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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

Embedded Special Lagrangian Submanifolds in Calabi-Yau Manifolds