Riemannian Surfaces with Simple Singularities
Abstract
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 \emph{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.
Cite
@article{arxiv.2201.03359,
title = {Riemannian Surfaces with Simple Singularities},
author = {Marc Troyanov},
journal= {arXiv preprint arXiv:2201.03359},
year = {2022}
}
Comments
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 Fran\c{c}ois Fillastre and Dmitriy Slytskiy and to be published by Springer and the Centre de recherches math\'ematiques (CRM) in Montr\'eal