$\displaystyle|A|^{2}\,$ $\displaystyle\leq\,k^{2}+C(n,G,M)r^{-2}$
$\displaystyle\leq\left(\big{|}\vec{H}\big{|}+\frac{C(n)}{r}\right)^{2}+\frac{C(n,G,M)}{r^{2}}$
$\displaystyle\leq 2\big{|}\vec{H}\big{|}^{2}+\frac{C(n,G,M)}{r^{2}},$

therefore (28) implies the theorem.

To begin, let $\Delta=\Delta^{L_{t}}$ denote the Laplacian on $L$ given by the
metric induced from $F_{t}$, and define the symmetric $2$-tensor
$a_{ij}:=g^{kl}\alpha_{l}h_{ijk}=\langle\vec{H},A(e_{i},e_{j})\rangle$. Using
[43] Proposition 1.8.6 and Proposition 4.13, we calculate the following
evolution equations:

$\displaystyle\left(\frac{\partial}{\partial
t}-\Delta\right)\big{|}\vec{H}\big{|}^{2}\,$
$\displaystyle=\,2|a|^{2}-2|\nabla H|^{2}$ $\displaystyle\leq
2(k^{2}+(n-1)p^{2})\big{|}\vec{H}\big{|}^{2}-2|\nabla\big{|}\vec{H}\big{|}|^{2}$
$\displaystyle\leq
2\big{|}\vec{H}\big{|}^{4}+\frac{C(n)}{r^{2}}\big{|}\vec{H}\big{|}^{2}-2|\nabla\big{|}\vec{H}\big{|}|^{2},$
$\displaystyle\left(\frac{\partial}{\partial t}-\Delta\right)\theta\,$
$\displaystyle=\,0,$ $\displaystyle\left(\frac{\partial}{\partial
t}-\Delta\right)\psi\,$
$\displaystyle=\,-\Delta^{\mathbb{R}^{n}}\psi+\text{tr}^{\perp}(\mbox{Hess}(\psi)),$

and the following Laplacians:

$\displaystyle\log(\Gamma)$
$\displaystyle=\log(\big{|}\vec{H}\big{|}^{2})+\log(f)+\log(\psi)$
$\displaystyle\Delta\log(\big{|}\vec{H}\big{|}^{2})\,$
$\displaystyle=\,\frac{\Delta\big{|}\vec{H}\big{|}^{2}}{\big{|}\vec{H}\big{|}^{2}}-\frac{|\nabla\big{|}\vec{H}\big{|}^{2}|^{2}}{\big{|}\vec{H}\big{|}^{4}}$
$\displaystyle\Delta\log(f)\,$
$\displaystyle=\,\frac{f^{\prime}\Delta\theta}{f}+\left(\frac{f^{\prime\prime}}{f}-\frac{(f^{\prime})^{2}}{f^{2}}\right)|\nabla\theta|^{2}.$

We first bound $\Gamma$ at an increasing maximum — i.e. a spacetime point
$(y,s)\in L\times[0,T)$ such that $\Gamma(y,s)=\max_{x\in L}\Gamma(x,s)$ and
$\frac{\partial\Gamma}{\partial t}(y,s)\geq 0$. At such a point,
$\nabla\Gamma=0$ and $(\frac{\partial}{\partial t}-\Delta)\Gamma\geq 0$.
Noting that such a point is also an increasing maximum of $\log\Gamma$, it
follows that

$\displaystyle\nabla\log(\big{|}\vec{H}\big{|}^{2})+\nabla\log
f+\nabla\log\psi=0,$

whence for any $\varepsilon>0$,

$\displaystyle\left|\nabla\log(\big{|}\vec{H}\big{|}^{2})\right|^{2}$
$\displaystyle=\left|\frac{f^{\prime}}{f}\nabla\theta+\frac{\nabla\psi}{\psi}\right|^{2}\,\leq\,(1+\varepsilon)\frac{(f^{\prime})^{2}}{f^{2}}\big{|}\vec{H}\big{|}^{2}+(1+\tfrac{1}{\varepsilon})\frac{|\nabla\psi|^{2}}{\psi^{2}}.$

It follows that:

$\displaystyle 0$ $\displaystyle\leq\left(\frac{\partial}{\partial
t}-\Delta\right)\log(\Gamma)$
$\displaystyle=\,\frac{1}{\big{|}\vec{H}\big{|}^{2}}\left(\frac{\partial}{\partial
t}-\Delta\right)\big{|}\vec{H}\big{|}^{2}+\frac{f^{\prime}}{f}\left(\frac{\partial}{\partial
t}-\Delta\right)\theta+\frac{1}{\psi}\left(\frac{\partial}{\partial
t}-\Delta\right)\psi$
$\displaystyle\quad\quad\quad+|\nabla\log(\big{|}\vec{H}\big{|}^{2})|^{2}-\left(\frac{f^{\prime\prime}}{f}-\frac{(f^{\prime})^{2}}{f^{2}}\right)\big{|}\vec{H}\big{|}^{2}+\frac{|\nabla\psi|^{2}}{\psi^{2}}$
$\displaystyle\leq\left(2+\frac{f^{\prime\prime}}{f}-\frac{(f^{\prime})^{2}}{f^{2}}\right)\big{|}\vec{H}\big{|}^{2}+\frac{C(n)}{r^{2}}+\frac{1}{2}|\nabla\log(\big{|}\vec{H}\big{|}^{2})|^{2}$
$\displaystyle\quad\quad\quad-\frac{\Delta^{\mathbb{R}^{n}}\psi-\mbox{tr}^{\perp}\mbox{Hess}(\psi)}{\psi}+\frac{|\nabla\psi|^{2}}{\psi^{2}}$
$\displaystyle\leq\left[2+(1+\tfrac{1}{2}(1+\varepsilon))\frac{(f^{\prime})^{2}}{f^{2}}-\frac{f^{\prime\prime}}{f}\right]\big{|}\vec{H}\big{|}^{2}$
$\displaystyle\quad\quad\quad+C(n,\varepsilon)\left(-\frac{\Delta^{\mathbb{R}^{n}}\psi+\mbox{tr}^{\perp}\mbox{Hess}(\psi)}{\psi}+\frac{|\nabla\psi|^{2}}{\psi^{2}}+\frac{1}{r^{2}}\right).$
(29)

We now choose our $f$ to simplify this inequality. Writing
$\Psi:=f^{-\frac{1}{2}(1+\varepsilon)}$, the square bracket is equal to

$2+\frac{\Psi^{\prime\prime}}{b\Psi},$

where $b=\frac{1}{2}(1+\varepsilon)$. Then, solving

$2+\frac{\Psi^{\prime\prime}}{b\Psi}=-\frac{\varepsilon}{b},$

we find that a suitable function $\Psi$ is

$\displaystyle\Psi(\theta)=\cos((\theta-\overline{\theta})\sqrt{1+2\varepsilon})$
$\displaystyle\implies
f(\theta)=\cos^{-\frac{2}{1+\varepsilon}}((\theta-\overline{\theta})\sqrt{1+2\varepsilon}),$

where we now fix $\varepsilon$ sufficiently small so that $f$ is bounded on
$(\overline{\theta}-\frac{\pi}{2}+\varepsilon_{\theta},\overline{\theta}+\frac{\pi}{2}-\varepsilon_{\theta})$.
Therefore, the square bracket in (29) is now equal to
$\frac{-\varepsilon}{\frac{1}{2}(1+\varepsilon)}$, so

$0\leq\left[\frac{-\varepsilon}{\frac{1}{2}(1+\varepsilon)}\right]\big{|}\vec{H}\big{|}^{2}+C(n,\varepsilon)\left(-\frac{\Delta^{\mathbb{R}^{n}}\psi+\mbox{tr}^{\perp}\mbox{Hess}(\psi)}{\psi}+\frac{|\nabla\psi|^{2}}{\psi^{2}}+\frac{1}{r^{2}}\right).$
(30)

We now continue estimating from (30). Define $\psi_{R}=r^{2}\rho_{R}(r)$,
where $\rho_{R}\colon\mathbb{R}\to[0,1]$ is a smooth cutoff function
satisfying:

$\displaystyle\rho_{R}(r)=\begin{cases}1\quad\text{ for }r\leq R,\\\
0\quad\text{ for }r\geq 2R,\end{cases}\quad|\rho_{R}|\leq
1,\quad\left|\frac{\partial\rho_{R}}{\partial
x_{i}}\right|\leq\frac{C}{R}\sqrt{\rho_{R}},\quad\left|\frac{\partial^{2}\rho_{R}}{\partial
x_{i}\partial x_{j}}\right|\leq\frac{C}{R^{2}}$

(for example
$\rho_{R}(r):=\left(g(2-\frac{r}{R})(g(2-\frac{r}{R})+g(\frac{r}{R}-1))^{-1}\right)^{2}$,
where $g(x):=e^{-\tfrac{1}{x}}$). Note that if we choose $\psi=\psi_{R}$, then
the increasing maximum $(y,s)$ of $\Gamma$ must be inside $B_{2R}$. At
$(y,s)$, we may therefore estimate using (30):

$\displaystyle-\Delta^{\mathbb{R}^{n}}\psi+\mbox{tr}^{\perp}\mbox{Hess}(\psi)\,\leq\,C(n)$
$\displaystyle,\quad\quad\frac{|\nabla\psi|^{2}}{\psi}\,\leq\,C(n),$

so

$\displaystyle\psi\left[\frac{\varepsilon}{\tfrac{1}{2}(1+\varepsilon)}\right]\big{|}\vec{H}\big{|}^{2}$
$\displaystyle\leq C(n)$ $\displaystyle\implies\Gamma\,$ $\displaystyle\leq
C(n,\varepsilon_{\theta}).$ (31)

We may now estimate $\Gamma$ at an arbitrary point $(x,t)\in L\times[0,T)$.
Choose $R=r(F_{t}(x))$. Since $\Gamma$ has compact support it follows that the
supremum $\sup_{y\in L,\,s\in[0,t]}\Gamma$ is attained at a space-time point
$(\overline{y},\overline{s})$. If $\overline{s}=0$, then since $L_{0}$ has
bounded mean curvature, there exists a constant $C(L_{0})$ such that

$\Gamma(x,t)\leq\Gamma(\overline{y},0)\leq C(L_{0})R^{2}.$

If $\overline{s}\neq 0$, then $(\overline{y},\overline{s})$ is an increasing
maximum, and so by (31),

$\Gamma(x,t)\leq\Gamma(\overline{y},\overline{s})\leq
C(n,\varepsilon_{\theta})$

It follows that in general,

$\displaystyle\Gamma(x,t)\leq C(n,\varepsilon_{\theta},L_{0})(1+R^{2}),$

and hence

$\displaystyle|\vec{H}(x,t)|^{2}\leq
C(n,\varepsilon_{\theta},L_{0})\left(1+\frac{1}{r(x,t)^{2}}\right).$

∎

The location of singularities theorem follows as a corollary of these
estimates.

######  Theorem 6.2 (Location of Singularities Theorem).

Let $L_{t}\subset M$ be a connected $G$-invariant almost-calibrated Lagrangian
MCF, for $t\in(0,T)$. Then a singularity occurs at time $T$ if and only if
$(O,T)$ is the unique singular space-time point for $L_{t}$.

