# Isolated singularities of flat metrics on Riemann surfaces

## TL;DR

This paper classifies isolated singularities of flat metrics on Riemann surfaces, showing they are conical under certain area conditions, extending Bryant's results on positive curvature to flat metrics.

## Contribution

It provides a complete local classification of isolated singularities of flat metrics with area growth conditions, including finite area cases.

## Key findings

- Isolated singularities of flat metrics with finite area are conical.
- All local models for such singularities are characterized.
- The results extend Bryant's theorem to flat metrics.

## Abstract

Robert Bryant (Theorie des varietes minimales et applications, 1988, 154: 321-347) proved that an isolated singularity of a conformal metric of positive constant curvature on a Riemann surface is a conical one. Using Complex Analysis, we find all of the local models for an isolated singularity of a flat metric whose area satisfies some polynomial growth condition near the singularity. In particular, we show that an isolated singularity of a flat metric with finite area is also a conical one.