Let f : M 2n → R 2n+p denote an isometric immersion of a Kaehler manifold of complex dimension n ≥ 2 into Euclidean space with codimension p. If 2p ≤ 2n − 1, we show that generic rank conditions on the second fundamental form of the submanifold imply that f has to be a minimal submanifold. In fact, for codimension p ≤ 11 we prove that f must be holomorphic with respect to some complex structure in the ambient space.

Let f : M 2n → R 2n+4 be an isometric immersion of a Kaehler manifold of complex dimension n ≥ 5 into Euclidean space with complex rank at least 5 everywhere. Our main result is that, along each connected component of an open dense subset of M 2n , either f is holomorphic in R 2n+4 ∼ = C n+2 or it is in a unique way a composition f = F • h of isometric immersions. In the latter case, we have that h : M 2n → N 2n+2 is holomorphic and F : N 2n+2 → R 2n+4 belongs to the class, by now quite well understood, of non-holomorphic Kaehler submanifold in codimension two. Moreover, the submanifold F is minimal if and only if f is minimal.

We show that generic rank conditions on the second fundamental form of an isometric immersion f : M 2n → R 2n+p of a Kaehler manifold of complex dimension n ≥ 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.