BLACK-HOLES AND CRITICAL-POINTS IN MODULI SPACE

We study the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potent ial and background metric. When the relevant 4-dimensional theory is d escribed by special geometry, the generic properties of the critical p oints of this potential can be studied. We find that the extremal valu e 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. This is a property of a regular specia l geometry. We also study the critical points in all N greater than or equal to 2 supersymmetric theories. We relate these ideas to the Wein hold and Ruppeiner metrics introduced in the geometric approach to the rmodynamics and used for the study of critical phenomena. (C) 1997 Els evier Science B.V.