There has been much recent activity in the area of conformal geometry and conformally compact Einstein manifolds centered around volume renormalization [?], [?]; there one considers a complete Einstein metric g+ on the interior Ω of a compact manifold with boundary Ω = Ω ∪partial derivΩ and a conformal structure [g] on partial derivΩ, which is obtained as a scaling limit of g+. For a choice of a denning function ρ such that Ω = {ρ ＞ 0}, one can consider the volume expansion of subdomain Ω_ε = {ρ ＞ ε} with respect to g+. If n = dim partial derivΩ is even, it takes the form Vol(Ω_ε) = Σ_(j=0)~(n/2-1) C_jε~(2j-n) + L(partial derivΩ)log ε+ (bounded term), where C_j are constants (which depend on the choice of p) and L is a conformal invariant of (partial derivΩ,[g]). Moreover, it is shown that this conformal invariant L can be expressed as the integral of Branson's Q-curvature [?], a local Riemannian invariant which naturally arises from conformally invariant differential operators. Both L and Q have been studied extensively.
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