sense, if $\varphi_{j}\to\varphi_{\infty}$ in $\mathbb{B}^{I}$, then there are
$R_{j}\to+\infty$ such that

$\mathbb{B}_{1/R_{j}}(T[\Sigma_{\varphi_{\infty}}])\supset
T[\Sigma_{\varphi_{j}}]\cap\mathbb{B}_{R_{j}}.$

Moreover, let $\varpi$ be the $\mathbb{R}$-action on the space of pl-simple
ends introduced in Section 3.5, then
$\\{T[\varpi(a)[\Sigma_{e}]]\\}_{a\in\mathbb{R}}$ is a decomposition of
$\mathbb{R}^{n+1}$, i.e.

(4.4)
$\displaystyle\mathbb{R}^{n+1}=\coprod_{a\in\mathbb{R}}T[\varpi(a)[\Sigma_{e}]].$

To save the notation, for every pl-simple translating end $\Sigma_{e}$ over
$S\times\mathbb{R}_{+}$, we let $\mathcal{T}[\Sigma_{e}]$ to be the set of all
$\mathcal{I}$-minimizing translators exponentially asymptotic to $\Sigma_{e}$.
Note that since $S$ is closed and connected, any such
$T\in\mathcal{T}[\Sigma_{e}]$ has connected support, and there’s a unique
connected component $\Omega_{T}$ of $\mathbb{R}^{n+1}\setminus\text{spt}(T)$,
referred to as the inner region bounded by $T$, such that
$\Omega_{T}\cap\mathbb{R}^{n}\times\\{z\\}$ is bounded for $z>>1$. Here we
abuse the notation to identify an integral cycle $T$ with its support, since
everything is multiplicity one and smooth away from a low dimensional closed
subset.

###### Proof of Theorem 4.3..

We start with an observation that disjoint ends has disjoint
$\mathcal{I}$-minimizing translators exponentially asymptotic to them.
Claim 1. For every pl-simple end $\Sigma_{e}\in\mathcal{E}_{p}(S)$ and every
real number $a>0$, if $T_{0}\in\mathcal{T}[\Sigma_{e}]$ and
$T_{a}\in\mathcal{T}[\varpi(a)[\Sigma_{e}]]$, then
$\Omega_{T_{a}}\supset\text{Clos}(\Omega_{T_{0}})$.

###### Proof of Claim 1..

Suppose $\Sigma_{e}\cap\mathbb{R}^{n}\times\mathbb{R}_{\geq
z_{0}}=\mathrm{ES}[w_{0}]$ and $w_{a}$ is given by Lemma 3.27 so that
$\mathrm{ES}[w_{a}]\in\varpi(a)[\Sigma_{e}]$. Then by definition of
exponentially asymptotic, $T_{0}\cap\mathbb{R}^{n}\times\mathbb{R}_{\geq
z_{0}^{\prime}}=:\mathrm{ES}[u_{0}]$ and
$T_{a}\cap\mathbb{R}^{n}\times\mathbb{R}_{\geq
z_{0}^{\prime}}=:\mathrm{ES}[u_{a}]$ satisfies

$\displaystyle\ |(u_{a}-u_{0})(\cdot,z)\cdot z^{-\mu_{1}}-a\psi_{1}|$
$\displaystyle=$ $\displaystyle\ |(w_{a}-w_{0})(\cdot,z)\cdot
z^{-\mu_{1}}-a\psi_{1}|+|(u_{0}-w_{0})(\cdot,z))\cdot
z^{-\mu_{1}}|+|(u_{a}-w_{a})(\cdot,z))\cdot z^{-\mu_{1}}|\to 0.$

And since $a>0$ and $\psi_{1}>0$, this implies $u_{a}>u_{0}$ for $z\geq
z_{0}^{\prime\prime}>>1$, which means
$\text{Clos}(\Omega_{T_{0}})\cap\mathbb{R}^{n}\times\mathbb{R}_{\geq
z_{0}^{\prime\prime}}\subset\Omega_{T_{a}}$. But since
$\partial\Omega_{T_{0}}$ and $\partial\Omega_{T_{a}}$ are both
$\mathcal{I}$-minimizing, by comparing $\mathcal{I}$-functional and strong
maximum principle [SW89, Ilm96], this implies
$\text{Clos}(\Omega_{T_{0}})\subset\Omega_{T_{a}}$. ∎

Now let $\Sigma_{e}\in\mathcal{E}_{p}(S)$ be a pl-simple end in the Theorem.
For every $a\in\mathbb{R}$, we pick
$T_{a}\in\mathcal{T}[\varpi(a)[\Sigma_{e}]]$ to be an $\mathcal{I}$-minimizing
translator constructed in Theorem 4.1, and let $\Omega_{a}:=\Omega_{T_{a}}$ be
the inner region bounded by $T_{a}$. Recall that Claim 1 guarantees that
$\\{\Omega_{a}\\}_{a\in\mathbb{R}}$ is monotone increasing in $a$.

Define the closed subset,

$\Omega^{+}=\Omega^{+}[\Sigma_{e}]:=\bigcap_{a>0}\text{Clos}(\Omega_{a})=\bigcap_{a>0}\Omega_{a}.$

First notice that by Claim 1, $\Omega^{+}[\Sigma_{e}]$ is independent to the
choice of $T_{a}$. Also, $\Omega^{+}$ is the limit of $\Omega_{a}$ in the
measure sense, hence has $\mathcal{I}$-minimizing boundary by the compactness
of minimizing boundary [Sim83]. And since we have $C^{2}$-estimates for the
graphical functions of $\partial\Omega_{a}=T_{a}$ over $\mathrm{ES}$ outside a
uniform compact subset in Lemma 3.27 and Theorem 4.1, combined with Lemma
3.30, we have $\partial\Omega^{+}\in\mathcal{T}[\Sigma_{e}]$. Moreover, if
$\Sigma_{e}$ satisfies the assumption in Theorem 4.1, then
$\partial\Omega^{+}$ satisfies the uniform estimate (4.1). Similarly,

$\Omega^{-}=\Omega^{-}[\Sigma_{e}]:=\bigcup_{a<0}\Omega_{a},$

is an open subset with connected $\mathcal{I}$-minimizing boundary, which
falls in $\mathcal{T}[\Sigma_{e}]$.

Let $T[\Sigma_{e}]:=\Omega^{+}[\Sigma_{e}]\setminus\Omega^{-}[\Sigma_{e}]$.
Since $\Omega^{-}\subset\Omega^{+}$, by strong maximum principle [SW89,
Ilm96], either $\partial\Omega^{-}=\partial\Omega^{+}$, in which case (i)
holds in Theorem 4.3, or
$\text{Clos}(\Omega^{-})\subset\text{Int}(\Omega^{+})$, which means (ii)
holds. And again by Claim 1, for every $T\in\mathcal{T}[\Sigma_{e}]$,
$T\subset\Omega^{+}$ and $T\cap\Omega^{-}=\emptyset$, which means $T\subset
T[\Sigma_{e}]$. The upper-semi-continuity of $T[\Sigma_{\varphi}]$ in
$\varphi$ follows directly from (4.1), Theorem 3.7 (ii) and that
$T[\Sigma_{\varphi}]$ contains all $\mathcal{I}$-minimizing translator
exponentially asymptotic to $\Sigma_{\varphi}$.

To prove (4.4), first by Claim 1,
$\\{T[\varpi(a)[\Sigma_{e}]]\\}_{a\in\mathbb{R}}$ are pairwise disjoint; And
by its definition,

$\bigcup_{a\in\mathbb{R}}T[\varpi(a)[\Sigma_{e}]]=\bigcup_{a>0}\Omega_{a}\setminus\bigcap_{a<0}\Omega_{a}.$

Therefore, it suffices to show that
$\Omega_{+\infty}:=\bigcup_{a>0}\Omega_{a}=\mathbb{R}^{n+1}$ and
$\Omega_{-\infty}:=\bigcap_{a<0}\Omega_{a}=\emptyset$. In the following, we
shall only prove one of them, while the other one is similar.

