  We consider deformations of metrics in a given conformal class such that the
smallest eigenvalue of the Ricci tensor to be a constant. It is related to the
notion of minimal volumes in comparison geometry. Such a metric with the
smallest eigenvalue of the Ricci tensor to be a constant is an extremal metric
of volume in a suitable sense in the conformal class. The problem is reduced to
solve a Pucci type equation with respect to the Schouten tensor. We establish a
local gradient estimate for this type of conformally invariant fully nonlinear
uniform elliptic equations. Combining it with the theory of fully nonlinear
equations, we establish the existence of solutions for this equation.
