Spherical metrics with conical singularities on 2-spheres

Abstract

Suppose that ${\theta }_1,{\theta }_2,\dots ,{\theta }_n$ are positive numbers and $n\ge 3$. Does there exist a sphere with a spherical metric with $n$ conical singularities of angles $2\pi {\theta }_1,2\pi {\theta }_2,\dots ,2\pi {\theta }_n$? A sufficient condition was obtained by Gabriele Mondello and Dmitri Panov (arXiv:1505.01994). We show that it is also necessary when we assume that ${\theta }_1,{\theta }_2,\dots ,{\theta }_n\notin \mathbb{N}$.

In this updated version, the proof of the main result has been changed, and is now more geometric in nature. 9 pages, 1 figure

Cited by in corpus (5)

• Conical metrics on Riemann surfaces, I: the compactified configuration space and regularity
• Conical metrics on Riemann surfaces, II: spherical metrics
• Rigidity of a family of spherical conical metrics
• Polyhedral Kähler cone metrics on ${\mathbb{C}}^n$ singular at hyperplane arrangements
• Spectral properties of reducible conical metrics