Embedded Special Lagrangian Submanifolds in Calabi-Yau Manifolds

摘要

A Calabi-Yau manifold is a Kahler 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 Kahler class. Therefore, we have two special forms ω and Ω in an n-dimensional Calabi-Yau manifold N, where u is the Kahler 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 TV 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 ReΩ. 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.