###  6.2 Convergence Theorems

We now prove two key propositions regarding convergence of blowup sequences.
Proposition 6.3 allows us to pass from convergence of the $G$-invariant
Lagrangian $L^{i}$ to convergence of their profile curves $l^{i}$ when
considering the Type I blowup. Proposition 6.4 meanwhile will be used to rule
out certain bad behaviour for Type II blowups.

######  Proposition 6.3 (Convergence of $G$-invariant Lagrangians).

Let $L^{i}\subset M$ be a sequence of connected, $G$-invariant, almost-
calibrated Lagrangian submanifolds with Lagrangian angles $\theta_{i}$, let
$L^{\infty}_{j}\subset M\cup\\{0\\}$ be connected smooth Lagrangians or
Lagrangian cones with connected link which are pairwise disjoint for $1\leq
j\leq N$, with Lagrangian angles $\overline{\theta}_{j}$, and let $m_{j}$ be
integer multiplicities. Assume that for any $\phi\in
C^{\infty}_{c}(\mathbb{C}^{n})$, $f\in C^{2}(\mathbb{R})$,

$\int_{L^{i}}f(\theta_{i})\phi\,d\mathcal{H}^{n}\longrightarrow\sum_{j=1}^{N}m_{j}\int_{L^{\infty}_{j}}f(\overline{\theta}_{j})\phi\,d\mathcal{H}^{n}\,\text{
as }i\rightarrow\infty.$

Then:
(a) $L^{\infty}_{j}\cap M$ are connected, $G$-invariant Lagrangians,
(b) For any $\psi\in C^{\infty}_{c}(\mathbb{C})$, $f\in C^{2}(\mathbb{R})$,

$\int_{l^{i}}f(\theta_{i})\psi\,d\mathcal{H}^{1}\longrightarrow\sum_{j=1}^{N}m_{j}\int_{l^{\infty}_{j}}f(\overline{\theta}_{j})\psi\,d\mathcal{H}^{1}\,\text{
as }i\rightarrow\infty.$

###### Proof.

We work with the underlying Radon measures. Denoting the indicator function of
a subset $A\subset\mathbb{C}^{n}$ by $\chi_{A}$, we make the definitions

$\displaystyle\Theta_{i}:\mathbb{C}^{n}\rightarrow\mathbb{R},\,\,\Theta_{i}$
$\displaystyle:=f(\theta_{i})\chi_{L^{i}},\quad\quad\quad\Theta_{\infty}:\mathbb{C}^{n}\rightarrow\mathbb{R},\,\,\Theta_{i}:=\sum_{j=1}^{N}m_{j}\,f(\overline{\theta}_{j})\chi_{L^{\infty}_{j}},$
$\displaystyle\mu_{i}$ $\displaystyle:=\mathcal{H}^{n}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Theta_{i},\quad\quad\quad\mu_{\infty}:=\mathcal{H}^{n}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\Theta_{\infty}.$

Then, for $\phi\in C^{\infty}_{c}(\mathbb{C}^{n})$, and denoting by
$L_{g}\colon\mathbb{C}^{n}\to\mathbb{C}^{n}$ the action of $g\in G$ on
$\mathbb{C}^{n}$, the assumptions of the proposition imply that

$\mu_{i}(\phi)\rightarrow\,\mu_{\infty}(\phi)\implies\mu_{\infty}(\phi\circ
L_{g})=\lim_{i\rightarrow\infty}\mu_{i}(\phi\circ
L_{g})=\lim_{i\rightarrow\infty}\mu_{i}(\phi)=\mu_{\infty}(\phi).$

Note that since $L^{\infty}_{j}$ is a smooth Lagrangian, this measure-
theoretic $G$-invariance of $\mu_{\infty}$ implies $G$-invariance of the
supporting set $L^{\infty}:=\bigcup_{j=1}^{N}L^{\infty}_{j}$, and therefore of
the connected components $L^{\infty}_{j}$. This proves (a). Defining in the
same way the profile curve measures,

$\displaystyle\overline{\Theta}_{i}:\mathbb{C}\rightarrow\mathbb{R},\,\,\overline{\Theta}_{i}$
$\displaystyle:=f(\theta_{i})\chi_{l^{i}},\quad\quad\quad\overline{\Theta}_{\infty}:\mathbb{C}^{n}\rightarrow\mathbb{R},\,\,\overline{\Theta}_{i}:=\sum_{j=1}^{N}m_{j}\,f(\overline{\theta}_{j})\chi_{l^{\infty}_{j}},$
$\displaystyle\overline{\mu}_{i}$
$\displaystyle:=\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\overline{\Theta}_{i},\quad\quad\quad\overline{\mu}_{\infty}:=\mathcal{H}^{1}\mathbin{\vrule
height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule
height=0.55974pt,depth=0.0pt,width=5.59721pt}\overline{\Theta}_{\infty},$

our aim is to prove that for all $\psi\in C^{\infty}_{c}(\mathbb{C})$, we have
$\overline{\mu}_{i}(\psi)\rightarrow\overline{\mu}_{\infty}(\psi)$.

Consider $\psi\in C^{\infty}_{c}(\mathbb{C})$ supported in $B_{\delta}(x)$ for
some $x\in\mathbb{C}\setminus\\{0\\}$ and $\delta<|x|$. For each
$y\in\mathbb{C}^{n}$, by Proposition 4.11 there exists $h\in G$ such that
$h\cdot y\in P_{z}$. Picking such an $h$ for each $y$ we may define the
$G$-invariant function $\phi:M\rightarrow\mathbb{R}$,
$\phi(y):=\sum_{\alpha\in C_{m}}\psi(\alpha\cdot h\cdot y)$ (this definition
is independent of the choices of $h$). We may then extend $\phi$ to a smooth,
$G$-invariant compactly supported function $\phi\in
C^{\infty}_{c}(\mathbb{C}^{n})$.

Now, by $G$-invariance, the co-area formula, and noting that for $x\in P_{z}$,
$\mathcal{H}^{n-1}(\mathcal{O}_{x})=|x|^{n-1}\mathcal{H}^{n-1}(\mathcal{O}_{1})$,
we calculate the following upper and lower bounds:

$\displaystyle\mu_{\infty}(\phi)\,=\,\lim_{i\rightarrow\infty}\int_{L^{i}}f(\theta_{i})\phi\,d\mathcal{H}^{n}\,$
$\displaystyle=\,\lim_{i\rightarrow\infty}\int_{l^{i}}\left(\int_{\mathcal{O}_{l^{i}(s)}}f(\theta_{i})\phi\,d\mathcal{H}^{n-1}\right)\,d\mathcal{H}^{1}$
$\displaystyle=\,\mathcal{H}^{n-1}(\mathcal{O}_{1})\cdot
m\cdot\lim_{i\rightarrow\infty}\int_{l^{i}}|l^{i}(s)|^{n-1}f(\theta_{i})\psi\,\,d\mathcal{H}^{1}$
$\displaystyle\geq\left||x|-\delta\right|^{n-1}\cdot\mathcal{H}^{n-1}(\mathcal{O}_{1})\cdot
m\cdot\lim_{i\rightarrow\infty}\overline{\mu}_{i}(\psi),$ (32)
$\displaystyle\mu_{\infty}(\phi)=\sum_{j=1}^{N}m_{j}\,\int_{L^{\infty}_{j}}f(\overline{\theta}_{j})\,\phi\,d\mathcal{H}^{n}\,$
$\displaystyle\geq\,||x|-\delta|^{n-1}\cdot\mathcal{H}^{n-1}(\mathcal{O}_{1})\cdot
m\cdot\overline{\mu}_{\infty}(\psi),$ (33)

and

$\displaystyle\mu_{\infty}(\phi)\,\,$
$\displaystyle\leq\,\left||x|+\delta\right|^{n-1}\cdot\mathcal{H}^{n-1}(\mathcal{O}_{1})\cdot
m\cdot\lim_{i\rightarrow\infty}\overline{\mu}_{i}(\psi),$ (34)
$\displaystyle\mu_{\infty}(\phi)$
$\displaystyle\leq||x|+\delta|^{n-1}\cdot\mathcal{H}^{n-1}(\mathcal{O}_{1})\cdot
m\cdot\overline{\mu}_{\infty}(\psi).$ (35)

The inequality (32) implies that the Radon measures $\overline{\mu}_{i}$ are
uniformly bounded on compact sets. Thus, by compactness for Radon measures,
after passing to a subsequence there exists a Radon measure $\overline{\mu}$
such that for all $\psi\in C^{\infty}_{c}(\mathbb{C})$,

$\lim_{i\rightarrow\infty}\overline{\mu}_{i}(\psi)=\overline{\mu}(\psi).$

We now show $\overline{\mu}=\overline{\mu}_{\infty}$. By considering $\psi$
with $\text{supp}(\psi)\subset B_{\delta}(x)\setminus(l^{\infty}\cap\\{0\\})$,
the bound (32) implies that $\text{supp}(\overline{\mu})\subset l^{\infty}$.
Therefore, $\overline{\mu}$ is a $1$-rectifiable Radon measure supported in
$l^{\infty}$, and so there exists
$\overline{\Theta}:\mathbb{C}\rightarrow\mathbb{R}$ supported on $l^{\infty}$
such that

$\overline{\mu}(\psi)=\int_{l^{\infty}}\overline{\Theta}\,\psi\,d\mathcal{H}^{1},$

where for $\mathcal{H}^{1}$-almost-every $y\in\text{supp}(\overline{\mu})$,
$\overline{\Theta}(y)=\lim_{r\rightarrow
0}\frac{\overline{\mu}(B(y,r))}{\omega_{n}r^{n}}$ (see for example [21,
§1.3]). Similarly, for $\mathcal{H}^{1}$-almost-every
$y\in\text{supp}(\overline{\mu}_{\infty})$,
$\overline{\Theta}_{\infty}(y)=\lim_{r\rightarrow
0}\frac{\overline{\mu}_{\infty}(B(y,r))}{\omega_{n}r^{n}}$. By (32 \- 35),

$\displaystyle\frac{||x|-\delta|^{n-1}}{||x|+\delta|^{n-1}}\overline{\mu}_{\infty}(\psi)\geq\overline{\mu}(\psi)\geq\frac{||x|+\delta|^{n-1}}{||x|-\delta|^{n-1}}\overline{\mu}_{\infty}(\psi),$

which implies that $\overline{\Theta}=\overline{\Theta}_{\infty}$ on
$\mathcal{H}^{1}$-almost-every $x\in\mathbb{C}$, and consequently that
$\overline{\mu}=\overline{\mu}_{\infty}$. This proves that
$\lim_{i\rightarrow\infty}\overline{\mu}_{i}(\phi)=\overline{\mu}_{\infty}(\phi)$
as required.

