Boundary operator associated to $\sigma_k$ curvature

On a Riemannian manifold $(M, g)$, the $\sigmak$ curvature is the $k$-th elementary symmetric function of the eigenvalues of the Schouten tensor $Ag$. It is known that the prescribing $\sigmak$ curvature equation on a closed manifold without boundary is variational if k=1, 2 or $g$ is locally conformally flat; indeed, this problem can be studied by means of the energy $\int \sigmak(Ag) dvg$. We construct a natural boundary functional which, when added to this energy, yields as its critical points solutions of prescribing $\sigma_k$ curvature equations with general non-vanishing boundary data. Moreover, we prove that the new energy satisfies the Dirichlet principle. If time permits, I will also discuss applications of our methods. This is joint work with Jeffrey Case.