Universal recursive formula for GJMS-operators in the locally conformally flat case

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Let (M,g)(M,g) be a locally conformally flat Riemannian manifold of dimension n≥3n\geq3. For N≥1N\geq1, let M2N{\mathcal M}_{2N} denote the operator defined from the recursive construction, let [⋅]0[\cdot]^0 denote the non-constant part, let P{\sf P} be the Schouten tensor, and let δ\delta be the negative divergence. Universal recursive formula.

M2N0=−cNδ(PN−1#d),cN=2N−1N!(N−1)!.{\mathcal M}_{2N}^0=-c_N\delta({\sf P}^{N-1}\#d),\qquad c_N=2^{N-1}N!(N-1)!.

This conjecture seeks a recursive description of all subcritical GJMS-operators in the locally conformally flat category. The formula is established in the first few orders, but is not known in general.

References

Primary source

Andreas Juhl, “On conformally covariant powers of the Laplacian”, arXiv:0905.3992 (2010).

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