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# 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

#### Abstract

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 Mα of *X*α such that Mα 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.