Chern-Gauss-Bonnet formula for singular Yamabe metrics in dimension four

We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic four dimensional manifold-with-boundary in terms geometry Loewner-Nirenberg singular Yamabe metric prescribed conformal class. The involves renormalized volume and boundary integral. It is shown that if umbilic, then sum integral invariant. Analogous results are proved asymptotically hyperbolic metrics dimension which second elementary symmetric function eigenvalues Schouten tensor constant. Extensions generalizations these discussed. Finally, general result identifying infinitesimal anomaly an its coefficients, used to outline alternate proofs invariance plus