Asymptotic analysis of energy functionals in anti-de Sitter spacetimes

Abstract

Conformal Gravity (CG) is a Weyl--invariant metric theory whose action is free from divergences for generic asymptotically anti-de Sitter spaces. For Neumann boundary conditions, it reduces to renormalized Einstein--AdS gravity at tree level. By evaluating CG's action on a replica orbifold, one obtains a codimension-2 local conformal invariant functional, $L_Σ$, which reduces to the renormalized area, the reduced Hawking mass and the Willmore Energy, for a surface $Σ$. Although there is evidence supporting the idea that this functional should be finite, a detailed analysis of its asymptotic behavior near the conformal boundary is still lacking. In this work, the finiteness of the conformal surface functional $L_Σ$ is shown for any boundary--anchored surface embedded in an arbitrary ambient spacetime which is a solution to CG. This conclusion is drawn regardless the fact the surface is minimal or not. This result implies that Conformal Renormalization method not only applies to the bulk action, but also codimension--2 functionals.