∎

######  Proposition 6.4 (Convergence of Translated $G$-invariant Lagrangians).

Let $L^{i}\subset\mathbb{C}^{n}$ be a sequence of connected, almost-calibrated
Lagrangian submanifolds with $O\in L^{i}$, and let
$L^{\infty}\subset\mathbb{C}^{n}$ be a connected Lagrangian submanifold,
$\nu\in S^{2n-1}$ and $y_{i}\in M$ such that

  * •

$L^{i}+y_{i}$ is a $G$-invariant Lagrangian in $M$,

  * •

$y_{i}\rightarrow\infty$ and $\frac{y_{i}}{|y_{i}|}\rightarrow\nu$,

  * •

$L^{i}$ converges to $L^{\infty}$ in $C^{\infty}_{\emph{loc}}$.

Then $\nu\in M$, and $T_{\nu}\mathcal{O}_{\nu}\subset L^{\infty}$. Since
$\mathcal{O}_{\nu}$ is isotropic, it follows that $L^{\infty}$ contains an
isotropic $(n-1)$-plane.

###### Proof.

Recall $z_{0}\in M$, so that $\frac{z_{0}}{|z_{0}|}\in M\cap S^{2n-1}$. By
Proposition 4.11, $M\cap S^{2n-1}=(\mathrm{U}(1)\times
G)\cdot\frac{z_{0}}{|z_{0}|}$, and so $M\cap S^{2n-1}$ is closed. Therefore,
since $y_{i}\in M$, we have $\hat{y}_{i}:=\frac{y_{i}}{|y_{i}|}\in M\cap
S^{2n-1}$, and hence $\nu\in M$.
Now by Proposition 4.11, there exist $g_{i}\in G$ such that
$g_{i}\hat{y}_{i}\in P_{\nu}$. Since $G$ is compact, by passing to a
subsequence we may assume that $g_{i}\rightarrow\overline{g}$ for
$\overline{g}\in G$ such that $\overline{g}\nu\in P_{\nu}$, from which it
follows that $\overline{g}^{-1}g_{i}\hat{y}_{i}\in P_{\nu}$,
$\overline{g}^{-1}g_{i}\rightarrow\text{Id}$, and
$\overline{g}^{-1}g_{i}\hat{y}_{i}\rightarrow\nu$. Furthermore, defining
$\alpha_{i}\in\mathbb{C}^{*}$ such that
$\overline{g}^{-1}g_{i}\hat{y}_{i}=\alpha_{i}\cdot\nu$, it follows that
$\alpha_{i}\to 1$. Then, since $L^{i}+y_{i}$ is $G$-invariant,

$\displaystyle\mathcal{O}_{y_{i}}\subset L^{i}+y_{i}$
$\displaystyle\implies|y_{i}|\left(\mathcal{O}_{\hat{y}_{i}}-\hat{y}_{i}\right)\subset
L^{i}$
$\displaystyle\implies|y_{i}|g_{i}^{-1}\overline{g}\left(\mathcal{O}_{\overline{g}^{-1}g_{i}\hat{y}_{i}}-\overline{g}^{-1}g_{i}\hat{y}_{i}\right)\subset
L^{i}$ $\displaystyle\implies
g_{i}^{-1}\overline{g}\alpha_{i}\cdot|y_{i}|\left(\mathcal{O}_{\nu}-\nu\right)\subset
L^{i}.$ (36)

Since $\mathcal{O}_{\nu}$ is a smooth $(n-1)$-dimensional submanifold and
$|y_{i}|\rightarrow\infty$, it follows that
$|y_{i}|\left(\mathcal{O}_{\nu}-\nu\right)$ converges locally smoothly to
$T_{\nu}\mathcal{O}_{\nu}$. Since
$g_{i}^{-1}\overline{g}\alpha_{i}\rightarrow\text{Id}$, it follows that the
left hand side of the inclusion of (36) smoothly converges to
$T_{\nu}\mathcal{O}_{\nu}$, which along with the smooth convergence of $L^{i}$
to $L^{\infty}$ implies that $T_{\nu}\mathcal{O}_{\nu}\subset L^{\infty}$, as
required. ∎

######  Remark 6.5.

In fact, more is true: given the assumptions of Proposition 6.4, $L^{\infty}$
is translation invariant with respect to the additive action of the abelian
group $T_{\nu}\mathcal{O}_{\nu}$. However, we will not require the stronger
result in this work.

###  6.3 The Type I Blowup

In this subsection, we prove Theorem 6.7 characterising Type I blowups of
cohomogeneity-one almost-calibrated Lagrangian mean curvature flow, which we
show to be unit-density pairs of Theorem 5.1’s special Lagrangian cones. A
sketch of the proof is as follows. Applying Theorem A of [36] and Proposition
6.3, a Type I blowup must be a union of $G$-invariant special Lagrangian
cones, and the Lagrangian angle converges in an integral sense. In Lemma 6.6,
we use this convergence along with the curvature bounds of Theorem 6.1 to rule
out higher-multiplicity cones in the blowup, demonstrating unit-density. By
Lemma 4.16 and Theorem 5.1, for each Lagrangian angle $\overline{\theta}$
there are only two possible cones that may occur in the blowup, and we may
argue that the blowup is their union.

######  Lemma 6.6 (Multiplicity-One Lemma).

Let $l^{i}\subset\mathbb{C}\setminus\\{0\\}$ be a sequence of complete
connected embedded smooth curves. Let $l^{\infty}\subset\mathbb{C}$ be a half-
line from the origin, $\overline{x}\in l^{\infty}$, $0<d<|\overline{x}|$, and
define
$\theta^{i}=\emph{arg}\\!\left((l^{i})^{\prime}\right)+(n-1)\emph{arg}\\!\left(l^{i}\right)$.
Assume further that:

  * (a)

There exist $m\in\mathbb{N}$,
$\overline{\theta}\equiv\emph{arg}(l^{\infty})\,\,(\emph{mod }\pi)$ such that
for all $\psi\in C^{\infty}_{c}(B_{d}(\overline{x}))$ and $f\in
C^{2}(\mathbb{R})$,

$\int_{l^{i}}f(\theta^{i})\psi\,d\mathcal{H}^{1}\longrightarrow
m\cdot\int_{l^{\infty}}f(\overline{\theta})\psi\,d\mathcal{H}^{1},$

  * (b)

There exists $c$ such that $|\vec{k}(l^{i})|\leq c$ in $B_{d}(\overline{x})$
for all $i$.

Then $m$ = 1.

###### Proof.

Without loss of generality (for example, by applying a rotation and a scaling)
we may assume that $l^{\infty}$ is the positive $x$-axis, $\overline{x}=1$,
and $\overline{\theta}\equiv 0\,\,(\text{mod }\pi)$. We will make the
assumption that $\overline{\theta}\equiv 0\,\,(\text{mod }2\pi)$, since the
other case is identical. Throughout the proof, all balls are centred at
$\overline{x}$. For a curve $\eta\subset\mathbb{C}\setminus\\{0\\}$, define
$T_{\varepsilon,\delta}(\eta):=\\{x\in\eta\cap
B_{\delta}\,:\,|\theta(x)-\overline{\theta}|>\varepsilon\\}$. We make three
claims:

_Claim 1._ _(Uniform convergence of angle)_
For all $\delta<\tfrac{d}{2}$ and $\varepsilon>0$, there exists
$N=N(\varepsilon,\delta)$ such that for all $i>N$,
$T_{\varepsilon,\delta}(l^{i})=\O$.

_Claim 2._ _(Single connected component)_
There exists $\overline{\delta}$ with $d>\overline{\delta}>0$ such that if
$\delta,\varepsilon<\overline{\delta}$ satisfy $l^{i}\cap
B_{{\delta}/{2}}\neq\O$ and $T_{\varepsilon,2\delta}(l^{i})=\O$, then there is
only one connected component of $l^{i}\cap B_{\delta}$.

_Claim 3._ _(Convergence of Length)_
For any $\delta\leq d$, $\mathcal{H}^{1}(l^{i}\cap B_{\delta})\rightarrow
2\delta m.$

From these claims, we may deduce the result as follows. Choose
$\overline{\delta}>0$ as in Claim 2, and let
$0<\delta,\varepsilon<\overline{\delta}$ be small enough that
$\cos(\varepsilon+(n-1)\delta)\geq\frac{2}{3}$ and $2\delta<\frac{d}{2}$. By
Claim 1, there exists $N=N(\varepsilon,2\delta)$ such that $i>N$ implies
$T_{\varepsilon,2\delta}(l^{i})=\O$. Choosing $N$ larger if necessary,
assumption (a) implies that for $i>N$ we have $l^{i}\cap B_{\delta/2}\neq\O$.
Therefore, by Claim 2, there is only one connected component of $l^{i}\cap
B_{\delta}$, which we label $\eta^{i}$ (note that therefore $l^{i}\cap
B_{\delta}=\eta^{i})$. In particular, we have
$T_{\varepsilon,\delta}(\eta^{i})=\O$, so that every $x\in\eta^{i}$ satisfies
$|\theta(x)-\overline{\theta}|\leq\varepsilon$.
Now, using that $\eta\subset B_{\delta}$, and then invoking Lemma 4.15,
suppressing the index $i$ for notational clarity, we have bounds:

$\displaystyle\arg(\eta)$
$\displaystyle\in[-\sin^{-1}(\delta)\,,\,\sin^{-1}(\delta)]\subset[-\delta,\delta],$
$\displaystyle\implies\arg(\eta^{\prime})$
$\displaystyle\in\left[\,\overline{\theta}-\varepsilon-(n-1)\delta\,,\,\overline{\theta}+\varepsilon+(n-1)\delta\right].$

Therefore, parametrising $\eta$ by unit speed (remembering that
$\overline{\theta}=2k\pi$), we have that
$\eta^{\prime}(s)=e^{i(\lambda(s)+\overline{\theta})}=e^{i\lambda(s)}$ for
some
$\lambda(s)\in\left[-\varepsilon-(n-1)\delta\,,\,\varepsilon+(n-1)\delta\right]$,
and hence

$\displaystyle
2\delta\geq\left|\int_{\eta}\eta^{\prime}\,ds\right|\geq\left|\int_{\eta}\cos(\lambda(s))\,ds\right|\geq\mathcal{H}^{1}(\eta)\cos(\varepsilon+(n-1)\delta)\geq\mathcal{H}^{1}(\eta)\cdot\frac{2}{3}$

from which we obtain

$\displaystyle\mathcal{H}^{1}(l^{i}\cap
B_{\delta})=\mathcal{H}^{1}(\eta^{i})\leq 2\delta\cdot\frac{3}{2}.$

Finally, by Claim 3, it follows that $m=1$, as required. It therefore remains
to prove the three claims.

