Conical metrics on Riemann surfaces, I: the compactified configuration space and regularity

TL;DR

This paper develops a compactified framework for studying conical metrics on Riemann surfaces, focusing on regularity and deformations as conic points merge, with implications for spherical metrics with large cone angles.

Contribution

It introduces a new compactification of the divisor space and analyzes the regularity of conical metrics under deformation and coalescence of conic points.

Findings

Established a sharp regularity theorem for conical metrics near coalescence

Constructed a compactified space of divisors and punctured surfaces as real manifolds with corners

Provided groundwork for studying spherical conic metrics with large cone angles

Abstract

We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then study the space of constant curvature metrics on this Riemann surface with prescribed conical singularities at these divisors. Our interest here is in the local deformation for these metrics, and in particular the behavior as conic points coalesce. We prove a sharp regularity theorem for this phenomenon in the regime where these metrics are known to exist. This setting will be used in a subsequent paper to study the space of spherical conic metrics with large cone angles, where the existence theory is still incomplete.