Communications in Analysis and Geometry

Volume 11 (2003)

Number 3

Embedded Special Lagrangian Submanifolds in Calabi-Yau Manifolds

Pages: 391 – 423

DOI: http://dx.doi.org/10.4310/CAG.2003.v11.n3.a1

Author

Yng-Ing Lee

Abstract

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 ImΩ 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

Theorem 3. Suppose that L is a closed, connected, and immersed special Lagrangian submanifold in a closed Calabi-Yau manifold N of complex dimension 3. Assume that L has only isolated transversal self-intersection points. Then L is the limit of a family of embedded closed special Lagrangian submanifolds in N.

Theorem 4. Suppose that L is a closed, connected, and immersed special Lagrangian submanifold in a closed Calabi-Yau manifold N of complex dimension n > 3. Moreover, assume that L has only isolated transversal self-intersection of two sheets and the two tangent planes at each intersection point satisfy the angle condition θ1 + . . . +θn = π / 2 (see section 2). Then L is the limit of a family of embedded closed special Lagrangian submanifolds in N.

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