Compatibility complexes for the conformal-to-Einstein operator
Igor Khavkine, Josef Šilhan
TL;DR
The paper addresses the problem of characterizing Einstein metrics within a conformal class by studying the conformal-to-Einstein operator $E_0$ and constructing a compatibility complex under natural genericity assumptions on the Weyl curvature in dimensions $n\ge 4$. It develops invariant differential operators and a curvature-driven obstruction framework, then applies an explicit lifting method (kh-compat) to produce a complete compatibility complex for $E_0$ in the generic case (and a shorter complex when the solution space is trivial), with a parallel projective analogue for the projective-to-Ricci-flat operator. The work connects to Bernstein-Gelfand-Gelfand (BGG) sequences, showing how, in the locally flat case, BGG sequences provide compatibility complexes, and extends these ideas to curved geometries via invariant operators such as $\widetilde{\nabla}$, $\tilde{d}$, $D_1$, $C_1$, and $\overline{D}$. The results yield precise obstructions to the existence of solutions and deliver a structured, conformally (and projectively) invariant framework for understanding when Einstein metrics can arise in a given conformal (or projective) class, with potential implications for curvature obstructions and parabolic-geometric PDEs.
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.