Uniformization of $S^2$ and flat singular surfaces

TL;DR

This paper develops methods to construct flat metrics with prescribed singularities on closed surfaces, providing an elementary proof of the sphere's uniformization theorem and extending classical results to more general singularities.

Contribution

It introduces a new approach to uniformization with prescribed singularities, including conical and cylindrical types, on real surfaces.

Findings

Constructed flat metrics with specific singularities on surfaces.

Provided an elementary proof of the sphere's uniformization theorem.

Extended uniformization results to include cylindrical and large conical singularities.

Abstract

We construct flat metrics in a given conformal class with prescribed singularities of real orders at marked points of a closed real surface. The singularities can be small conical, cylindrical, and large conical with possible translation component. Along these lines we give an elementary proof of the uniformization theorem for the sphere.