Compatibility complexes for the conformal-to-Einstein operator

TL;DR

This paper constructs compatibility complexes for the conformal-to-Einstein operator, revealing conditions under which solutions correspond to Einstein metrics, and explores analogous results for the projective-to-Ricci-flat operator, using symmetry-based methods.

Contribution

It introduces a method to build compatibility complexes for these operators under generic conditions, extending previous techniques and providing new insights into their geometric properties.

Findings

At most one independent solution under generic Weyl curvature conditions

Construction of conformally and projectively invariant compatibility complexes

Interpretation of Bernstein-Gelfand-Gelfand sequences as compatibility complexes in flat cases

Abstract

The conformal-to-Einstein operator is a conformally invariant linear overdetermined differential operator whose non-vanishing solutions correspond to Einstein metrics within a conformal class. We construct compatibility complexes for this operator under natural genericity assumptions on the Weyl curvature in dimension $n\ge 4$, which implies at most one independent solution. An analogous result for the projective-to-Ricci-flat operator is obtained as well. The construction is based on a method, previously proposed by one of the authors, that leverages existing symmetries and geometric properties of the starting operator. In this case the compatibility complexes consist of, respectively, conformally and projectively invariant operators. We also make some comments on how Bernstein-Gelfand-Gelfand sequences can be interpreted as compatibility complexes in the locally flat case, which may be of general interest.