Asymptotic analysis of energy functionals in anti-de Sitter spacetimes

Abstract: Conformal Gravity (CG) is a Weyl--invariant metric theory free of divergences that with generalized Newman boundary conditions, reduces to renormalized Einstein--AdS gravity. By evaluating CG's action on a replica orbifold, one obtains a codimension-2 local conformal invariant functional, $L_\Sigma$, which recovers known results as the renormalized area, the reduced Hawking mass and the Willmore Energy. Although it was anticipated that $L_\Sigma$ would be finite, a general analysis of its asymptotic behavior near the conformal boundary remains unexplored. In this work, we show that $L_\Sigma$ is finite for a boundary--anchored surface $\Sigma$ embedded in an arbitrary ambient spacetime, including surfaces that are non--minimal and are anchored at arbitrary angles. This result extends the conformal renormalization prescription to codimension--2 functionals.