Similarly, if $\bx \in B_{\gamma_{0}}(\Gamma) \cap Q_{\tau}$ and $\by \in \Gamma$ is a closest point, then

(2a) $\by \not \in \partial \Gamma$, and

(2b) $(\by - \bx) \cdot \be < 0$.

Consider $\eta\to 0$ and $T_1 \to -\infty$ along some sequence (not labeled). The corresponding parameter $R\to \infty$. Hence $\partial \Xi_{\tau} -  R\be$ converges in $C^{\infty}_\textrm{loc}$ to $\hat\Sigma_{h_0} \times \{0\}$ uniformly with respect to $\tau \in (-\infty,T_1]$ (thanks to  (<ref>)). On the other hand, recall that $\Gamma^\lambda_\delta - \lambda t \be$ converges weakly to the flow defined by $\partial^*\hat \Omega_\delta(t)\times \RR$ and $[0,3] \to \hat\Sigma_\delta(t)$ is smooth we find that

\[
\Gamma - R\be = \Gamma^{R/2}_\delta - \tfrac R 2 2\be \to \partial^*\hat \Omega_\delta(2) \times \RR
\]

again in $C^{\infty}_{\textrm{loc}}$ (and $\Gamma$ is independent of $\tau$) so

\[
Q_{\tau} -  R\be \to ((\hat\Omega \setminus {\overline{ \hat \Omega_{\delta}(2)}}) \times \RR)
\]

in the local Hausdorff sense (along with $C^{\infty}_{\textrm{loc}}$ convergence of the boundary). By (<ref>) we can thus choose $T_1 \ll 0$, $\gamma_{0}>0$, and $\eta_{0}>0$ small so that

\[
B_{\gamma_{0}}(\partial\Xi_{\tau} \cap \{z>\tfrac R2 \}) \cap Q_{\tau} = \emptyset
\]

for all $\eta \in (0,\eta_0)$, $\tau \in (-\infty,T_{1}]$. This ensures (1a) holds. Similarly, we note that as $T_1\to-\infty$, $\eta \to 0$, $\Xi_{\tau} - \tfrac R 2 \be$ converges to $\hat\Sigma_{2^{-\alpha}h_0} \times \RR$ uniformly with respect to $\tau$. Thus, after taking $T_1\ll0$ more negative and $\gamma_{0}>0,\eta_{0}>0$ smaller, condition (2a) holds (thanks to (<ref>)).

We now consider condition (1b). Observe that if $\bx \in Q_{\Xi_{\tau}}$ then (1b) holds by the graphicality of $\Xi_{\tau}$ over (a portion of) $\hat\Sigma\times \RR$. Observe that

\[
Q_{\tau} \setminus  Q_{\Xi_{\tau}} = Q_{\Gamma} \cap \{z \geq R\}.
\]

On the other hand, as $\eta\to 0$, $T_1\to-\infty$, we find that $B_{\gamma_{0}}(\Xi_{\tau}) - R\be$ converges to $B_{\gamma_{0}}(\hat\Sigma_{h_{0}} \times (-\infty,0])$ (uniformly in $\tau$) and $Q_{\Gamma} \cap \{z\geq R\} - R\be$ converges to $((\hat\Omega \setminus {\overline{\hat\Omega_{\delta}(2)}}) \times [0,\infty))$ (where both convergence statements hold in the local Hausdorff sense). These sets are disjoint by choice of $\gamma_{0}$ above, so we have verified (1b).

We finally consider (2b). As above, $\be$-graphicality of $\Gamma$ over $\hat\Omega_{\delta} \times \{\tfrac R2\}$ allows us to consider only the case of $\bx \in B_{\gamma_{0}}(\Gamma) \cap (Q_{\tau} \setminus Q_{\Gamma})$. As before, we have already guaranteed this intersection is empty. This completes the proof.

We can now complete the proof of the main barrier construction.

Fix $\eta_{0},\gamma_{0},T_1$ as in Proposition <ref>.

Suppose that for $T_{2}\leq T_{1}$, $(-\infty,T_{2}]\ni \tau\mapsto \cM(\tau)$ is a rescaled Brakke flow in $\Omega \cap \{z > Z_{1}\} \subset \RR^{n+1}$ so that for $\Xi_{\tau},Q_{\tau}$ defined based on some $\eta\in (0,\eta_{0})$, it holds that

* for all $\tau \in (-\infty,T_{2}]$, it holds that

\begin{multline*}
\supp \cM(\tau) \cap (B_{2\rho}(\bOh) \times  (Z_{1},2Z_{1})) \subset\\
\{ (\bomega,z) + (\tilde v_{\tau}(\bomega,z) + s) \nu_{\hat\Sigma}(\bomega) : z \in [Z_{1},2Z_{1}], s \in (0,\eta z^{\alpha}) \} ,
\end{multline*}

* there is some $r\geq 0$ so that

\[
\supp \cM(\tau) \subset B_{r}(\bOh) \cup \{ tx : \bx \in \partial B_{1}(\bOh), d(\be,\bx) < \theta, t\geq 0\}
\]

for all $\tau \in (-\infty, T_{2}]$,

* $\supp \cM(\tau)$ converges in the local Hausdorff sense to $\Sigma \cap (B_{2\rho}(\bOh) \times [Z_{1},\infty))$ as $\tau \to -\infty$.

We claim that $\supp \cM(\tau) \subset Q_{\tau}$ for all $\tau \in (-\infty,T_{2}]$. Note that since $\eta^{-\kappa} \ll R$, it holds that

\begin{equation*}
Q_{\tau} \cap \{Z_{1}<z<\eta^{-\kappa}\} =
\{ (\bomega,z) + (\tilde v_{\tau}(\bomega,z) + s) \nu_{\hat\Sigma}(\bomega) : z \in [Z_{1},\eta^{-\kappa}], s \in (0,\eta z^{\alpha}) \} ,
\end{equation*}

so this will suffice to prove Theorem <ref>.

To prove that $\supp \cM(\tau) \subset Q_{\tau}$ for all $\tau \in (-\infty,T_{2}]$ we can modify the argument in <cit.> as follows. First, by combining properties (2) and (3) with Lemma <ref> (linear growth of translators) we find that there is some maximal $\tau_{2} \in (-\infty,T_{2}]$ so that $\supp \cM(\tau) \subset Q_{\tau}$ for all $\tau \in (-\infty,\tau_{2})$ and moreover, if $\tau_{2} < T_{2}$ there exists

\[
\bx_{2} \in \supp\cM(\tau_{2}) \cap (\partial Q_{\tau_{2}} \cap \Omega \cap \{z > Z_{1}\}).
\]

(Note that small spherical barriers rule out points of $\supp\cM(\tau_{2})$ lying in the interior of the complement of $Q_{\tau_{1}}$). Note that $\be \cdot \bx_{2} > 2Z_{1}$ by (1).

It must hold that $\bx_{2} \in \Xi_{\tau_{2}}$ or $\Gamma$. In each case we reach a contradiction using a similar argument. We first consider $\bx_{2} \in \Xi_{\tau_{2}}$. Consider

\[
\phi(\bx,\tau) = \begin{cases}
(\gamma - d(\bx,\Xi_{\tau}))^{\beta} & d(\bx,\Xi_{\tau}) < \gamma \\
0 & d(\bx,\Xi_{\tau}) \geq \gamma
\end{cases}
\]

for $\gamma \in (0,\gamma_{0})$ and $\beta>2$. Taking $\gamma \in (0,\gamma_{0})$ sufficiently small, we can arrange that for $\tau \in [\tau_{2}-1,\tau_{2}]$, $\supp \phi(\cdot,\tau) \cap \supp \cM(\tau) \cap \{z \in [Z_{1},2Z_{1}]\} = \emptyset$ by property (1). Thus, even though $\phi$ is not compactly supported in $\Omega$, it is an allowable test function in the Brakke flow condition for $\cM(\tau)$. Furthermore, (after passing to the non-rescaled flow) the calculations in <cit.> carry over essentially without change. The only potential problem is that $\Xi_{\tau}$ has nontrivial boundary and is a supersolution to (rescaled) mean curvature flow (see Proposition <ref>) as opposed to being a solution. To handle this, we can rely on Proposition <ref>. Indeed, for $\tau \in [\tau_{2}-1,\tau_{2}]$, at any point of

\[
\bx \in \supp \phi(\cdot,\tau) \cap \supp\cM(\tau),
\]

the closest point $\by \in \Xi_{\tau}$ is not in $\partial\Xi_{\tau}$ and $\bx$ lies “below” $y$ with respect to $\nu_{\Xi_{\tau}}$ (in the sense of Proposition <ref>). Using this, we readily check that the remainder of the argument in <cit.> yields a contradiction. Similarly, when $\bx_{2} \in \Gamma$, we can use (2) and Lemma <ref> to show $\Gamma$ and $\supp\cM(\tau)$ are well separated at infinity. The argument is then essentially the same as before (replacing $\Xi_{\tau}$ by $\Gamma$ in the definition of $\phi$). This completes the proof.

§ UNIQUENESS IN THE ANCIENT PAST

The goal of this section is to show that if we are given $(\Sigma,\Omega)\in\cS_{n}''$ and an arbitrary one-sided ancient Brakke flow converging to $\Sigma$ with multiplicity one as $\tau \to -\infty$, then the Brakke flow satisfies the hypothesis of Theorem <ref>, thus proving an unconditional uniqueness result for such flows.

Because certain parts of the argument below hold without the nice, stable end hypothesis on $\Sigma$, we will specify the specific situation considered at the beginning of each subsection. However, we will never weaken a hypothesis in later sections after imposing it earlier, so any result derived earlier in the section will always be applicable.

§.§ Nearly sharp upper bound in the compact region
In this section consider a smooth shrinker $(\Sigma, \Omega) \in \cS_{n}$ (with no condition at infinity). Consider $(\check \cM(\tau))_{\tau <T}$ an ancient (integral unit regular) rescaled Brakke flow with $(\supp \check \cM)_{\tau} \subset \Omega$ for all $\tau <T$. Assume that $\check \cM(\tau)$ limits smoothly to $\Sigma$ with multiplicity one as $\tau\to -\infty$.

Choose $\Sigma_{0} \subset \Sigma$ compact domain with smooth boundary. Let $\mu_{0}$ denote the first Dirichlet eigenvalue of $L_{\Sigma}$ on $\Sigma_{0}$. Note that $\inf \mu_{0} = \mu \leq -1$, where $\inf \mu_{0}$ denotes the the infimum of $\mu_0$ over all such $\Sigma_{0}$ and $\mu$ is the first eigenvalue of $L_{\Sigma}$ on the entire $\Sigma$.