Suppose for contradiction that $\Omega_{+\infty}\neq\mathbb{R}^{n+1}$, then
again by Compactness Theorem of minimizing boundary [Sim83],
$\partial\Omega_{+\infty}$ is an $\mathcal{I}$-minimizing translator in
$\mathbb{R}^{n+1}$. By Lemma 2.3, suppose $\\{\sqrt{-t}V\\}_{t<0}$ is a
tangent flow of $\partial\Omega_{+\infty}$ at $-\infty$, where $V$ splits in
$z$-direction. And since $\Omega_{+\infty}\supset\Omega_{1}$, whose boundary
has a simple end over $S\times\mathbb{R}_{+}$, we then know that the self-
shrinking integral varifold $V$ is supported in
$\text{Clos}(D)\times\mathbb{R}$, where $D$ is the unbounded component of
$\mathbb{R}^{n}\setminus S$. By strong maximum principle [SW89],
$V=m|S\times\mathbb{R}|+V_{+}$ for some $m\in\mathbb{Z}_{\geq 1}$ and $V_{+}$
is another self-shrinking integral varifold supported in $D\times\mathbb{R}$.
By avoidance principle of MCF, $V_{+}=0$ (see also [CCMS20, Appendix C]).
Also, by Corollary 2.4, the entropy
$\lambda[\partial\Omega_{a}]=\lambda[S\times\mathbb{R}]$ for every
$a\in\mathbb{R}$. Then the entropy of $V$ satisfies,

$m\lambda[S\times\mathbb{R}]=\lambda[V]=\lambda[\partial\Omega_{+\infty}]\leq\liminf_{a\to+\infty}\lambda[\partial\Omega_{a}]=\lambda[S\times\mathbb{R}].$

This forces $m=1$, and hence $\partial\Omega_{+\infty}$ is also an
$\mathcal{I}$-minimizing translator with simple end over
$S\times\mathbb{R}_{+}$. Then by Lemma 3.29, there exists $a\geq 0$ such that
$\partial\Omega_{+\infty}\in\varpi(a)[\Sigma_{e}]$. But by Claim 1, this means
$\partial\Omega_{+\infty}\subset\text{Int}(\Omega_{a+1})\subset\text{Int}(\Omega_{+\infty})$,
which is impossible. This finish the proof of
$\Omega_{+\infty}=\mathbb{R}^{n+1}$. ∎

######  Definition 4.4.

Case (ii) in Theorem 4.3 is called fattening. In contrast, case (i) in Theorem
4.3 is called non-fattening

In Section 5, we will show that fattening indeed happens. Nevertheless, the
following Corollary suggests that fattening is a rare phenomenon.

######  Corollary 4.5.

For a fixed $z_{0}\geq C_{0}(S)$, let
$\\{\Sigma_{\varphi}:=\mathrm{ES}[u_{\varphi,z_{0}}]\\}_{\varphi\in\mathbb{B}^{I}}$
be the family of pl-simple end constructed in Theorem 3.7. Then there exists a
meager subset $Z\subset\mathbb{B}^{I}$ such that
$\forall\varphi\in\mathbb{B}^{I}\setminus Z$, $T[\Sigma_{\varphi}]$ is non-
fattening.

###### Proof.

Define

$Z_{j}:=\\{\varphi\in\mathbb{B}^{I}:T[\Sigma_{\varphi}]\cap\overline{\mathbb{B}_{j}^{n+1}}\text{
contains a ball of radius }1/j\\}.$

Then by upper-semi-continuity of $T[\Sigma_{\varphi}]$,
$Z_{j}\subset\mathbb{B}^{I}$ is a closed subset. Moreover, let $\varpi$ be the
restriction on $\mathbb{B}^{I}$ of the $\mathbb{R}$-action introduced in
Section 3.5. By (4.4), for every $\varphi\in\mathbb{B}^{I}$,
$\\{a\in\mathbb{R}:\varpi(a)[\varphi]\in Z_{j}\\}$ is a countable subset of
$\mathbb{R}$. Hence by Corollary 3.28, $Z_{j}$ has no interior, and
$Z:=\bigcup_{j\geq 1}Z_{j}$ is the desired meager subset. ∎

Since the complete translators constructed in Theorem 4.3 have finite entropy,
they can possibly arise as the type-II blow-up limit of mean curvature flows
of closed hypersurfaces. We propose the following conjecture.

######  Conjecture 4.6.

The complete translators constructed in Theorem 4.3 are the type-II blow-up
limit of mean curvature flows of closed hypersurfaces.

In [AAG95, AV97], there are explicit examples of mean curvature flow of closed
surfaces that have type-II singularities, and the type-II blow-up limit is the
bowl soliton. This implies that if the self-shrinker $S$ is the round sphere
$\mathbb{S}^{2}_{2}$, then the translators in Theorem 4.3 can arise as the
type-II blow-up limit of mean curvature flows of closed hypersurfaces.

##  5\. Topology Change and Fattening

Theorem 4.1 establishes the existence of a complete embedded translator that
is asymptotic to a given simple translating end. Meanwhile, Theorem 4.3
suggests a possible phenomenon that is called fattening, and the fattening
implies non-uniqueness of the complete embedded translators that are
exponentially asymptotic to the same simple end.

In this section, we show that fattening indeed can happen, and it is due to
the topology change of a family of mean curvature flows. Recall that the
Angenent torus $S_{A}\subset\mathbb{R}^{3}$ introduced in [Ang92] is a closed
rotational symmetric self-shrinker, and topologically it is a torus. From
Theorem 3.7 and Lemma 3.27, there exists $\Sigma_{e}\subset\mathbb{R}^{4}$
that is a rotational symmetric simple translating end over
$S_{A}\times\mathbb{R}_{+}$, and such that all its one-sided deformations
$T[\varpi(a)[\Sigma_{e}]]$ are rotationally symmetric. In the following, we
will fix such an end.

######  Theorem 5.1.

There exists $a\in\mathbb{R}$ such that $T[\varpi(a)[\Sigma_{e}]]$ fattens. In
other words, case (ii) in Theorem 4.3 happens when $S=S_{A}$.

To prove the theorem, we need the following lemma concerning certain rescaled
limits of one-sided deformations of translators. Let $S\subset\mathbb{R}^{n}$
be an arbitrary closed embedded smooth self-shrinker. $\mathbb{R}^{n}\setminus
S=E_{+}\sqcup E_{-}$, where $E_{+}$ is unbounded. For every constant
$0<\kappa\leq\kappa_{S}<<1$, the argument in [CCMS20] showed the existence of
a unique smooth ancient RMCF $\\{S^{\kappa}_{\pm}(\tau)\\}_{\tau\leq 0}$
asymptotic to $S$ near $-\infty$ such that $S^{\kappa}_{\pm}(\tau)\subset
E_{\pm}$, $\forall\tau\leq 0$, and the Hausdorff distance
$dist_{\mathbb{R}^{n}}(S,S^{\kappa}_{\pm}(0))=\kappa$.

######  Lemma 5.2.

Let $\mathrm{ES}[u_{0}]\subset\mathbb{R}^{n+1}$ be a pl-simple translating end
over $S\times\mathbb{R}_{+}$, $\kappa\in(0,\kappa_{S}]$. For $a\in\mathbb{R}$,
$u_{a}\in\varpi(a)[u_{0}]$ be the one-sided deformations introduced in Section
3.5. Suppose that $T[\mathrm{ES}[u_{a}]]$ is nonfattening for $|a|>>1$ (and
hence is an $\mathcal{I}$-minimizing translator by Theorem 4.3). Then there
exist functions $a_{\pm}(z)$ defined on $z\in\mathbb{R}_{+}$ such that when
$z\to+\infty$,

  1. (i)

$\pm a_{\pm}(z)\to+\infty$;

  2. (ii)

The ancient RMCF $\\{\mathcal{R}_{\tau-\log
z}(T[\mathrm{ES}[u_{a_{\pm}(z)}]])\\}_{\tau\leq 0}$
$C^{\infty}_{loc}$-converges to the one-sided flow
$\\{S^{\kappa}_{\pm}(\tau)\times\mathbb{R}\\}_{\tau\leq 0}$ above with
Hausdorff distance $\kappa$ to $S$ at $\tau=0$, where recall

$\mathcal{R}_{\tau}(T):=e^{\tau/2}\cdot(T-e^{-\tau}\partial_{z}).$

###### Proof.

WLOG we work in $E_{+}$. Define

$a_{+}(z):=\sup\left\\{a\in\mathbb{R}:T[\mathrm{ES}[u_{a}]]\cap\mathbb{R}^{n}\times\\{z\\}\subset\sqrt{z}\cdot\mathbb{B}_{\kappa}(S)\times\\{z\\}\right\\}.$

Note that by (4.4), for every $z>>1$, $a_{+}(z)<+\infty$, and
$T_{z}^{+}:=T[\mathrm{ES}[u_{a_{+}(z)}]]$ satisfies

(5.1) $\displaystyle\\{x\in\mathbb{R}^{n}:(\sqrt{z}x,z)\in
T_{z}^{+}\\}\subset\text{Clos}(\mathbb{B}_{\kappa}(S)),$
$\displaystyle\\{x\in\mathbb{R}^{n}:(\sqrt{z}x,z)\in
T_{z}^{+}\\}\cap\partial\mathbb{B}_{\kappa}(S)\neq\emptyset.$

Also by the uniform estimate in Lemma 3.27 and Theorem 4.1, we know that
$a_{+}(z)\to+\infty$ as $z\to\infty$.
Claim. There exists a function $\tilde{\rho}(s)\to+\infty$ as $s\to+\infty$
such that for every $z\in[s,\tilde{\rho}(s)\cdot s]$,

$\\{x\in\mathbb{R}^{n}:(\sqrt{z}x,z)\in T_{s}^{+}\\}\subset E_{+}.$

Note that once the Claim is proved, we immediately see that in
$\mathbb{R}^{n}\times\\{0\\}$ slice, the RMCF $\\{\mathcal{R}_{\tau-\log
z}(T_{z}^{+})\\}_{\tau\leq 0}$ sits inside $E_{+}$ in a larger and larger time
interval in $(-\infty,0]$, and hence by Lemma 2.3 converges to a splitting
one-sided RMCF of $S\times\mathbb{R}$ and has Hausdorff distant $\kappa$ to
$S$ at $\tau=0$ slice by (5.1), which finishes the proof of (ii) of the Lemma.

