Some constructions of formally self-adjoint conformally covariant polydifferential operators (Q2131759)

A conformally variational invariant (CVI) on a manifold equipped with a conformal structure is a natural Riemannian scalar invariant homogeneous in the metric and variational within the conformal class of the metric. Given a CVI denoted by \(L\), the authors show in the main theorem (Theorem 1.6 in the paper) that there is a self-adjoint conformally covariant polydifferential operator \(D\) associated to \(L\). The definitions are established in the introduction of the paper. The construction of these operators uses the ambient space of \textit{C. Fefferman} and \textit{C. R. Graham} [The ambient metric. Princeton, NJ: Princeton University Press (2012; Zbl 1243.53004)]. Examples of polydifferential operators presented in the paper include the operator of rank \(2k\) associated to the renormalised volume coefficient \(v_k\), which is a CVI of weight \(-2k\); a two-parameter family of tridifferential operators of rank 4 and homogeneous of degree \(-6\); a subfamily of Ovsienko-Radou bidifferential operators on the round sphere and their curved versions and a family of differential operators with critical order.