On the prescribing ${\sigma }_2$ curvature equation on ${\mathbb{S}}^4$

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 $K$.