###### Proof of the Claim..

We fix $\gamma:=\mu^{-}/2$. By Lemma 3.24, there exists $\bar{z}(S,u_{0})>>1$
such that

$\sup_{R\geq\bar{z}}R^{-\gamma}\cdot\|u_{0}\|_{C^{2,1/2}_{\star},S,R}<1.$

For every $s\geq s_{0}(\bar{z},S)>>\bar{z}$, Lemma 3.27 applies by taking
$\delta(s):=(\log s)^{-1}$, $z_{0}(s):=s$ and $a(s):=\delta(s)s^{-\mu_{1}}$,
which produces $v^{(s)}:=u_{a(s)}$ solving $\mathscr{T}(v^{(s)})=0$ on
$S\times\mathbb{R}_{\geq s}$, with the following asymptotic estimate, for
every $R\geq s$,

(5.2)
$\displaystyle\|v^{(s)}-u_{0}-\delta(s)\cdot\left(\frac{z}{s}\right)^{\mu_{1}}\psi_{1}\|_{C^{2}_{\star},S,R}\leq
2\cdot\delta(s)^{3/2}\left(\frac{R}{s}\right)^{\gamma/2+\mu_{1}}.$

In particular, we have for every $R\geq s$,

(5.3)
$\displaystyle\|v^{(s)}\|_{C^{2}_{\star},S,R}\leq\|u_{0}\|_{C^{2}_{\star},S,R}+C(S)(\log
s)^{-1}\cdot\left(\frac{R}{s}\right)^{\mu_{1}}\leq\|u_{0}\|_{C^{2}_{\star},S,R}+C(S)(\log
R)^{-1};$

We assert that the RHS above is independent of $s$ and tends to $0$ as
$R\to+\infty$; And since $|u_{0}(\cdot,z)|\leq z^{\gamma}$, there exists some
$\rho(s)$ tending to $+\infty$ as $s\to+\infty$ such that for every
$s<z<\rho(s)s$,

(5.4) $\displaystyle
v^{(s)}(\cdot,z)\geq-\|u_{0}(\cdot,z)\|_{C^{0},S}+C^{\prime}(S)(\log
s)^{-1}\cdot\left(\frac{z}{s}\right)^{\mu_{1}}\geq C^{\prime\prime}(S)(\log
s)^{-1}\cdot\left(\frac{z}{s}\right)^{\mu_{1}}.$

By (5.3) and Theorem 4.1, when $s>>1$, there exists constant
$\Lambda(S,u_{0})>>1$ such that
$T[\mathrm{ES}[v^{(s)}]]\cap\mathbb{R}^{n}\times[\Lambda
s,+\infty)=\mathrm{ES}[u^{(s)}]\cap\mathbb{R}^{n}\times[\Lambda s,+\infty)$
and that

(5.5) $\displaystyle|(u^{(s)}-v^{(s)})(\cdot,z)|\leq C(S,u_{0})e^{-(z-\Lambda
s)/2}.$

Combine this with (5.4) and take $s>>1$ we have,

$u^{(s)}>0\ \ \ \ \text{ on }S\times[\Lambda s,\rho(s)s].$

In other words, for every $z\in[\Lambda s,\rho(s)s]$, we have,

$\\{x\in\mathbb{R}^{n}:(\sqrt{z}x,z)\in T[\mathrm{ES}[v^{(s)}]]\\}\subset
E_{+}.$

On the other hand notice that $T[\mathrm{ES}[v^{(s)}]]\in\varpi(a(s))[u_{0}]$,
and by (5.3) and (5.5), we see that

$|u^{(s)}(\cdot,z)|\leq C(S,u_{0})(\log z)^{-1}\ \ \ \ \text{ on
}S\times\mathbb{R}_{\geq\Lambda s},$

which implies that for $s>>1$,
$T[\mathrm{ES}[v^{(s)}]]\cap\mathbb{R}^{n}\times\\{\Lambda
s\\}\subset\sqrt{\Lambda s}\cdot\mathbb{B}_{\kappa/2}(S)\times\\{\Lambda
s\\}$, and hence by its definition, $a_{+}(\Lambda s)>a(s)$ and
$T^{+}_{\Lambda s}$ lies outside $T[\mathrm{ES}[v^{(s)}]]$. In particular for
every $z\in[\Lambda s,\rho(s)s]$,

$\\{x\in\mathbb{R}^{n}:(\sqrt{z}x,z)\in T^{+}_{\Lambda s}\\}\subset E_{+}.$

This finishes the proof of the Claim. ∎

∎

######  Lemma 5.3.

Let $n\geq 2$, $O\subset\mathbb{R}^{n}$ be a bounded smooth strictly mean
convex domain, in other words, the mean curvature vector $\vec{H}_{\partial
O}$ points inward. Then there exists a constant $\rho_{O}>>1$ such that for
every $\rho\geq\rho_{O}$, if $T\subset\mathbb{R}^{n+1}$ is a compact
$\mathcal{I}$-minimizing hypersurface with boundary $\partial
T\subset\rho\cdot O\times\\{0\\}$, then
$T\subset\rho\cdot\text{Clos}(O)\times\mathbb{R}_{\leq 0}$.

###### Proof.

Suppose $\Lambda>>1$ such that $O\subset\mathbb{B}^{n}_{\Lambda}$. For every
$\rho>1$, let $T_{\rho}\subset\mathbb{R}^{n+1}$ be an $\mathcal{I}$-minimizing
hypersurface with boundary $\partial T_{\rho}=\rho\cdot\partial
O\times\\{0\\}$. Since the translator-mean-curvature vector of every
hyperplane $\mathbb{R}^{n}\times\\{z\\}$ points downward and the translator-
mean-curvature vector of every paraboloid
$\\{(x,|x|^{2}-M):x\in\mathbb{R}^{n}\\}$ points inward, by strong maximum
principle [SW89], we know that $T\setminus\partial T,T_{\rho}\setminus\partial
T_{\rho}\subset\mathbb{B}_{\rho\Lambda}^{n}\times[-\rho^{2}\Lambda^{2},0)$;
Let $U_{\rho}\subset\mathbb{R}^{n+1}$ be the open region enclosed by
$T_{\rho}\cup(\rho\cdot O\times\\{0\\})$. Then by comparing
$\mathcal{I}$-functional, we have $T\subset\text{Clos}(U_{\rho})$.

Since $O$ is strictly mean convex, $\exists\ \eta_{0}\in(0,1)$ such that for
every $\eta\in(0,\eta_{0}]$, $\mathbb{B}_{\eta}(O)$ is also a smooth strictly
mean convex domain. Since the translator-mean-curvature vector of every
$\rho\cdot\partial\mathbb{B}_{\eta}(O)\times\mathbb{R}$ points inward, again
by strong maximum principle [SW89], we know that if

(5.6) $\displaystyle
U_{\rho}\subset\rho\cdot\mathbb{B}_{\eta_{0}}(O)\times\mathbb{R}_{\leq 0},$

then
$T\subset\text{Clos}(U_{\rho})\subset\rho\cdot\text{Clos}(O)\times\mathbb{R}_{\leq
0}$, which finish the proof of the Lemma.

To show that (5.6) holds for all sufficiently large $\rho$, suppose for
contradiction that there exist $\rho_{j}\nearrow+\infty$ and

(5.7)
$\displaystyle(x_{j},z_{j})\in(\rho_{j}^{-1}T_{\rho_{j}})\setminus(\mathbb{B}_{\eta_{0}}(O)\times\mathbb{R}_{\leq
0})\subset\left(\mathbb{B}_{\Lambda}^{n}\setminus\mathbb{B}_{\eta_{0}}(O)\right)\times[-\rho_{j}\Lambda^{2},0].$

By Ilmanen’s elliptic regularization [Ilm94],
$\\{T_{j}(t):=\rho_{j}^{-1}T_{\rho_{j}}+\rho_{j}t\cdot\partial_{z}\\}_{t\geq
0}$ are mean curvature flows which converges in the Brakke sense in
$\mathbb{R}^{n}\times\mathbb{R}_{\leq 0}$ to some Brakke motion
$\\{\mu_{t}\times\mathbb{R}\\}_{t\geq 0}$ starts at $O\times\mathbb{R}$. In
particular, $(x_{j},0)\in\text{spt}(T_{j}(-z_{j}/\rho_{j}))$ will subconverge
in $\mathbb{R}^{n}\times\mathbb{R}$ to some point in the support of some
$\mu_{t}\times\mathbb{R}$, which by (5.7) lies in
$(\mathbb{B}_{\Lambda}^{n}\setminus\mathbb{B}_{\eta_{0}}(O))\times\\{0\\}$.
This implies for some $t\geq 0$,

$\text{spt}(\mu_{t})\setminus\mathbb{B}_{\eta_{0}}(O)\neq\emptyset.$

On the other hand, since $O$ is mean convex, any Brakke flow starts from $O$
will move inward, which means $\text{spt}(\mu_{t})\subset\text{Clos}(O)$ for
every $t\geq 0$. This becomes a contradiction. ∎

Now we go back to Theorem 5.1. Let us recall some basic facts of the Angenent
torus $S_{A}$. The Angenent torus is rotational symmetric in $\mathbb{R}^{3}$,
which is homeomorphic to $\mathbb{S}^{1}\times\mathbb{S}^{1}$. In the
following, when we say something is rotational symmetry in $\mathbb{R}^{3}$,
we mean it has the same rotational axis as the Angenent torus. From now on, we
fix a $\kappa\in(0,\kappa_{S})$. Since the one-sided RMCF
$\\{S_{\pm}(\tau):=S^{\kappa}_{\pm}(\tau)\\}$ is unique and asymptotic to
$S_{A}$ as $\tau\to-\infty$, when $\tau$ is sufficiently negative,
$S_{\pm}(\tau)$ are rotational symmetric and isotopic to $S_{A}$.

