Geodesic
A Classification of Twisted Austere $3$-Folds
Symmetry, integrability and geometry: methods and applications, Tome 17 (2021) Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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     author = {Thomas A. Ivey and Spiro Karigiannis},
     title = {A {Classification} of {Twisted} {Austere} $3${-Folds}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2021},
     volume = {17},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a22/}
}
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Thomas A. Ivey; Spiro Karigiannis. A Classification of Twisted Austere $3$-Folds. Symmetry, integrability and geometry: methods and applications, Tome 17 (2021). http://geodesic.mathdoc.fr/item/SIGMA_2021_17_a22/

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