By time translation we can assume that $\check \cM(\tau)$ is the normal graph of a positive function $\check u : \Sigma' \times (-\infty,0] \to\RR$ for some $\Sigma'\Supset \Sigma_{0}$ with $\Sigma ' \supset \Sigma \cap B_{2R}(\bOh)$, where $R$ is sufficiently large depending on $\Sigma$. To be precise, $R$ should satisfy (<ref>), (<ref>), and (<ref>).

Our goal is to prove the following nearly sharp upper bound for the graphical function. We expect this result and the method of proof to be useful in other contexts.

There are $C>0, \bar{\tau}<0$ depending on $\Sigma,\Sigma_{0},R,\check \cM$ satisfying

\[
\check u(\bx,\tau) \leq C e^{-\mu_{0} \tau }
\]

for $(\bx,\tau) \in \Sigma_{0} \times (-\infty, \bar{\tau})$.

Denote by $\varphi_{0}>0$ the first Dirichlet eigenfunction of $L_{\Sigma}$ on $\Sigma_{0}$ with $\Vert \varphi_{0}\Vert_{\infty} = 1$. Note that the Hopf boundary lemma implies that

\begin{equation}\label{eq:Hopf_lemma}
\inf_{\partial\Sigma_{0}} |\nabla \varphi_{0}| : = \delta_{0} > 0.
\end{equation}

Consider $\Sigma^a_{-}(\tau)$ defined to be the normal graph of

\[
v_{-}^a(\bx,\tau)=   ae^{-\mu_{0}\tau} (1 - aM  e^{-\mu_{0}\tau}) \varphi_{0}(\bx)
\]

over $\Sigma_0$, where  $M=M(\Sigma_{0})$ is the  constant given in the next proposition.

There are constants $\eta = \eta(\Sigma_{0}) \in (0,\tfrac 12)$ and $M=M(\Sigma_{0})\geq 1$ with the following significance. Suppose $a>0$ and $ \tau_0\leq 0$ are chosen to satisfy

* $a M e^{-\mu_{0}\tau_{0}} \leq \eta$, and

* $v_{-}^a(\bx,\tau_{0}) < \check u(\bx,\tau_{0})$ for $\bx \in \Sigma_{0}$.

Then, for $\tau_0 \leq \tau \leq 0$ with $a M e^{-\mu_{0}\tau} \leq \eta$, it holds that $v_{-}(\bx,\tau) \leq \check u(\bx,\tau)$ for $\bx \in \Sigma_{0}$.

Let $C_{2} = 20nR^2$ denote the constant in Proposition <ref>. We claim there is $\delta = \delta(\Sigma,\Sigma_{0},\Sigma')$ so that

\begin{equation}\label{eq:interior-decay.1}
|\nabla \log v_{-}^a|^{2} - 2 \partial_{\tau} \log v_{-}^a > C_{2}
\end{equation}

on $(\partial\Sigma_{0})_{2\delta} \cap \Sigma_{0}$, where $(\partial\Sigma_{0})_{2\delta}$ denotes the $2\delta$-neighborhood of $\partial \Sigma_0$, for all times $\tau$ satisfying $a M e^{-\mu_{0}\tau} \leq \eta$. Indeed, remembering  $a M e^{-\mu_{0}\tau} \leq \eta < \tfrac 12$, we can compute

\[
|\nabla \log v_{-}^a|^{2} - 2 \partial_{\tau} \log v_{-}^a = \varphi_{0}^{-2} |\nabla \varphi_{0}|^{2} + 2\mu_{0} \left( 1 - \frac{a M e^{-\mu_{0}\tau}}{1- a M e^{-\mu_{0}\tau}} \right)>\varphi_{0}^{-2} |\nabla \varphi_{0}|^{2}.
\]

In addition, $|\nabla \varphi_{0}|$ is bounded away from zero at $\partial\Sigma_{0}$ by the Hopf boundary point lemma (<ref>), but $\varphi_{0}\to 0$ as $\bx \to \partial\Sigma_0$. Thus, we can take $\delta$ small enough (independent of $M$) so that (<ref>) holds. This fixes $\delta$.

Next, we can observe

\[
(\partial_{\tau} - L) v_{-}^a =  a^2 M \mu_{0} e^{-2\mu_{0}\tau} \varphi_{0}.
\]

Thus, by Corollary <ref>, the graph $\Sigma^a_-(\tau)$ satisfies

\[
(\nu_{-}^a\cdot\nu_{\Sigma})^{-1}(\partial_{\tau}\bx  - \bH - \tfrac 12 \bx )\cdot\nu_{-}^a  = (\partial_{\tau} - L) v_{-}^a  + E =  a^2 M e^{-2\mu_{0}\tau}\mu_{0} \varphi_{0} + E,
\]

where $\nu_{-}^a$ is a unit normal vector to $\Sigma^a_-(\tau)$ and

\[
|E(\bx)| \leq C a^2 e^{-2\mu_{0}\tau}(1 + a^2 M^{2}e^{-2\mu_{0}\tau}) (\varphi_{0}(\bx)^{2} + |\nabla \varphi_{0}(\bx)|^{2} + |\nabla^{2} \varphi_{0}(\bx)|^{2})
\]

as long as $v_{-}^a \leq \eta' $ for small $\eta' = \eta'(\Sigma_{0})$ and large $C=C(\Sigma_0)$. Hence, there is $N=N(\Sigma_0)$ so that

\[
|E(\bx)| \leq N  a^2 e^{-2\mu_{0}\tau}(1 + a^2 M^{2}e^{-2\mu_{0}\tau}).
\]

\[
(\nu_{-}^a\cdot\nu_{\Sigma})^{-1}(\partial_{\tau}\bx  - \bH - \tfrac 12 \bx )\cdot\nu_{-}^a\leq ( \mu_{0}   M  \varphi_{0} + N   + N a^2 M^{2} e^{-2\mu_{0}\tau})a^2 e^{-2\mu_{0}\tau}.
\]

We now fix $M=M(\Sigma_{0})\geq 1$ by the requirement that[Note that we make essential use of the fact that $\delta>0$ here, since $\varphi_{0}\to 0$ as  $\bx\to \partial\Sigma_{0}$. ]

\[
\mu_{0} M \varphi_{0} + N \leq -1
\]

on $\Sigma_{0}\setminus (\partial\Sigma_{0})_{\delta}$. Finally we fix $\eta = \eta(\Sigma_{0})$ by

\[
\eta = \min\{ N^{-\frac12}, \tfrac{1}{2}, \eta'\}.
\]

Note that $ a M e^{-\mu_{0}\tau} \leq \eta$  and $M\geq 1$ imply that $v_{-}^a\leq \eta \leq \eta'$, so the above estimates are valid. Combining the above inequalities yields

\begin{equation}\label{eq:interior-decay.2}
(\nu_{-}^a\cdot\nu_{\Sigma})^{-1}(\partial_{\tau}\bx  - \bH - \tfrac 12 \bx )\cdot\nu_{-}^a=\partial_{\tau} - L) v_{-}^a  + E\leq (-1  + N\eta^2)a^2 e^{-2\mu_{0}\tau}\leq 0
\end{equation}

over $\Sigma_{0}\setminus (\partial\Sigma_{0})_\delta$.

Now consider $(\bx,\tau)$ the first time of contact with $a M e^{-\lambda_{0}\tau} \leq \eta$ towards a contradiction. We have

\[
\check u = v_{-}^a, \qquad \nabla \check u = \nabla v_{-}^a, \qquad \partial_{\tau} \check u \leq \partial_{\tau} v_{-}^a
\]

at $(\bx,\tau)$. This implies that  $\bx \in \Sigma_{0}\setminus (\partial\Sigma_{0})_{\delta}$. Indeed, if $\bx \in \Sigma_{0} \cap (\partial\Sigma_{0})_{2\delta}$, then thanks to (<ref>) we find that

\[
|\nabla \log \check u|^{2} - 2 \partial_{\tau} \log \check u \geq |\nabla \log v_{-}^a|^{2} - 2 \partial_{\tau} \log v_{-}^a > C_{2}
\]

at $(\bx,\tau)$. This contradicts Proposition <ref>.

On the other hand, by (<ref>), we see that at the point of contact $\bx \in \Sigma_{0}\setminus (\partial\Sigma_{0})_{\delta}$, the graph of $v_{-}^a$ is a subsolution. This is a contradiction.

We first note that by the parabolic Harnack inequality it suffices to prove that there is $C>0,\tau_{0}<0$ so that

\[
\inf_{\bx \in \Sigma_{0}} \check u(\bx,\tau) \leq C e^{-\mu_{0} \tau }
\]

for $\tau \leq \tau_{0}$.

Assume this fails. Then, there are $\tau_{k}\to-\infty,C_{k}\to\infty$ so that

\begin{equation}\label{eq:interior-decay.3}
\inf_{\bx \in\Sigma_{0}} \check u(\bx,\tau_{k}) \geq C_{k} e^{-\mu_{0} \tau_{k}}.
\end{equation}

\[
a_{k} :=\min\{ \tfrac 12  C_{k} ,  \eta M^{-1} e^{\mu_0\tau_k}\} \to \infty,
\]

where $C_{k}$ is defined as in (<ref>) and $M,\eta$ are fixed in Proposition <ref>.  Consider $v_{k}(\bx,\tau): = v_{-}^{a_k}(\bx,\tau)$. Note that $v_{k}(\bx,\tau_k) \leq \tfrac 12 C_{k} e^{-\mu_{0}\tau_{k}} < \inf_{\bx\in\Sigma_{0}} \check u(\bx,\tau_{k})$ by (<ref>). Using Proposition <ref>, we thus find that $v_{k}(\bx,\tau) < u(\bx,\tau)$ as long as $\tau_k \leq \tau \leq 0$ and $a_{k} M e^{-\mu_{0}\tau} \leq \eta$. Note that

\[
a_{k} M e^{-\mu_{0}\tau} \leq \eta \qquad  \Leftrightarrow \qquad \tau \leq \tfrac{1}{\mu_{0}} (\log M + \log a_{k} - \log \eta).
\]

\[
\tilde \tau_{k} = \tfrac{1}{\mu_{0}} (\log M + \log a _{k} - \log \eta) \to -\infty,
\]

namely $a_k M e^{-\mu_0\tilde \tau_k} = \eta$. Note that $\tilde\tau_k \geq \tau_k$ by definition of $a_k$.

