A Classification of Twisted Austere 3-Folds
| Autor: | Spiro Karigiannis, Thomas A. Ivey |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Euclidean space Second fundamental form 010102 general mathematics Order (ring theory) Submanifold 01 natural sciences Generalized helicoid Base (group theory) Bundle 0103 physical sciences Calibrated geometry Mathematics::Differential Geometry 010307 mathematical physics Geometry and Topology 0101 mathematics Mathematical Physics Analysis Mathematics |
| Zdroj: | Symmetry, Integrability and Geometry: Methods and Applications. |
| ISSN: | 1815-0659 |
| DOI: | 10.3842/sigma.2021.023 |
| Popis: | 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' $N^* M + d \mu$ 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 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 $\mathbb R^n$. |
| Databáze: | OpenAIRE |
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