Wolfson [72] constructed an example of a Calabi–Yau 2-fold $(M,J,g,\Omega)$ (a $K3$ surface) with the following properties:

• (i)

  There exists $\alpha\in H_{2}(M,{\mathbin{\mathbb{Z}}})$ with $\alpha\cdot\alpha=-4$, such that every compact, immersed Lagrangian $L$ in $M$ has $[L]\in{\mathbin{\mathbb{Z}}}\cdot\alpha\subset H_{2}(M,{\mathbin{\mathbb{Z}}})$.

• (ii)

  There exists an immersed Lagrangian two-sphere $L$ in $M$ with $[L]=\alpha$.

• (iii)

  There does not exist a compact, immersed SL 2-fold $L^{\prime}$ in $M$ with homology class $\alpha$ (even if one allows branch point singularities in $L^{\prime}$).

Here (iii) is proved as follows: $L^{\prime}$ must be connected, as we cannot split $\alpha=\beta+\gamma$ for $\beta\neq 0\neq\gamma$ homology classes represented by SL 2-folds. Suppose $L^{\prime}$ has genus $g$, and for simplicity has $k$ transverse self-intersection points. An easy calculation shows that $[L^{\prime}]\cdot[L^{\prime}]=2g+2k-2\geqslant-2$. But $[L^{\prime}]=\alpha$ and $\alpha\cdot\alpha=-4$.

So we can ask: what happens to Lagrangian MCF $\{L^{t}:t\in[0,T)\}$ in $M$ with $L^{0}=L$? I expect that $L$ has $HF^{*}$ obstructed, and that a finite time singularity develops at $t=T$ after which one cannot continue the (graded) flow, as in Principle 3.29. As evidence for this, note that if Lagrangian MCF with surgeries $\{L^{t}:t\in[0,\infty)\}$ existed for all time, one would expect $L^{\prime}=\lim_{t\rightarrow\infty}L^{t}$ to be an SL 2-fold in homology class $\alpha$, which is excluded by (iii).

Wolfson uses his example to prove something different. Schoen and Wolfson [61] show that by minimizing volume amongst (not necessarily graded) compact, immersed, oriented Lagrangians $L$ in a Calabi–Yau 2-fold in a fixed homology class $\alpha$ and taking a limit, one can construct a singular Lagrangian $L^{\prime}$ with minimal volume in homology class $\alpha$, such that $L^{\prime}$ is Hamiltonian stationary and has finitely many singular points of two kinds:

• (a)

  Branch points, like those of Riemann surfaces, and

• (b)

  Singularities modelled on certain Lagrangian cones $C_{p,p+1}$ in ${\mathbin{\mathbb{C}}}^{2}$ for $p\geqslant 1.$ These $C_{p,p+1}$ are Hamiltonian stationary, but not Maslov zero, or graded.

If there are only singular points of type (a), then $L^{\prime}$ is special Lagrangian. Wolfson deduces [72, Th. 3.3] that in his example, the minimizer $L^{\prime}$ must have singular points of type (b). But then $L^{\prime}$ is not graded, so it is not a possible limit $\lim_{t\rightarrow\infty}L^{t}$ for graded Lagrangian MCF.

Example 2.16 described a family of Lagrangian MCF translators $L$ in ${\mathbin{\mathbb{C}}}^{m}$ given in equation (2.10), asymptotic to the union of two Lagrangian planes $\Pi_{0},\Pi_{\boldsymbol{\phi}}\cong{\mathbin{\mathbb{R}}}^{m}$ intersecting in ${\mathbin{\mathbb{R}}}$. We have sketched $L$ in Figure 3.9 (not easy to draw in only two dimensions).

We indicate the intersection of $L$ with the $z_{m}$-axis, the curve

which bounds a noncompact $J$-holomorphic curve $\Sigma$ in the $z_{m}$-axis as shown.

We will try and describe a type II singularity of Lagrangian MCF $\{L^{t}:t\in[0,T)\}$ with a singularity at $x\in M$ modelled on these LMCF translators $L$, using Principle 3.9(b). Identifying $M$ with $T_{x}M\cong{\mathbin{\mathbb{C}}}^{m}$ near $x\in M$, each $L^{t}$ should to ‘first order’ approximate an LMCF translator $L$ from Example 2.16, and as $t\rightarrow T$ these LMCF translators should slowly shrink homothetically, as well as translate. What interests us is the ‘second order’ changes to $L$ which cause this shrinking.

Far to the right in Figure 3.9, the LMCF translator $L$ approximates two non-intersecting affine Lagrangian planes $\tilde{\Pi}_{0},\tilde{\Pi}_{\boldsymbol{\phi}}$ in ${\mathbin{\mathbb{C}}}^{m}$ from (2.11), just as far to the right in Figure 2.1, the ‘grim reaper’ approximates two non-intersecting parallel lines in ${\mathbin{\mathbb{C}}}$. I suggest that to ‘second order’ in $L^{t}$, the two planes $\tilde{\Pi}_{0},\tilde{\Pi}_{\boldsymbol{\phi}}$ should be bent towards each other by a small angle, introducing a new immersed self-intersection point, and so that the noncompact $J$-holomorphic curve $\Sigma$ becomes a compact ‘teardrop’ as in Figure 2.3, which makes $HF^{*}$ obstructed. This modification $\tilde{L}$ of $L$ is sketched in Figure 3.10.

I expect that this ‘bending’ of $\tilde{\Pi}_{0},\tilde{\Pi}_{\boldsymbol{\phi}}$ towards one another is both the ‘outside influence’ in Principle 3.9(b) which makes $L$ shrink and causes the finite time singularity, and also the cause of the self-intersection point, the ‘teardrop’ curve $\Sigma$, and the obstructions to $HF^{*}$.