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We show that a real Kähler submanifold in codimension 6 6 is essentially a holomorphic submanifold of another real Kähler submanifold in lower codimension if the second fundamental form is not sufficiently degenerated. We also give a shorter proof of this result when the real Kähler submanifold is minimal, using recent results about isometric rigidity.

In this work, we prove that any two free boundary minimal hypersurfaces in the unit Euclidean ball have an intersection point in any half-ball. This is a strong version of the Frankel property proved by A. Fraser and M. Li [5]. As a consequence, we obtain the two-piece property for free boundary minimal hypersurfaces in the unit ball: every equatorial disk divides any compact minimal hypersurface with free boundary in the unit ball in two connected pieces.

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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.

We show that a real Kähler submanifold in codimension 6 is essentially a holomorphic submanifold of another real Kähler submanifold in lower codimension if the second fundamental form is not sufficiently degenerated. We also give a shorter proof of this result when the real Kähler submanifold is minimal, using recent results about isometric rigidity.