Holomorphic triples and the prescribed curvature problem on $S^2$

TL;DR

This paper establishes new existence results for the prescribed Gaussian curvature problem on the sphere by linking it with holomorphic triples theory, suggesting potential applications to other elliptic equations on R^2.

Contribution

It introduces a novel approach connecting prescribed curvature problems with holomorphic triples theory on Riemann surfaces, expanding the toolkit for solving semi-linear elliptic equations.

Findings

Proved existence of solutions for the prescribed Gaussian curvature problem on S^2.

Linked the curvature problem with holomorphic triples theory.

Suggested applicability of the method to other elliptic equations.

Abstract

We prove new results on existence of solutions for the prescribed gaussian curvature problem on the euclidean sphere S^2. Those results are achieved by relating this problem with the holomorphic triples theory on Riemann surfaces. We think this approach might be applied to study some other semi-linear elliptic equations of 2nd order on the sphere.