Isoperimetric flow and convexity of H-graphs

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.