For every Böhm Einstein metric g on a manifold of the form S^k+1× S^l or S^k+l+1, the Lichnerowicz Laplacian on transverse-traceless symmetric 2-tensors has a negative eigenvalue; equivalently, there exist a nonzero transverse-traceless tensor h and a number lambda<0 such that Delta_L^g h=lambda h. This predicts the corresponding instability of the generalized black-hole spacetimes constructed from these Böhm metrics.

Gibbons–Hartnoll–Pope conjecture on Böhm-metric instability

The Gibbons–Hartnoll–Pope conjecture predicts instability associated with Böhm Einstein metrics and their related black-hole spacetimes. Earlier work established instability for several families, while the full conjecture remains open.

Known results

Gibbons, Hartnoll, and Pope (2002) proved negative transverse-traceless Lichnerowicz modes for $\mathrm{Bohm}(2,2)_{2m+1}$ and $\mathrm{Bohm}(2,3)_{2m+1}$ when $m \ge 1$ .
Their numerical work found negative modes on $S^5$ , with increasingly negative bounds along specified sequences.
A 2003 instability analysis reported increasingly negative eigenvalues for Böhm metrics on $S^2 \times S^3$ and $S^5$ , while treating arbitrarily negative values for some sequences as expected rather than proved.

August 26, 2026 instability theorem

A new arXiv report claims that the number of negative transverse-traceless Lichnerowicz eigenvalues tends to infinity for specified Böhm-metric sequences. This supplies a rigorous instability mechanism relevant to associated black-hole spacetimes, but it does not claim to settle the full conjecture; the result is unverified here.

Current status (as of August 2026): Instability is proved for several Böhm-metric families, and a new report claims divergence of the negative-mode count for specified sequences, but the full Gibbons–Hartnoll–Pope conjecture remains open.

Gibbons–Hartnoll–Pope conjecture on Böhm-metric instability

Primary source

Additional references

Progress summary

Solutions 0