_Proof of Claim 1:_ Parametrise $l^{i}$ by arclength, say
$l^{i}(s)=r(s)e^{i\alpha(s)}$. A calculation shows that within $B_{d}$, by
assumption (b):

$\displaystyle\theta$
$\displaystyle=\tan^{-1}(\tfrac{r\alpha^{\prime}}{r^{\prime}})+n\alpha$
$\displaystyle\implies\theta^{\prime}$ $\displaystyle=\langle
l^{\prime\prime},il^{\prime}\rangle+(n-1)\alpha^{\prime}$
$\displaystyle\implies|\theta^{\prime}|\,$
$\displaystyle\leq\,|\vec{k}|+\frac{n-1}{r}\,\leq\,c+\frac{n-1}{1-d}\,=:\,\widetilde{c}.$
(37)

Now choose $\delta<\tfrac{d}{2}$ and
$\varepsilon\leq\overline{\varepsilon}_{\delta}$, where
$\overline{\varepsilon}_{\delta}:=\frac{1}{4}(d-2\delta)(1+\widetilde{c})$.
Furthermore, define $\varepsilon^{\prime}=\frac{\varepsilon}{1+\widetilde{c}}$
and choose smooth functions $\psi:\mathbb{C}\to[0,1]$, $f:\mathbb{R}\to[0,1]$
so that $\psi=1$ on $B_{\tfrac{d}{2}}$ and $\psi=0$ outside of $B_{d}$, and
$f=0$ on
$(\overline{\theta}-\frac{\varepsilon^{\prime}}{2},\overline{\theta}+\frac{\varepsilon^{\prime}}{2})$
and $f=1$ outside of
$(\overline{\theta}-\varepsilon^{\prime},\overline{\theta}+\varepsilon^{\prime})$.
By assumption (a), we have

$\mathcal{H}^{1}\\!\left(T_{\varepsilon^{\prime},\delta+\varepsilon^{\prime}}(l^{i})\right)\leq\int_{l^{i}}f(\theta)\psi\,d\mathcal{H}^{1}\rightarrow
m\cdot\int_{l^{\infty}}f(\overline{\theta})\psi\,d\mathcal{H}^{1}=0,$

and therefore there exists $N=N(\varepsilon,\delta)$ such that for all $i>N$,
$\mathcal{H}^{1}\\!\left(T_{\varepsilon^{\prime},\delta+\varepsilon^{\prime}}(l^{i})\right)\leq\varepsilon^{\prime}$.

Now, since
$\mathcal{H}^{1}\left(T_{\varepsilon^{\prime},\delta+\varepsilon^{\prime}}(l^{i})\right)\leq\varepsilon^{\prime}$,
for $i>N$ any connected component of $l^{i}\cap
B_{\delta+\varepsilon^{\prime}}$ intersecting $B_{\delta}$ must include a
point $x$ such that
$\theta(x)\in(\overline{\theta}-\varepsilon^{\prime},\overline{\theta}+\varepsilon^{\prime})$.
Therefore by (37), on $B_{\delta}$,
$\theta\in(\overline{\theta}-\varepsilon^{\prime}-\widetilde{c}\varepsilon^{\prime},\overline{\theta}+\varepsilon^{\prime}+\widetilde{c}\varepsilon^{\prime})=(\overline{\theta}-\varepsilon,\overline{\theta}+\varepsilon)$
as required.

Finally, since $\varepsilon<\tilde{\varepsilon}$ implies
$T_{\delta,\varepsilon}(l^{i})\supset T_{\delta,\tilde{\varepsilon}}(l^{i})$,
we note that given $\delta<\tfrac{d}{2}$ and $\varepsilon>0$, for all
$i>N(\min(\varepsilon,\overline{\varepsilon}_{\delta}),\delta)$ it follows
that $T_{\varepsilon,\delta}(l^{i})\subset
T_{\min(\overline{\varepsilon}_{\delta},\varepsilon),\delta}(l^{i})=\O$.

_Proof of Claim 2:_ Consider $\delta,\varepsilon<\overline{\delta}$, where
$\overline{\delta}<\frac{\pi}{4n}$ will be chosen during the course of the
proof. Choose $x\in l^{i}\cap B_{\frac{\delta}{2}}$, let $\eta^{i}$ be the
connected component of $l^{i}\cap B_{\delta}$ containing $x$, and let
$\overline{\eta}^{i}$ be the connected component of $l^{i}\cap B_{2\delta}$
containing $x$. Since $T_{\varepsilon,2\delta}(l^{i})=\O$, within
$B_{2\delta}$ we have the following bound by Lemma 4.15 (suppressing the index
$i$ for notational clarity):

$\displaystyle\text{arg}(l^{\prime})$
$\displaystyle\in\left[\,\overline{\theta}-\varepsilon-(n-1)\delta\,,\,\overline{\theta}+\varepsilon+(n-1)\delta\right]$
$\displaystyle\equiv\left[-\varepsilon-(n-1)\delta\,,\,\varepsilon+(n-1)\delta\right]\,\,(\text{mod
}2\pi)$
$\displaystyle\subset\left[-n\overline{\delta}\,,\,n\overline{\delta}\right]\,\,(\text{mod
}2\pi).$

Note that since $n\overline{\delta}<\frac{\pi}{4}$, this bound along with the
fact that $l^{i}\cap B_{\frac{\delta}{2}}\neq\O$ implies that
$\overline{\eta}$ must enter $B_{2\delta}$ through the left semicircle of
$B_{2\delta}$ and leave $B_{2\delta}$ through the right semicircle. Moreover,
since $\eta$ intersects $B_{\frac{\delta}{2}}$, if $\overline{\delta}$ is
taken sufficiently small, then it also implies that $\overline{\eta}\cap
B_{\delta}=\eta$, so that $\overline{\eta}$ contains only one connected
component of $l\cap B_{\delta}$. We fix $\overline{\delta}$ such that both of
these facts hold.

Now, for a contradiction, assume that there is a second connected component
$\xi\subset l\cap B_{\delta}$, contained in a connected component
$\overline{\xi}\subset l\cap B_{2\delta}$. Then by the same argument,
$\overline{\xi}$ enters on the left of $B_{2\delta}$ and leaves on the right.
By the embeddedness assumption, $\overline{\xi}$ and $\overline{\eta}$ are
disjoint. However, this behaviour is topologically impossible, since
$\overline{\eta}$ and $\overline{\xi}$ are connected components of a complete
connected embedded curve $l$, the ends of which must diverge to infinity.

_Proof of Claim 3:_ For $0<a<d-\delta$, we define smooth compactly supported
test functions $\check{\psi},\hat{\psi}\colon\mathbb{C}\to[0,1]$ satisfying

$\displaystyle\check{\psi}:=\begin{cases}1\quad\text{ on }B_{\delta-a}\\\
0\quad\text{ outside
}B_{\delta},\end{cases}\quad\hat{\psi}:=\begin{cases}1\quad\text{ on
}B_{\delta}\\\ 0\quad\text{ outside }B_{\delta+a}.\end{cases}$

Then

$\int_{l^{i}}\check{\psi}\,d\mathcal{H}^{1}\,\leq\,\mathcal{H}^{1}(l^{i}\cap
B_{\delta})\,\leq\,\int_{l^{i}}\hat{\psi}\,d\mathcal{H}^{1},$

and as $i\rightarrow\infty$, by assumption (a) (choosing $f\equiv 1$),

$\displaystyle\int_{l^{i}}\check{\psi}\,d\mathcal{H}^{1}\rightarrow
m\cdot\int_{l^{\infty}}\check{\psi}\,d\mathcal{H}^{1}\,\geq
2(\delta-a)m,\quad\int_{l^{i}}\hat{\psi}\,d\mathcal{H}^{1}\rightarrow
m\cdot\int_{l^{\infty}}\hat{\psi}\,d\mathcal{H}^{1}\,\leq 2(\delta+a)m.$

Since $a>0$ may be chosen arbitrarily small, the result follows.

∎

Having established this, we now characterise the Type I blowups of
cohomogeneity-one $G$-equivariant Lagrangian mean curvature flow, using the
notation of Theorem 5.1.

######  Theorem 6.7 (Structure of Type I Blowups).

Let $L_{t}\subset M$ be a connected $G$-invariant almost-calibrated Lagrangian
MCF for $t\in[0,T)$, with $T$ the singular time.

Then there exist $\overline{\theta}$ and $k\in\\{1,\ldots,\frac{2n}{m}\\}$
such that any Type I blowup at time $T$ is the special Lagrangian cone
$L^{\infty}$ with profile curve
$l^{\infty}=C_{m}\cdot(\tilde{c}_{k-1,\overline{\theta}}\cup\tilde{c}_{k,\overline{\theta}})$.

###### Proof.

Without loss of generality, assume $T=0$. By Theorem 6.2, $(O,0)$ is the
unique singular space-time point of the flow. Let $W\subset\mathbb{C}$ be the
wedge given by Lemma 4.16, and note that by Theorem 5.1, for each
$\overline{\theta}$ there exist at most two $G$-invariant special Lagrangian
cones $\widetilde{C}\subset M$ with connected link, profile curve
$\widetilde{c}$ contained in $C_{m}\cdot W$ and Lagrangian angle
$\overline{\theta}$, which have consecutive values of the parameter $k$.

Consider a sequence $L_{s}^{i}$ of Type I rescalings, defined by
$L_{s}^{i}:=\lambda_{i}L_{\lambda_{i}^{-2}s}$ with $\lambda_{i}\to\infty$. By
Theorem 2.4 (A), there exist $N$ special Lagrangian cones $L^{\infty}_{j}$
with connected link, Lagrangian angles $\overline{\theta}_{j}$ and
multiplicities $m_{j}\in\mathbb{Z}^{+}$ such that for $\phi\in
C^{\infty}_{c}(\mathbb{C}^{n})$, $f\in C^{2}(\mathbb{R})$, $s<0$,

$\lim_{i\rightarrow\infty}\int_{L_{s}^{i}}f(\theta_{i,s})\phi\,d\mathcal{H}^{n}\,=\,\sum_{j=1}^{N}m_{j}f(\overline{\theta}_{j})\int_{L^{\infty}_{j}}\phi\,d\mathcal{H}^{n}.$

In other words, the Type I blowup corresponding to the sequence $L^{i}_{s}$ is
$L^{\infty}:=\sum_{j=1}^{N}m_{j}\cdot L^{\infty}_{j}$. We will show that each
$m_{j}=1$ and that $N=2$.
By the almost-calibrated condition, the special Lagrangian cones
$L^{\infty}_{j}$ are distinct from one another. By Proposition 6.3,
$L^{\infty}_{j}$ is $G$-invariant for each $1\leq j\leq N$, and by Theorem 5.1
the profile curves $l^{\infty}_{j}$ are $C_{m}$-orbits of half-lines from the
origin. Moreover, for all $s<0$ the sequence $l^{i}_{s}$ converges in the
sense of Proposition 6.3 to the profile curves $l^{\infty}_{j}$ with
multiplicities $m_{j}$. Choosing $\overline{x}_{j}\in l^{\infty}_{j}$ and
$d>0$ such that $B_{d}(\overline{x}_{j})\cap l^{\infty}_{k}=\O$ for each
$k\neq j$, we note that the conditions of Lemma 6.6 are satisfied for each $j$
(in particular, the curvature bound follows from Theorem 6.1), and we may
therefore conclude that $m_{j}=1$ for all $j$.

