Connected Sums of Special Lagrangian Submanifolds

Abstract

Let $M_1$ and $M_2$ be special Lagrangian submanifolds of a compact Calabi-Yau manifold $X$ that intersect transversely at a single point. We can then think of $M_1\cup M_2$ as a singular special Lagrangian submanifold of $X$ with a single isolated singularity. We investigate when we can regularize $M_1\cup M_2$ in the following sense: There exists a family of Calabi-Yau structures $X_\alpha$ on $X$ and a family of special Lagrangian submanifolds $M_\alpha$ of $X_\alpha$ such that $M_\alpha$ converges to $M_1\cup M_2$ and $X_\alpha$ converges to the original Calabi-Yau structure on $X$. We prove that a regularization exists in two key cases: (1) when the complex dimension of $X$ is three, $\Hol(X)=\SU(3)$, and $[M_1]$ is not a multiple of $[M_2]$ in $H_3(X)$, and (2) when $X$ is a torus with complex dimension at least three, $M_1$ is flat, and the intersection of $M_1$ and $M_2$ satisfies a certain angle criterion. One can easily construct examples of the second case, and thus as a corollary we construct new examples of non-flat special Lagrangian submanifolds of Calabi-Yau tori.