The Nirenberg Problem for Conical Singularities

TL;DR

This paper introduces a new method for prescribing Gaussian curvature with conical singularities on the 2-sphere, avoiding traditional variational techniques and providing sufficient conditions for such metrics.

Contribution

It offers a novel approach to the Nirenberg problem with conical singularities, including a general precompactness theorem, differing from previous variational methods.

Findings

Established sufficient conditions for Gaussian curvature with conical singularities.

Proved a precompactness theorem for Riemann surfaces with singularities.

Developed a method not relying on the Moser-Trudinger inequality.

Abstract

We propose a new approach to the question of prescribing Gaussian curvature on the 2-sphere with at least three conical singularities and angles less than $2\pi$, the main result being sufficient conditions for a positive function of class at least $C^2$ to be the Gaussian curvature of such a conformal conical metric on the round sphere. Our methods particularly differ from the variational approach in that they don't rely on the Moser-Trudinger inequality. Along the way, we also prove a general precompactness theorem for compact Riemann surfaces with at least three conical singularities and angles less than 2$\pi$.