Curvature flow to Nirenberg problem

TL;DR

This paper investigates a curvature flow approach to solving the Nirenberg problem on the sphere, establishing existence results under specific conditions on the prescribed Gaussian curvature function.

Contribution

It provides new existence results for the Nirenberg problem using curvature flow, especially when the prescribed curvature has non-degenerate critical points with positive Laplacian at saddle points.

Findings

Existence of solutions when positive part of curvature has non-degenerate critical points.

Conditions on the Laplacian at saddle points ensure solvability.

Extension of Brendle and Struwe's curvature flow approach to new cases.

Abstract

In this note, we study the curvature flow to Nirenberg problem on $S^2$ with non-negative nonlinearity. This flow was introduced by Brendle and Struwe. Our result is that the Nirenberg problems has a solution provided the prescribed non-negative Gaussian curvature $f$ has its positive part, which possesses non-degenerate critical points such that $\Delta_{S^2} f>0$ at the saddle points.