On the prescribing $\sigma_2$ curvature equation on $\mathbb S^4$

TL;DR

This paper analyzes solutions to the $\sigma_2$ curvature equation on the 4-sphere, establishing asymptotic behavior, ruling out blow-up solutions under certain conditions, and proving existence of solutions via degree theory.

Contribution

It provides the first asymptotic profile analysis and blow-up classification for the $\sigma_2$ curvature equation on $S^4$, and proves existence results using degree theory under non-degeneracy conditions.

Findings

Asymptotic profile analysis for solutions with potential blow-up.

Blow-up solutions are ruled out under a non-degeneracy condition on $K$.

Existence of solutions is established via degree theory for a deformation of $K$.

Abstract

Prescribing $\sigma_k$ curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. Given a positive function $K$ to be prescribed on the 4-dimensional round sphere. We obtain asymptotic profile analysis for potentially blowing up solutions to the $\sigma_2$ curvature equation with the given $K$; and rule out the possibility of blowing up solutions when $K$ satisfies a non-degeneracy condition. We also prove uniform a priori estimates for solutions to a family of $\sigma_2$ curvature equations deforming $K$ to a positive constant under the same non-degeneracy condition on $K$, and prove the existence of a solution using degree argument to this deformation involving fully nonlinear elliptic operators under an additional, natural degree condition on a finite dimensional map associated with $K$.