Conformally compact metrics and the Lovelock tensors

Abstract: We study conformally compact metrics satisfying the Lovelock equations, which generalize the Einstein equation. We show that these metrics admit polyhomogeneous expansions, thereby naturally realizing the Fefferman-Graham expansion, which is an important tool in conformal geometry and the AdS/CFT correspondence. In even dimensions, we identify a boundary obstruction to smoothness near the boundary that generalizes the ambient obstruction tensor in the Einstein setting. Under appropriate regularity and curvature conditions, we also construct a formal solution to the singular Yamabe-(2q) problem and provide an index obstruction for the conformally compact Lovelock filling problem of spin manifolds.