solutions into two branches which we term the *small* and *large* localized black holes.[^3] The Euclidean negative modes of the localized and non-uniform solutions are computed in the class of eigenfunctions which preserve the isometry of the background, and we find small localized solutions have a single negative mode, and the large localized and non-uniform strings computed have two. These negative modes are consistent with merger of the large localized and the non-uniform string branches, with one eigenvalue remaining finite and the other apparently diverging to minus infinity at the merger point. We also provide additional evidence for merger by considering low lying positive eigenvalues of various operators. Perhaps the most intriguing result is the observation of a rather narrow window of small localized solutions that appear to have positive specific heat capacity, but possess a negative mode. This will require more rigorous numerical work to confirm. Assuming it holds true, this provides the first static counter-example to any claim that one might be able to reverse the link between local thermodynamic stability and the existence of negative modes discussed in [@Whiting:1988qr; @Whiting:1988ge; @Prestidge:1999uq; @Reall:2001ag; @Gregory:2001bd], complementing the interesting recent stationary counter-example of [@Monteiro:2009tc].

The Einstein-DeTurck equation {#sec:setup}
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We 