Regularizing a singular special Lagrangian variety

TL;DR

This paper proves that certain singular special Lagrangian varieties with a transverse intersection can be smoothed into smooth minimal Lagrangian submanifolds by a gluing and perturbation technique, under specific angle conditions.

Contribution

It introduces a method to regularize singular special Lagrangian varieties with isolated intersection points by gluing a neck and perturbing to achieve smoothness and minimality.

Findings

Singular special Lagrangian varieties with transverse intersections are regularizable.

A gluing and perturbation method constructs smooth minimal Lagrangian submanifolds.

The angle condition ensures the regularization process is feasible.

Abstract

Suppose $M_{1}$ and $M_{2}$ are two special Lagrangian submanifolds of $\Rtn$ with boundary that intersect transversally at one point $p$. The set $M_{1} \cup M_{2}$ 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 condition (which always holds in dimension $n=3$). Then, $M_{1} \cup M_{2}$ is regularizable; in other words, there exists a family of smooth, minimal Lagrangian submanifolds $M_{\alpha}$ with boundary that converges to $M_{1} \cup M_{2}$ in a suitable topology. This result is obtained by first gluing a smooth neck into a neighbourhood of $M_{1} \cap M_{2}$ and then by perturbing this approximate solution until it becomes minimal and Lagrangian.