On parametrization, optimization and triviality of test configurations

Uniform stability of twisted constant scalar curvature Kähler metrics

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature Kähler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. We characterise the triviality of test configurations, by showing that a test configuration has zero minimum norm if and only if it has zero $L^2$-norm, if and only if it is almost trivial. We prove that the existence of a twisted constant scalar curvature Kähler metric implies u…