Next, we derive further topological properties of $S_{\pm}(\tau)$. The
following two lemmas are essentially proved in [CCMS20, Corollary 8.8].

######  Lemma 5.4.

There exists $\tau_{-}>0$, and for any $\eta>0$ there exists
$\epsilon_{-}(\eta)>0$ with the following significance:

  1. (1)

For $\tau\in(-\infty,\tau_{-})$, $S_{-}(\tau)$ is a regular rotational
invariant torus, and $S_{-}(\tau_{-})$ is a rotational invariant
$\mathbb{S}^{1}$ curve.

  2. (2)

For $\tau\in[\tau_{-}-\epsilon_{-},\tau_{-})$, $S_{-}(\tau)$ is a strictly
mean convex surface sitting inside an $\eta$-neighbourhood of the curve
$S_{-}(\tau_{-})$.

  3. (3)

After $\tau_{-}$, $S_{-}(\tau)$ is empty.

###### Proof.

From [CIMW13, CCMS20], $\\{S_{-}(\tau)\\}_{\tau\in\mathbb{R}}$ is self-
shrinker-mean-convex and it moves inwards. [CIMW13] implies that $S_{-}$ has a
finite time singularity. Then we can consider the profile curves
$\gamma^{-}(\tau)$ of the flow $S_{-}(\tau)$ and $\gamma^{-}(\tau)$ is a
circle in the upper half plane $\mathbb{R}^{2}$ which moves inwards. By
[CIMW13, CCMS20], the tangent flow of a self-shrinker mean convex RMCF has to
be either a cylinder or a sphere, which implies that the profile curve must
shrink to a point in finite time $\tau_{-}$, and this is the only singular
time. Because by our scaling, $S_{-}(\tau)$ does not vanish at $\tau=0$, so
$\tau_{-}>0$. This shows items (1) and (3).

Item (2) follows from the fact that the tangent flows of $S_{-}(\tau)$ at the
singular time are cylinders. Because $S_{-}(\tau)$ is rotationally symmetric,
as $\tau\to\tau_{-}$, the profile curve becomes round, which implies that the
profile curve is convex when $\tau$ is sufficiently close to $\tau_{-}$. This
allows us to choose $\epsilon_{-}$ that satisfies item (2). ∎

######  Lemma 5.5.

There exists $\tau_{+}\in(0,+\infty)$ with the following significance: for
$\tau\in[\tau_{+},+\infty)$, $S_{+}(\tau)$ is a regular star shape sphere
enclosing the origin, i.e. $\langle X,\nu_{S_{+}(\tau)}\rangle>0$.

Here we say a surface $\Sigma$ enclosing a point if the point is in the
bounded connected component of $\mathbb{R}^{3}\backslash\Sigma$.

###### Proof.

In [CCMS20, Theorem 9.1 (10)], it was proved that the one-sided ancient MCF is
regular and star-shaped in the time interval $(-\delta,\delta)$ for some
$\delta>0$. Notice that star-shapedness is preserved by rescaling, and for the
time $t$ slice of a MCF as $t\nearrow 0$ corresponds to the time $\tau$ slice
of the corresponding RMCF as $\tau\nearrow+\infty$, hence rescaling to RMCF
gives the desired property. ∎

The next lemma shows how the topology of
$\\{S_{\pm}(\tau)\\}_{\tau\in\mathbb{R}}$ implies the topology of
$T[\mathrm{ES}[u_{a_{\pm}(z)}]]$. From now on we use $(x,z)=(x_{1},x_{2},y,z)$
to denote the coordinate of $\mathbb{R}^{3}\times\mathbb{R}=\mathbb{R}^{4}$,
where $y$ denotes the direction of the rotation axis of the Angenent torus.
Because the Angenent torus is a rotational symmetric torus, there must be a
constant $\overline{\kappa}>0$, such that
$\\{(x_{1},x_{2},y)|\sqrt{x_{1}^{2}+x_{2}^{2}}<\overline{\kappa}\\}$ is
disjoint from the bounded region $E_{-}$ enclosed by Angenent torus.

######  Lemma 5.6.

Suppose the conditions in Lemma 5.2. Then for $z$ sufficiently large,

  1. (1)

$T[\mathrm{ES}[u_{a_{-}(z)}]]$ does not intersect with the generalized
cylinder
$\mathcal{C}_{\overline{\kappa}}:=\\{(x_{1},x_{2},y,z)|y,z\in\mathbb{R},\sqrt{x_{1}^{2}+x_{2}^{2}}\leq\overline{\kappa}\\}$.

  2. (2)

$T[\mathrm{ES}[u_{a_{+}(z)}]]$ must intersect with the subspace
$\\{(0,0,y,z)|y,z\in\mathbb{R}\\}$.

###### Proof.

Lemma 5.2 shows that the RMCF $\\{\mathcal{R}_{\tau-\log
z}(T[\mathrm{ES}[u_{a_{\pm}(z)}]])\\}_{\tau\leq 0}$
$C_{loc}^{\infty}$-converges to the one-sided flow
$\\{S_{\pm}(\tau)\times\mathbb{R}\\}_{\tau\leq 0}$. Moreover, by the
uniqueness of this one-sided RMCF proved in [CCMS20, Theorem 9.2], the RMCF
$\\{\mathcal{R}_{\tau-\log
z}(T[\mathrm{ES}[u_{a_{\pm}(z)}]])\\}_{\tau\in\mathbb{R}}$ converges to the
one-sided flow $\\{S_{\pm}(\tau)\times\mathbb{R}\\}_{\tau\in\mathbb{R}}$ in
the Brakke sense, and by Brakke’s regularity theorem, $\mathcal{R}_{\tau-\log
z}(T[\mathrm{ES}[u_{a_{\pm}(z)}]])$ would $C_{\text{loc}}^{\infty}$-converges
to $S_{\pm}(\tau)\times\mathbb{R}$ whenever $S_{\pm}(\tau)$ is regular. In the
following proof we use $T^{\pm}_{z}$ to denote the translators
$T[\mathrm{ES}[u_{a_{\pm}(z)}]]$.

Let us first prove the item (1). From Lemma 5.4, fix an $\eta\in(0,1)$ such
that $\mathbb{B}^{3}_{\eta}(S_{-}(\tau_{-}))$ is strictly mean convex and
disjoint from $\\{(x_{1},x_{2},y):x_{1}^{2}+x_{2}^{2}\leq\bar{\kappa}^{2}\\}$;
Then we can choose $\epsilon_{-}=\epsilon_{-}(\eta)$ such that item (2) of
Lemma 5.4 is satisfied. $S_{-}(\tau)$ is regular for all $\tau<\tau_{-}$, and
$S_{-}(\tau)\subset E_{-}$ for all $\tau\in(-\infty,\tau_{-}]$. By the
compactness of RMCF, for any $\delta\in(0,1)$ and sufficiently large $z$
depending on $\delta$,

$\mathcal{R}_{\tau-\log z}(T^{-}_{z})\cap\mathbb{R}^{3}\times\\{0\\}\subset
E_{-}\times\\{0\\},$

for $\tau\in[-\delta^{-1},\tau_{-}-\delta]$. In particular, it does not
intersect with a cylinder
$\\{(x_{1},x_{2},y,0)|x_{1}^{2}+x_{2}^{2}\leq\overline{\kappa}^{2}\\}$ that is
disjoint from $E_{-}\times\\{0\\}$. Moreover, if we rescale the flow back to
get the translator $T^{-}_{z}$, we have
$T^{-}_{z}\cap\mathbb{R}^{3}\times[c_{1}(z),c_{2}(z)]$ does not intersect with
$\mathcal{C}_{\overline{\kappa}}$ provided $z>>1$, where
$c_{1}(z):=ze^{-(\tau_{-}-\delta)}$, $c_{2}(z):=ze^{1/\delta}$.

