  Let X be a smooth subvariety of CP^N. We study a flow, called balancing flow,
on the space of projectively equivalent embeddings of X, which attempts to
deform the given embedding into a balanced one. If L->X is an ample line
bundle, considering embeddings via H^0(L^k) gives a sequence of balancing
flows. We prove that, provided these flows are started at appropriate points,
they converge to Calabi flow for as long as it exists. This result is the
parabolic analogue of Donaldson's theorem relating balanced embeddings to
metrics with constant scalar curvature [JDG 59(3):479-522, 2001]. In our proof
we combine Donaldson's techniques with an asymptotic result of Liu-Ma
[arXiv:math/0601260v2] which, as we explain, describes the asymptotic behaviour
of the derivative of the map FS\circ Hilb whose fixed points are balanced
metrics.
