have a special Lagrangian with phase $i$ asymptotic to $P(-\frac{\theta}{2})\cup P(\frac{\theta}{2})$.

The simplest case is when $\theta_1=\ldots=\theta_n=\frac{\pi}{n}$: here $N$ is called the *Lagrangian catenoid*. When $n=2$, under a coordinate change the Lagrangian catenoid becomes the complex curve $\{(z,\frac{1}{z})\in\C^2\,:\,z\in\C\setminus\{0\}\}$ that we saw before. When $n=3$, the only possibilities for the angles are $\sum_i\theta_i=\pi,2\pi$, but if $\sum_i\theta_i=2\pi$ we can rotate coordinates and change the order of the planes so that $P(0)\cup P(\theta)$ becomes $P(0)\cup P(\theta')$ where $\sum_i\theta_i'=\pi$. Hence, given any pair of transverse special Lagrangian planes in $\C^3$, there exists a Lawlor neck asymptotic to their union.

Using complex geometry it is easy to classify all of the smooth special Lagrangians in $\C^2$ asymptotic to a pair of transverse planes, and one sees that the Lawlor necks in $\C^2$ are the unique exact special Lagrangians with this property. It is now known that the Lawlor necks are the unique smooth exact special Lagrangian asymptotic to a pair of planes in all dimensions [@IJO].

We can find special Lagrangians in Calabi–Yau manifolds using the following easy result.

\[SLfixedpt.prop\] Let $(M,\omega,\Upsilon)$ be a Calabi–Yau manifold and let $\sigma:M\rightarrow M$ be such that $\sigma^2=\text{\emph{Id}}$, $\sigma^*(\omega)=-\omega$, $\sigma^*(\Upsilon)=\overline{\Upsilon}$. Then $\text{Fix}(\sigma)$ is special Lagrangian, if it is non-empty.

