Embedded Special Lagrangian Submanifolds in Calabi-Yau Manifolds
Embedded Special Lagrangian Submanifolds in Calabi-Yau Manifolds
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Calabi-Yau 流形中嵌入特殊拉格朗日子流形
DOI：
10.4310/cag.2003.v11.n3.a1
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发表时间：
2003
影响因子：
0.7
通讯作者：
Ynging Lee
中科院分区：
文献类型：
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作者：
Ynging Lee
卡拉比-丘流形是具有平凡标准线丛的凯勒流形。这是由S.T. Yau [24]证明了在Calabi-Yau流形中存在唯一的Ricci平坦度量。因此，我们在n维Calabi-Yau流形N中有两种特殊形式ω和Ω，其中ω是Ricci平坦度量g的Kähler形式，Ω是关于g的单位长度的平行全纯（n，0）形式。N中的一个真实的n维子流形L称为拉格朗日子流形，如果ω对L的限制为零。另外，如果Im Ω对L的限制也为零，则L称为特殊拉格朗日量。这相当于用Re Ω校准L。经过校准的子歧管始终使体积最小化。(See[7]或本文第1节。特别地，特殊拉格朗日子流形是中维的极小子流形。这激发了我们对特殊拉格朗日子流形或更一般的拉格朗日极小子流形的研究（[11]，[12]，[20]）。另一个动机来自镜像对称。在[23]中，A. Stominger，S.T. Yau和E. Zaslow提出利用特殊拉格朗日环面的模空间及其平坦联络构造Calabi-Yau流形的镜像流形。对于这个猜想的发展和修改，我们参考[9]，[5]，[17]等。以及其中的引用。本文是一个尝试，在采用微扰方法来研究这个方向的问题。特别是，我们证明
A Calabi-Yau manifold is a Kähler manifold with trivial canonical line bundle. It is proved by S.T. Yau [24] that in a Calabi-Yau manifold there exists a unique Ricci flat metric in its Kähler class. Therefore, we have two special forms ω and Ω in an n-dimensional Calabi-Yau manifold N , where ω is the Kähler form of the Ricci flat metric g and Ω is a parallel holomorphic (n, 0) form of unit length with respect to g. A real n-dimensional submanifold L in N is called Lagrangian if the restriction of ω on L vanishes. If in addition, the restriction of Im Ω on L also vanishes, then L is called special Lagrangian. This is equivalent to that L is calibrated by Re Ω. A calibrated submanifold is always volume minimizing. (See [7] or section 1 in this paper.) In particular, special Lagrangian submanifolds are minimal submanifolds of middle dimension. This motivates our study on special Lagrangian submanifolds or more generally on Lagrangian minimal submanifolds ([11], [12], [20]). Another motivation comes from mirror symmetry. In [23], A. Stominger, S.T. Yau, and E. Zaslow proposed to construct the mirror manifold of a Calabi-Yau manifold by the moduli space of special Lagrangian tori together with their flat connections. For development and modification of this conjecture, we refer to [9], [5], [17] etc., and the reference therein. The current paper is an attempt in employing the perturbation method to study problems in this direction. In particular, we prove