As explained above, Proposition <ref> thus yields

\[
M^{-1} \eta (1-  \eta) \varphi_0(\bx) = v_{k}(\bx,\tilde\tau_{k}) \leq \check u(\bx,\tilde\tau_k)
\]

On the other hand, since $\tau_k\to-\infty$, we have

\[
\sup_{\bx\in\Sigma_0} \check u(\bx,\tilde\tau_k)  \to 0
\]

as $k\to\infty$. This is a contradiction, completing the proof.

§.§ Spatial sub-linear growth of ancient solutions

In this section  we consider $(\Sigma,\Omega)\in \cS_n'$. We will prove that an ancient one-sided Brakke flow has a weak sublinear growth property along the ends of $\Sigma$. (Compare with the stronger conclusion <cit.> in the conical case.)

By Lemma <ref>, we can fix $R>0$ sufficiently large so that $\Sigma \setminus B_R(\bOh) = \Sigma_\textrm{cyl}\cup\Sigma_\textrm{con}$. For $\vartheta$ small, set

\[
U_\textrm{con}^{\vartheta} : = \{ t \bx : \bx \in \partial B_1(\bOh), d(\bx, \sqrt{0} \Sigma_\textrm{con})<\vartheta, t\geq 0 \}
\]

\[
U_\textrm{cyl}^{\vartheta} : = \cup_{j=1}^{N_\textrm{cyl}} \{ t\bx : \bx \in \partial B_1(\bOh), d(\bx, \bx_j)<\vartheta, t\geq 0\}
\]

where $\bx_j$ are the asymptotic directions of the cylindrical ends.

Fix an ancient unit regular integral Brakke flow $\check \cM(t)$ so that $(\supp \check \cM)_t \subset \sqrt{-t}\Omega$ for all $t<0$ and $\Theta(\check\cM,-\infty)< \infty$. Then, for any $\vartheta>0$, there is $R = R(\vartheta) \geq R_0$ so that for $t \leq -1$,

\[
\frac{1}{\sqrt{-t}} (\supp\check\cM)_t \subset B_R(\bOh) \cup U_\textnormal{con}^{\vartheta} \cup U_\textnormal{cyl}^{\vartheta}.
\]

By the Frankel property for self-shrinkers, all tangent flows $\tilde \cM$ at $t=-\infty$ to $\check \cM$ satisfy

\[
\tilde \cM(t) =  k \cH^n\lfloor \sqrt{-t} \Sigma
\]

for all $t<0$, where[Density considerations imply that $k$ is independent of the tangent flow, implying uniqueness of the tangent flow, although this will not be relevant in the sequel.] $k\in \NN$. In particular, if $\bx \in \RR^{n+1}$ has $\Theta((\bx,0),\tilde \cM) > 0$ then $\bx \in \sqrt{0}\Sigma$.

Now, we assume that the assertion fails. That is, there is $(\bx_k,t_k) \in \supp\check \cM$ with $t_{k}\leq -1$, so that $\bx_k \not \in U_\textrm{con}^\vartheta\cup U_\textrm{cyl}^\vartheta$ and $|\bx_k|/\sqrt{-t_k}\to\infty$. Note that $|\bx_{k}|\to\infty$, since $t_{k}\leq -1$. Define $\tilde \cM_k(t)$ and $(\tilde \bx_k,\tilde t_k)$ by dilating around $(\bOh,0)$ so that $|\tilde\bx_k|=1$. Passing to a subsequence, $\tilde \cM_k(t) \rightharpoonup \tilde\cM(t)$ a tangent flow to $\check\cM$ at $t=-\infty$ and $\tilde \bx_{k}\to\tilde \bx$. Note that $\tilde t_{k}\to 0$ by construction. In particular, we find that

\[
\Theta((\bx,0),\tilde \cM) > 0,
\]

so $\bx \in \sqrt{0}\Sigma$. However, by construction,

\[
\sqrt{0}\Sigma \setminus\{\bOh\} \subset U_\textnormal{con}^{\vartheta} \cup U_\textnormal{cyl}.
\]

This is a contradiction.

§.§ Graphicality and estimates along the conical ends

In this section we consider $(\Sigma,\Omega)\in \cS_n'$ and  $(\check \cM(\tau))_{\tau}$ an ancient (integral unit regular) rescaled Brakke flow with $(\supp \check \cM)_{\tau} \in \Omega$ for all $t<0$. Assume that $ \check \cM(\tau)$ limits in $C^{\infty}_{\textrm{loc}}$ to $\Sigma$ with multiplicity one as $\tau \to -\infty$.

By Lemma <ref>, we can fix $R_0>0$ sufficiently large so that for $R\geq R_0$, $\Sigma \setminus B_R(\bOh) = \Sigma_\textrm{cyl}\cup\Sigma_\textrm{con}$. For $\vartheta$ small, set

\[
U_\textrm{con}^{\vartheta} : = \{ s \bx : \bx \in \partial B_1(\bOh), d(\bx, \sqrt{0} \Sigma_\textrm{con})<\vartheta, s\geq 0 \}
\]

\[
U_\textrm{cyl}^{\vartheta} : = \cup_{j=1}^{N_\textrm{cyl}} \{ s\bx : \bx \in \partial B_1(\bOh), d(\bx, \bx_j)<\vartheta, s\geq 0\}
\]

where $\bx_j$ are the asymptotic directions of the cylindrical ends. Assume that $\vartheta$ is chosen sufficiently small and $R$ sufficiently large so that (i) there is exactly one component of $\Sigma_{\textrm{con}}$ in each component of $U^{\vartheta}_{\textrm{con}}\setminus B_{R}(\bOh)$ and (ii) for $\bx \in \Sigma_{\textrm{con}}$ it holds that

\[
\bx \pm \tfrac 12 |A_{\Sigma}| \nu_{\Sigma}(\bx) \not \in U_{\textrm{con}}.
\]

(This second condition can be verified by considering the blow-down of $\Sigma_{\textrm{con}}$, a smooth cone by assumption, and it will guarantee that graphs over $\Sigma_{\textrm{con}}$ that remain in $U_{\textrm{con}}^{\theta}$ lie within a tubular neighborhood of $\Sigma_{\textrm{con}}$).

We first recall the weighted norms from [22] along the conical ends are defined as follows:

\begin{align*}
\Vert f \Vert^{(d)}_{k;\Sigma_{\textrm{con}}} & : = \sum_{i=0}^{k} \sup_{\bx \in\Sigma_{\textrm{con}}} |\bx|^{-d+i} |\nabla^{i}_{\Sigma} f(\bx)| \\
[f]^{(d)}_{\alpha;\Sigma_{\textrm{con}}} & : = \sup_{\bx\neq \by \in \Sigma_{\textrm{cyl}}} \frac{1}{|\bx|^{d-\alpha}+|\by|^{d-\alpha}} \frac{|f(\bx)-f(\by)|}{d_{\Sigma}(\bx,\by)^{\alpha}}\\
\Vert f \Vert^{(d)}_{k,\alpha;\Sigma_{\textrm{con}}} & : = \Vert f \Vert^{(d)}_{k;\Sigma_{\textrm{cyl}}} + [\nabla^{k}_{\Sigma}f]^{(d-k)}_{\alpha;\Sigma_{\textrm{cyl}}}.
\end{align*}

Fix $\vartheta$ satisfying (i) and (ii) above. For $R \geq R_0$, there is $T<0$ so that for $\tau<T$,

\[
\check\cM(\tau) \lfloor (U_{\textnormal{con}}^{\vartheta} \cup B_{R})
\]

agrees with the multiplicity-one graph of $\check u(\cdot,\tau)$ defined on some subset of $\Sigma$. Furthermore, for any integer $k\geq 1$,

\[
\lim_{\tau\to-\infty} (\Vert \check u(\cdot,\tau)\Vert_{k;\Sigma_{\textnormal{con}}}^{(1)} + \Vert \check u (\cdot,\tau)\Vert_{C^{k}(B_{2R}(\bOh))}) = 0
\]

As in <cit.>, this follows by rescaling the fact that the tangent cone to the un-rescaled flow at $-\infty$ is $t\mapsto\sqrt{-t}\Sigma$ (with multiplicity-one).

There exists $R_1\geq R_0$ and $C>0$ such that for all $R\geq R_1$ and $\tau_0\leq T$,

\[ \Vert \check u(\cdot,\tau_0)\Vert_{2,\alpha;\Sigma_{\textnormal{con}}}^{(1)} \leq C \sup_{\tau\leq \tau_0} \Vert \check u(\cdot, t)\Vert_{C^0(\Sigma_\textnormal{con}\cap B_{2R}(\bOh))}\, .\]

We construct barriers for the rescaled flow. To do this, we consider a smooth function $\varphi$ defined on $\Sigma_\textrm{con} \cap \{|\bx| \geq R\}$ to be chosen below. Set

\[\Gamma_\varphi = \{ \bx + \varphi(\bx)  \nu_{\Sigma'}(\bx) : \bx \in \Sigma_\textrm{con}, |\bx| \geq R\}\, .\]

Note that if $|\varphi(\bx)| |A_\Sigma|(\bx) < 1$ for all $\bx \in \Sigma_\textrm{con} \cap \{|\bx| \geq R\}$, then $\Gamma_\varphi$ will be a smooth hypersurface.

We would like to choose $\varphi$ so that $\bH + \bx^\perp/2$ points towards $\Sigma$, or equivalently, we would like that $H + \tfrac{1}{2} \bx \cdot \nu \leq 0$. Expanding this equation as in Appendix <ref>, we can write this as

\[
L\varphi + E(\varphi) \leq 0, \,
\]

where $E$ is as in Lemma <ref> (cf. Corollary <ref> and <cit.>). Choosing $\varphi:= \alpha |\bx| -\beta$ as in <cit.> we have

\begin{equation}\label{eq:est.barr.est.lin.term.con.ends}
L\varphi \leq -\tfrac{1}{2} (1 + O(r^{-2}))\beta + O(r^{-1}) \alpha\, .
\end{equation}

Using Lemma <ref>, we can write

\begin{align*}
|E(\varphi)(\bx)| & \leq C(|\bx|^{-2}|\varphi(\bx)| + |\bx|^{-1}|\nabla \varphi(\bx)|)(|\bx|^{-1}|\varphi(\bx)| + |\nabla \varphi(\bx)| + |\bx| |\nabla^2\varphi(\bx)|)\\
& \leq C |\bx|^{-1} (\alpha + \beta |\bx|^{-1})^2
\end{align*}

For $R >0$ large we choose
$\beta = \alpha (R-1)$. Thus $\varphi(\bx) = \alpha$ on $\Sigma_\textnormal{con} \cap \{|\bx| =R \}$, and $|\varphi(\bx)| \leq \alpha (|\bx| +1 - R)$ on $\Sigma_\textnormal{con} \cap \{|\bx|  \geq R \}$. In particular, since $|A_{\Sigma_\textrm{con}}| = O(|\bx|^{-1})$ (by scaling), we see that $\Gamma_\varphi$ is indeed a smooth hypersurface, for $\alpha \leq \alpha_0$ small.

