Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 023, 31 pages      arXiv:2006.15119

Twisted-Austere Submanifolds in Euclidean Space

Thomas A. Ivey a and Spiro Karigiannis b
a) Department of Mathematics, College of Charleston, USA
b) Department of Pure Mathematics, University of Waterloo, Canada

Received October 13, 2020, in final form March 02, 2021; Published online March 10, 2021; Misprints fixed March 12, 2021

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

Key words: calibrated geometry; special Lagrangian submanifolds; austere submanifolds; exterior differential systems.

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