Suppose two S-Lagrangian submanifolds intersect transversally at one point. Then their union is a singular special Lagrangian variety with an isolated singularity at the point of intersection. Suppose further that the tangent planes at the intersection satisfy an angle criterion (which always holds in dimension 3). Then, provided one is allowed to deform the ambient CY structure, this variety is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds with boundary that converges to the singular variety in a suitable topology. This result is obtained by first gluing a smooth neck into a neighbourhood of the intersection point and then by perturbing this approximate solution until
it becomes minimal and Lagrangian. The resulting S-Lagrangian submanifold is topologically the direct sum of its two constituents.
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