In particular, we find that

\begin{equation}\label{eq:est.barr.est.err.term.con.ends}
|E(\varphi)(\bx)| = O(\alpha^2|\bx|^{-1})
\end{equation}

on $\Sigma_\textnormal{con} \cap \{|\bx|  \geq R \}$. Combining (<ref>) and (<ref>) we find that for $R$ sufficiently large (independent of $\alpha \leq \alpha_0$),

\[
L\varphi + E(\varphi) \leq 0
\]

on $\Sigma_\textnormal{con} \cap \{|\bx|  \geq R \}$, as desired.
Combining the fact that $  \check \cM(\tau)$ limits smoothly to $\Sigma$ together with Proposition <ref> this implies

\[ \Vert \check u(\cdot,\tau_0)\Vert_{0;\Sigma_{\textnormal{con}}}^{(1)} \leq C \sup_{\tau\leq \tau_0} \Vert \check u(\cdot, \tau)\Vert_{C^0(\Sigma_\textnormal{con}\cap B_{2R}(\bOh))}\, .\]

Finally, the assertion follows from the Schauder estimates stated in Proposition <ref> below.

The following estimates are proven in <cit.>.

For $\check u(\cdot,\tau)$ as defined in Lemma <ref>, it holds that

\[
\Vert \check u(\cdot, \tau) \Vert_{2,\alpha;\Sigma_\textnormal{con}}^{(1)} \leq  \sup_{\tau\leq \tau_0}  C\left(  \Vert \check u(\cdot, \tau)\Vert_{C^0(\Sigma_\textnormal{con}\cap B_{2R}(\bOh))} + \Vert \check u(\cdot,\tau_0)\Vert_{0;\Sigma_{\textnormal{con}}}^{(1)} \right)
\]

We give the references to the relevant results from <cit.>. By <cit.> (Schauder estimates for the linearized equation $Lu=h$), we find that

\begin{equation}\label{eq:schauder.nonlinearity.from.gen.1}
\Vert \check u(\cdot, \tau) \Vert_{2,\alpha;\Sigma_\textnormal{con}}^{(1)} \leq C\sup_{\tau\leq \tau_0} \left(    \Vert \check u(\cdot, \tau)\Vert_{C^0(\Sigma_\textnormal{con}\cap B_{2R}(\bOh))} + \Vert E(\check u)(\cdot,\tau_0) \Vert_{0,\alpha;\Sigma_{\textnormal{con}}}^{(-1)} \right)
\end{equation}

where $E(\check u)$ is the nonlinear error term satisfying the following weighted Hölder estimates proven in <cit.>:

\[
\Vert E(\check u)(\cdot,\tau_0) \Vert_{0,\alpha;\Sigma_{\textnormal{con}}}^{(-1)} \leq C \Vert \check u(\cdot,\tau_0) \Vert_{1,\alpha;\Sigma_{\textnormal{con}}}^{(1)} \Vert \check u(\cdot,\tau_0) \Vert_{2,\alpha;\Sigma_{\textnormal{con}}}^{(1)}.
\]

Using Lemma <ref>, we can thus absorb this term into the left-hand side of (<ref>), completing the proof.

§.§ Upgrading closeness to graphicality along the cylindrical ends

In this section we consider $(\Sigma,\Omega)\in \cS_n'$ and  $(\check \cM(t))_{t}$ an ancient (integral unit regular) non-rescaled Brakke flow with $(\supp \check \cM)_{t} \in \sqrt{-t} \Omega$ for all $t<0$. Assume that $\frac{1}{\sqrt{-t}} \check \cM(t)$ limits in $C^{\infty}_{\textrm{loc}}$ to $\Sigma$ with multiplicity one as $t\to -\infty$.

Fix $\vartheta$ as in Lemma <ref>.
For $T<0$ fixed, consider smooth positive strictly monotone functions $b,\eps:(-\infty,T) \to (0,\infty)$ with $\lim_{t\to-\infty} b(t) = \infty$ and $\lim_{t\to-\infty} \eps(t) = 0$. We say that $(b(t),\eps(t))$ forms a barrier estimate for $\check \cM$ along the cylindrical ends if

\[
\tfrac{1}{\sqrt{-t}} (\supp\check \cM)_t\cap \big(U^\vartheta_\text{cyl}\setminus B_{2R_0}(\bOh)\big) \cap B_{b(t)}(\bOh) \subset U_{\eps(t)}(\Sigma_\text{cyl}),
\]

for $t \in (-\infty,T)$. Here, $U_r(\Sigma_\text{cyl})$ is the $r$-neighborhood of $\Sigma_\text{cyl}$.

For $\eps_0>0$ to be fixed in Proposition <ref>, define the graphical radius $\rho(t)$ to be supremum of $\rho>0$ so that $(\frac{1}{\sqrt{-t}} \check \cM(t)) \lfloor \big((U^\vartheta_\text{cyl}\setminus B_{2R_0}(\bOh)) \cap B_{\rho}(\bOh)\big)$ is the graph of some function defined on a subset of $\Sigma_\text{cyl}$, with $C^5$-norm $\leq \eps_0$.

Note that we do not claim that $\rho(t)$ is continuous (it would be possible to prove that $\rho(t)$ has certain partial continuity properties, but we will not need that below). However, it is clear that in the setting considered here $\lim_{t\to-\infty}\rho(t) = \infty$.

Suppose that $(b(t),\eps(t))$ forms a barrier estimate for $\check \cM$. Then, we can take $\eps_0=\eps_0(\Sigma)>0$ small and $T=T(\Sigma,\check \cM,b(t),\eps(t))<0$ sufficiently negative so that $\rho(t) \geq b(t)$ for $t<T$.

Assume (for contradiction) that there is $t_i \to -\infty$ so that $\rho(t_i) \leq b(t_i)$. Because $\rho(t) \to \infty$ as $t\to-\infty$,

\[
\inf\{ s \leq t_i : \rho(s) \leq \rho(t_i) \}  > -\infty.
\]

Without some continuity property of $\rho(t)$ we cannot ensure that the infimum is attained, but we can find

\[
t_i' \in \{ s \leq t_i : \rho(s) \leq \rho(t_i) \}
\]

\[
t_i' \leq \inf \{ s \leq t_i : \rho(s) \leq \rho(t_i) \} + 1.
\]

Note that for $s \leq t_i' -1$, we have $\rho(s) \geq \rho(t_i) \geq \rho(t_i')$. Furthermore, we have $\rho(t_i') \leq \rho(t_i) \leq b(t_i) \leq b(t_i')$ (since $b(t)$ is strictly decreasing) and $t_i'\to-\infty$ (since $\rho(t) \to \infty$ as $t\to-\infty$).

By assumption, there are points $\bx_i \in \supp\check\cM \cap \overline{B_{\rho(t_i')}(\bOh)}$ so that $\check \cM(t_i')$ is not locally written as a graph over a subset of $\Sigma$ with $C^5$-norm $\leq \eps_0$ (if this held at each point in $\supp\check\cM \cap \overline{B_{\rho(t_i')}(\bOh)}$ we could patch these graphs together to write $\supp\check\cM \cap B_{\rho(t_i')+\delta_0}(\bOh)$—for some $\delta_0>0$ small—as a graph over a subset of $\Sigma$ with $C^5$-norm $\leq \eps_0$; note that there can only be a single sheet by the multiplicity one covergence). It is clear that $\bx_i\to\infty$ as $i\to\infty$.

We first suppose that (after passing to a subsequence), $\bx_i \in U_\textrm{con}^{\vartheta}$, where $\vartheta$ is fixed as in Lemma <ref>. Applying Lemma <ref> with $k=5$, we find a contradiction.

Thus, we may pass to a subsequence and assume that $\bx_i \in U_\textrm{cyl}^{\vartheta}$. Up to passing to a subsequence and rotating we assume that $\bx_i/|\bx_i |\to \be$ and $\Sigma$ has a cylindrical end (modeled on $\hat \Sigma \times \RR$) in the $\be$ direction.

Write $\bx_i = (\bomega_i,z_i)$. Consider the flows

\[
\check\cM_i(t):= \tfrac{1}{\sqrt{|t_i'|}}\check \cM(|t_i'| t) - z_i \be.
\]

Passing to a subsequence, these flows converge to a Brakke flow $\tilde\cM(t)$ with

\[
\tfrac{1}{\sqrt{-t}} (\supp\tilde\cM)_t\subset \overline{\hat \Omega\times \RR}.
\]

Now, fix $t_0 < -1$ and consider $t \leq t_0$. Observe that by definition of the graphical radius,

\[
\Big(\tfrac{1}{\sqrt{-|t_i'|t}} \check\cM(|t_i'|t)\Big) \lfloor B_{\rho(|t_i'|t)} (\bOh)
\]

is graphical over some subset of $\Sigma$ (with $C^5$ norm $\leq \eps_0$). Note that (since $t\leq t_0<-1$)

\[
|t_i'| t < t_i' - 1
\]

for $i$ sufficiently large (depending on $t_0$), so for such $i$, we find that

\[
\rho(|t_i'|t) \geq \rho(t_i').
\]

\[
\Big(\tfrac{1}{\sqrt{-|t_i'|t}} \check\cM(|t_i'|t)\Big) \lfloor B_{\rho(t_i')} (\bOh)
\]

is graphical over some subset of $\Sigma$ (with $C^5$ norm $\leq \eps_0$). Translating in the $\be$ direction, we find

\[
\tfrac{1}{\sqrt{-t}} \check \cM_{i}(t) \lfloor B_{\rho(t_i')} (-\tfrac{1}{\sqrt{-t}}z_i\be)
\]

is graphical over some subset of $\Sigma-\tfrac{1}{\sqrt{-t}}z_i\be$ (with $C^5$ norm $\leq \eps_0$). Because $t<-1$, we note that $B_{\rho(t_i')} (-\tfrac{1}{\sqrt{-t}}z_i\be)$ limits to $\RR^{n+1}$ as $i\to\infty$. Passing this information to the limit (taking $\eps_0$ sufficiently small so that we can use interior estimates) and then sending $t_0\nearrow -1$, we thus find that $\frac{1}{\sqrt{-t}} \tilde\cM(t)$ is graphical over $\hat \Sigma \times \RR$ with $C^5$ norm $\leq \eps_0$ for all $t < -1$.

