Mode Stability of Hermitian Instantons

In this note, we prove the Riemannian analog of black hole mode stability for Hermitian, non-self-dual gravitational instantons which are either asymptotically locally flat (ALF) and Ricci-flat, or compact and Einstein with positive cosmological constant. We show that the Teukolsky equation on any such manifold is a positive definite operator. We also discuss the compatibility of the results with the existence of negative modes associated to variational instabilities.