Minimal K\"ahler submanifolds in product of space forms

TL;DR

This paper investigates minimal K"ahler submanifolds in products of space forms, establishing conditions for their existence and characterizing their geometric properties, especially focusing on restrictions imposed by curvature and immersion types.

Contribution

It characterizes minimal K"ahler immersions into product space forms, identifying when such immersions are possible and describing their geometric restrictions.

Findings

Only minimal immersions of Riemannian surfaces into certain products exist.

Existence of such immersions constrains Ricci and scalar curvatures.

Characterization of immersions with parallel second fundamental form or anti-pluriharmonic property.

Abstract

In this article, we study minimal isometric immersions of K\"ahler manifolds into product of two real space forms. We analyse the obstruction conditions to the existence of pluriharmonic isometric immersions of a K\"ahler manifold into those spaces and we prove that the only ones into $\mathbb{S}^{m-1}\times\mathbb{R}$ and $\mathbb{H}^{m-1}\times \mathbb{R}$ are the minimal isometric immersions of Riemannian surfaces. Futhermore, we show that the existence of a minimal isometric immersion of a K\"ahler manifold $M^{2n}$ into $\mathbb{S}^{m-1}\times\mathbb{R}$ and $\mathbb{S}^{m-k}\times \mathbb{H}^k$ imposes strong restrictions on the Ricci and scalar curvatures of $M^{2n}$. In this direction, we characterise some cases as either isometric immersions with parallel second fundamental form or anti-pluriharmonic isometric immersions.