SPECIAL LAGRANGIAN 4-FOLDS WITH $SO(2)\rtimes S_3$-SYMMETRY IN COMPLEX SPACE FORMS

Abstract

In this article we obtain a classification of special Lagrangian submanifolds in complex space forms subject to a pointwise $SO(2)\rtimes S_3$-symmetry on the second fundamental form. The algebraic structure of this form has been obtained by Marianty Ionel in [8]. However, the classification of special Lagrangian submanifolds in $\mathbb{C}^4$ having this $SO(2)\rtimes S_3$ symmetry in [8] is incomplete. In this paper we give a complete classification of such submanifolds, and extend the classification to special Lagrangian submanifolds of arbitrary complex space forms with a pointwise $SO(2)\rtimes S_3$-symmetry in the second fundamental form.