Connected Sums of Special Lagrangian Submanifolds

Let M₁ and M₂ be special Lagrangian submanifolds of a compact Calabi-Yau manifold X that intersect transversely at a single point. We can then think of M₁ ⋃ M₂ as a singular special Lagrangian submanifold of X with a single isolated singularity. We investigate when we can regularize M₁ ⋃ M₂ in the following sense: There exists a family of Calabi-Yau structures Xα on X and a family of special Lagrangian submanifolds Mα of Xα such that Mα converges to M₁ ⋃ M₂ and Xα converges to the original Calabi-Yau structure on X. We prove that a regularization exists in two important cases: (1) when dim_C X = 3, Hol(X) = SU(3), and [M₁] is not a multiple of [M₂] in H₃(X), and (2) when X is a torus with dim_CX ≥ 3, M₁ is flat, and the intersection of M₁ and M₂ satisfies a certain angle criterion. One can easily construct examples of the second case, and thus as a corollary we construct new examples of non-flat special Lagrangian submanifolds of Calabi-Yau tori.

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Published 1 January 2004