Let X CPN be a smooth subvariety. 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 H0(Lk) 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 Donaldsons theorem relating balanced
embeddings to metrics with constant scalar curvature [12]. In
our proof we combine Donaldson乫s techniques with an asymptotic
result of Liu and Ma [17] which, as we explain, describes the asymptotic
behavior of the derivative of the map FS . Hilb whose
fixed points are balanced metrics.