Negative Lichnerowicz modes on Einstein warped products and spheres

Every nonhomogeneous compact Böhm metric on a sphere or a product of spheres in dimensions five through nine has a negative eigenvalue of the full Lichnerowicz Laplacian on transverse-traceless tensors. More generally, for a nonround doubly warped Einstein sphere with factor dimensions p, q≥2, the inequalities (p−1)(q−4)≤4 and (q−1)(p−4)≤4 suffice. They also apply to Wang's SO(3)×SO(9)-invariant Einstein metric on S¹¹. For a positive Einstein warped product of closed connected manifolds, with base dimension k≥3, fibre dimension m≥2, and positive nonconstant warping function, the condition (k−2)(m−4)≤4 gives the same conclusion. Fibre dimensions two through four are therefore covered without a bound on the total dimension. The variational tool is a sharp divergence penalty on smooth trace-free tensors: on a closed connected positive Einstein manifold of dimension n≥3, subtracting n/(n−1) times the squared divergence from the full Lichnerowicz quadratic form preserves its negative transverse-traceless index. For the sphere problem, this penalty cancels the quadratic radial-derivative term in a smooth interface construction; a weighted-radius rigidity theorem supplies the sign of the remaining interface contribution. Applications include finite quotients and ancient Ricci flows obtained from Kröncke's nonlinear instability theorem. An appendix treats negative spectrum on Ricci-flat asymptotically conical manifolds.