2022/01/10 by Marc Troyanov, Troyanov, Marc
Mathematics · #52.a55 #53.c20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Point processes and geometric inequalities #math.DG #msc:52.a55 #msc:53.c20
paper · pdf · doi:10.48550/arxiv.2201.03359
This article is a translation of the paper \cite{Troyanov1990}, to be included in the forthcoming book "Reshetnyak's Theory of Subharmonic Metrics", edited by François Fillastre and Dmitriy Slytskiy and to be published by Springer and the Centre de recherches mathématiques (CRM) in Montréal
arxiv created 2022/01/10 · openalex publication_date 2022/01/10 · arxiv updated 2022/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note we discuss the geometry of Riemannian surfaces having a discrete set of singular points. We assume the conformal structure extends through the singularities and the curvature is integrable. Such points are called simple singularities. We first describe them locally and then globally using the notion of (real) divisor. We formulate a Gauss-Bonnet formula and relate it to some asymptotic isoperimetric ratio. We prove a classifications theorem for flat metrics with simple singularities on a compact surface and discuss the Berger--Nirenberg Problem on surfaces with a divisor. We finally discuss the relation with spherical polyhedra.