  We study the stabilization of scalars near a supersymmetric black hole
horizon using the equation of motion of a particle moving in a potential and
background metric. When the relevant 4-dimensional theory is described by
special geometry, the generic properties of the critical points of this
potential can be studied. We find that the extremal value of the central charge
provides the minimal value of the BPS mass and of the potential under the
condition that the moduli space metric is positive at the critical point. We
relate these ideas to the Weinhold and Ruppeiner metrics introduced in the
geometric approach to thermodynamics and used for study of critical phenomena.
