Relative K-stability for K\"ahler manifolds

We study the existence of extremal K"ahler metrics on K"ahler manifolds. After introducing a notion of relative K-stability for K"ahler manifolds, we prove that K"ahler manifolds admitting extremal K"ahler metrics are relatively K-stable. Along the way, we prove a general $L^p$ lower bound on the Calabi functional involving test configurations and their associated numerical invariants, answering a question of Donaldson. When the K"ahler manifold is projective, our definition of relative K-stability is stronger than the usual definition given by Sz'ekelyhidi. In particular our result strengthens the known results in the projective case (even for constant scalar curvature K"ahler metrics), and rules out a well known counterexample to the “na"ive” version of the Yau-Tian-Donaldson conjecture in this setting.