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

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