It suffices to show that
$T^{-}_{z}\cap\mathbb{R}^{3}\times\mathbb{R}_{<c_{1}(z)}$ and
$T^{-}_{z}\cap\mathbb{R}^{3}\times\mathbb{R}_{>c_{2}(z)}$ also does not
intersect $\mathcal{C}_{\overline{\kappa}}$. For the first set, notice that by
Lemma 5.4 item (2), when $\delta=\epsilon_{-}$ and $z>>1$, we have

$T^{-}_{z}\cap\mathbb{R}^{3}\times\\{c_{1}(z)\\}\subset\sqrt{c_{1}(z)}\mathbb{B}^{3}_{\eta}(S_{-}(\tau_{-}))\times\\{c_{1}(z)\\}.$

Because $T^{-}_{z}$ is an $\mathcal{I}$-minimizing translator, Lemma 5.3
implies that when $z>>1$,

$T^{-}_{z}\cap\mathbb{R}^{3}\times\mathbb{R}_{<c_{1}(z)}\subset\sqrt{c_{1}(z)}\mathbb{B}^{3}_{\eta}(S_{-}(\tau_{-}))\times\mathbb{R}.$

In particular by the choice of $\eta$,
$T^{-}_{z}\cap\mathbb{R}^{3}\times\mathbb{R}_{<c_{1}(z)}$ is disjoint from
$\mathcal{C}_{\overline{\kappa}}$.

For the second set, first let $T_{0}$ be an $\mathcal{I}$-minimizing
translator exponentially asymptotic to $\mathrm{ES}[u_{0}]$, and let
$\Omega_{0}\subset\mathbb{R}^{n+1}$ be the inner region bounded by $T_{0}$.
Then since $a_{-}(z)\to-\infty$ as $z\nearrow+\infty$, by Theorem 4.3, we have
$T^{-}_{z}\subset\Omega_{0}$ when $z$ is sufficiently large. And because
$T_{0}=\partial\Omega_{0}$ has a simple translating end over
$S_{A}\times\mathbb{R}_{+}$, we know that
$\Omega_{0}\cap\mathbb{R}^{3}\times\mathbb{R}_{\geq\tilde{z}}\cap\mathcal{C}_{\overline{\kappa}}=\emptyset$
for some sufficiently large $\tilde{z}>1$. Therefore,
$T^{-}_{z}\cap\mathbb{R}^{3}\times\mathbb{R}_{\geq c_{2}(z)}$ doesn’t
intersect $\mathcal{C}_{\overline{\kappa}}$ when $z>>1$.

Next, we prove the item (2). Because $\mathcal{R}_{\tau-\log z}(T_{z}^{+})$
$C_{loc}^{\infty}$-converges to the time slice of the one-sided flow
$S_{+}(\tau)\times\mathbb{R}$ whenever $S_{+}(\tau)\times\mathbb{R}$ is
smooth, $\mathcal{R}_{\tau_{+}-\log
z}(T_{z}^{+})\cap\mathbb{R}^{3}\times\\{0\\}$ smoothly converges to
$S_{+}(\tau_{+})\times\\{0\\}$ as $z\to+\infty$. By Lemma 5.5, this implies
that when $z$ is sufficiently large, $\mathcal{R}_{\tau_{+}-\log
z}(T_{z}^{+})\cap\\{(x,z^{\prime}):z^{\prime}=0\\}$ is a smooth star-shaped
surface enclosing the origin in $\\{(x,z^{\prime}):z^{\prime}=0\\}$, in
particular it does intersect with the line $\\{(0,0,y,0)|y\in\mathbb{R}\\}$.
This implies that a slice of $T_{z}^{+}$ intersects with the subspace
$\\{(0,0,y,z)|y,z\in\mathbb{R}\\}$. ∎

Now we are willing to prove Theorem 5.1.

###### Proof of Theorem 5.1.

We prove this by contradiction. Let $\mathrm{ES}[u_{0}]$ be a rotational
symmetric simple translating end. Suppose no fattening happens, then for any
$a\in\mathbb{R}$, $T[\varpi(a)[u_{0}]]$ is the support of a complete embedded
$\mathcal{I}$-minimizing translator, and we denote it by $T_{a}$. Then by the
compactness of minimizing surface in any compact region, as well as the
convergence of the ends, as $a_{i}\to a$, $T_{a_{i}}\to T_{a}$ smoothly.

Because by Theorem 3.7, 4.3 and Lemma 3.27, all $T_{a}$ are rotation
symmetric, we can consider the rotation profile surface $\Theta_{a}$ of
$T_{a}$, where $\Theta_{a}$ is a surface in the upper half space
$\\{(w,y,z)|w\geq 0\\}\subset\mathbb{R}^{3}$. The item (1) of Lemma 5.6 shows
that when $A>0$ is sufficiently large, $\Theta_{A}$ is disjoint from
$\\{(w,y,z)|w\leq\kappa\\}$. Therefore, any rotational symmetric closed curve
generated by a given point on $\Theta_{A}$ is non-contractible in $T_{A}$,
since it’s the generator of the fundamental group
$\pi_{1}(\mathbb{R}^{4}\setminus\mathcal{C}_{\overline{\kappa}})=\mathbb{Z}$.

On the other hand, when $A>0$ is sufficiently large, the item (2) of Lemma 5.6
shows that $T_{-A}$ intersects with $\\{(w,y,z)|w=0\\}$. In particular, for
any rotational symmetric closed curve $\sigma_{p}$ generated by a given point
$p\in\Theta_{-A}$, the path on $\Theta_{-A}$ connecting $p$ to a point on
$\\{(w,y,z)|w=0\\}$ generates a disc on $T_{-A}$ bounded by $\sigma_{p}$,
which means $\sigma_{p}$ is contractible in $T_{-A}$.

From the item (2) of Theorem 3.7 and Theorem 4.1, for each $a\in[-A,A]$, there
exists $z_{a}>0$ and $\delta_{a}>0$ such that for
$a^{\prime}\in(a-\delta,a+\delta)\cap[-A,A]$,
$T_{a^{\prime}}\cap\mathbb{R}^{3}\times\mathbb{R}_{>z_{a}}$ is a graph over
$T_{a}\cap\mathbb{R}^{3}\times\mathbb{R}_{>z_{a}}$, and they are all
diffeomorphic to $S\times[0,+\infty)$, and
$T_{a}\cap\mathbb{R}^{3}\times\\{z_{a}\\}$ varies smoothly as $a$ changes.
Then by the compactness of $[-A,A]$, there exists a uniform $\bar{z}>0$, such
that for every $a\in[-A,A]$,
$T_{a}\cap\mathbb{R}^{3}\times\mathbb{R}_{\geq\bar{z}}$ is diffeomorphic to
$S\times[0,+\infty)$, and when $a$ varies,
$T_{a}\cap\mathbb{R}^{3}\times\\{\bar{z}\\}$ varies smoothly as hypersurfaces
in $\mathbb{R}^{3}\times\\{\bar{z}\\}$. Therefore,
$\\{T_{a}\cap\mathbb{R}^{3}\times\mathbb{R}_{\leq\bar{z}}\\}_{a\in[-A,A]}$ is
an isotopy in $\mathbb{R}^{3}\times\mathbb{R}_{\leq\bar{z}}$ relative to the
boundary in $\mathbb{R}^{3}\times\\{\bar{z}\\}$, and
$\\{\partial[T_{a}\cap\mathbb{R}^{3}\times\mathbb{R}_{\leq\bar{z}}]\\}_{a\in[-A,A]}$
gives an isotopy of smooth tori in $\mathbb{R}^{3}\times\\{\bar{z}\\}$. It is
natural to identify a rotational symmetric closed curve on the boundary via
this boundary isotopy. However, such a closed curve is contractible in
$T_{-A}\cap\mathbb{R}^{3}\times\mathbb{R}_{\leq\bar{z}}$ , but non-
contractible in $T_{A}\cap\mathbb{R}^{3}\times\mathbb{R}_{\leq\bar{z}}$, which
yields a contradiction. ∎

We conclude this section by proposing the following conjecture.

######  Conjecture 5.7.

If $T[\Sigma_{0}]$ fattens, then either there exists an unstable complete
translator that is in the interior of $T[\Sigma_{0}]$, or $T[\Sigma_{0}]$
admits a foliation by complete $\mathcal{I}$-stable translators.

If $T[\Sigma_{0}]$ fattens, then the two connected components of
$\partial[T[\Sigma_{0}]]$ are $\mathcal{I}$-minimizing complete translators.
In the minimal surface theory, once we have two area-minimizing hypersurfaces,
it is natural to use the min-max method to produce the third one, and the
third one is usually unstable. Such an idea was adapted by Bernstein-Wang
[BW20] to construct unstable self-expanders between two strictly stable self-
expanders. Self-expanders is a class of mean curvature flow solitons, and we
conjecture that a similar result holds for translators.

