Interior C2 regularity of convex solutions to prescribing scalar curvature equations

TL;DR

This paper proves interior second derivative and curvature estimates for convex solutions to scalar curvature and Hessian equations, advancing understanding of geometric PDE regularity.

Contribution

It introduces new interior $C^2$ and curvature estimates for convex solutions of scalar curvature and $\sigma_2$-Hessian equations under weakened conditions.

Findings

Established interior $C^2$ estimates for convex solutions.

Proved interior curvature estimates for hypersurfaces with positive scalar curvature.

Results apply under less restrictive assumptions.

Abstract

We establish interior $C^2$ estimates for convex solutions of scalar curvature equation and $\sigma_2$-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces $(M^n,g)\subset \mathbb R^{n+1}$ with positive scalar curvature. These estimates are consequences of an interior estimate for these equations obtained under a weakened condition.