Associative submanifolds of the 7-sphere
Associative submanifolds of the 7-sphere
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7 球体的关联子流形
DOI：
10.1112/plms/pds029
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发表时间：
2012
影响因子：
1.8
通讯作者：
Lotay J
中科院分区：
文献类型：
--
作者：
Lotay J
7球的关联子流形𝒮7are 3维最小子流形是在被称为Cayley锥的4维标定锥的连接。因此，结合3 -褶皱的例子是由复和特殊拉格朗日锥的连杆以及近似Kähler 6 -球的拉格朗日子流形给出的。通过对结合群轨道进行分类，我们展示了已知的第一个明确的例子，即𝒮7which中的结合3 - fold不是来自其他几何形状。然后，我们研究了满足曲率约束（称为Chen等式）的关联3 -折叠，该约束等价于第二个基本形式上的自然点向条件，并使用一组新的伪全纯曲线（在2平面的Grassmannian中）和𝒮6中的各向同性最小曲面来描述它们。我们还证明了由测地线圆控制的结合3 -褶皱，就像空间形式中的最小曲面一样，允许局部等距变形族。最后，我们利用𝒮6中的调和2 -球构造了满足Chen等式的具有𝒮1 -族全局等长变形的关联3 -折叠。
Associative submanifolds of the 7‐sphere 𝒮7are 3‐dimensional minimal submanifolds which are the links of calibrated 4‐dimensional cones in ℝ8called Cayley cones. Examples of associative 3‐folds are thus given by the links of complex and special Lagrangian cones in ℂ4, as well as Lagrangian submanifolds of the nearly Kähler 6‐sphere. By classifying the associative group orbits, we exhibit the first known explicit example of an associative 3‐fold in 𝒮7which does not arise from other geometries. We then study associative 3‐folds satisfying the curvature constraint known as Chen's equality, which is equivalent to a natural pointwise condition on the second fundamental form, and describe them using a new family of pseudoholomorphic curves in the Grassmannian of 2‐planes in ℝ8and isotropic minimal surfaces in 𝒮6. We also prove that associative 3‐folds which are ruled by geodesic circles, like minimal surfaces in space forms, admit families of local isometric deformations. Finally, we construct associative 3‐folds satisfying Chen's equality which have an 𝒮1‐family of global isometric deformations using harmonic 2‐spheres in 𝒮6.