We consider some boundary value problems motivated by four dimentional Chern-Gauss-Bonnet formula for manifolds with boundary. We present existence results for some fully nonlinear PDEs with Neumann boundary condition. As an application, we prove that given a conformally compact Einstein 4-manifold with some conditions on the conformal infinity and the renormalized volume, there exists a conformal compactification such that the \sigma_2 curvature is constant.

Lecture #12229

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