Regularizing a singular special Lagrangian variety

Suppose M-1 and M-2 are two special Lagrangian submanifolds with boundary of R-2n, n greater than or equal to 3, that intersect transversally at one point p. The set M-1 boolean OR M-2 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, M-1 boolean OR M-2 is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds Malpha, with boundary that converges to M-1 boolean OR M-2 in a suitable topology. This result is obtained by first gluing a smooth neck into a neighbourhood of M-1 boolean AND M-2 and then by perturbing this approximate solution until it becomes minimal and Lagrangian.