论文标题
在Alexandrov的表面上，有界的积分曲率
On Alexandrov's Surfaces with Bounded Integral Curvature
论文作者
论文摘要
在1940 - 1970年期间，亚历山德罗夫和“列宁格勒学校”对奇异表面的几何形状进行了深入研究。该学校开发的理论是关于具有内在度量的拓扑表面，我们可以为其定义曲率概念，这是ra的量度。这类表面具有良好的收敛特性，并且相对于各种几何构建体（胶合等）非常稳定。它包括$ c^2 $的多面体表面以及riemannian表面，这两个类都是亚历山德罗夫表面的密集家族。任何可以合理地想到的奇异表面都是Alexandrov表面，而平滑表面的许多几何特性则扩展并推广到该类别。 本文的目的是介绍亚历山德罗夫的理论，提供一些例子并说明该理论的一些基本事实。我们讨论了Yuri G. Reshetnyak引入的共形观点，并解释了它如何导致Alexandrov表面的分类。
During the years 1940-1970, Alexandrov and the "Leningrad School" have investigated the geometry of singular surfaces in depth. The theory developed by this school is about topological surfaces with an intrinsic metric for which we can define a notion of curvature, which is a Radon measure. This class of surfaces has good convergence properties and is remarkably stable with respect to various geometrical constructions (gluing etc.). It includes polyhedral surfaces as well as Riemannian surfaces of class $C^2$, and both of these classes are dense families of Alexandrov's surfaces. Any singular surface that can be reasonably thought of is an Alexandrov surface and a number of geometric properties of smooth surfaces extend and generalize to this class. The goal of this paper is to give an introduction to Alexandrov's theory, to provide some examples and state some of the fundamental facts of the theory. We discuss the conformal viewpoint introduced by Yuri G. Reshetnyak and explain how it leads to a classification of compact Alexandrov's surfaces.
