Boundary operators associated to the ${\sigma }_k$-curvature

Abstract

We study conformal deformation problems on manifolds with boundary which include prescribing ${\sigma }_k\equiv 0$ in the interior. In particular, we prove a Dirichlet principle when the induced metric on the boundary is fixed and an Obata-type theorem on the upper hemisphere. We introduce some conformally covariant multilinear operators as a key technical tool.