# Conical metrics on Riemann surfaces, II: spherical metrics

## TL;DR

This paper investigates the existence and deformation of spherical conic metrics on Riemann surfaces with certain cone angles, revealing conditions for smooth moduli spaces and connecting to geometric constructions by Mondello and Panov.

## Contribution

It extends previous work by analyzing deformation obstructions related to the Laplacian spectrum and demonstrates the possibility of smooth moduli spaces through cone point splitting.

## Key findings

- Deformations are obstructed when 2 is in the Laplacian spectrum.
- A smooth local moduli space exists when cone points split.
- Connections established with geometric constructions by Mondello and Panov.

## Abstract

We continue our study, initiated in our earlier paper, of Riemann surfaces with constant curvature and isolated conic singularities. Using the machinery developed in that earlier paper of extended configuration families of simple divisors, we study the existence and deformation theory for spherical conic metrics with some or all of the cone angles greater than $2\pi$. Deformations are obstructed precisely when the number $2$ lies in the spectrum of the Friedrichs extension of the Laplacian. Our main result is that, in this case, it is possible to find a smooth local moduli space of solutions by allowing the cone points to split. This analytic fact reflects geometric constructions in papers by Mondello and Panov.