Mode Stability of Hermitian Instantons

TL;DR

The paper addresses the problem of mode stability for Hermitian gravitational instantons in Riemannian signature, focusing on ALF Ricci-flat and compact Einstein–Hermitian four-manifolds with $\lambda>0$. It develops a spinor-based, conformally invariant GHP framework to derive and analyze the Teukolsky equation, proving the Teukolsky operator $L$ is positive definite on these backgrounds and that perturbations satisfy $\vartheta\Psi_0=0$. The key contribution is establishing mode stability for both ALF and compact cases by a weighted energy identity and establishing $\Psi_2>0$ in the relevant backgrounds, ensuring positivity and boundary-term control. The work also clarifies how potential negative modes from variational instability can coexist with mode stability, by showing such perturbations are conformally half-flat and do not drive $\vartheta\Psi_0$ away from zero, thereby preserving the stability conclusion with respect to the Teukolsky equation.

Abstract

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.

Theorems & Definitions (18)

• Remark 1.1
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Conjecture 1.7
• Remark 1.8
• Remark 2.1
• Definition 2.2: 2023arXiv230614567A