On the Regularity of Alexandrov Surfaces with Curvature Bounded Below

Luigi Ambrosio 1  and Jérôme Bertrand 2
  • 1 Scuola Normale Superiore, Piazza dei Cavalieri 7 56126 Pisa, Italy
  • 2 Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Toulouse III, 31062 Toulouse cedex 9, France

Abstract

In this note, we prove that on a surface with Alexandrov’s curvature bounded below, the distance derives from a Riemannian metric whose components, for any p ∈ [1, 2), locally belong to W1,p out of a discrete singular set. This result is based on Reshetnyak’s work on the more general class of surfaces with bounded integral curvature.

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