论文标题
欧几里得空间中的扭曲 - 严重的子曼菲尔德
Twisted-Austere Submanifolds in Euclidean Space
论文作者
论文摘要
一个扭曲的$ k $ -fold $（m，m，μ）$ in Mathbb r^n $由a $ k $ dimensional submanifold $ m $ m $ of $ \ mathbb r^n $一起，封闭的$ 1 $ - form $ - form $ - form-form $ - $ m $，使得``twisted conmoral'$ $ n^$ n^$ n^$ n^* m +μ $ \ mathbb c^n $。 $ m $的1型$μ$和第二个基本形式必须满足特定的耦合非线性二阶PDE系统。我们首先回顾了这些扭曲的扭曲条件，并举一个明确的例子。然后，我们将重点放在扭曲的3倍上，当基本$ m $是一个圆柱体时，当$ m $ austere时，给出了所有解决方案的几何描述。最后，我们证明，除了Bryant发现的$ \ Mathbb r^5 $中的广义螺旋体外，没有其他可能性的基础$ M $。这给出了$ \ mathbb r^n $的扭曲扭曲$ 3 $ folds的完整分类。
A twisted-austere $k$-fold $(M, μ)$ in $\mathbb R^n$ consists of a $k$-dimensional submanifold $M$ of $\mathbb R^n$ together with a closed $1$-form $μ$ on $M$ such that the `twisted conormal bundle' $N^* M + μ$ is a special Lagrangian submanifold of $\mathbb C^n$. The 1-form $μ$ 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 $\mathbb R^n$.