On the other hand, we have that

\[
\Big(\tfrac{1}{\sqrt{-t_i'}} (\supp \cM)_{t_i'}\Big) \cap B_{b(t_i')}(\bOh) \subset U_{\eps(t_i')}(\Sigma),
\]

because $(b(t),\eps(t))$ forms a barrier estimate. Translating this as above, we find $t<-1$,

\[
\tfrac{1}{\sqrt{-t}}(\supp \check\cM_i)_t \cap B_{b(t_i')}(-\tfrac{1}{\sqrt{-t}}z_{i}\be) \subset U_{\eps(t_i')} (\Sigma -\tfrac{1}{\sqrt{-t}}z_{i}\be).
\]

Since $b(t_i') \geq \rho(t_i') \geq z_{i} $ and $\eps(t_i')\to 0$, we can conclude as above that $\frac{1}{\sqrt{-t}}(\supp \tilde \cM)_t \subset \hat\Sigma\times \RR$ for all $t<-1$. Using the integral unit regular property of $\tilde \cM$, we thus find that

\[
\tilde \cM(t) = \cH^n \lfloor \sqrt{-t} (\hat\Sigma\times \RR)
\]

for all $t<0$. Note that we have unit multiplicity here, due to the fact that $\frac{1}{\sqrt{-t}} \tilde\cM(t)$ is graphical over $\hat \Sigma \times \RR$ with $C^5$ norm $\leq \eps_0$ for all $t < -1$.

By unit regularity, we thus find that for $i$ sufficiently large, $\check\cM_i(-1) = \frac{1}{\sqrt{|t_i'|}} \check \cM(t_i') - z_i\be$ is graphical near $(\bomega_i,0)$ (with $C^5$ norm $\leq o(1)$). This contradicts our assumption about $\bx_i$, completing the proof.

§.§ Global decay estimates

We now impose the stable end hypothesis on $\Sigma$. More precisely, in this section we consider $(\Sigma,\Omega)\in \cS_n''$ and  $(\check \cM(\tau))_{\tau}$ an ancient (integral unit regular) rescaled Brakke flow with $(\supp \check \cM)_{\tau} \in \Omega$ for all $\tau < T$. Assume that $\check \cM(\tau)$ limits in $C^{\infty}_{\textrm{loc}}$ to $\Sigma$ with multiplicity one as $\tau\to -\infty$.

We define the following norm, adapted to the graphical radius along the cylindrical ends

\begin{equation}\label{eq:defi.partial.seminorm.1}
\Vert \cdot \Vert_{k,\alpha; \text{cyl}}(\rho)=   \Vert  \cdot \Vert_{k,\alpha;\Sigma \cap B_{2R_0}(\bOh)} + \Vert \cdot \Vert_{k,\alpha;\Sigma_\textnormal{cyl} \cap B_{\rho}(\bOh)}\, .\end{equation}

We now assume there is $(-\infty,\tau_{0}] \ni \tau \mapsto \tilde v_{\tau} \in C^{\infty}_\textrm{loc}(\Sigma)$ so that

\[
\Vert \tilde v_\tau\Vert_{C^3;\hat\Sigma\times [Z_0,\infty)} = o(1) \textrm{ as $\tau\to-\infty$},
\]

and so that

\[
\tilde \Sigma_\tau = \Graph_{\Sigma} \tilde v_{\tau}(\cdot)\textrm{ is a rescaled mean curvature flow}.
\]

(An important special case is $\tilde v_{\tau}\equiv 0$, so that $\tilde \Sigma_\tau \equiv \Sigma$, but we will also consider the case that $\tilde v_{\tau}$ is the ancient flow constructed in Proposition <ref>).

Let $\tilde \Omega_\tau \subset \Omega$ be the open set such that $\partial \tilde \Omega_\tau = \Sigma_\tau$. We assume that $\check \cM$ lies on one side of $\tilde \Sigma_\tau$, i.e. $\check \cM(\tau) \subset \tilde \Omega_\tau$ for $\tau \leq \tau_0$. We have the following graphicality and decay estimate. Fix $\alpha>0$ as in Theorem <ref>.

There exists $\hat\delta>0, \vartheta>0, R_1\geq 2R_0$ and depending only $\tilde \Sigma_\tau, \Sigma$ with the following property: for every $R\geq R_1$ there exists $\tau_1\leq \tau_0$ such that for $\tau \leq \tau_1$, $\check\cM(\tau) \lfloor (B_{8R}(\bOh))$ is the graph of $\check u(\cdot,\tau)$ defined on some subset of $\Sigma$ (including $\Sigma\cap B_{4R}(\bOh)$) with

$$\delta(\tau) := \sup_{\tau' \leq \tau} \|\check u(\cdot, \tau')- \tilde v(\cdot, \tau') \|_{4,\alpha; \Sigma\cap B_{4R}(\bOh)} \leq \hat\delta\, .$$

Then either $\delta(\tau') = 0$ for some $\tau'<\tau_1$ and $\check{M}(\tau) = \tilde\Sigma_\tau$ with multiplicity one for all $\tau \leq \tau_1$, or
$\delta(\tau)>0$ for all $\tau<\tau_1$ and there exists $\bar\tau\leq \tau_1$ such that for $\tau \leq \bar\tau$

\[
\check\cM(\tau) \lfloor \big((U^\vartheta_\text{cyl}\setminus B_{2R_0}(\bOh))\cap B_{\tfrac{1}{2}\delta(\tau)^{-\frac{1}{1+\alpha}}}(\bOh)\big)
\]

is the graph of $\check u(\cdot,\tau)$ defined on some subset of $\Sigma_{\text{cyl}}$ such that

* $\Vert \check u(\cdot, \tau) - \tilde v(\cdot, \tau) \Vert_{4,\alpha; \textnormal{cyl}}( \tfrac{1}{2}\delta(\tau)^{-\frac{1}{1+\alpha}}) \leq C\delta(\tau)^{\theta}$ for $\tau\leq \bar\tau$ and any $\theta \in (0,\frac{1}{1+\alpha})$ where $C$ depends on $\theta$, and

* rotating $\Sigma$ so that some cylindrical end $\Sigma'$ points in the $\be$ direction, then

$$\check u(\cdot,\tau) - \tilde v(\cdot, \tau) \leq C  \delta(\tau) z^\alpha$$

for $\tau \leq \bar \tau, R\leq z \leq \tfrac{1}{2}\delta(\tau)^{-\frac{1}{1+\alpha}}$.

By taking $R$ sufficiently large, we can assume that the following property holds. For any cylindrical end $\Sigma'$, after rotating to point in the $\be$-direction, applying Theorem <ref> (with $w_{\tau}\equiv w$), then for $Z_{1}$ as defined in Theorem <ref>, it holds that $\Sigma' \cap \{z=Z_1\} \subset B_{R/2}(\bOh)$.

We assume first that $\delta(\tau)>0$ for all $\tau<\tau_1$. We claim that by increasing $R$ if necessary, for $\tau_1$ sufficiently negative and $\delta_0$ sufficiently small, Theorem <ref> (with $\overline v_{\tau}\equiv \overline v$ the graphical function of the end $\Sigma'$ over $\hat\Sigma\times \RR$, cf. (<ref>), or $\overline v_{\tau}$ the graphical function of the end of $\Sigma_\tau$ over $\hat\Sigma\times \RR$), with $\eta' = 2\delta(\tau')$ and $T_{2} = \tau'$ applies to

\[
\check\cM'(\tau) : = \check\cM(\tau) \lfloor (\Omega \cap \{z > Z_{1}\} \cap \{t\bx : \bx \in \partial B_{1}(\bOh), d(\bx,\be) < \vartheta_{0}, t>0\})
\]

for $\vartheta_{0}$ defined as in (<ref>). Note that (<ref>) and Proposition <ref> guarantee that after taking $\tau_1$ even more negative, $\check\cM'(\tau) $ is[The point here being that there might be other components of $\check\cM(\tau)$ in $\Omega \cap \{z > Z_{1}\}$ corresponding to some other conical/cylindrical end that extends into $\{z>Z_{1}\}$, but for sufficiently negative times these components are disjoint and can be dropped without affecting the Brakke flow property.] a (rescaled) Brakke flow in $\Omega \cap \{z > Z_{1}\}$.

Property (1) in Theorem <ref> follows from the choice of $\eta'$ by taking $Z_1$ sufficiently large (i.e. increasing $R$) and choosing $\delta_0>0$ sufficiently small. Property (2) in Theorem <ref> follows from (<ref>) and Proposition <ref>. Finally, (3) in Theorem <ref>  holds by assumption. Thus, we find that

\begin{multline}\label{eq:barrier.app.semi.globa.est}
\supp \check\cM'(\tau') \cap \{Z_{1} < z < (\eta')^{-\kappa}\}
%\cap (B_{2\rho}(\bOh) \times  [Z_{1},\eta^{-\kappa}])
\subset\\
\{ (\bomega,z) + (w_{\tau'}(\bomega,z) + s) \nu_{\hat\Sigma}(\bomega) : z \in [Z_{1},(\eta')^{-\kappa}], s \in [0,(\eta') z^{\alpha}) \} ,
\end{multline}

where $\kappa = (1+\alpha)^{-1}$ is as in Theorem <ref> and this holds for all $\tau' \leq \bar \tau$.

\[
b(-e^{-\tau}) = (2\delta(\tau))^{-\frac{1}{1+\alpha}}, \qquad \eps(-e^{-\tau}) = (2\delta(\tau))^{\frac{1}{1+\alpha}},
\]

we claim that $(b(t),\eps(t))$ forms a barrier estimate for $\check \cM$ in the sense of Definition <ref>. Indeed, we observe that as $t\to -\infty$, $b(t) \to \infty$ and $\eps(t)\to 0$ since $\delta(\tau)\to 0$ as $\tau\to -\infty$. Finally, the assumption

\[
\supp\check\cM(\tau) \cap \big(U^\vartheta_\text{cyl}\setminus B_{2R_0}(\bOh)\big) \cap B_{b(t)}(\bOh) \subset U_{\eps(t)}(\Sigma_\text{cyl})
\]

follows from (<ref>).  As such, Proposition <ref> yields $\bar\tau< 0$ so that along the cylindrical ends $\check\cM(\tau) \cap B_{\tfrac{1}{2} \delta(\tau)^{-\frac{1}{1+\alpha}}}(\bOh)$ is graphical with $C^{5}$ norm bounded by $\eps_{0}$ (fixed in Proposition <ref>) for all $\tau \leq \bar\tau$.

