  We give a moment map interpretation of some relatively balanced metrics. As
an application, we extend a result of S. K. Donaldson on constant scalar
curvature K\"ahler metrics to the case of extremal metrics. Namely, we show
that a given extremal metric is the limit of some specific relatively balanced
metrics. As a corollary, we recover uniqueness and splitting results for
extremal metrics in the polarized case.
