Special Lagrangians in nearly K\"ahler $\mathbb{CP}^3$

TL;DR

This paper investigates special Lagrangian submanifolds in nearly K"ahler $ ext{CP}^3$, providing new homogeneous examples, classifying totally geodesic cases, and showing symmetry implies homogeneity.

Contribution

It introduces new homogeneous examples of special Lagrangians, classifies totally geodesic cases, and proves symmetry leads to homogeneity in nearly K"ahler $ ext{CP}^3$.

Findings

Described new homogeneous special Lagrangian examples.

Classified all totally geodesic special Lagrangians.

Proved symmetry implies homogeneity in the studied setting.

Abstract

This article explores special Lagrangian submanifolds in $\mathbb{CP}^3$, viewed as a nearly K\"ahler manifold, from two different perspectives. Intrinsically, using a moving frame set-up, and extrinsically, using $\mathrm{SU}(2)$ moment-type maps. We describe new homogeneous examples, from both perspectives, and classify totally geodesic special Lagrangian submanifolds. We show that every special Lagrangian in $\mathbb{CP}^3$, or the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^3)$ admitting a symmetry of an $\mathrm{SU}(2)$ subgroup of nearly K\"ahler automorphisms is automatically homogeneous.