| The primary goal of this work is to explain a gluing construction for compact special Lagrangian submanifolds of Calabi-Yau manifolds. We summarize our results as follows:; Let M1 and M2 be compact 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 M 1 ∪ M2 in the following sense: There exists a family of Calabi-Yau structures Xt on X and a family of compact special Lagrangian submanifolds Mt of Xt such that Mt converges to M1 ∪ M2 and Xt converges to the original Calabi-Yau structure on X as t → 0. We prove that a regularization exists in two important cases: (1) when dimC 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 dimC X ≥ 3, M1 is flat, and the intersection of M1 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 special Lagrangian submanifolds of Calabi-Yau tori.; These results build on earlier work by A. Butscher and Y.-I. Lee. |
