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

TL;DR

This paper establishes explicit correspondences between minimal $ ext{delta}(2)$-ideal Lagrangian submanifolds in complex space, minimal totally complex surfaces in quaternionic projective space, and minimal Lagrangian surfaces in complex projective space, revealing deep geometric links.

Contribution

It constructs explicit maps linking minimal $ ext{delta}(2)$-ideal Lagrangian submanifolds to quaternionic and complex projective geometries, providing new classification results.

Findings

Explicit map from Lagrangian submanifolds to quaternionic projective space

One-to-one correspondence for $n=3$ between minimal $ ext{delta}(2)$-ideal Lagrangians and complex surfaces

Correspondence between complex surfaces in $ ext{HP}^2$ and Lagrangian surfaces in $ ext{CP}^2

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 $\mathbb{C}P^2$.