We now show that $N=2$. Choose $s<0$ such that (perhaps on passing to a
subsequence) the conclusion of Theorem 2.4 (B) holds for the sequence
$(L^{i}_{s})$ and $R=1$. Therefore, choosing a convergent sequence
$\Sigma^{i}$ of connected components of $L^{i}_{s}\cap B_{1}(O)$ intersecting
$B_{\frac{1}{4}}(O)$, there exists a special Lagrangian cone
$\Sigma^{\infty}\subset\bigcup_{i=1}^{N}L^{\infty}_{j}$ with Lagrangian angle
$\overline{\theta}$ such that for all $\phi\in
C^{\infty}_{c}(\mathbb{C}^{n})$, $f\in C^{2}(\mathbb{R})$,

$\lim_{i\rightarrow\infty}\int_{L_{s}^{i}}f(\theta_{i,s})\phi\,d\mathcal{H}^{n}\,=\,f(\overline{\theta}_{j})\int_{\Sigma^{\infty}}\phi\,d\mathcal{H}^{n}.$

By the same argument as above, $\Sigma^{\infty}$ is $G$-invariant, and the
profile curve $\sigma^{\infty}$ is a union of $C_{m}$-orbits of half-lines
$\bigcup_{k=1}^{M}\sigma^{\infty}_{k}$, for $1\leq M\leq N$. Note that since
$\Sigma^{i}\cap B_{\frac{1}{4}}(O)\neq\O$, it follows that
$\mathcal{H}^{1}\left(\sigma^{\infty}\cap B_{1}(O)\right)\geq\frac{3}{2}$, so
$M\geq 2$, and since there are only two $G$-invariant special Lagrangian cones
of angle $\overline{\theta}$ with profile curve in $W$, we have $M=2$, and
hence $N\geq 2$. Explicitly, there exist $k,\overline{\theta}$ such that
$\sigma^{\infty}=C_{m}\cdot(\tilde{c}_{k-1,\overline{\theta}}\cup\tilde{c}_{k,\overline{\theta}})$.
Finally, $N>2$ would imply that there exists a second sequence of connected
components $\widehat{\Sigma}^{i}$ of $L^{i}\cap B_{1}(O)$ intersecting
$B_{\frac{1}{4}}(O)$, converging to $\widehat{\Sigma}^{\infty}$ with profile
curve
$\widehat{\sigma}^{\infty}=C_{m}\cdot(\widetilde{c}_{\hat{k}-1,\hat{\theta}}\cup\widetilde{c}_{\hat{k},\hat{\theta}})$.
Since $m_{j}=1$ for all $j$, it follows that
$\hat{\theta}\neq\overline{\theta}$. However, the convergence of
$\sigma^{i}\to\sigma^{\infty}$ and
$\widehat{\sigma}^{i}\to\widehat{\sigma}^{\infty}$ are only possible if
$\sigma$ and $\widehat{\sigma}$ intersect for sufficiently large $i$,
contradicting embeddedness of $l^{i}$. Therefore, $N=2$ and
$l^{\infty}=\sigma^{\infty}=C_{m}\cdot(\widetilde{c}_{k-1,\overline{\theta}}\cup\widetilde{c}_{k,\overline{\theta}})$.

Finally, by Theorem 2.4 (A), the angle $\overline{\theta}$ does not depend on
the choice of blowup sequence. ∎

###  6.4 The Type II Blowup

In this subsection, we prove Theorem 6.9 characterising Type II blowups of
cohomogeneity-one Lagrangian mean curvature flow, which we show to be the
unique special Lagrangians of Theorem 5.2. Our first step is to note that if
the Type I blowups at a singular space-time point all have Lagrangian angle
$\overline{\theta}$, then the Type II blowups are special Lagrangians of the
same angle. This follows as a corollary of the following theorem, proven in
[50].

######  Theorem 6.8 (Convergence of Lagrangian Angle for the Type II
Rescalings).

Let $L_{t}$ be an almost-calibrated Lagrangian MCF in $\mathbb{C}^{n}$ with
Lagrangian angle $\theta_{t}$, which forms a singularity at time $T$ with
singular space-time point $(O,0)$. Assume that any convergent sequence of Type
I rescalings $L^{\sigma_{i}}_{s}$ converge to the same special Lagrangian cone
$C$, with angle $\overline{\theta}$.

Let $X_{i}=(x_{i},t_{i})$ be a sequence of space-time points such that
$(x_{i},t_{i})\rightarrow(O,0)$, let $\lambda_{i}\in\mathbb{R}$ satisfy
$-\lambda_{i}^{2}t_{i}\rightarrow\infty$, and define the rescalings

$L^{X_{i},\lambda_{i}}_{\tau}\,:=\,\lambda_{i}\left(L_{t_{i}+\lambda_{i}^{2}\tau}-x_{i}\right)$

with Lagrangian angle $\theta_{\tau}^{i}$.

Then for any bounded parabolic region $\Omega\times
I\subset\mathbb{C}^{n}\times\mathbb{R}$,
$\theta^{i}_{\tau}\rightarrow\overline{\theta}$ uniformly in $\Omega\times I$.
In particular, if the singularity is Type II and the rescalings are Type II
rescalings, then the Type II blowup is a special Lagrangian with angle
$\overline{\theta}$.

We may now conclude that any Type II blowup of a $G$-equivariant Lagrangian
mean curvature flow $L_{t}\subset M$ must be one of the $G$-invariant special
Lagrangians in $M$ given by Theorem 5.2.

######  Theorem 6.9 (Structure of Type II Blowups).

Let $L_{t}\subset M$ be a connected $G$-invariant almost-calibrated Lagrangian
MCF for $t\in[0,T)$, with $T$ the singular time. Let
$\overline{\theta}\in\mathbb{R}$ and $k\in\\{1,\ldots,\frac{2n}{m}\\}$ be the
constants given by Theorem 6.7.

Then there exists $B>0$ such that any Type II blowup at time $T$ is a
translation of the connected $G$-invariant special Lagrangian
$\widetilde{L}^{\infty}$ with profile curve
$\widetilde{l}^{\infty}=C_{m}\cdot\widetilde{l}_{B,k,\overline{\theta}}$. In
particular, the asymptotes of the Type II blowup are given by the unique Type
I blowup of Theorem 6.7.

###### Proof.

Let $L_{t}\subset M$ be a connected $G$-invariant almost-calibrated LMCF for
$t\in[0,T)$, with a singularity at time $T$. By Theorem 6.2, $(O,T)$ is the
unique singular space-time point of the flow. Let $\overline{\theta}$, $k$ be
the constants given by Theorem 6.7. Let $W\subset\mathbb{C}$ be the wedge
given by Lemma 4.16, and note that by Theorem 5.2 there is a unique special
Lagrangian $\widetilde{L}$ with profile curve $\widetilde{l}\subset C_{m}\cdot
W$, $\sup_{\widetilde{L}}|A|=1$, and Lagrangian angle $\overline{\theta}$,
given by $\widetilde{l}_{B,k,\overline{\theta}}$ for some unique $B>0$. We
will prove that any Type II blowup is given by this special Lagrangian.

Consider then a Type II blowup
$L_{s}^{(p_{i},t_{i})}\rightarrow\widetilde{L}_{s}^{\infty}$, denoting by
$A_{i}$ the scaling factors. By the construction of Type II blowups,
$\widetilde{L}^{\infty}_{s}$ is a smooth mean curvature flow. Moreover, since
any Type I blowup is the same special Lagrangian cone with angle
$\overline{\theta}$ by Theorem 6.7, it follows by Proposition 6.8 that
$\widetilde{L}^{\infty}_{s}$ is a special Lagrangian with angle
$\overline{\theta}$, and so a static mean curvature flow. We may therefore
work with a particular time slice. Fix $s\in\mathbb{R}$, and define

$\displaystyle x_{i}:=F_{t_{i}}(p_{i}),\quad\quad
L^{i}:=L_{s}^{(p_{i},t_{i})}=A_{i}L_{A_{i}^{-2}s+t_{i}}-A_{i}x_{i},\quad\quad
y_{i}:=A_{i}x_{i},$

so that $L^{i}+y_{i}$ is a sequence of $G$-invariant Lagrangian submanifolds,
and if $\widetilde{L}^{\infty}:=\widetilde{L}^{\infty}_{s}$,

$\int_{L^{i}}\phi\,d\mathcal{H}^{n}\longrightarrow\int_{\widetilde{L}^{\infty}}\phi\,d\mathcal{H}^{n}\,\text{
as }i\rightarrow\infty.$

Passing to a subsequence, we may assume
$\frac{y_{i}}{|y_{i}|}\rightarrow\nu\in S^{2n-1}$, and also that either
$|y_{i}|\rightarrow\infty$ or there exists $y_{\infty}$ such that
$y_{i}\rightarrow y_{\infty}$.

Assume that $|y_{i}|\rightarrow\infty$. Then Proposition 6.4 implies that
$\widetilde{L}^{\infty}$ contains an isotropic $(n-1)$-plane. It follows from
[17, Thm. III.5.5] that $\widetilde{L}^{\infty}$ is a special Lagrangian
$n$-plane. This is a contradiction, since by the properties of Type II
blowups, $\sup_{\widetilde{L}^{\infty}}|A|=1$.

We must therefore have that there exists $y_{\infty}$ such that
$y_{i}\rightarrow y_{\infty}$. Then

$A_{i}L_{A_{i}^{-2}s+t_{i}}\to\widetilde{L}^{\infty}+y_{\infty},$

and $\widetilde{L}^{\infty}+y_{\infty}$ is a $G$-invariant submanifold by
Proposition 6.3, with $\sup_{\widetilde{L}^{\infty}+y_{\infty}}|A|=1$ and with
profile curve contained in $C_{m}\cdot W$. As already noted, there is only one
such special Lagrangian of angle $\overline{\theta}$, with profile curve
$C_{m}\cdot\widetilde{l}_{B,k,\overline{\theta}}$, and so we are done. ∎

###  6.5 Existence of Singularities with Prescribed Models

Finally, we consider the behaviour of cohomogeneity-one Lagrangian mean
curvature flow for a particular class of initial conditions which form finite-
time singularities. In the process, we demonstrate existence of singularities
modelled on the examples of Theorems 5.1 and 5.2. We restrict to the case of
connected Lagrangian flows $L_{t}\subset M$ such that each connected component
$\eta_{t}$ of the profile curve $l_{t}$ is asymptotically linear, i.e. there
exist distinct half-lines $c_{1},c_{2}$ and a ball $B_{R}$ such that for all
$t$, $\eta_{t}\setminus B_{R}$ may be expressed as a graph over $c_{1}\cup
c_{2}$ that decays to $0$ in $C^{0}$ at infinity. We will denote the spanning
angle between the asymptotes $c_{1},c_{2}$ of each connected component of $l$
by $\alpha_{\text{span}}(l)$.

Our key tool is the following avoidance principle for equivariant LMCF [50,
Thm. 4.2], which allows us to employ barrier arguments.

######  Theorem 6.10 (Preservation of Embeddedness / Avoidance Principle).

Let $f_{t},\tilde{f}_{t}:\mathbb{R}\rightarrow\mathbb{C}\setminus\\{0\\}$ be
asymptotically linear solutions of (26) for $t\in[0,T)$ with images
$l_{t},\tilde{l}_{t}$ respectively, such that $l_{0}$ and $\tilde{l}_{0}$ are
embedded and disjoint and the asymptotes of $l_{t}$ and $\tilde{l}_{t}$ are
different. Then $l_{t}$, $\tilde{l}_{t}$ are embedded and disjoint for all
$t\in[0,T)$.

