The $\sigma_{2}$-curvature equation on a compact manifold with boundary

TL;DR

This paper develops local $C^2$ estimates for solutions to the $\sigma_2$-curvature equation with nonlinear boundary conditions and proves existence of conformal metrics with prescribed curvature and boundary mean curvature on certain manifolds.

Contribution

It establishes new local estimates and existence results for the $\sigma_2$-curvature equation on compact manifolds with boundary in dimensions three and four.

Findings

Established local $C^2$ estimates for solutions.

Proved existence of conformal metrics with prescribed $\sigma_2$-curvature.

Analyzed blow-up behavior using local estimates.

Abstract

We first establish local $C^2$ estimates of solutions to the $\sigma_2$-curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the mean curvature of a background metric is nonnegative on totally non-umbilic boundary, for dimensions three and four there exists a conformal metric having a prescribed positive $\sigma_2$-curvature and a prescribed nonnegative boundary mean curvature. The local estimates play an important role in the blow up analysis for the latter existence result.