On Alexandrov's Surfaces with Bounded Integral Curvature

TL;DR

This paper introduces Alexandrov's theory of surfaces with bounded integral curvature, highlighting their properties, examples, and a classification approach via conformal methods, extending classical geometry to singular surfaces.

Contribution

It provides an accessible overview of Alexandrov's theory, including examples, fundamental facts, and a classification scheme for compact surfaces using conformal techniques.

Findings

Alexandrov surfaces include polyhedral and smooth Riemannian surfaces.

The class of Alexandrov surfaces is stable under geometric operations.

Conformal viewpoint aids in classifying compact Alexandrov surfaces.

Abstract

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.