We first consider the case where $\alpha_{\text{span}}>\frac{\pi}{n}$. In this
case, we may use a barrier flow constructed by Neves that collapses to the
origin to ‘force’ a singularity to occur. The proof is identical to that of
[50, Thm. 4.11].

######  Theorem 6.11 (Finite-Time Singularity for Wide Asymptotic Angles).

Let $L_{t}\subset M$ be a connected, embedded $G$-invariant Lagrangian mean
curvature flow such that the profile curve $l_{t}$ is asymptotically linear,
and $\alpha_{\text{span}}(l_{t})>\frac{\pi}{n}.$ Then $L_{t}$ forms a finite-
time singularity.

###### Proof.

Let $\eta_{t}$ be the family of asymptotically linear curves with
$\alpha_{\text{span}}(\eta_{t})=\beta$ solving (26) with initial condition

$\eta_{0}(s):=\left(\sin\left(\tfrac{\pi
s}{\beta}\right)\right)^{\frac{-\beta}{\pi}}e^{is}.$

This flow exists by an argument of Neves [36, Thm. 4.1], and forms a finite-
time singularity at the origin if $\frac{2\pi}{n}>\beta>\frac{\pi}{n}$ [50,
Thm. 4.10].

Now consider a connected component $\gamma_{t}$ of $l_{t}$, choose
$\beta<\alpha$, and rotate and scale the curve $\eta_{0}$ so that $\gamma_{0}$
lies in between $\eta_{0}$ and the origin, and $\eta_{0}$ and $\gamma_{0}$ are
disjoint. Let $T$ be the singular time of $\eta_{t}$. It follows by Theorem
6.10 that $\gamma_{t}$ approaches the origin by time $T$, and (24) then
implies that the curvature of $\gamma_{t}$ becomes unbounded by time $T$. ∎

As a corollary, we note that for each of the special Lagrangians of Theorem
5.2, there exists a Lagrangian mean curvature flow forming a finite-time
singularity modelled on that special Lagrangian.

######  Theorem 6.12 (Existence of Singularities with Prescribed Models).

Let $L^{\infty}$ be a complete connected $G$-invariant special Lagrangian of
constant isotropy type such that $L^{\infty}\subset\mu^{-1}(0)$.

Then $L^{\infty}$ is asymptotically conical, and there exists a Lagrangian
mean curvature flow $L_{t}$ forming a Type II singularity such that:

  * •

Any Type I blowup is the asymptotic cone of $L^{\infty}$,

  * •

Any Type II blowup is $L^{\infty}$.

###### Proof.

Let $(H)$ be the isotropy type of $L^{\infty}$, so there exists a connected
component $M\subset\mu^{-1}(0)\cap\mathbb{C}^{n}_{(H)}$ with
$L^{\infty}\subset M$. It follows from Theorems 5.1 and 5.2 that there exist
$B>0,\,k\in\mathbb{Z},\,\overline{\theta}\in\mathbb{R}$ such that $L^{\infty}$
has profile curve $C_{m}\cdot\tilde{l}_{B,K,\overline{\theta}}$, and is
asymptotically conical to a special Lagrangian cone with profile curve
$C_{m}\cdot(\tilde{c}_{k-1,\overline{\theta}}\cup\tilde{c}_{k,\overline{\theta}}).$

Now, the Neves example $\eta_{t}\subset\mathbb{C}$ provides an example of a
flow in $M$ with a finite-time singularity. By Theorems 6.7 and 6.9, the Type
I and Type II blowups are of the required forms, after applying a
$\mathrm{U}(1)_{\Delta}$ rotation to the flow. ∎

######  Remark 6.13.

Consider the $T^{n-1}$-action on $\mathbb{C}^{n}$, as discussed in Examples
3.8 and 4.7(b) and Remark 5.6. Let $L_{t}$ be an almost-calibrated Lagrangian
mean curvature flow in $M_{0}:=\mu^{-1}(0)\setminus\\{0\\}$; by Propositions
4.9 and 4.10 $L_{t}$ must be homeomorphic to $\mathbb{R}\times T^{n-1}$.
Assuming that $T$ is the singular time of the flow, it follows by Theorem 6.2
that the unique singular point is the origin. By Theorems 6.7 and 6.9, there
exist $\overline{\theta}\in\mathbb{R}$ and $k\in\\{1,2\\}$ such that:

  * •

Any Type I blowup at time $T$ is the pair of Harvey-Lawson $T^{n-1}$-invariant
special Lagrangian cones whose profile curve is
$C_{n}\cdot(\widetilde{c}_{k-1,\overline{\theta}}\cup\widetilde{c}_{k,\overline{\theta}})$.

  * •

There is a unique $B>0$ such that any Type II blowup at time $T$ is a
translation of the connected $T^{n-1}$-invariant special Lagrangian with
profile curve $C_{n}\cdot\widetilde{l}_{B,k,\overline{\theta}}$.

Moreover, by Theorem 6.12, there exists a flow exhibiting such a finite-time
singularity. This contrasts with the result of Lambert-Lotay-Schulze [29, Thm.
1.2] that there is no singularity of almost-calibrated Lagrangian mean
curvature flow in a Calabi-Yau 3-fold such that the blowdown of the Type II
blowup is given by a single Harvey-Lawson $T^{2}$-cone.

##  7 Examples

Above, we studied $G$-invariant Lagrangians and $G$-equivariant LMCF, where
$G\leq\mathrm{SU}(n)$ is a compact connected Lie subgroup acting linearly on
$\mathbb{C}^{n}$. Our analysis focused on the cohomogeneity-one case, in which
there exists an orbit type with $(n-1)$-dimensional isotropic orbits. In this
section, we provide examples of compact connected Lie subgroups
$G\leq\mathrm{SU}(n)$ admitting such an orbit type.

In $\S$7.1, we explain how, if $G$ is semisimple, $(n-1)$-dimensional
isotropic orbits in $\mathbb{C}^{n}$ may be recast as Lagrangian orbits in
$\mathbb{CP}^{n-1}$. Then, in $\S$7.2, we quote Bedulli and Gori’s
classification [6] of the unitary representations of compact _simple_ Lie
groups admitting a Lagrangian orbit in $\mathbb{CP}^{n-1}$. In this way, their
list — consisting of 7 infinite families and 14 sporadic exceptions — also
classifies the compact simple Lie subgroups $G\leq\mathrm{SU}(n)$ that admit
$(n-1)$-dimensional isotropic orbits in $\mathbb{C}^{n}$. Finally, in $\S$7.3,
we mention a further example in which the compact Lie group $G$ is not
semisimple.

###  7.1 Preliminaries on the Semisimple Case

Let $G\leq\mathrm{U}(n)$ be a compact semisimple Lie group, so that
$\mathfrak{z}(\mathfrak{g})=0$. For $z\in\mathbb{C}^{n}\setminus 0$, let
$\mathcal{O}_{z}\subset\mathbb{C}^{n}$ be the $G$-orbit of $z$, and let $H$ be
the stabiliser of $z$. By Lemma 3.6, the orbit
$\mathcal{O}_{z}\subset\mathbb{C}^{n}$ is isotropic if and only if
$z\in\mu^{-1}(0)$.
As discussed in $\S$4.2, the $G$-action on $\mathbb{C}^{n}\setminus 0$ induces
a $G$-action on $\mathbb{CP}^{n-1}$ via $g\cdot[z]:=[gz]$, where we write
$[z]=P_{z}\in\mathbb{CP}^{n-1}$ for the complex line through
$z\in\mathbb{C}^{n}\setminus 0$. We let

$\mathcal{O}_{[z]}=\\{[w]\in\mathbb{CP}^{n-1}\colon
w\in\mathcal{O}_{z}\\}\subset\mathbb{CP}^{n-1}$

denote the $G$-orbit of $[z]$, and let $\widetilde{H}$ be the stabiliser of
$[z]$. Note that the $G$-equivariant map
$q_{z}\colon\mathcal{O}_{z}\to\mathcal{O}_{[z]}$ via $w\mapsto[w]$ is a
principal $(\widetilde{H}/H)$-bundle. In particular,
$\dim(\mathcal{O}_{z})=\dim(\mathcal{O}_{[z]})+\dim(\widetilde{H}/H)$.
Let $\tau\colon\mathbb{C}^{n}\setminus 0\to\mathbb{CP}^{n-1}$ be the usual
$\mathrm{U}(n)$-equivariant surjective submersion $\tau(z)=[z]$, and equip
$\mathbb{CP}^{n-1}$ with its standard (Fubini-Study) Kähler structure
($g_{\text{FS}},J_{\text{FS}},\omega_{\text{FS}})$. With respect to the
Fubini-Study structure, the $G$-action on $\mathbb{CP}^{n-1}$ is Kähler and
Hamiltonian. To see that the action is Hamiltonian, we recall a well-known
formula for the moment map (see, e.g., [27, pg. 12-14]). Let
$\widetilde{\mu}\colon\mathbb{C}^{n}\setminus 0\to\mathfrak{g}^{*}$ be

$\widetilde{\mu}(z)=\frac{1}{|z|^{2}}\mu(z).$

Then $\widetilde{\mu}$ is $\mathbb{C}^{*}$-invariant, so descends to a
function $\nu\colon\mathbb{CP}^{n-1}\to\mathfrak{g}^{*}$, meaning
$\widetilde{\mu}=\tau^{*}\nu$. In the sequel, we use the same symbol $\rho$ to
denote the infinitesimal $G$-action on both $\mathbb{C}^{n}$ and
$\mathbb{CP}^{n-1}$.

######  Lemma 7.1.

The map $\nu\colon\mathbb{CP}^{n-1}\to\mathfrak{g}^{*}$ is a moment map for
the $G$-action on $\mathbb{CP}^{n-1}$, i.e. :

$\displaystyle d\nu^{X}$ $\displaystyle=-\iota_{\rho(X)}\omega_{\mathrm{FS}},\
\ \text{ for all }X\in\mathfrak{g},$ $\displaystyle\nu(g\cdot[z])$
$\displaystyle=\mathrm{Ad}_{g}^{*}\\!\left(\nu([z])\right),\ \ \text{ for all
}g\in G,\,[z]\in\mathbb{CP}^{n-1}.$

###### Proof.

Before beginning, we set
$\gamma:=\partial(|z|^{2})=\sum\overline{z}_{j}dz_{j}$ for convenience. Noting
that
$\text{Re}(\gamma)=\frac{1}{2}(\gamma+\overline{\gamma})=\frac{1}{2}d(|z|^{2})$,
we see that every $X\in\mathfrak{g}$ satisfies $\text{Re}(\gamma)(\rho(X))=0$,
so that $\gamma(\rho(X))=-\overline{\gamma}(\rho(X))$. Recalling the Liouville
$1$-form

$\lambda=\frac{1}{2}\sum
x_{j}dy_{j}-y_{j}dx_{j}=-\frac{i}{4}\sum\overline{z}_{j}dz_{j}-z_{j}d\overline{z}_{j}=-\frac{i}{4}(\gamma-\overline{\gamma}),$

we use Lemma 4.1 to observe

$\mu^{X}(z)=\lambda(\rho(X))=-\frac{i}{4}\left[\gamma(\rho(X))-\overline{\gamma}(\rho(X))\right]=-\frac{i}{2}\gamma(\rho(X)).$

