The Harvey Lawson $T^{2}$-cone is a special Lagrangian cone inside $\mathbb{C}^{3}$ with link $T^{2}$, invariant under the diagonal $T^{2}\subset SU(3)$. Explicitly,

$$L_{HL}=\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}:|z_{1}|=|z_{2}|=|z_{3}|,\quad\text{Im}(z_{1}z_{2}z_{3})=0,\quad\text{Re}(z_{1}z_{2}z_{3})\geq 0\}.$$

Haskins [36, Thm 1] proved that up to unitary transformations, this is the only strictly stable⁵⁷⁵⁷ 57 Strict stability here is a condition on the Laplacian spectrum of the link. Unfortunately, the word ‘stable’ is overloaded with many standard meanings in the literature. special Lagrangian cone with smooth embedded link diffeomorphic to $T^{2}$.

The Harvey-Lawson cone admits three different 1-parameter deformations into smooth embedded special Lagrangians $L_{s}^{1},L_{s}^{2},L_{s}^{3}$ for $s>0$. Here

$$L_{s}^{1}=\{(z_{1},z_{2},z_{3})\in\mathbb{C}^{3}:|z_{1}|^{2}-s=|z_{2}|^{2}=|z_{3}|^{2},\quad\text{Im}(z_{1}z_{2}z_{3})=0,\quad\text{Re}(z_{1}z_{2}z_{3})\geq 0\},$$

and $L_{s}^{2}$, $L_{s}^{3}$ arise via cyclic permutations of $z_{1},z_{2},z_{3}$. Notably, there is a holomorphic disc $D_{t}^{1}$ of area $\pi s$ with boundary on $L_{s}^{1}$ (and similarly for $L_{s}^{2},L_{s}^{3}$),

$$D_{t}^{1}=\{(z_{1},0,0):|z_{1}|^{2}\leq s\}.$$

In particular, $L_{s}^{a}$ for $a=1,2,3$ cannot be exact Lagrangians, but have nonzero Lagrangian flux. The $s\to 0$ limit corresponds to the holomorphic discs shrinking to zero area, or equivalently the Lagrangian flux tends to zero.

Now on a compact special Lagrangian inside an almost Calabi-Yau manifold, the Harvey-Lawson cone can arise as a local model for conical singularities. The gluing results of Joyce [44, section 10] shows that when certain homological conditions are satisfied, then there exist desingularisation families of special Lagrangians locally modelled on $L_{s}^{a}$, such that the singular special Lagrangians carrying the $T^{2}$-cone singularity arise in codimension one, so in this case the $T^{2}$-cone is an index one singularity in Joyce’s sense.⁵⁸⁵⁸ 58 Joyce’s gluing result is quite subtle. Under certain homological conditions, the smoothing can be forbidden, in which case the $T^{2}$-cone is an index zero singularity. In other cases, due to some linear dependence of certain homology classes, two $T^{2}$-cone singularity may not behave independently, but together behave like an index one singularity. See [44, section 10]. This gluing result is not sensitive to varying $\Omega$. On the other hand, if one restricts to deformations with Lagrangian flux zero, which can be regarded as the analogue of exact isotopies in the mildly singular case, then an isolated local $T^{2}$-cone singularity cannot be desingularized, but instead keeps the singularity as it deforms.