Curved versions of the Ovsienko-Redou operators

TL;DR

This paper classifies certain curved bidifferential operators related to the Laplacian on n-dimensional manifolds, extending known conformally invariant operators on spheres, and constructs new self-adjoint conformally invariant operators.

Contribution

It provides a complete classification of curved tangential bidifferential operators of bounded order expressed via the Laplacian, extending prior conformal invariance results.

Findings

Classified all such bidifferential operators of order at most n

Constructed a large class of self-adjoint conformally invariant operators

Extended the conformal invariance classification to curved manifolds

Abstract

We give a complete classification of tangential bidifferential operators of total order at most $n$ which are expressed purely in terms of the Laplacian on the ambient space of an $n$-dimensional manifold. This gives a curved analogue of the classification, due to Ovsienko--Redou and Clerc, of conformally invariant bidifferential operators on the sphere. As an application, we construct a large class of formally self-adjoint conformally invariant differential operators.