Until recently it was believed there are no compact, graded, embedded Lagrangians in ${\mathbin{\mathbb{C}}}^{m}$. However, Ekholm, Eliashberg, Murphy and Smith [15, Cor. 1.6] found an example of a compact, graded, embedded Lagrangian ${\mathbin{\cal S}}^{1}\times{\mathbin{\cal S}}^{2}$ in ${\mathbin{\mathbb{C}}}^{3}$, and products give Lagrangian $({\mathbin{\cal S}}^{1}\times{\mathbin{\cal S}}^{2})^{n}$’s in ${\mathbin{\mathbb{C}}}^{3n}$. These all have $HF^{*}$ obstructed, as they are not strictly immersed.

Writing ${\mathbin{\cal S}}^{m}=\bigl\{(x_{0},\ldots,x_{m})\in{\mathbin{\mathbb{R}}}^{m+1}:x_{0}^{2}+\cdots+x_{m}^{2}\bigr\}$, the Whitney sphere $L=\iota({\mathbin{\cal S}}^{m})$ is the Lagrangian immersion $\iota:{\mathbin{\cal S}}^{m}\rightarrow{\mathbin{\mathbb{C}}}^{m}$ given by

It has the special property of having conformal Maslov form. It has one transverse self-intersection point at $p=(0,\ldots,0)=\iota(1,0,\ldots,0)=\iota(-1,0,\ldots,0)$, with $\mu_{L_{-},L_{+}}(p)=-1$, $\mu_{L_{+},L_{-}}(p)=m+1$. Thus if $m>2$, Lemma 2.23 shows that $L$ has $HF^{*}$ unobstructed, so as in (B), $(L,E,b)\cong 0$ in $D^{b}{\mathbin{\mathscr{F}}}({\mathbin{\mathbb{C}}}^{m})$.

Ekholm Eliashberg, Murphy and Smith [15, §1] construct Lagrangian immersions $\jmath:{\mathbin{\cal S}}^{m}\rightarrow{\mathbin{\mathbb{C}}}^{m}$ for $m$ odd, with one transverse self-intersection point $p$ with $\mu_{L_{+},L_{-}}(p)=2$. If $m\geqslant 3$ it has $HF^{*}$ obstructed, as in (A).

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.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold for $m\geqslant 2$, $L$ a compact Lagrangian in $M$, and $p\in L$. In [57], Neves defines another Lagrangian $\tilde{L}$ in $M$, which is Hamiltonian isotopic to $L$ and coincides with $L$ except in a small open neighbourhood of $p$. Here $L,\tilde{L}$ are locally $\mathop{\rm SO}\nolimits(m)$ surfaces of revolution on the curves in sketched in Figure 3.7. (Actually Neves restricts to $m=2$, but the same ideas should work for all $m\geqslant 2$.)

Neves’ main result [57, Th. A] is that Lagrangian MCF starting from $\tilde{L}$ develops finite time singularities. This is important, as it shows that finite time singularities in Lagrangian MCF are unavoidable in many situations (although note that $\tilde{L}$ has phase variation greater than $\pi$, so this does not show that almost calibrated Lagrangian MCF has finite time singularities).

What actually happens in Lagrangian MCF starting from $\tilde{L}$? Neves’ proof does not tell us, as he assumes for a contradiction that no finite time singularity occurs. The author expects a Lagrangian MCF with surgeries $\{L^{t}:t\in[0,T)\}$ in $M$ with $L^{0}=\tilde{L}$, with two singular times $0<T_{1}<T_{2}<T$. For $t\in[0,T_{1}),$ $L^{t}$ looks much like $\tilde{L}$, but as $t\rightarrow T_{1}$ in $[0,T_{1})$, the region marked with crosses ‘$\times$’ in Figure 3.7 undergoes a ‘neck pinch’. At $t=T_{1}$, as sketched in Figure 3.8, $L^{T_{1}}$ decomposes as $L^{T_{1}}_{1}\amalg L^{T_{1}}_{2}$, where $L^{T_{1}}_{1}$ is a small immersed ${\mathbin{\cal S}}^{m}$ near $p$ with one transverse self-intersection point with $\mu_{L_{+},L_{-}}(p)=m+1$, a ‘Whitney sphere’ as in Example 3.23, and $L^{T_{1}}_{2}$ looks quite like the original $L$.

Then as $t$ increases from $T_{1}$ to $T_{2}$, the component $L^{t}_{1}$ should shrink to a point, until at the second singular time $t=T_{2}$ it undergoes ‘collapsing a zero object’ as in Principle 3.25. Meanwhile, the Lagrangian MCF of $L^{t}_{2}$ looks quite like that of the original $L$, and continues for $t>T_{2}$. Thus, at least conjecturally, Neves’ examples [57] are not counterexamples to the programme of §3.2.