Regularizing a singular special Lagrangian variety

Communications in Analysis and Geometry


Consider two special Lagrangian submanifolds with boundary in 2n-dimensional Euclidean space, with n ≥ 3, that intersect transversally at one point. 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 n = 3). Then, this union is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds with boundary that converges to the union in a suitable topology. This result is obtained by first gluing a smooth neck into a neighbourhood of singularity and then by perturbing this approximate solution until it becomes minimal and Lagrangian.

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