Conformally compact metrics and the Lovelock tensors

TL;DR

This paper explores conformally compact metrics satisfying Lovelock equations, extending the Fefferman-Graham expansion and identifying boundary obstructions, with implications for conformal geometry and the AdS/CFT correspondence.

Contribution

It generalizes the Fefferman-Graham expansion to Lovelock metrics and identifies boundary obstructions in even dimensions, advancing understanding of conformally compact Einstein-like geometries.

Findings

Polyhomogeneous expansions for Lovelock metrics

Boundary obstruction generalizing the ambient obstruction tensor

Construction of formal solutions to the singular Yamabe-(2q) problem

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.