  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 $2\ell+1$,
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 $a=0$.
