A parabolic flow of balanced metrics

We prove a general criterion to establish existence and uniqueness of a shorttime solution to an evolution equation involving "closed" sections of a vector bundle, generalizing a method used by Bryant and Xu [8] for studying the Laplacian flow in G(2)-geometry. We apply this theorem in balanced geometry introducing a natural extension of the Calabi flow to the balanced case. We show that this flow has always a unique short-time solution belonging to the same Bott-Chern cohomology class of the initial balanced structure and that it preserves the Kuhler condition. Finally, we study explicitly the flow on the Iwasawa manifold.