Estimates for modified $\sigma_{2}$ curvature equation on compact manifolds with boundary

TL;DR

This paper derives global and local boundary second-derivative estimates for solutions to a modified sigma-2 curvature equation on compact manifolds with boundary, advancing understanding of geometric PDEs in this context.

Contribution

It provides the first comprehensive $C^2$-estimates for the modified sigma-2 curvature equation with boundary conditions on three-manifolds.

Findings

Established global $C^2$-estimates for the equation.

Derived local boundary $C^2$-estimates on three-manifolds.

Enhanced the analytical tools for geometric PDEs with boundary conditions.

Abstract

We establish the global $C^2$-estimates for the modified $\sigma_2$ curvature equation with prescribed boundary mean curvature, and particularly, the local boundary $C^2$ estimates on three-manifolds.