The estimate $\check u (\cdot,\tau) - \tilde v(\cdot, \tau) \leq C \delta(\tau) z^{\alpha}$ then follows from the height bounds from the barrier (<ref>). This proves (2). Interpolation yields (1).

Now consider the case $\delta(\tau') = 0$ for some $\tau'<\tau_1$. Note that we  can repeat the first steps of the previous barrier arguments with a sequence $\eta_i>0$, $\eta_i \to 0$ to see that the support of $\check \cM(\tau)$ is contained in $\tilde\Sigma_\tau$ for all $\tau \leq \tau'$. Since $\check \cM(\tau)$ agrees with $\tilde\Sigma_\tau$ on $B_{2R}(\bOh)$ for all $\tau \leq \tau'$, the constancy Lemma implies that $\check \cM(\tau) = \tilde\Sigma_\tau$ with multiplicity one for a.e. $\tau\leq \tau'$. Unit regularity then implies that $\check{M}(\tau) = \tilde\Sigma_\tau$ with multiplicity one for all $\tau \leq \tau_1$.

We extend (<ref>) to the following semi-global norm

\begin{equation}\label{eq:defi.partial.seminorm.2} \Vert \cdot \Vert_{k,\alpha}(\rho)=   \Vert  \cdot \Vert_{k,\alpha; \text{cyl}}(\rho) + \Vert  \cdot \Vert_{k,\alpha;\Sigma_{\textnormal{con}}}^{(1)}\, .\end{equation}

The previous proposition and the decay along the core yields the following graphicality and decay estimate. Recall that $\mu$ is the first eigenvalue of $L_{\Sigma}$.

For $\eps\in (0,|\mu|)$, there is $\delta_0>0$, $\vartheta>0$, $\tau_1\ll0$ depending on $\Sigma,\check\cM,\eps$ so that for $\tau \leq \tau_1$,

\[
\check\cM(\tau) \lfloor (B_{3e^{-2\delta_0 \tau}}(\bOh) \cup U^\vartheta_\textnormal{con})
\]

is the graph of $\check u(\cdot,\tau)$ defined on some subset of $\Sigma$ with:

* $\Vert \check u(\cdot, \tau) \Vert_{4,\alpha}(2 e^{-\delta_0 \tau}) \leq e^{2\delta_0\tau}$, and

* rotating $\Sigma$ so that some cylindrical end $\Sigma'$ points in the $\be$ direction, then $\check u(\cdot,\tau) \leq C e^{-(\mu-\eps)\tau} z^\alpha$ for $\tau \leq \tau_1, R\leq z \leq 2e^{-\delta_0\tau}$.

By Lemma <ref>, for any $R>0$ we can find $\tau_1\leq \tau_0$ sufficiently negative and a function $\check u$ defined on $\Sigma_\textrm{con} \cup B_{4R}(\bOh)$ for $\tau \leq \tau_1 < 0$ so that

\begin{equation}\label{eq:con-weak-decay-graphicality-first-lemma}
\lim_{\tau\to-\infty} \big(\Vert \check u(\cdot,\tau)\Vert_{k;\Sigma_{\textnormal{con}}}^{(1)} + \Vert \check u (\cdot,\tau)\Vert_{C^{k}(B_{5R}(\bOh))}\big) = 0
\end{equation}

for all $k \in \NN$.

On the other hand, taking $R$ even larger and $\tau_1$ even more negative, Theorem <ref>, together with interpolation, yields

\begin{equation}\label{eq:inner-decay.2}
\delta(\tau) \leq e^{(-\mu - \tfrac \varepsilon 8) \tau}  %\leq e^{\frac \tau 2}
\end{equation}

on $\Sigma \cap B_{4R}(\bOh)$ for $\tau \leq \tau_1$.  We combine (<ref>) with Proposition <ref> to see that (taking $ \tau_1$ even more negative) for $\tau \leq \tau_1$

\[
\Vert \check u(\cdot,\tau)\Vert_{2,\alpha;\Sigma_{\textnormal{con}}}^{(1)}   \leq  e^{(-\mu - \tfrac \varepsilon 4) \tau} \, .\]

Interpolating with (<ref>) we thus find (taking $\bar \tau$ even more negative) for $\tau \leq \bar \tau$

\begin{equation}\label{eq:semi.global.decay.interior}
\Vert \check u(\cdot,\tau)\Vert_{4,\alpha;\Sigma \cap B_{2R}(\bOh)} + \Vert \check u(\cdot,\tau)\Vert_{4,\alpha;\Sigma_{\textnormal{con}}}^{(1)}   \leq e^{(-\mu - \tfrac \varepsilon 2) \tau} \, .
\end{equation}

After increasing $R$ such that $R\geq R_1$ (where $R_1$ is from Proposition <ref>) we can decrease $\tau_1$ further such that $\tau_1 \leq \bar{\tau}$ where $\bar\tau$ is as in Proposition <ref>. The estimate (<ref>) combined with Proposition <ref> (applied with $\tilde v_\tau \equiv 0$) yield the asserted bounds. This completes the proof.

We furthermore record the following decay estimate for the weighted $L^2$-norm. We denote

$$\Sigma_\rho = \Sigma_\textnormal{con} \cup (\Sigma \cap B_{2R}(\bOh)) \cup (\Sigma_\textnormal{cyl} \cap B_{\rho}(\bOh))\, .$$

For every $\varepsilon >0$ there is $C>0$ and $\bar \tau \ll 0$ such that

$$ \int _{\Sigma_{2e^{-\delta_0\tau}}} \check u^2 e^{-\tfrac{1}{4} |\bx|^2} \leq C e^{2(-\mu - \varepsilon)\tau}\, .$$

Using Theorem <ref> and Proposition <ref> (decreasing $\bar\tau$ if necessary) we see that for $\tau \leq \bar \tau$

\[ \Vert \check u(\cdot,\tau)\Vert_{0;\Sigma \cap B_{2R}(\bOh)} + \Vert \check u(\cdot,\tau)\Vert_{0;\Sigma_{\textnormal{con}}}^{(1)}   \leq C e^{(-\mu -\varepsilon) \tau}\, .\]

By Corollary <ref> we have along each cylindrical end $\Sigma'$ that (after rotating to point in the $\be$-direction),

$$ \check u(\cdot,\tau) \leq C e^{(-\mu -\varepsilon) \tau} z^\alpha $$

for $\tau \leq \bar \tau$, $R\leq z \leq 2 e^{-\delta_0\tau}$. Combining these estimates yields the stated integral estimate.

§.§ Spectral dynamics

In this section we continue to consider $(\Sigma,\Omega)\in \cS_n''$ and  $(\check \cM(\tau))_{\tau}$ an ancient integral unit regular rescaled Brakke flow with $(\supp \check \cM)_{\tau} \subset \Omega$ for all $\tau < T<0$. Assume that $\check \cM(\tau)$ limits in $C^{\infty}_{\textrm{loc}}$ to $\Sigma$ with multiplicity one as $\tau\to -\infty$.

Recall that we denote with $\varphi>0$ the first eigenfunction (corresponding to the eigenvalue $\mu$) of the operator $L$ along $\Sigma$, normalized such that $\|\varphi \|_{W} =1$. The following is the main result of this section.

There exists a unique constant $\check a=\check a(\check\cM) \geq 0$ with the following significance. Given any $R\gg 1$, there exist $C=C(\check\cM,R) >0$ and $\bar\tau=\bar \tau(\check\cM,R)  \ll-1$ such that

\[
\sup_{\Sigma \cap B_R(\bOh)} \left| \check u(\cdot, \tau) - \check a e^{-\mu \tau}\varphi(\cdot) \right| \leq C e^{\big(-\mu + \tfrac{\delta_0}{4}\big)\tau}
\]

holds for all $\tau \leq \bar\tau$, where $\check u$ and $\delta_0>0$ are as in Corollary <ref>.

Recall that from Corollary <ref>  we have that $\check u$ satisfies

\begin{equation}\label{eq:evolution-check-u}
\partial_\tau \check u = L\check u + E
\end{equation}

where $E = \check u E_1 + E_2(\nabla \check u, \nabla \check u)$ satisfy the estimates given in Corollary <ref> and in Lemma <ref> along the conical ends.

We consider a smooth cut-off function $\eta\geq0$ satisfying $\eta(r) =1 $ for $|r|\leq 1, \eta(r) =0$ for $|r|\geq 2, |\eta'|\leq 2$, and $|\eta''| \leq 2$. We define on $\Sigma$

\begin{equation}\label{eq:defn.hat.u}
\hat u(\bx,\tau) = \check u (\bx, \tau) \,\hat\eta(\bx,\tau)\, ,
\end{equation}

where $\hat \eta =1$ on $\Sigma \setminus \Sigma_\textnormal{cyl}$, and $\hat \eta(\bx,\tau) = \eta(e^{\delta_0 \tau} |\bx|)$ on $\Sigma_\textnormal{cyl}$. Then we have

\begin{equation}\label{eq:evolution-hat-u}
\partial_\tau \hat u = L\hat u + \hat E =  L\hat u + \hat u E_1 + E_2(\nabla \hat u, \nabla \check u) + F
\end{equation}

\begin{equation*}\label{eq:error-hat-u}
F = - 2 \nabla \hat \eta \cdot \nabla \check u - \check u\, \Delta \hat \eta + \tfrac{1}{2} \check u\, (\bx^T\cdot \nabla \hat \eta) - \check u\, E_2(\nabla \hat \eta, \nabla \check u) + \check u\, \partial_\tau \hat \eta\, .
\end{equation*}

There exists $\bar\tau = \bar \tau(\check\cM) \ll -1$ so that for  $\tau \leq \bar\tau$

$$ \|F(\cdot,\tau)\|_W \leq C e^{5\delta_0 \tau} \exp \left( -\tfrac{1}{4} e^{-2\delta_0 \tau} \right) \, .$$

Combining Corollary <ref> and Proposition <ref> we have
$ E_2(\nabla \hat \eta, \nabla \check u) \leq C |\nabla \hat \eta| |\nabla \check u|$ and thus

$$ |F(\bx,\tau)| \leq C \|\check u(\cdot, \tau) \|_{C^1(\Sigma_\textnormal{cyl} \cap \{ e^{-\delta_0 \tau} \leq |\bx| \leq 2 e^{-\delta_0 \tau}\})} \chi(\bx, \tau)\, ,$$

where $\chi =1$ for $\Sigma_\textnormal{cyl} \cap \{ e^{-\delta_0 \tau} \leq |\bx| \leq 2 e^{-\delta_0 \tau}\}$ and $\chi = 0$ otherwise. Thus Proposition <ref> yields

$$ \|F(\cdot, \tau)\|_W \leq C e^{4\delta_0 \tau} \int_\Sigma  \chi^2 e^{-\tfrac{1}{4} |\bx|^2} \leq C   e^{4\delta_0 \tau} \int_{e^{-\delta_0\tau}}^\infty e^{-\tfrac{1}{4} r^2}\, dr \leq C e^{5\delta_0 \tau} \exp \left( -\tfrac{1}{4} e^{-2\delta_0 \tau} \right)\, .$$

This completes the proof.

