Interior C2 regularity of convex solutions to prescribing scalar curvature equations

Abstract

We establish interior $C^2$ estimates for convex solutions of the scalar curvature equation and the ${\sigma }_2$-Hessian equation. We also prove interior curvature estimates for isometrically immersed hypersurfaces $(M^n,g)\subset {\mathbb{R}}^{n+1}$ with positive scalar curvature. These estimates are consequences of interior estimates for these equations obtained under a weakened condition.