Symmetries and mode stability of gravitational instantons

We study two aspects of gravitational instantons: mode stability of gravitational instantons, and T²-symmetric (toric) gravitational instantons. We study Ricci-flat perturbations of gravitational instantons of Petrov type D. Analogously to the Lorentzian case, the Weyl curvature scalars of extreme spin weight satisfy a Riemannian version of the separable Teukolsky equation. As a step toward infinitesimal rigidity of the type D Kerr and Taub-bolt families of instantons, we prove mode stability, that is, that the Teukolsky equation admits no solutions compatible with regularity and asymptotic (local) flatness. For an asymptotically locally Euclidean (ALE) or ALF gravitational instanton (M, g) with toric symmetry, we express the signature of (M, g) directly in terms of its rod structure. Applying Hitchin–Thorpe-type inequalities for Ricci-flat ALE/ALF manifolds, we formulate, as a step toward a classification of toric ALE/ALF instantons, necessary conditions that the rod structures of such spaces must satisfy. Finally, we apply these results to the study of rod structures with three turning points.