There exists $\bar\tau = \bar \tau(\check\cM) \ll -1$ so that for  $\tau \leq \bar\tau$

$$ \big| \langle \hat u(\cdot,\tau), \hat E(\cdot,\tau) \rangle_W\big| \leq C e^{\delta_0 \tau} \| \hat u(\cdot,\tau)\|_W^2 +  C e^{5\delta_0 \tau} \exp \left( -\tfrac{1}{4} e^{-2\delta_0 \tau} \right) \|\hat u(\cdot, \tau)\|_W\, .$$

Combining Corollary <ref> and Proposition <ref> we find

$$ \big| \langle \hat u(\cdot,\tau), \hat u(\cdot,\tau)\, E_1(\cdot,\tau) \rangle_W\big| \leq C \| \check u(\cdot, \tau) \|_{4}(2 e^{-\delta_0 \tau}) \|\hat u(\cdot, \tau)\|^2_W \leq C e^{\delta_0\tau}  \|\hat u(\cdot, \tau)\|^2_W\, .$$

To control the second error term, we note that using again Corollary <ref>, and Proposition <ref>

\begin{equation*}
\begin{split}
\big| \langle \hat u(\cdot,\tau),\,   &E_2(\nabla \hat u(\cdot,\tau), \nabla \check u(\cdot,\tau)) \rangle_W\big| = \frac{1}{2} \left| \int_\Sigma E_2(\nabla \hat u^2(\cdot,\tau), \nabla \check u(\cdot,\tau))\ e^{-\frac{1}{4} |\bx|^2}\right|\\
& = \frac{1}{2} \left| \int_\Sigma \hat u^2\Div \!\Big({E}^\#_2(\nabla \check u(\cdot,\tau)) \, e^{-\frac{1}{4} |\bx|^2}\Big)\right|\\
& \leq \frac{1}{2}  \left| \int_\Sigma \hat u^2\Div \!\big({E}^\#_2(\nabla \check u(\cdot,\tau))\big) \, e^{-\frac{1}{4} |\bx|^2}\right|
+ \frac{1}{4}  \left| \int_\Sigma \hat u^2 {E}_2(\bx^T,\nabla \check u(\cdot,\tau))\big) \, e^{-\frac{1}{4} |\bx|^2}\right|\\
& \leq C e^{2\delta_0 \tau} \|\hat u\|_W^2 + \frac{1}{4}  \left| \int_\Sigma \hat u^2 {E}_2(\bx^T,\nabla \check u(\cdot,\tau))\big) \, e^{-\frac{1}{4} |\bx|^2}\right|
\end{split}
\end{equation*}

where ${E}^\#_2$ is the $(1,1)$-tensor obtained via the natural isomorphism $\#:T^*\Sigma \to T\Sigma$ induced via $g$, acting on the first entry of $E_2(\cdot,\cdot)$. Note that along the conical ends we have from Lemma <ref> and Proposition <ref> that

$$ |E_2(\bx^T, \nabla \check u(\cdot,\tau))| \leq C |\nabla \check u(\cdot,\tau)| \leq C e^{2\delta_0 \tau}\, , $$

whereas along the cylindrical ends, since $|\bx^T|\leq 2 e^{-\delta_0\tau}$ on the support of $\hat u$, we have with Corollary <ref> and (2) in Proposition <ref> (recalling that $\delta_{0} < \tfrac{|\mu|-\eps}{\alpha}$).

$$ |E_2(\bx^T, \nabla \check u(\cdot,\tau))| \leq C e^{-\delta_0 \tau} |\nabla \check u(\cdot,\tau)| \leq C e^{\delta_0 \tau}\, .$$

This implies

$$\big| \langle \hat u(\cdot,\tau),\,   E_2(\nabla \hat u(\cdot,\tau), \nabla \check u(\cdot,\tau)) \rangle_W\big|  \leq C e^{\delta_0 \tau} \|\hat u\|_W^2\, .$$

Using Lemma <ref> together with

$$\big| \langle \hat u(\cdot,\tau), F(\cdot,\tau) \rangle_W\big| \leq \|\hat u(\cdot,\tau)\|_W \|F(\cdot,\tau)\|_W$$

yields the desired result.

We define orthogonal projections $P_i:L^2_W(\Sigma) \to L^2_W(\Sigma)$ via

$$ P_1(v) := \langle v, \varphi \rangle_W \varphi \qquad \text{and} \qquad P_2(v) := v - P_1(v)\, .$$

There exists $\bar\tau=\bar\tau(\check\cM) \ll -1$ such that for  $\tau \leq \bar\tau$ and $i=1,2$

\begin{equation}\label{eq:spectral-control-error.2}
\big| \langle P_i\hat u(\cdot,\tau), \hat E(\cdot,\tau) \rangle_W\big|  \leq C e^{\delta_0 \tau} \| \hat u(\cdot,\tau)\|_W^2 +  C e^{5\delta_0 \tau} \exp \left( -\tfrac{1}{4} e^{-2\delta_0 \tau} \right) \|\hat u(\cdot, \tau)\|_W\, .
\end{equation}

Due to Lemma <ref> it is sufficient to prove (<ref>) for $i=1$. Note that as in the proof of Lemma <ref> we have

$$ \big| \langle P_1\hat u(\cdot,\tau), \hat u(\cdot,\tau)\, E_1(\cdot,\tau) \rangle_W\big| \leq C e^{\delta_0\tau}  \|P_1 \hat u(\cdot, \tau)\|^2_W $$

$$\big| \langle P_1\hat u(\cdot,\tau), F(\cdot,\tau) \rangle_W\big| \leq C e^{5\delta_0 \tau} \exp \left( -\tfrac{1}{4} e^{-2\delta_0 \tau} \right) \|\hat P_1u(\cdot, \tau)\|_W\, .$$

Furthermore, we have

\begin{equation*}
\begin{split}
\langle P_1\hat u(\cdot,\tau), E_2(\nabla \hat u(\cdot,\tau), \nabla \check u(\cdot,\tau)) \rangle_W &= - \int_\Sigma \hat u( \cdot,\tau) \Div\!\Big(P_1\hat u( \cdot,\tau) \, E_2^\#(\nabla \check u(\cdot,\tau))\ e^{-\frac{1}{4} |\bx|^2}\Big)\\
&=  - \int_\Sigma \hat u( \cdot,\tau) P_1\hat u(\cdot,\tau) \, \Div\!\Big(E_2^\#(\nabla \check u(\cdot,\tau))\ e^{-\frac{1}{4} |\bx|^2}\Big)\\
&\quad - \int_\Sigma \hat u( \cdot,\tau) E_2(\nabla P_1\hat u(\cdot,\tau), \nabla \check u( \cdot,\tau))e^{-\frac{1}{4} |\bx|^2}
\end{split}
\end{equation*}

The first term on the right hand side can be estimates as in the proof of Lemma <ref>

$$\left|\int_\Sigma \hat u( \cdot,\tau) P_1\hat u \, \Div\!\Big(E_2^\#(\nabla \check u(\cdot,\tau))\ e^{-\frac{1}{4} |\bx|^2}\Big)\right| \leq Ce^{\delta_0\tau} \|P_1\hat u(\cdot, \tau)\|_W  \|\hat u(\cdot, \tau)\|_W \, .$$

The second term satisfies

$$ \left|\int_\Sigma \hat u( \cdot,\tau) E_2(\nabla P_1\hat u(\cdot,\tau), \nabla \check u( \cdot,\tau))e^{-\frac{1}{4} |\bx|^2}\right| \leq C e^{\delta_0\tau} \|\nabla P_1 \hat u(\cdot,\tau)\|_W \|\hat u(\cdot,\tau)\|_W\, .$$

$$ \|\nabla P_1 \hat u(\cdot,\tau)\|_W \leq \| \nabla \varphi\|_W   \|P_1 \hat u(\cdot,\tau)\|_W\leq C \|\hat u(\cdot,\tau)\|_W$$

for some constant $C$ depending on $\varphi$, combining these inequalities yields the stated estimate.

We recall that the operator $L$ has a discrete spectrum on $H^1_W(\Sigma)$ and that we denote by $\mu = \mu_1< -1$ the bottom of the spectrum. We write $-1\geq \mu_2>\mu$ for the next eigenvalue.

There exists $\bar\tau = \bar\tau(\check \cM) \ll -1$ so that for  $\tau \leq \bar\tau$ the quantities $\hat U_1:= \|P_1\hat u\|_W^2 + \exp(-\tfrac{1}{4}e^{-2\delta_0\tau})$ and $\hat U_2:= \|P_2\hat u\|_W^2$ satisfy

\begin{align*}
\tfrac{d}{d\tau} \hat U_1 &\geq -2 \mu \hat U_1 - C e^{\delta_0 \tau} (\hat U_1 +\hat U_2)\\
\tfrac{d}{d\tau} \hat U_2 &\leq -2 \mu_2 \hat U_2 + C e^{\delta_0 \tau} (\hat U_1 +\hat U_2)\, .
\end{align*}

We compute

\begin{equation}\label{eq:evol_U1-1}
\tfrac{1}{2} \tfrac{d}{d\tau} \hat U_1 = \langle P_1 \hat u,\partial_\tau \hat u\rangle_W + \tfrac{\delta_0}{2}e^{-2\delta_0\tau} \exp(-\tfrac{1}{4}e^{-2\delta_0\tau})\, ,
\end{equation}

\begin{equation}\label{eq:evol_U1-2}
\langle P_1 \hat u,\partial_\tau \hat u\rangle_W = \langle P_1 \hat u, L \hat u + \hat E\rangle_W = -\mu \|P_1 \hat u\|_W^2 + \langle P_1 \hat u,\hat E\rangle_W\, .
\end{equation}

Applying Lemma <ref> yields the first inequality. Similarly we can obtain

$$ \tfrac{1}{2} \tfrac{d}{d\tau} \hat U_2 = \langle P_2 \hat u,\partial_\tau \hat u\rangle_W =  \langle P_2 \hat u, L \hat u + \hat E\rangle_W  \leq -\mu_2 \|P_2 \hat u\|_W^2 + \langle P_2 \hat u,\hat E\rangle_W\, . $$

Hence Lemma <ref> also yields the second inequality.

