Communications in Analysis and Geometry

Volume 12 (2004)

Number 3

Connected Sums of Special Lagrangian Submanifolds

Pages: 553 – 579

DOI: http://dx.doi.org/10.4310/CAG.2004.v12.n3.a3

Author

Dan A. Lee

Abstract

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 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 dimCX ≥ 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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