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On Alexandrov’s Surfaces with Bounded Integral Curvature

In: Reshetnyak's Theory of Subharmonic Metrics

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  • Marc Troyanov

    (École Polytechnique Féderale de Lausanne, Section de Mathématiques)

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.

Suggested Citation

  • Marc Troyanov, 2023. "On Alexandrov’s Surfaces with Bounded Integral Curvature," Springer Books, in: François Fillastre & Dmitriy Slutskiy (ed.), Reshetnyak's Theory of Subharmonic Metrics, chapter 0, pages 9-34, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-24255-7_2
    DOI: 10.1007/978-3-031-24255-7_2
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