A Classification of Twisted Austere $3$-Folds

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 second fundamental form $A$ of $M$ and the $1$-form $\mu$ satisfy a particular system of coupled nonlinear second order PDE. Given such an object, the “twisted conormal bundle” $N^* M + \mathrm{d} \mu$ is a special Lagrangian submanifold of ${\mathbb C}^n$. We review the twisted-austere condition and give an explicit example. Then we focus on twisted-austere $3$-folds. We give 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 ${\mathbb R}^n$.