipped with a degenerate metric whose kernel has dimension 1. Although the metric is degenerate, the hanging chain and hanging surface problems are well-posed if we employ the relative arc length and relative area to measure the weight. Here, the concepts of relative arc length and relative area emerge by seeing the simply isotropic geometry as a relative geometry. In addition to characterizing the simply isotropic catenary, i.e., the solutions of the hanging chain problem, we also prove that it is the generating curve of a minimal surface of revolution in the simply isotropic space. Finally, we obtain the 2-dimensional analog of the catenary, the so-called singular minimal surfaces, and determine the shape of a hanging surface of revolution in the simply isotropic space.
