On Alexandrov’s Surfaces with Bounded Integral Curvature

Marc Troyanov ()
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Marc Troyanov: École Polytechnique Féderale de Lausanne, Section de Mathématiques

Chapter Chapter 2 in Reshetnyak's Theory of Subharmonic Metrics, 2023, pp 9-34 from Springer

Abstract: 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 C2, 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.

Keywords: Alexandrov Surfaces; Surfaces with bounded integral curvature; Subharmonic metrics (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-031-24255-7_2

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DOI: 10.1007/978-3-031-24255-7_2

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