Extremal Kähler metrics, when they exist, are ''canonical'' representatives of their Kähler class. Their existence is
conjecturaly equiavlent to the stability of the underlying polarised variety. Via quanitsation, there is a strong connection between
extremal metrics and balanced projective embeddings. In addition to these aspects we also consider the production of extremal metrics via
geometric analysis. One tool for this is the Calabi flow which attempts to deform a given Kähler metric to an extremal one. The
quantization of this flow is balancing flow, a certain flow on the space of projective embeddings. We are intereseted in better
understanding Calabi flow via the projective geometry of balancing flow.