  Computing heavy quark-antiquark potentials within the AdS/CFT correspondence
often leads to behaviors that differ from what one expects on general physical
grounds and field-theory considerations. To isolate the configurations of
physical interest, it is of utmost importance to examine the stability of the
string solutions dual to the flux tubes between the quark and antiquark. Here,
we formulate and prove several general statements concerning the perturbative
stability of such string solutions, relevant for static quark-antiquark pairs
in a general class of backgrounds, and we apply the results to N=4 SYM at
finite temperature and at generic points of the Coulomb branch. In all cases,
the problematic regions are found to be unstable and hence physically
irrelevant.