##  Appendix A Geometry of Auxiliary Ends

Let $S\subset(\mathbb{R}^{n},g)$ be a closed self-shrinker,
$\\{x^{1},\dots,x^{n-1},y\\}$ be local Fermi coordinates at $p\in S$ (hence
$S=\\{y=0\\}$); Let $g_{ij}$ be the induced Riemannian metrics on $S$,
$[g^{ij}]:=[g_{ij}]^{-1}$ be the inverse matrix, $\nu:=\partial_{y}$ be the
unit normal field, $A_{ij}=A_{S}(\partial_{i},\partial_{j})$ be the second
fundamental form of $S\subset\mathbb{R}^{n}$.

Let $u\in C^{2}(S\times\mathbb{R}_{+})$ be such that for every $z>0$,

$\|u(\cdot,z)\|_{C^{1},S}\leq\delta_{S},$

where $\delta_{S}\in(0,1)$ be a small geometric constant. Consider the
parametrization

$\Phi_{u}:S\times\mathbb{R}_{+}\to\mathbb{R}^{n+1},\ \ \
(x,z)\mapsto(\sqrt{z}(x+u(x,z)\nu_{x}),z).$

We denote $\mathrm{ES}_{u}:=\Phi_{u}(S\times\mathbb{R}_{+})$ to be the image.
Note that $\mathrm{ES}=\mathrm{ES}_{0}$. Let
$\bar{g}^{u}:=\Phi_{u}^{*}g_{\text{Euc}}$. Then under coordinates
$\\{x^{1},,\dots,x^{n-1},z\\}$,

(A.1)
$\displaystyle\begin{split}\bar{g}^{u}_{ij}&=z(g_{ij}-2uA_{ij}+u^{2}A_{ik}A^{k}_{j}+u_{i}u_{j})=:zg^{u}_{ij},\\\
\bar{g}_{iz}&=\frac{1}{2}(X_{i}-uA_{i}^{k}X_{k}+u_{i}(X^{\perp}+u+2zu_{z}))=:\frac{X_{i}-\beta^{u}_{i}}{2},\\\
\bar{g}_{zz}&=1+\frac{1}{4z}|x+(u+2zu_{z})\nu|^{2}.\end{split}$

where $X^{\perp}:=\langle x,\nu\rangle$, $X_{i}:=\langle
x,\partial_{i}\rangle$; Also let $[g_{u}^{ij}]:=[g^{u}_{ij}]^{-1}$. The volume
density is given by

(A.2)
$\displaystyle\bar{G}^{u}:=\det[\bar{g}^{u}]=z^{n-1}\det[g^{u}]\cdot\left(1+\frac{1}{4z}\left(|x+(u+2zu_{z})\nu|^{2}-g_{u}^{ij}(X_{i}-\beta_{i}^{u})(X_{j}-\beta_{j}^{u})\right)\right).$

Define functions $F\in
C^{\infty}(S_{x}\times\mathbb{R}_{p}\times\mathbb{R}^{n-1}_{\xi})$, $E\in
C^{\infty}(S_{x}\times\mathbb{R}_{z}\times\mathbb{R}_{p}\times\mathbb{R}^{n-1}_{\xi}\times\mathbb{R}_{\eta})$
by

(A.3)
$\displaystyle\begin{split}F(x,p,\xi)&:=\sqrt{\det[g_{ij}^{p,\xi}]\det[g_{ij}]^{-1}};\\\
E(x,z,p,\xi,\eta)&:=\left(1+\frac{1}{4z}\left(|x+(p+2\eta)\nu|^{2}-g_{p,\xi}^{ij}(X_{i}-\beta_{i}^{p,\xi,\eta})(X_{j}-\beta_{j}^{p,\xi,\eta})\right)\right)^{1/2};\\\
\tilde{F}(x,z,p,\xi,\eta)&:=F(x,p,\xi)\cdot E(x,z,p,\xi,\eta).\end{split}$

where

$\displaystyle g_{ij}^{p,\xi}$
$\displaystyle:=g_{ij}-2pA_{ij}+p^{2}A_{ik}A^{k}_{j}+\xi_{i}\xi_{j};$
$\displaystyle[g^{ij}_{p,\xi}]:=[g_{ij}^{p,\xi}]^{-1};$
$\displaystyle\beta_{i}^{p,\xi,\eta}$
$\displaystyle:=pA_{i}^{k}X_{k}-\xi_{i}(X^{\perp}+p+2\eta).$

Note that under this,

$\bar{G}^{u}=z^{n-1}\tilde{F}(x,z,u,\nabla u,zu_{z})^{2}\cdot\det[g_{ij}].$

Here $\nabla$ is derivative in $x$. Moreover, for every $k\geq 0$,
$\nabla^{k}_{x}F$ and $\nabla_{x}^{k}E$ are power series in $p,\xi,1/z,\eta$,
whose convergence radius depends only on $k,S$, provided $S$ is compact.

The Ilmanen’s area functional for $\mathrm{ES}_{u}$ is,

$\displaystyle\mathcal{I}[u]$
$\displaystyle:=\mathcal{I}[\mathrm{ES}_{u}]=\int_{S\times\mathbb{R}_{+}}e^{z}\sqrt{\det[\bar{G}^{u}]}$
$\displaystyle=\int_{S\times\mathbb{R}_{+}}e^{z}z^{(n-1)/2}\tilde{F}(x,z,u,\nabla
u,zu_{z})\ d\mathrm{vol}_{S\times\mathbb{R}_{+}}.$

Hence we define $\mathscr{T}$ to be the E-L operator of $\mathcal{I}$, i.e.
given by

(A.4) $\displaystyle\begin{split}\mathscr{T}(u)=&\
-\text{div}_{S}\left(\tilde{F}_{\xi}(x,z,u,\nabla
u,zu_{z})\right)-z\partial_{z}\left(\tilde{F}_{\eta}(x,z,u,\nabla
u,zu_{z})\right)\\\ &\ -(z+\frac{n+1}{2})\tilde{F}_{\eta}(x,z,u,\nabla
u,zu_{z})+\tilde{F}_{p}(x,z,u,\nabla u,zu_{z})\end{split}$

######  Lemma A.1.

Let $S$ be a closed self-shrinker. Then for every $j,k\geq 0$,

$z^{k+1}|\nabla^{j}\partial_{z}^{k}\mathscr{T}(0)|\leq C(S,j,k).$

###### Proof.

Notice that $H_{S}=-X^{\perp}/2$ and,

$\displaystyle\tilde{F}(x,z,0,\mathbf{0},0)$
$\displaystyle=\sqrt{1+\frac{|X^{\perp}|^{2}}{4z}}=:E_{0};$ $\displaystyle\
\tilde{F}_{p}(x,z,0,\mathbf{0},0)$
$\displaystyle=-H_{S}E_{0}+\frac{X^{\perp}}{4zE_{0}},$
$\displaystyle\tilde{F}_{\xi}(x,z,0,\mathbf{0},0)$
$\displaystyle=\frac{-1}{4zE_{0}}g^{ij}X_{i}X^{\perp},$ $\displaystyle\
\tilde{F}_{\eta}(x,z,0,\mathbf{0},0)$
$\displaystyle=\frac{X^{\perp}}{2zE_{0}}.$

Therefore, there exists a function $h(x,\hat{z})$ $C^{\infty}$ in $x\in S$ and
real-analytic in $\hat{z}$ near $0$ such that
$\mathscr{T}(0)(x,z)=z^{-1}h(x,z^{-1})$, which directly implies the Lemma. ∎

We also need the following error estimates for $\mathscr{T}$. Recall that we
are working with the following norms (see (3.20) and (3.22)), for
$\Omega\subset S\times\mathbb{R}_{>1}$ and $R>1$,

$\displaystyle[f]_{\alpha;\Omega}^{\star}$
$\displaystyle:=\sup\left\\{\frac{|f(x,z)-f(x^{\prime},z^{\prime})|}{|x-x^{\prime}|^{\alpha}+R^{-\alpha/2}|z-z^{\prime}|^{\alpha}}:(x,z),(x^{\prime},z^{\prime})\in
S\times[R,2R]\right\\};$ $\displaystyle\|f\|_{C^{\alpha}_{\star};S,R}$
$\displaystyle=\|f\|_{C^{\alpha}_{\star};S\times[R,2R]}:=\sup_{S\times[R,2R]}|f|+[f]_{\alpha;S\times[R,2R]}^{\star};$
$\displaystyle\|u\|_{C^{2,\alpha}_{\star},S,R}$
$\displaystyle:=\Big{(}\|u\|_{C^{\alpha}_{\star};S,R}+\|\nabla
u\|_{C^{\alpha}_{\star};S,R}+R\|\partial_{z}u\|_{C^{\alpha}_{\star};S,R}$
$\displaystyle\ \ \
+\|\nabla^{2}u\|_{C^{\alpha}_{\star};S,R}+\sqrt{R}\|\partial_{z}\nabla
u\|_{C^{\alpha}_{\star};S,R}+R\|\partial_{z}^{2}u\|_{C^{\alpha}_{\star};S,R}\Big{)}.$
$\displaystyle\|u\|_{C^{d}_{\sharp},S,R}$ $\displaystyle:=\sum_{0\leq k+l\leq
d}\sup_{S\times[R,R+1]}R^{-l/2}|\partial_{z}^{k}\nabla^{l}_{S}u|;$
$\displaystyle\|u\|_{C^{d,\alpha}_{\sharp},S,R}$
$\displaystyle:=\|u\|_{C^{d}_{\sharp},S,R}\ +$ $\displaystyle\
\sup_{\begin{subarray}{c}(x,z)\neq(x^{\prime},z^{\prime})\in S\times[R,R+1]\\\
|x-x^{\prime}|\leq R^{-1/2}\end{subarray}}\sum_{0\leq k+l\leq
d}\frac{R^{-l/2}|\partial_{z}^{k}\nabla^{l}_{S}u(x,z)-\partial_{z}^{k}\nabla^{l}_{S}u(x^{\prime},z^{\prime})|}{R^{\alpha/2}|x-x^{\prime}|^{\alpha}+|z-z^{\prime}|^{\alpha}}.$

