Twisted-Austere Submanifolds in Euclidean Space

Abstract

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' $N^{\ast }M+\mu$ 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 ${\mathbb{R}}^n$.

31 pages. Final published version. (A few minor misprints corrected)

References in corpus (1)

• Deformations of calibrated subbundles of Euclidean spaces via twisting by special sections