Computing renormalized curvature integrals on Poincar\'e-Einstein manifolds

We describe a general procedure for computing renormalized curvature integrals on Poincar\'e-Einstein manifolds. In particular, we explain the connection between the Gauss-Bonnet-type formulas of Albin and Chang-Qing-Yang for the renormalized volume, and explicitly identify a scalar conformal invariant in the latter formula. Our approach constructs scalar conformal invariants of weight $-n$ on $n$-manifolds, $n \geq 8$, that are natural divergences; these imply that the scalar invariant in the Chang-Qing-Yang formula is not unique in dimension $n \geq 8$. Our procedure also produces explicit conformally invariant Gauss--Bonnet-type formulas for compact Einstein manifolds.