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SIGMA, 2021 Volume 17, 023, 31 pp. (Mi sigma1706)

A Classification of Twisted Austere $3$-Folds

Thomas A. Iveya, Spiro Karigiannisb

a Department of Mathematics, College of Charleston, USA
b Department of Pure Mathematics, University of Waterloo, Canada

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 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 ${\mathbb R}^n$.

Keywords: calibrated geometry, special Lagrangian submanifolds, austere submanifolds, exterior differential systems.

MSC: 53B25, 53C38, 53C40, 53D12, 58A15

Received: October 13, 2020; in final form March 2, 2021; Published online March 10, 2021

Language: English

DOI: 10.3842/SIGMA.2021.023



Bibliographic databases:
ArXiv: 2006.15119


© Steklov Math. Inst. of RAS, 2024