Instability of Böhm metrics in higher dimensions

Establish that every Böhm metric in higher dimensions has at least one negative eigenvalue of the Lichnerowicz Laplacian, extending the known instability result for Böhm metrics on S^3×S^2, S^3×S^3, and S^5.

Background

Böhm metrics are cohomogeneity-one Einstein metrics constructed on certain low-dimensional spheres and products of spheres. Before this paper, Gibbons, Hartnoll, and Pope had shown that the Lichnerowicz Laplacian associated with every Böhm metric on S^3×S^2, S^3×S^3, and S^5 admits at least one negative mode.

The paper proves that, along the specific sequences of Böhm metrics on S^{k+1}×S^l and S^{k+l+1} with k,l≥2 and k+l≤8, the number of negative Lichnerowicz eigenvalues tends to infinity. This gives instability for all but possibly finitely many metrics in those sequences, but it does not settle the broader higher-dimensional conjecture stated by Gibbons, Hartnoll, and Pope.