Conformal operators on weighted forms; their decomposition and null space on Einstein manifolds

TL;DR

This paper studies a class of conformally invariant differential operators on differential forms on Einstein manifolds, providing explicit formulas and decompositions of their null spaces, extending understanding of conformal geometry and operator theory.

Contribution

It introduces explicit factored formulas for conformal operators on Einstein manifolds and decomposes their null spaces in non-Ricci-flat cases, advancing conformal differential geometry.

Findings

Explicit formulas for conformally invariant operators on Einstein manifolds.

Decomposition of null spaces in terms of second order factors.

Application to non-Ricci-flat Einstein manifolds.

Abstract

There is a class of Laplacian like conformally invariant differential operators on differential forms $L^\ell_k$ which may be considered the generalisation to differential forms of the conformally invariant powers of the Laplacian known as the Paneitz and GJMS operators. On conformally Einstein manifolds we give explicit formulae for these as explicit factored polynomials in second order differential operators. In the case the manifold is not Ricci flat we use this to provide a direct sum decomposition of the null space of the $L^\ell_k$ in terms of the null spaces of mutually commuting second order factors.