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

Author:

Xuezhang Chen, Wei Wei

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP)

journal:

--

date:

2023-07-25 16:00:00

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 background metric has nonnegative mean curvature on totally non-umbilic boundary, for dimensions three and four we prove the existence of a conformal metric with 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 in the latter existence result.