Instability of B\"ohm's Einstein metrics
B\"ohm metrics on $S^{k+1}\times S^l$ and $S^{k+l+1}$ ($k,l\geqslant 2$, $k+l\leqslant 8$) occur in sequences of Einstein metrics that converge to a cone. We prove that, along such a sequence, the number of negative eigenvalues of the Lichnerowicz Laplacian acting on transverse-traceless tensors tends to infinity: these metrics become increasingly unstable. This result gives a partial answer to a conjecture of Gibbons, Hartnoll and Pope concerning the instability of the generalised black hole spacetimes built from B\"ohm metrics.
math.DG↗