Extensions of extremal K\"ahler submanifolds of complex projective spaces

TL;DR

This paper proves that connected extremal K"ahler submanifolds of complex projective spaces can be extended to complete K"ahler manifolds with holomorphic isometric immersions, and explores applications to extremal K"ahler hypersurfaces.

Contribution

It establishes a natural extension for extremal K"ahler submanifolds and provides new insights into their structure within complex projective spaces.

Findings

Connected extremal K"ahler submanifolds can be extended to complete manifolds.

Such extensions admit holomorphic isometric immersions.

Application to extremal K"ahler hypersurfaces in projective spaces.

Abstract

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.