Lower bounds on the modified K-energy and complex deformations

Let (X,L) be a polarized K"ahler manifold that admits an extremal K"ahler metric in c1(L). We show that on a nearby polarized deformation that preserves the symmetry induced by the extremal vector field of (X,L), the modified K-energy is bounded from below. This generalizes a result of Chen, Sz'ekelyhidi and Tosatti to extremal metrics. Our proof also extends a convexity inequality on the space of K"ahler potentials due to X.X. Chen to the extremal metric setup. As an application, we compute explicit polarized 4-points blow-ups of CP1\times CP1 that carry no extremal metric but with modified K-energy bounded from below.