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 that are divergences at any Einstein manifold; these imply that the scalar invariant in the Chang-Qing-Yang formula is not unique in dimension at least eight. Our procedure also produces explicit conformally invariant Gauss-Bonnet-type formulas for compact Einstein manifolds.