Minimal Kähler submanifolds in product of space forms

Abstract

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