Regularizing a singular special Lagrangian variety

Suppose M₁ and M₂ are two special Lagrangian submanifolds with boundary of ℝ²ⁿ, 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 M_α 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.

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Published 1 January 2004