On Alexandrov's Surfaces with Bounded Integral Curvature

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.

This article is an updated and translated version of the paper \cite{Troyanov2009}, to be included in the forthcoming book "Reshetnyak's Theory of Subharmonic Metrics", edited by François Fillastre and Dmitriy Slutskiy and to be published by Springer and the Centre de recherches Mathématiques (CRM) in Montréal