######  Lemma A.2.

Let $S$ be a closed self-shrinker. Then there exists $\delta_{S}\in(0,1)$ and
$C_{S}>0$ such that if $u_{\pm}\in C^{2}(S\times\mathbb{R}_{>1})$ satisfies

$\sup_{S\times\mathbb{R}_{>1}}(|u_{\pm}|+|\nabla
u_{\pm}|+|z\partial_{z}u_{\pm}|+|z\partial_{z}^{2}u_{\pm}|+|\partial_{z}\nabla
u_{\pm}|+|\nabla^{2}u_{\pm}|)\leq\delta_{S}.$

Then $v:=u_{+}-u_{-}$ satisfies,

$\displaystyle\mathscr{T}(u_{+})-\mathscr{T}(u_{-})=$ $\displaystyle\
-\Big{(}z\partial^{2}_{z}v+z\partial_{z}v+\underbrace{(\Delta_{S}-\frac{x}{2}\cdot\nabla_{S}+|A_{S}|^{2}+\frac{1}{2})v}_{L_{S}v}\Big{)}$
$\displaystyle\ +\bar{\mathcal{E}_{1}}\cdot
z\partial_{z}^{2}v+(\bar{\mathcal{E}_{2}}+x^{S})\cdot\partial_{z}\nabla
v+\bar{\mathcal{E}_{3}}\cdot\nabla^{2}v+\bar{\mathcal{E}_{4}}\cdot
z\partial_{z}v+\bar{\mathcal{E}_{5}}\cdot\nabla v+\bar{\mathcal{E}_{6}}v;$

While for $1\leq l\leq 6$ we have pointwise estimates,

$|\bar{\mathcal{E}_{l}}|\leq C_{S}(z^{-1}+|u_{\pm}|+|\nabla
u_{\pm}|+|z\partial_{z}u_{\pm}|+|z\partial_{z}^{2}u_{\pm}|+|\partial_{z}\nabla
u_{\pm}|+|\nabla^{2}u_{\pm}|);$

and Hölder estimates,

$\displaystyle\|\bar{\mathcal{E}_{l}}\|_{C^{\alpha}_{\star},S,R}$
$\displaystyle\leq
C(S,\alpha)\left(R^{-1}+\|u_{+}\|_{C^{2,\alpha}_{\star},S,R}+\|u_{-}\|_{C^{2,\alpha}_{\star},S,R}\right),$
$\displaystyle\|\bar{\mathcal{E}_{l}}\|_{C^{\alpha}_{\sharp},S,R}$
$\displaystyle\leq
C(S,\alpha)\left(R^{-1}+R\|u_{+}\|_{C^{2,\alpha}_{\sharp},S,R}+R\|u_{-}\|_{C^{2,\alpha}_{\sharp},S,R}\right).$

In particular, if we write
$\mathscr{T}(u)=:\mathscr{T}(0)-z(\partial_{z}^{2}+\partial_{z}+z^{-1}L_{S})u+z\mathscr{R}(u)$,
then the error term $\mathscr{R}$ satisfies,

$\displaystyle\|\mathscr{R}(u_{+})-\mathscr{R}(u_{-})\|_{C^{\alpha}_{\star},S,R}$
$\displaystyle\leq
C(S,\alpha)R^{-1}\left(R^{-1}+\|u_{\pm}\|_{C^{2,\alpha}_{\star},S,R}\right)\|u_{+}-u_{-}\|_{C^{2,\alpha}_{\star},S,R};$
$\displaystyle\|\mathscr{R}(u_{+})-\mathscr{R}(u_{-})\|_{C^{\alpha}_{\sharp},S,R}$
$\displaystyle\leq
C(S,\alpha)\left(R^{-1}+\|u_{\pm}\|_{C^{2,\alpha}_{\star},S,R}\right)\|u_{+}-u_{-}\|_{C^{2,\alpha}_{\sharp},S,R}.$

###### Proof.

By analyticity of $\tilde{F}$ with respect to $1/z,p,\xi,\eta$, it suffices to
compute the corresponding derivatives of $\tilde{F}$ at
$(x,\infty,0,\mathbf{0},0)$.

$\displaystyle\tilde{F}_{\xi}(x,\infty,0,\mathbf{0},0)$ $\displaystyle=0;$
$\displaystyle(z\tilde{F}_{\eta})(x,\infty,0,\mathbf{0},0)$
$\displaystyle=\frac{X^{\perp}}{2};$
$\displaystyle\tilde{F}_{p}(x,\infty,0,\mathbf{0},0)$ $\displaystyle=-H_{S};$
$\displaystyle\tilde{F}_{\xi\xi}(x,\infty,0,\mathbf{0},0)$
$\displaystyle=g^{ij};$
$\displaystyle(z\tilde{F}_{\xi\eta})(x,\infty,0,\mathbf{0},0)$
$\displaystyle=-\frac{g^{ij}X_{i}}{2};$ $\displaystyle\tilde{F}_{\xi
p}(x,\infty,0,\mathbf{0},0)$ $\displaystyle=0;$
$\displaystyle(z\tilde{F}_{\eta\eta})(x,\infty,0,\mathbf{0},0)$
$\displaystyle=1;$ $\displaystyle(z\tilde{F}_{\eta
p})(x,\infty,0,\mathbf{0},0)$ $\displaystyle=H_{S}^{2}+\frac{1}{2};$
$\displaystyle\tilde{F}_{pp}(x,\infty,0,\mathbf{0},0)$
$\displaystyle=H_{S}^{2}-|A_{S}|^{2}.$

∎

We may also need the following improved error estimate for the divergence form
the equation.

######  Lemma A.3.

Let $S$ be a closed self-shrinker. Then there exists $\delta_{S}\in(0,1)$ and
$C_{S}>0$ such that if $u^{\pm}\in C^{2}(S\times\mathbb{R}_{>1})$ satisfies

$\sup_{S\times\mathbb{R}_{>z_{0}}}(|u^{\pm}|+|\nabla_{S}u^{\pm}|+|zu^{\pm}_{z}|+|\nabla^{2}_{S}u^{\pm}|+|\sqrt{z}\partial_{z}\nabla_{S}u^{\pm}|+|z\partial_{z}^{2}u^{\pm}|)\leq\delta<\delta_{S}.$

for some $z_{0}>1$. Then let $v:=u^{+}-u^{-}$, we have,

$\displaystyle\mathscr{T}(u^{+})-\mathscr{T}(u^{-})$
$\displaystyle=-\partial_{z}\left((1+b)z\partial_{z}v-\frac{X}{2}\cdot\nabla_{S}v\right)-(1+b)z\partial_{z}v-(\Delta_{S}-\frac{X}{2}\cdot\nabla_{S}+|A_{S}|^{2}+\frac{1}{2})v$
$\displaystyle\ -\text{div}_{S}E_{1}-\partial_{z}E_{2}-E_{3},$

where we have pointwise estimates,

$\displaystyle|E_{j}(v)|$ $\displaystyle\leq
C_{S}\cdot\big{(}(z^{-1}+\delta)\cdot(|\nabla v|+|v|)+|\partial_{z}v|\big{)},$
$\displaystyle|b|+|\nabla_{S}b|+|\sqrt{z}\partial_{z}b|$ $\displaystyle\leq
C_{S}\cdot(z^{-1}+\delta).$

