In this paper we showed that every connected extremal K\"ahler submanifold of a complex projetive space has a natural extension which is a complete K\"ahler manifold and admits a holomorphic isometric immersion into the same ambient space. We also give an application to study extremal K\"ahler hypersurfaces of complex projective spaces.PDF: Extensions of extremal K\"ahler submanifolds of complex projective spaces.pdf
