Special Lagrangian 4-folds with $SO(2)\rtimes S_3$-Symmetry in Complex Space Forms

TL;DR

This paper provides a complete classification of special Lagrangian submanifolds with $SO(2)\rtimes S_3$-symmetry in complex space forms, extending previous incomplete results and covering arbitrary complex space forms.

Contribution

It offers the first complete classification of these submanifolds and generalizes prior work to all complex space forms.

Findings

Complete classification of special Lagrangian submanifolds with $SO(2)\rtimes S_3$-symmetry in complex space forms.

Extension of classification from $\\mathbb{C}^4$ to all complex space forms.

Clarification of algebraic structure of the second fundamental form.

Abstract

In this article we obtain a classification of special Lagrangian submanifolds in complex space forms subject to an $SO(2)\rtimes S_3$-symmetry on the second fundamental form. The algebraic structure of this form has been obtained by Marianty Ionel. However, the classification of special Lagrangian submanifolds in $\mathbb{C}^4$ having this $SO(2)\rtimes S_3$ symmetry in that paper is incomplete. In the present paper we give a complete classification of such submanifolds, and extend the classification to special Lagrangian submanifolds of arbitrary complex space forms with $SO(2)\rtimes S_3$-symmetry.