2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129826In this note, we prove that on polarized toric manifolds the relative $K$-stability with respect to Donaldson's toric degenerations is a necessary condition for the existence of Calabi's extremal metrics, and also we show that the modified $K$-energy is proper in the space of $G_0$-invariant Kähler metrics for the case of toric surfaces which admit the extremal metrics.8 pagesDifferential GeometryAlgebraic Geometry53C55;32J15text