Minimal real Kaehler submanifolds

TL;DR

This paper demonstrates that under certain generic conditions, Kaehler submanifolds immersed in Euclidean space with low codimension must be minimal, revealing a fundamental geometric property of such submanifolds.

Contribution

It establishes a new link between rank conditions on the second fundamental form and minimality for Kaehler submanifolds in low codimension.

Findings

Generic rank conditions imply minimality of the submanifold.

Existence of a one-parameter family of minimal immersions for simply connected cases.

Holomorphic submanifolds are exceptions to the minimality result.

Abstract

We show that generic rank conditions on the second fundamental form of an isometric immersion $f\colon M^{2n}\to\mathbb{R}^{2n+p}$ of a Kaehler manifold of complex dimension $n\geq 2$ into Euclidean space with low codimension $p$ implies that the submanifold has to be minimal. If $M^{2n}$ if simply connected, this amounts to the existence of a one-parameter associated family of isometric minimal immersions unless $f$ is holomorphic.