Let f :M2n→R2n+4f:M2n→R2n+4 be an isometric immersion of a Kaehler manifold of complex dimension n≥5n≥5 into Euclidean space with complex rank at least 55 everywhere. Our main result is that, along each connected component of an open dense subset of M2nM2n, either ff is holomorphic in R2n+4≅Cn+2R2n+4≅Cn+2, or it is in a unique way a composition f=F∘hf=F∘h of isometric immersions. In the latter case, we have that h :M2n→N2n+2h:M2n→N2n+2 is holomorphic and F :N2n+2→R2n+4F:N2n+2→R2n+4 belongs to the class, by now quite well understood, of non-holomorphic Kaehler submanifolds in codimension two. Moreover, the submanifold FF is minimal if and only if ff is minimal.
Referencia
Chion, S. & Dajczer, M. (2023). Real Kaehler submanifolds in codimension up to four. Revista Matemática Iberoamericana. Advanced online publication. DOI 10.4171/RMI/1427 Real Kaehler submanifolds in codimension up to four
