Regularizing a Singular Special Lagrangian Variety

Suppose M1 and M2 are two special Lagrangian submanifolds with boundary of R , n ≥ 3, that intersect transversally at one point p. The set M1 ∪M2 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 n = 3). Then, M1 ∪ M2 is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds Mα with boundary that converges to M1 ∪M2 in a suitable topology. This result is obtained by first gluing a smooth neck into a neighbourhood of M1∩M2 and then by perturbing this approximate solution until it becomes minimal and Lagrangian.