  In this paper we consider a ``flow'' of nonparametric solutions of the volume
constrained Plateau problem with respect to a convex planar curve. Existence
and regularity is obtained from standard elliptic theory, and convexity results
for small volumes are obtained as an immediate consequence. Finally, the
regularity is applied to show a strong stability condition for all volumes
considered. This condition, in turn, allows us to adapt an argument of Cabr\'e
and Chanillo which yields that any solution enclosing a non-zero volume has a
unique nondegenerate critical point.