Now, for the first claim of the lemma, we note that
$\omega=\frac{i}{2}\partial\overline{\partial}(|z|^{2})=\frac{i}{2}\partial\overline{\gamma}$
to compute

$\displaystyle\tau^{*}\omega_{\text{FS}}=\frac{i}{2}\partial\overline{\partial}(\log|z|^{2})=\frac{i}{2}\partial\\!\left(|z|^{-2}\overline{\gamma}\right)$
$\displaystyle=-\frac{i}{2}|z|^{-4}\gamma\wedge\overline{\gamma}+|z|^{-2}\omega,$

and hence

$\displaystyle\tau^{*}(-\iota_{\rho(X)}\omega_{\text{FS}})=-\iota_{\rho(X)}(\tau^{*}\omega_{\text{FS}})$
$\displaystyle=\frac{i}{2}|z|^{-4}\iota_{\rho(X)}(\gamma\wedge\overline{\gamma})-|z|^{-2}\omega(\rho(X),\cdot)$
$\displaystyle=\frac{i}{2}|z|^{-4}\gamma(\rho(X))\left(\gamma+\overline{\gamma}\right)-|z|^{-2}d\mu^{X}$
$\displaystyle=-|z|^{-4}\mu^{X}(z)\left(\gamma+\overline{\gamma}\right)-|z|^{-2}d\mu^{X}.$

On the other hand, we may also compute

$\displaystyle\tau^{*}(d\nu^{X})=d(\tau^{*}\nu^{X})=d\widetilde{\mu}^{X}=d(|z|^{-2}\mu^{X})$
$\displaystyle=-|z|^{-4}\mu^{X}(z)d(|z|^{2})-|z|^{-2}d\mu^{X}$

from which we obtain

$\tau^{*}(d\nu^{X})=\tau^{*}(-\iota_{\rho(X)}\omega_{\text{FS}}).$

Noting that $\tau^{*}$ is injective gives the first claim. For the
$G$-equivariance, we simply note that

$\nu([gz])=\widetilde{\mu}(gz)=\frac{1}{|gz|^{2}}\mu(gz)=\text{Ad}_{g}^{*}\\!\left[\frac{1}{|z|^{2}}\mu(z)\right]=\text{Ad}_{g}^{*}(\nu([z])).$

∎

######  Lemma 7.2.

Let $G\leq\mathrm{U}(n)$ be a compact semisimple Lie group, and let
$z\in\mathbb{C}^{n}\setminus 0$.
(a) The $G$-orbit $\mathcal{O}_{z}\subset\mathbb{C}^{n}$ is isotropic if and
only if the $G$-orbit $\mathcal{O}_{[z]}\subset\mathbb{CP}^{n-1}$ is
isotropic.
(b) If $G\leq\mathrm{SU}(n)$ and $\mathcal{O}_{z}\subset\mathbb{C}^{n}$ is
isotropic, then $\dim(\mathcal{O}_{z})=\dim(\mathcal{O}_{[z]})$.

###### Proof.

(a) By definition of $\nu$, we have
$\mu(z)=|z|^{2}\widetilde{\mu}(z)=|z|^{2}\nu([z])$. Consequently,

$\mathcal{O}_{z}\text{ is isotropic }\iff
z\in\mu^{-1}(0)\iff[z]\in\nu^{-1}(0)\iff\mathcal{O}_{[z]}\text{ is
isotropic.}$

(b) Suppose that $G\leq\mathrm{SU}(n)$. If
$\mathcal{O}_{z}\subset\mathbb{C}^{n}$ is isotropic, then $z\in\mu^{-1}(0)$.
Now, Proposition 4.6 implies that $\widetilde{H}/H\cong C_{m}$ is finite, so
$q_{z}$ is a local diffeomorphism and
$\dim(\mathcal{O}_{[z]})=\dim(\mathcal{O}_{z})$. ∎

######  Corollary 7.3.

Let $G\leq\mathrm{SU}(n)$ be a compact semisimple Lie group, and let
$z\in\mathbb{C}^{n}\setminus 0$. The $G$-orbit
$\mathcal{O}_{[z]}\subset\mathbb{CP}^{n-1}$ is Lagrangian if and only if the
$G$-orbit $\mathcal{O}_{z}\subset\mathbb{C}^{n}$ is an isotropic submanifold
of dimension $(n-1)$.

######  Remark 7.4.

In fact, if $G\leq\mathrm{SU}(n)$ is a compact semisimple Lie group, each
Lagrangian $G$-orbit in $\mathbb{CP}^{n-1}$ is a minimal submanifold; see [6,
Prop. 3.1]. Consequently, the corresponding isotropic $G$-orbit in the unit
sphere $\mathbb{S}^{2n-1}(1)\subset\mathbb{C}^{n}$ is a special Legendrian
submanifold, and hence is the link of a special Lagrangian cone in
$\mathbb{C}^{n}$, which accords with our discussion in $\S$5.1.

###  7.2 The Classification for $G$ Simple

We now turn to Bedulli and Gori’s classification of compact simple groups in
$\mathrm{U}(n)$ that admit a Lagrangian orbit in $\mathbb{CP}^{n-1}$.

######  Theorem 7.5 ([6]).

Let $G$ be a compact simple Lie group. Suppose $G$ acts on $\mathbb{CP}^{n-1}$
via a unitary representation $\sigma\colon G\to\mathrm{U}(n)$. Then there
exists a Lagrangian $G$-orbit in $\mathbb{CP}^{n-1}$ if and only if $G$
appears in the table of Figure 3.

######  Remark 7.6.

If $G\leq\mathrm{U}(n)$ is a compact simple Lie group, then
$G\leq\mathrm{SU}(n)$. To see this, let $K$ be the kernel of the determinant
map $\det\colon G\to\mathrm{U}(1)$, and let $K^{0}$ be its identity component.
Since $K^{0}$ is a connected normal subgroup of $G$, we have
$K^{0}=\\{\mathrm{Id}\\}$ or $K^{0}=G$ by the simplicity of $G$. In the former
case, $K$ is finite, so $\dim(G)=\dim(G/K)=\dim(\det(G))\leq 1$, contradicting
the simplicity of $G$. Thus, $K^{0}=G$, so $K=G$, so $G\leq\mathrm{SU}(n)$.

The upshot is that, by Corollary 7.3, the table of Figure 3 classifies the
compact simple Lie subgroups $G\leq\mathrm{SU}(n)$ that admit
$(n-1)$-dimensional isotropic orbits in $\mathbb{C}^{n}$. For each such
subgroup $G$, the results of the preceding sections yield information about
$G$-equivariant Lagrangian mean curvature flow. For illustration, here is an
explicit example.

######  Example 7.7.

Let $G=\mathrm{SU}(2)$ act on
$\mathbb{C}^{4}=\mathrm{Sym}^{3}(\mathbb{C}^{2})$ in the standard way. That
is, for $g\in\mathrm{SU}(2)$ and a homogeneous cubic polynomial
$p\in\mathrm{Sym}^{3}(\mathbb{C}^{2})$ in two variables
$w=(w_{1},w_{2})\in\mathbb{C}^{2}$, we define

$(g\cdot p)(w):=p\\!\left(g^{T}w\right)=p\\!\left(g^{T}\begin{bmatrix}w_{1}\\\
w_{2}\end{bmatrix}\right).$

More explicitly, with respect to the basis
$\\{w_{1}^{3},\sqrt{3}w_{1}^{2}w_{2},\sqrt{3}w_{1}w_{2}^{2},w_{2}^{3}\\}$ of
$\mathrm{Sym}^{3}(\mathbb{C}^{2})$, the $\mathrm{SU}(2)$-action on
$\mathbb{C}^{4}$ is given by

$\begin{pmatrix}a&-\overline{b}\\\
b&\overline{a}\end{pmatrix}\cdot\begin{bmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\
z_{4}\end{bmatrix}:=\begin{pmatrix}a^{3}&-\sqrt{3}a^{2}\overline{b}&\sqrt{3}a\overline{b}^{2}&-\overline{b}^{3}\\\
\sqrt{3}a^{2}b&a(|a|^{2}-2|b|^{2})&-\overline{b}(2|a|^{2}-|b|^{2})&\sqrt{3}\overline{a}\overline{b}^{2}\\\
\sqrt{3}ab^{2}&b(2|a|^{2}-|b|^{2})&\overline{a}(|a|^{2}-2|b|^{2})&-\sqrt{3}\overline{a}^{2}\overline{b}\\\
b^{3}&\sqrt{3}\overline{a}b^{2}&\sqrt{3}\overline{a}^{2}b&\overline{a}^{3}\end{pmatrix}\begin{bmatrix}z_{1}\\\
z_{2}\\\ z_{3}\\\ z_{4}\end{bmatrix}$

where $a,b\in\mathbb{C}$ satisfy $|a|^{2}+|b|^{2}=1$. A calculation shows that
the moment map of the $\mathrm{SU}(2)$-action is the function
$\mu\colon\mathbb{C}^{4}\to\mathfrak{su}(2)\subset\mathfrak{su}(4)$ given by

$\mu(z)=\frac{i}{40}\begin{bmatrix}3p(z)&\sqrt{3}r(z)&0&0\\\
\sqrt{3}\,\overline{r(z)}&p(z)&2r(z)&0\\\
0&2\overline{r(z)}&-p(z)&\sqrt{3}r(z)\\\
0&0&\sqrt{3}\,\overline{r(z)}&-3p(z)\end{bmatrix},$

where

$\displaystyle p(z)$
$\displaystyle:=3|z_{1}|^{2}+|z_{2}|^{2}-|z_{3}|^{2}-3|z_{4}|^{2}$
$\displaystyle r(z)$
$\displaystyle:=2\sqrt{3}(z_{1}\overline{z}_{2}+z_{3}\overline{z}_{4})+4z_{2}\overline{z}_{3}.$

By Proposition 3.11, every $\mathrm{SU}(2)$-invariant Lagrangian submanifold
$L_{0}\subset\mathbb{C}^{4}$ lies in the level set

$\mu^{-1}(0)=\left\\{z\in\mathbb{C}^{4}\colon p(z)=0,\,r(z)=0\right\\}\\!.$

While the principal $\mathrm{SU}(2)$-orbits in $\mathbb{C}^{4}$, such as that
of $(0,1,i,0)$, have trivial stabiliser, the $\mathrm{SU}(2)$-action also
admits both exceptional and singular orbits. For example, the point
$z=(1,0,0,1)\in\mu^{-1}(0)$ has stabiliser

$\displaystyle H=\text{Stab}(z)$
$\displaystyle=C_{3}=\left\\{\begin{pmatrix}\zeta&0\\\
0&\overline{\zeta}\end{pmatrix}\colon\zeta^{3}=1\right\\},$
$\displaystyle\widetilde{H}=\text{Stab}([z])$ $\displaystyle=C_{3}\rtimes
C_{4}=\left\\{\begin{pmatrix}\zeta&0\\\
0&\overline{\zeta}\end{pmatrix}\begin{pmatrix}0&i\\\
i&0\end{pmatrix}^{k}\,:\,\zeta^{3}=1,\,k\in\\{0,1,2,3\\}\right\\}$

and the $\mathrm{SU}(2)$-orbit is an isotropic submanifold of $\mathbb{C}^{4}$
diffeomorphic to the Lens space $\mathbb{S}^{3}/C_{3}$. It follows that
$M:=\mathrm{SU}(2)\cdot(P_{z}\setminus 0)\subset\mu^{-1}(0)$ is a connected
component of $M_{0}:=\mu^{-1}(0)\cap\mathbb{C}^{4}_{(C_{3})}$, and in fact
further calculation yields that $M=M_{0}=\mu^{-1}(0)\setminus 0$ (see [34,
Lem. 3.14]). Therefore, any $\mathrm{SU}(2)$-invariant Lagrangian lies in $M$.

