Calabi-energy sublevels, K-stability and Calabi flow

In this paper, we prove path connectedness of every nonempty strict Calabi-energy sublevel and identify the infimum of the square root of the Calabi energy over smooth metrics with the infimum of the descending Mabuchi slope in the metric completion. In particular, there exist smooth metrics of arbitrarily small Calabi energy in every K-semistable real ample class. On compact Kähler surfaces, we prove the existence of constant scalar curvature Kähler metrics in the K-polystable classes of the controlled cone. In any class containing such a metric, the Calabi flow exists smoothly for all time with a uniform Sobolev bound whenever the initial metric is smooth and has sufficiently small Calabi energy.