Minimal $\delta (2)$-ideal Lagrangian submanifolds and the Quaternionic projective space

Abstract

We construct an explicit map from a generic minimal $\delta (2)$-ideal Lagrangian submanifold of ${\mathbb{C}}^n$ to the quaternionic projective space $\mathbb{H}{P}^{n-1}$, whose image is either a point or a minimal totally complex surface. A stronger result is obtained for $n=3$, since the above mentioned map then provides a one-to-one correspondence between minimal $\delta (2)$-ideal Lagrangian submanifolds of ${\mathbb{C}}^3$ and minimal totally complex surfaces in $\mathbb{H}{P}^2$ which are moreover anti-symmetric. Finally, we also show that there is a one-to-one correspondence between such surfaces in $\mathbb{H}{P}^2$ and minimal Lagrangian surfaces in $\mathbb{C}{P}^2$.

21 pages