Extended obstruction tensors and renormalized volume coefficients

The behavior under conformal change of the renormalized volume coefficients associated to a pseudo-Riemannian metric is investigated. It is shown that they define second order fully nonlinear operators in the conformal factor whose algebraic structure is elucidated via the introduction of "extended obstruction tensors". These together with the Schouten tensor constitute building blocks for the coefficients in the ambient metric expansion. The renormalized volume coefficients have recently been considered by Chang-Fang motivated by comparison with the elementary symmetric functions of the eigenvalues of the Schouten tensor.