Connected Sums of Special Lagrangian Submanifolds

摘要

Let M_1 and M_2 be special Lagrangian submanifolds of a compact Calabi-Yau manifold X that intersect transversely at a single point. We can then think of M_1 ∪ M_2 as a singular special Lagrangian submanifold of X with a single isolated singularity. We investigate when we can regularize M_1 ∪ M_2 in the following sense: There exists a family of Calabi-Yau structures X_α on X and a family of special Lagrangian submanifolds M_α of X_α such that M_α converges to M_1 ∪ M_2 and X_α 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 [M_1] is not a multiple of [M_2] in H_3(X), and (2) when X is a torus with dim_C X ≥ 3, M_1 is flat, and the intersection of M_1 and M_2 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.