Twisted-Austere Submanifolds in Euclidean Space

A twisted-austere $k$-fold $(M, \mu)$ in $\mathbb R^n$ consists of a $k$-dimensional submanifold $M$ of $\mathbb R^n$ together with a closed $1$-form $\mu$ on $M$ such that the `twisted conormal bundle' $N^* M + \mu$ is a special Lagrangian submanifold of $\mathbb C^n$. The 1-form $\mu$ and the second fundamental form of $M$ must satisfy a particular system of coupled nonlinear second order PDE. We first review these twisted-austere conditions and give an explicit example. Then we focus on twisted-austere 3-folds, giving a geometric description of all solutions when the base $M$ is a cylinder and when $M$ is austere. Finally, we prove that, other than the case of a generalized helicoid in $\mathbb R^5$ discovered by Bryant, there are no other possibilities for the base $M$. This gives a complete classification of twisted-austere $3$-folds in $\mathbb R^n$.