Regularizing a Singular Special Lagrangian Variety

Abstract

Suppose M1 and M2 are two special Lagrangian submanifolds of Rtn with boundary that intersect transversally at one point p. The set M1 cup 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 condition (which always holds in dimension n=3). Then, M1 cup M2 is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds M(alpha) with boundary that converges to M1cup M2 in a suitable topology. This result is obtained by first gluing a smooth neck into a neighbourhood of M1 cap M2 and then by perturbing this approximate solution until it becomes minimal and Lagrangia