Extended obstruction tensors and renormalized volume coefficients

Understand

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.