Two Kazdan–Warner-type identities for the renormalized volume coefficients and the Gauss–Bonnet curvatures of a Riemannian metric

Abstract

We prove two Kazdan‐Warner-type identities involving the renormalized volume coefficients v .2k/ of a Riemannian manifold .M n ; g/, the Gauss‐ Bonnet curvature G2r , and a conformal Killing vector field on .M n ; g/. In the case when the Riemannian manifold is locally conformally flat, we find v .2k/ D. 2/ k k and G2r.g/D 4 r .n r/WrW .n 2r/W r and our results reduce to earlier ones established by Viaclovsky in 2000 and the second author in 2006.