Calabi type functionals for coupled Kähler–Einstein metrics

Abstract We introduce the coupled Ricci–Calabi functional and H-functional which measure how far a Kähler metric is from Kähler–Einstein in sense of Hultgren–Witt Nyström. first give corresponding moment weight type inequalities estimate each terms algebraic invariants. Secondly, we Hessian formulas for these functionals at critical point, have an application to Matsushima obstruction theorem existence metric.