###### Proof.

By (A.4), it suffices to estimate

$\tilde{F}_{\sigma}(x,z,u^{+},\nabla
u^{+},zu_{z}^{+})-\tilde{F}_{\sigma}(x,z,u^{-},\nabla u^{-},zu_{z}^{-}).$

for variable $\sigma\in\\{\xi,\eta,p\\}$. By the proof of Lemma A.2,

$\displaystyle\ \tilde{F}_{\xi}(x,z,u^{+},\nabla
u^{+},zu_{z}^{+})-\tilde{F}_{\xi}(x,z,u^{-},\nabla u^{-},zu_{z}^{-})$
$\displaystyle=$ $\displaystyle\
(g^{ij}+\bar{o})\cdot(\partial_{i}u^{+}-\partial_{i}u^{-})\partial_{j}+\bar{o}\cdot(u^{+}-u^{-})+O(z^{-1})\cdot(zu_{z}^{+}-zu_{z}^{-1})$
$\displaystyle=:$ $\displaystyle\ \nabla_{S}v+E_{1}(v).$

where $\bar{o}$ are error terms in Taylor expansion, with estimate

$\bar{o}=O(z^{-1}+|u^{\pm}|+|\nabla_{S}u^{\pm}|+|zu^{\pm}_{z}|+|\nabla^{2}_{S}u^{\pm}|+|\sqrt{z}\partial_{z}\nabla_{S}u^{\pm}|+|z\partial_{z}^{2}u^{\pm}|)\leq
C_{S}(z^{-1}+\delta).$

Hence it gives the desired estimate on $E_{1}$. Similarly,

$\displaystyle\ \tilde{F}_{p}(x,z,u^{+},\nabla
u^{+},zu_{z}^{+})-\tilde{F}_{p}(x,z,u^{-},\nabla u^{-},zu_{z}^{-})$
$\displaystyle=$ $\displaystyle\
\bar{o}\cdot(\nabla_{S}u^{+}-\nabla_{S}u^{-})+(H_{S}^{2}-|A_{S}|^{2}+\bar{o})\cdot(u^{+}-u^{-})+O(z^{-1})\cdot(zu_{z}^{+}-zu_{z}^{-})$
$\displaystyle=:$ $\displaystyle\ (H_{S}^{2}-|A_{S}|^{2})v+E_{3,1}(v).$

where $E_{3,1}$ satisfies the desired estimate. On the other hand,

$\displaystyle\ z\left(\tilde{F}_{\eta}(x,z,u^{+},\nabla
u^{+},zu_{z}^{+})-\tilde{F}_{\eta}(x,z,u^{-},\nabla u^{-},zu_{z}^{-})\right)$
$\displaystyle=$ $\displaystyle\
(-\frac{X}{2}+\bar{o})\cdot(\nabla_{S}u^{+}-\nabla_{S}u^{-})+(H_{S}^{2}+\frac{1}{2}+\bar{o})\cdot(u^{+}-u^{-})$
$\displaystyle+$ $\displaystyle\
\left(\int_{0}^{1}z\tilde{F}_{\eta\eta}(x,z,u^{s},\nabla
u^{s},z\partial_{z}u^{s})\ ds\right)\cdot(zu_{z}^{+}-zu_{z}^{-}).$

where $u^{s}:=su^{+}+(1-s)u^{-}$. And since by its expression (A.3), we have

$\displaystyle\tilde{F}_{\eta\eta}(x,z,p,\xi,\eta)=\frac{F}{2E}\cdot\left((E^{2})_{\eta\eta}-2E_{\eta}^{2}\right)=\frac{F}{zE}\cdot(1-g_{p,\xi}^{ij}\xi_{i}\xi_{j})+O(z^{-2}).$

Therefore we have,

$\displaystyle\ z\left(\tilde{F}_{\eta}(x,z,u^{+},\nabla
u^{+},zu_{z}^{+})-\tilde{F}_{\eta}(x,z,u^{-},\nabla u^{-},zu_{z}^{-})\right)$
$\displaystyle=$ $\displaystyle\
-\frac{X}{2}\cdot\nabla_{S}v+(H_{S}^{2}+\frac{1}{2})v+\underbrace{\left(\int_{0}^{1}\frac{F}{E}\cdot(1-g_{p,\xi}^{ij}\xi_{i}\xi_{j})|_{(x,z,u^{s},\nabla
u^{s},z\partial_{z}u^{s})}\ ds\right)}_{=:\ 1+b}\cdot zv_{z}+E_{2}(v)$

where $b$ and $E_{2}$ satisfies the desired estimate. Combining (A.4) proves
the Lemma. ∎

##  Appendix B Rescaling of Graphs

Let $S\subset\mathbb{R}^{n}$ be a $C^{3}$ closed hypersurface with normal
field $\nu_{S}$. The goal for this Section is to prove the following.

######  Lemma B.1.

There exists a geometric constant $\vartheta_{S}\in(0,1)$ such that for every
$u\in C^{1}(S\times[0,1])$ with $\|u\|_{C^{1},S\times[0,1]}\leq\vartheta_{S}$,
there exists a unique $\check{u}\in
C^{1}(S\times[0,1]\times(1-\vartheta_{S},1+\vartheta_{S}))$ such that for
every $z\in[0,1]$ and $|a-1|<\vartheta$,

$\mathrm{graph}_{S}(\check{u}(\cdot,z,a))=a\cdot\mathrm{graph}_{S}(u(\cdot,z)).$

Moreover, for every such $a$, $\check{u}$ satisfies the estimate,

(B.1) $\displaystyle\|\check{u}(\cdot,\cdot,a)\|_{C^{1},S\times[0,1]}$
$\displaystyle\leq C(S)(\|u\|_{C^{1},S\times[0,1]}+|a-1|),$ (B.2)
$\displaystyle\|\partial_{a}\check{u}(\cdot,\cdot,a)\|_{C^{0},S\times[0,1]}$
$\displaystyle\leq C(S).$

###### Proof.

Let $\delta=\delta_{S}\in(0,1)$ such that
$\mathbf{G}:S\times(-\delta_{S},\delta_{S})\to\mathbb{R}^{n}$, $(x,y)\mapsto
x+y\nu_{S}(x)$ is a $C^{2}$ diffeomorphism onto its image;
$\Pi:\mathbb{B}_{\delta}(S)\to S$ be the $C^{2}$ nearest point projection onto
$S$. By implicit function theorem, there exists $0<\vartheta(S)<<1$ such that

  * •

For every $|a-1|\leq\vartheta$,
$a\cdot\mathbb{B}_{\vartheta_{S}}(S)\subset\mathbb{B}_{\delta/2}(S)$;

  * •

For every $|a-1|<\vartheta$ and every $u\in C^{1}(S\times[0,1])$ with
$\|u\|_{C^{1},S\times[0,1]}$, the following

$\displaystyle\mathbf{P}_{a,u}:S\times[0,1]\to S\times[0,1],\ \ \
(x,z)\mapsto(\Pi(a\cdot\mathbf{G}(x,z)),z),$

is a $C^{1}$ diffeomorphism and is $C^{2}$ in $a$.

Moreover, by choosing $\vartheta<<1$ we have,

$\displaystyle\|\mathbf{P}_{a,u}-id_{S\times[0,1]}\|_{C^{1},S\times[0,1]}+\|\mathbf{P}_{a,u}^{-1}-id_{S\times[0,1]}\|_{C^{1},S\times[0,1]}$
$\displaystyle\leq C(S)|a-1|;$
$\displaystyle\|\partial_{a}\mathbf{P}_{a,u}\|_{C^{0},S\times[0,1]}+\|\partial_{a}\mathbf{P}_{a,u}^{-1}\|_{C^{0},S\times[0,1]}$
$\displaystyle\leq C(S).$

Let $\mathbf{Q}_{a,u}\in C^{1}(S\times[0,1],S)$ be the $S$-component of
$\mathbf{P}^{-1}_{a,u}$. Under this notation, $\check{u}\in C^{1}$ is uniquely
defined by the explicit expression,

$\check{u}(x,z,a):=\rho_{S}\left(a\cdot\mathbf{G}\left(\mathbf{Q}_{a,u}(x,z),u\circ\mathbf{P}_{a,u}^{-1}(x,z)\right)\right),$

where $\rho_{S}:=dist_{\mathbb{R}^{n}}(S,\cdot)$. Since every map is $C^{1}$,
this immediately proves (B.2); To see (B.1), notice that,

$\displaystyle|\check{u}(\cdot,\cdot,a)|$
$\displaystyle\leq|\check{u}(\cdot,\cdot,1)|+C(S)|a-1|\leq
C(S)(\|u\|_{C^{0},S\times[0,1]}+|a-1|);$
$\displaystyle\|D\check{u}(\cdot,\cdot,a)\|$
$\displaystyle=\left\|D\rho_{S}\big{|}_{a\mathbf{G}}\cdot
a\left(\partial_{x}\mathbf{G}\cdot
D\mathbf{Q}_{a,u}+\partial_{y}\mathbf{G}\cdot Du\cdot
D\mathbf{P}_{a,u}^{-1}\right)\right\|$
$\displaystyle=a\left\|\left(D\rho_{S}\big{|}_{a\mathbf{G}}-D\rho_{S}\big{|}_{\mathbf{G}}\right)\cdot\partial_{x}\mathbf{G}\cdot
D\mathbf{Q}_{a,u}+D\rho_{S}\big{|}_{a\mathbf{G}}\cdot\partial_{y}\mathbf{G}\cdot
Du\cdot D\mathbf{P}_{a,u}^{-1}\right\|$ $\displaystyle\leq
C(S)(|a-1|+\|u\|_{C^{1},S\times[0,1]}).$

∎

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