Let $L$ be a graded, immersed Lagrangian in ${\mathbin{\mathbb{C}}}$ shaped like an $\infty$ sign, not necessarily symmetric, bounding two ‘teardrop’ $J$-holomorphic curves $\Sigma_{1},\Sigma_{2}$, as shown in Figure 3.4, and let $E\rightarrow L$ be a rank one ${\mathbin{\mathbb{F}}}$-local system, which is classified by its holonomy $\mathop{\rm Hol}\nolimits(\nabla_{E})[L]\in{\mathbin{\mathbb{F}}}^{*}$ around $L$.

Then $(L,E)$ has $HF^{*}$ obstructed if $\mathop{\rm area}(\Sigma_{1})\neq\mathop{\rm area}(\Sigma_{2})$. If $\mathop{\rm area}(\Sigma_{1})=\mathop{\rm area}(\Sigma_{2})$, there is a unique choice of $\mathop{\rm Hol}\nolimits(\nabla_{E})[L]=\pm 1$ which makes the obstructions to $HF^{*}$ due to $\Sigma_{1},\Sigma_{2}$ cancel, and then $(L,E)$ has $HF^{*}$ unobstructed.

Consider the immersed Lagrangian MCF (‘curve shortening flow’) $\{L^{t}:t\in[0,T)\}$ in ${\mathbin{\mathbb{C}}}$ starting from $L^{0}=L$ with first finite time singularity at $t=T$. The curve shortening flow is well understood, as in Abresch and Langer [1], Angenent [4, 5], Grayson [21], and others, and we can give a good description of the flow. The difference $\mathop{\rm area}(\Sigma_{1}^{t})-\mathop{\rm area}(\Sigma_{2}^{t})$ is constant during the flow, and both $\mathop{\rm area}(\Sigma_{1}^{t}),\mathop{\rm area}(\Sigma_{2}^{t})$ decrease until the smaller becomes zero at $t=T$.

In the case $\mathop{\rm area}(\Sigma_{1}^{t})>\mathop{\rm area}(\Sigma_{2}^{t})$, the flow is sketched in Figure 3.5. The loop bounding $\Sigma_{2}$ shrinks to a point at $t=T$, and the curve develops a cusp singularity. A type II blow up of this singularity sees only the small, highly curved regions indicated, and yields the ‘grim reaper’ translating soliton from Figure 2.1. Note that in this case, the type II blow up only gives a rather incomplete picture of what is happening.

Following Angenent [5], one can continue the flow for $t>T$ after a surgery at $t=T$ eliminating the self-intersection point, as in the last picture of Figure 3.5, but then the $L^{t}$ for $t>T$ are non-graded. From the point of view of this paper, this is the wrong thing to do, and only works as dimension $m=1$ is so simple. A better answer is that after the singularity at $t=T,$ one cannot continue the flow in graded Lagrangian MCF for $t>T$. This does not contradict the programme of §3.2, as the initial Lagrangian $L$ in Figure 3.4 has $HF^{*}$ obstructed in this case. We will discuss this phenomenon further in §3.8.

In the case $\mathop{\rm area}(\Sigma_{1}^{t})=\mathop{\rm area}(\Sigma_{2}^{t})$, the flow is sketched in Figure 3.6. The whole $\infty$ sign shrinks to a point at $t=T$. It is not a type I singularity modelled on a Lagrangian MCF shrinker, since this cannot happen in graded Lagrangian MCF as in §2.3. The curve does not rescale homothetically, but as in Figure 3.6 the curve shrinks faster in the vertical than in the horizontal directions. Type II blow ups at either end of the $\infty$ sign yield a ‘grim reaper’ translating soliton, as in Figure 2.1, as indicated. So, in this case of an immersed curve in ${\mathbin{\mathbb{C}}}$ with $HF^{*}$ unobstructed, the whole curve collapses to a point in finite time under Lagrangian MCF.