Eigenvalues of the MOTS stability operator for slowly rotating Kerr black holes

We study the eigenvalues of the MOTS stability operator for the Kerr black hole with angular momentum per unit mass $|a| ll M$. We prove that each eigenvalue depends analytically on $a$ (in a neighbourhood of $a=0$), and compute its first nonvanishing derivative. Recalling that $a=0$ corresponds to the Schwarzschild solution, where each eigenvalue has multiplicity $2ell+1$, we find that this degeneracy is completely broken for nonzero $a$. In particular, for $0 < |a| ll M$ we obtain a cluster consisting of $ell$ distinct complex conjugate pairs and one real eigenvalue. As a special case of our results, we get a simple formula for the variation of the principal eigenvalue. For perturbations that preserve the total area or mass of the black hole, we find that the principal eigenvalue has a local maximum at $a=0$. However, there are other perturbations for which the principal eigenvalue has a local minimum at $a=0$.