Now, let $L_{0}\subset M$ be a connected, immersed $\mathrm{SU}(2)$-invariant
Lagrangian of type $(C_{3})$. If $L_{0}$ is almost-calibrated, then $L_{0}$ is
exact, embedded, and homeomorphic to $\mathbb{R}\times(\mathbb{S}^{3}/C_{3})$
by Propositions 4.9 and 4.10, and any mean curvature flow starting at
$L_{0}\subset M$ must stay within $M$ by Proposition 3.20.

The $\mathrm{SU}(2)$-invariant special Lagrangians of type $(C_{3})$ are given
in Theorems 5.1 and 5.2, and were discussed by Marshall in [34]. Theorem 5.9
classifies the cohomogeneity-one $\mathrm{SU}(2)$-invariant Lagrangian
shrinkers and expanders in $\mathbb{C}^{4}$, while Theorem 5.10 rules out the
existence of cohomogeneity-one $\mathrm{SU}(2)$-invariant translators.

Let $L_{t}\subset M\subset\mathbb{C}^{4}$, $t\in[0,T)$, be a connected
$\mathrm{SU}(2)$-equivariant almost-calibrated Lagrangian mean curvature flow.
By Theorem 6.2, a singularity occurs at time $T$ if and only if $(O,T)$ is the
unique singular space-time point for $L_{t}$. Suppose that $T$ is, in fact,
the singular time. Then there exist $\theta\in\mathbb{R}$ and $k\in\\{1,2\\}$
such that, by Theorems 6.7 and 6.9:

  * •

Any Type I blowup at time $T$ is Marshall’s $\mathrm{SU}(2)$-invariant special
Lagrangian cone — namely, the special Lagrangian with profile curve
$C_{4}\cdot(\widetilde{c}_{k-1,\overline{\theta}}\cup\widetilde{c}_{k,\overline{\theta}})$.

  * •

There is a unique $B>0$ such that any Type II blowup at time $T$ is a
translation of the connected $\mathrm{SU}(2)$-invariant special Lagrangian
with profile curve $C_{4}\cdot\widetilde{l}_{B,k,\overline{\theta}}$. In
particular, the asymptotes of the Type II blowup are given by the unique Type
I blowup of Theorem 6.7.

Finally, Theorem 6.12 shows that there exists a mean curvature flow with such
a finite-time singularity, and therefore all $\mathrm{SU}(2)$-invariant smooth
special Lagrangians occur as Type II blowups of Lagrangian mean curvature
flows.

$G$ | $\sigma$ | $n$ | $\widetilde{H}^{0}$ | $\widetilde{H}/\widetilde{H}^{0}$
---|---|---|---|---
$\mathrm{SU}(p)$ | $2\Lambda_{1}$ | $\frac{1}{2}p(p+1)$ | $\mathrm{SO}(p)$ | $C_{p}$
$\mathrm{SU}(p)$ | $\Lambda_{1}\oplus\Lambda_{1}^{*}$ | $2p$ | $\mathrm{SU}(p-1)$ | $C_{2}$
$\mathrm{SU}(p)$ | $\Lambda_{1}\oplus\cdots\oplus\Lambda_{1}$ | $p^{2}$ | $1$ | $C_{p}$
$\mathrm{SU}(2p)$ | $\Lambda_{2}$ | $p(2p-1),\ p\geq 3$ | $\mathrm{Sp}(p)$ | $C_{2p}$
$\mathrm{SU}(2p+1)$ | $\Lambda_{2}\oplus\Lambda_{1}$ | $2p^{2}+3p+2,\ p\geq 2$ | $\mathrm{Sp}(p)$ | $C_{p+1}$
$\mathrm{SU}(2)$ | $3\Lambda_{1}$ | $4$ | $1$ | $C_{3}\rtimes C_{4}$
$\mathrm{SU}(6)$ | $\Lambda_{3}$ | $20$ | $\mathrm{SU}(3)\times\mathrm{SU}(3)$ | $C_{4}$
$\mathrm{SU}(7)$ | $\Lambda_{3}$ | $35$ | $\text{G}_{2}$ | $C_{7}$
$\mathrm{SU}(8)$ | $\Lambda_{3}$ | $56$ | $\text{Ad}(\mathrm{SU}(3))$ | $C_{16}$
$\mathrm{Sp}(p)$ | $\Lambda_{1}\oplus\Lambda_{1}$ | $4p$ | $\mathrm{Sp}(p-1)$ | $C_{2}$
$\mathrm{Sp}(3)$ | $\Lambda_{3}$ | $14$ | $\mathrm{SU}(3)$ | $C_{4}$
$\mathrm{SO}(p)$ | $\Lambda_{1}$ | $p,\ p\geq 3$ | $\mathrm{SO}(p-1)$ | $C_{2}$
$\mathrm{Spin}(7)$ | spin rep. | $8$ | $\text{G}_{2}$ | $C_{2}$
$\mathrm{Spin}(9)$ | spin rep. | $16$ | $\mathrm{Spin}(7)$ | $C_{2}$
$\mathrm{Spin}(10)$ | $\Lambda_{\text{even}}\oplus\Lambda_{\text{even}}$ | $32$ | $\text{G}_{2}$ | –
$\mathrm{Spin}(11)$ | spin rep. | $32$ | $\mathrm{SU}(5)$ | $C_{4}$
$\mathrm{Spin}(12)$ | $\Lambda_{\text{even}}$ | $32$ | $\mathrm{SU}(6)$ | $C_{4}$
$\mathrm{Spin}(14)$ | $\Lambda_{\text{even}}$ | $64$ | $\text{G}_{2}\times\text{G}_{2}$ | $C_{8}$
$\text{E}_{6}$ | $\Lambda_{1}$ | $27$ | $\text{F}_{4}$ | $C_{3}$
$\text{E}_{7}$ | $\Lambda_{1}$ | $56$ | $\text{E}_{6}$ | –
$\text{G}_{2}$ | $\Lambda_{2}$ | $7$ | $\mathrm{SU}(3)$ | $C_{2}$
Figure 3: Bedulli-Gori’s classification of compact simple Lie group actions on
$\mathbb{CP}^{n-1}$ with a Lagrangian orbit. In the table, we identify a
representation $\sigma\colon G\to\mathrm{U}(n)$ with the highest weights of
its irreducible components. Here, $\Lambda_{1},\Lambda_{2},\ldots$ are the
fundamental dominant weights of $G$, $\Lambda_{\text{even}}$ denotes the even
half-spin representation of $\mathrm{Spin}(2p)$, and $\widetilde{H}^{0}$ is
the identity component of $\widetilde{H}$.

###  7.3 A Non-Semisimple Example

To conclude, we mention a further example involving a non-semisimple group,
discussed in [9, pg. 4].

######  Example 7.8.

For $p,q\geq 3$, consider the action of
$G=\mathbb{S}^{1}\times\mathrm{SO}(p)\times\mathrm{SO}(q)$ on
$\mathbb{C}^{p+q}$ via

$(e^{i\theta},A,B)\cdot(z,w):=(e^{qi\theta}Az,e^{-pi\theta}Bw).$

For $p\neq q$, there are at least $9$ orbit types. In the following table we
use the shorthand “$\\{x,y\\}$ (in)dependent” to mean that the set of vectors
$\\{x,y\\}$ is an $\mathbb{R}$-linearly (in)dependent set.

$z=x+iy$ | $w=u+iv$ | dim of $G$-orbit $\mathcal{O}_{(z,w)}$
---|---|---
$\\{x,y\\}$ independent |  $\\{u,v\\}$ independent | $2p+2q-5$
$\\{x,y\\}$ independent |  $\\{u,v\\}$ dependent, $w\neq 0$ | $2p+q-3$
$\\{x,y\\}$ independent | $0$ | $2p-2$
$\\{x,y\\}$ dependent, $z\neq 0$ |  $\\{u,v\\}$ independent | $p+2q-3$
$\\{x,y\\}$ dependent, $z\neq 0$ |  $\\{u,v\\}$ dependent, $w\neq 0$ | $p+q-1$
$\\{x,y\\}$ dependent, $z\neq 0$ | $0$ | $p$
$0$ |  $\\{u,v\\}$ independent | $2q-2$
$0$ |  $\\{u,v\\}$ dependent, $w\neq 0$ | $q$
$0$ | $0$ | $0$

In particular, we note that the points $(z,w)=(x+iy,u+iv)$ having $z\neq 0$,
$w\neq 0$, and $\\{x,y\\}$ and $\\{u,v\\}$ both $\mathbb{R}$-linearly
dependent have $G$-orbits of real dimension $p+q-1$. Moreover, one can check
that these orbits are isotropic submanifolds of $\mathbb{C}^{p+q}$.
The moment map is the function

$\displaystyle\mu\colon\mathbb{C}^{p+q}$
$\displaystyle\to\mathfrak{u}(1)\oplus\mathfrak{so}(p)\oplus\mathfrak{so}(q)$
$\displaystyle\mu(z,w)$
$\displaystyle=\left(i(q|z|^{2}-p|w|^{2}),\,\text{Im}(z\overline{z}^{T}),\,\text{Im}(w\overline{w}^{T})\right)\\!.$

Note that
$\mathfrak{z}(\mathfrak{u}(1)\oplus\mathfrak{so}(p)\oplus\mathfrak{so}(q))=\\{(ic,0,0)\colon
c\in\mathbb{R}\\}$. The $\mu$-level sets at the central values $(ic,0,0)$ are

$\displaystyle\mu^{-1}(ic,0,0)$
$\displaystyle=\left\\{\left(\zeta\mathbf{r},\eta\mathbf{s}\right)\in\mathbb{C}^{p}\oplus\mathbb{C}^{q}\colon\mathbf{r}\in\mathbb{S}^{p-1},\mathbf{s}\in\mathbb{S}^{q-1}\text{
and }\zeta,\eta\in\mathbb{C}\text{ s.t.
}q|\zeta|^{2}-p|\eta|^{2}=c\right\\}\\!.$

By Proposition 3.11, every $G$-invariant Lagrangian submanifold
$L_{0}\subset\mathbb{C}^{p+q}$ lies in one of the level sets
$\mu^{-1}(ic,0,0)$. Further, by Proposition 3.20, any mean curvature flow
starting at $L_{0}$ must stay within its initial level set $\mu^{-1}(ic,0,0)$.
In the case of $\mu^{-1}(0,0,0)$, the results of $\S$6 follow as in the
previous examples.

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