In 1982 Calabi proposed studying gradient flow of the L^2 norm
of the scalar curvature (now called Calabi flow) as a tool for finding
canonical metrics within a given Kahler class. The main motivating
conjecture behind this flow (due to Calabi-Chen) asserts the smooth long
time existence of this flow with arbitrary initial data. By exploiting
aspects of the Mabuchi-Semmes-Donaldson metric on the space of Kahler
metrics I will construct a kind of weak solution to this flow, known as a
minimizing movement, which exists for all time.