There exists $\bar\tau =\bar\tau(\check\cM) \ll -1$ so that for  $\tau \leq \bar\tau$ it holds that

$$\| P_2 \hat u\|^2_W \leq Ce^{\delta_0 \tau} \left( \|P_1\hat u\|^2_W + \exp( -\tfrac{1}{4} e^{-2\delta_0 \tau})\right)\, $$

and $\|\hat u\|_W \leq Ce^{-\mu \tau}$.

Consider again $\hat U_1:= \|P_1\hat u\|_W^2 + \exp(-\tfrac{1}{4}e^{-2\delta_0\tau})$ and $\hat U_2:= \|P_2\hat u\|_W^2$ and define

$$ f(\tau) = e^{(\mu + \mu_2)\tau} \hat U_2 - M e^{(\mu + \mu_2 + \delta_0)\tau} \hat{U}_1\, ,$$

where $M$ is a positive constant to be determined. Proposition <ref> implies that

\begin{equation*}
\begin{split}
f'(\tau) &\leq -(\mu_2-\mu) e^{(\mu+\mu_2)\tau} \hat U_2 - M (\delta_0+\mu_2-\mu) e^{(\mu+\mu_2 +\delta_0)\tau} \hat U_1\\
&\quad + C(1+M e^{\delta_0 \tau}) e^{(\mu+\mu_2+\delta_0)\tau} (\hat U_1 +\hat U_2)\, .
\end{split}
\end{equation*}

We now choose $\bar \tau$ such that

$$ 2Ce^{\delta_0\tau} \leq \mu_2-\mu $$

for $\tau\leq \bar\tau$ and let $M = 2C(\mu_2-\mu)^{-1}$. This implies

$$ f'(\tau) \leq -(\mu_2-\mu) e^{(\mu+\mu_2)\tau} \hat U_2 - 2C e^{(\mu+\mu_2 +\delta_0)\tau} \hat U_1 + 2C e^{(\mu+\mu_2+\delta_0)\tau} (\hat U_1 +\hat U_2) \leq 0\, .$$

By Proposition <ref> we have $\lim_{\tau \to -\infty} f(\tau) =0$ and thus $f(\tau) \leq 0$ for $\tau \leq \bar\tau$, which implies

$$ \hat U_2 \leq 2C (\mu_2-\mu)^{-1} e^{\delta_0 \tau} \hat U_1 $$

for $\tau \leq \bar\tau$. This yields the first estimate.

Returning to Proposition <ref> we thus find

\[
\tfrac{d}{d\tau} \log e^{2\mu \tau}\hat U_1 \geq  - (C+1) e^{\delta_0 \tau}
\]

Because the right hand side is integrable on $(-\infty,\bar\tau]$, this implies that $\hat U_{1}\leq C' e^{-2\mu \tau}$. Combining this with the estimate for $\hat U_{2}$ yields the second estimate.

There exists $\check a \geq 0$, $C>0$, and $\bar{\tau}=\bar \tau(\check\cM) \ll -1$ such that for $\tau \leq \bar{\tau}$

$$\|\hat u - \check a e^{-\mu \tau} \varphi\|_W^2 \leq C e^{(2|\mu| + \delta_0)\tau}\, .$$

As in (<ref>), (<ref>), we find

\[
\left | \tfrac{d}{d\tau} \left(e^{2\mu \tau} \|P_1 \hat u\|^2_W \right)\right| \leq 2 e^{2\mu\tau} \langle P_1 \hat u,\hat E\rangle_W.
\]

Combining Lemma <ref> with Lemma <ref>, we thus find that

$$ \left | \tfrac{d}{d\tau} \left(e^{2\mu \tau} \|P_1 \hat u\|^2_W \right)\right| \leq C e^{\delta_0\tau}\, $$

for $\tau\leq \bar\tau\ll0$. Thus, the limit

$$ \lim_{\tau \to -\infty} e^{2\mu \tau} \|P_1 \hat u\|^2_W = \check a^2$$

exists, and

$$ \left| e^{2\mu \tau} \|P_1 \hat u\|^2_W - \check a^2 \right| \leq C e^{\delta_0 \tau}\, $$

for $\tau\leq \bar\tau\ll0$. Combined with Lemma <ref>, this proves the assertion.

This follows from Theorem <ref> and interpolation.

We emphasize that for fixed $\check \cM$, the constant $\check a$ is the same in both Theorem <ref> and Theorem <ref>.

§.§ Global lower barrier
In this section we continue to consider $(\Sigma,\Omega)\in \cS_n''$ and  $(\check \cM(\tau))_{\tau}$ an ancient (integral unit regular) rescaled Brakke flow with $(\supp \check \cM)_{\tau} \subset \Omega$ for all $\tau < T<0$. Assume that $\check \cM(\tau)$ limits in $C^{\infty}_{\textrm{loc}}$ to $\Sigma$ with multiplicity one as $\tau\to -\infty$.

In this section, we additionally impose the requirement that Theorem <ref> holds with constant $\check a >0$. Because we are concerned with an optimal lower barrier, the case that $\check a = 0$ will be trivially handled by choosing $\bar M(\tau) \equiv \Sigma$ for all $\tau$.

We denote by $\bar v:\Sigma \times (-\infty, \tau_0)\to \RR_+$ the function such that

$$\tau \mapsto \bar M(\tau)=\Graph_\Sigma(\bar v(\cdot,\tau))$$

is the smooth ancient rescaled mean curvature flow on one side of  $\Sigma$ constructed in Proposition <ref>. By shifting time, we can assume that

\begin{equation}\label{eq:entire-sol-cal}
\lim_{\tau \to -\infty} e^{\mu \tau} \langle \bar v(\cdot,\tau),\varphi \rangle_W =1\, .
\end{equation}

Note that this condition together with Proposition <ref> implies uniqueness under the decay assumption (<ref>). We also consider the general solution $\cM$ on one side of $\cM_\Sigma$, and denote the corresponding rescaled flow by $\tau \mapsto M(\tau)$.

Since we assumed that $\check a> 0$, we can shift (rescaled) time in the given flow $\check\cM$ to ensure that $\check a = 1$. In other words, we arrange that

\begin{equation}\label{eq:sol-cal}
\lim_{\tau \to -\infty} e^{\mu \tau} \langle \hat u,\varphi \rangle_W =1\, ,
\end{equation}

where $\hat u$ is the cutoff graphical function defined in (<ref>).

Applying Theorem <ref> to both[We use that the constant $\check a$ is the same in the point-wise estimate and integrated estimate.] the unknown flow $\check\cM$ and the graphical flow $\bar M$, we can choose a sequence $\tau_m \to -\infty$ such that

\begin{equation}\label{eq:lower-barrier}
(1-\tfrac{1}{m}) e^{-\mu\tau_m}\varphi \leq \min\{\bar v(\cdot, \tau_m), \hat u(\cdot, \tau_m)\}
\end{equation}

on $B_{2m}(\bOh) \cap \Sigma$. We consider a smooth cut-off function $\eta$ with values in $[0,1]$ such that

$$ \eta(r) = 1 \quad\text{for}\ r\leq 1, \qquad \eta(r) = 0\quad\text{for}\ r\geq 2\, ,$$

and define $\bar w^m \in C_c^\infty(\Sigma)$ by

$$ \bar w^m(\bx) = (1-\tfrac{1}{m})  e^{-\mu\tau_m} \eta(|\bx|/m)  \varphi(\bx)\, .$$

Recall (cf. Proposition <ref>) that we may assume that $\bar v(\cdot,\tau) \leq C e^{-\mu\tau}$ for some $C>0$ and $\tau \in  (-\infty, \tau_0)$.

There exists $\bar\tau = \bar \tau(\check\cM) \in (-\infty, \tau_0)$ and $C>0$ such that for all $m$ sufficiently large there exists a smooth rescaled ancient mean curvature flow

$$[\tau_m, \bar \tau]\ni \tau \mapsto \bar M^m(\tau)=\Graph_\Sigma(\bar v^m(\cdot,\tau))$$
such that

(1)  $\bar v^m(\cdot,\tau) \in C^\infty(\Sigma)$ and $\bar v^m(\cdot,\tau)\geq 0$ for all $\tau \in [\tau_m, \bar \tau]$,

(2) $\bar v^m(\cdot,\tau_m) = \bar w^m$,

(3) $\bar v^m \leq v$ holds for $\tau \in [\tau_m, \bar \tau]$  ,

(4) $(\supp\check\cM)_{\tau}$ avoids $\bar M^m(\tau)$ in the sense that

\[
(\supp\check\cM)_{\tau} \subset \Omega\setminus\{\bx + s \bar v^{m}(\bx,\tau): \bx \in \Sigma,s\in [0,1]\}
\]

for $\tau \in [\tau_m, \bar \tau]$,

(5) $\|\bar v^m(\cdot, \tau)\|_{C^4(\Sigma)} \leq C e^{-\mu\tau}$ for $\tau \in [\tau_m, \bar \tau]$.

Note that we can apply Corollary <ref> with initial datum $\bar w^m$ to obtain a solution $\tau \mapsto \bar v^m(\cdot, \tau)$ to graphical rescaled mean curvature flow for $\tau \in [\tau_m,\bar \tau_m]$. We note that the Ecker-Huisken maximum principle implies that

\begin{equation}\label{eq:upper-bound-vm}
v^m(\cdot, \tau) \leq v(\cdot, \tau)\leq   C e^{-\mu\tau}
\end{equation}

for some $C>0$ for $\tau \in [\tau_m,\bar \tau_m]$. Corollary <ref> then ensures the existence of a uniform lower bound $\bar \tau\leq \bar\tau_m$, when $m$ is sufficiently large. This establishes points (1) through (3). Point (4) follows from Proposition <ref>. Note that Proposition <ref> implies that $\|\bar v^m(\cdot,\tau)\|_{C^3(\Sigma)}$ is uniformly controlled for all $\tau \in [\tau_n,\bar\tau]$. Point (5) then follows by considering the unrescaled mean curvature flow and applying standard interior parabolic Schauder estimates to the mean curvature flow equation for graphs with controlled, small $C^{1,\alpha}$-norm over $\Sigma$, combined with (<ref>).

Under the assumptions (<ref>) and (<ref>) it holds that $\check\cM(\tau)$ lies above $\tau \mapsto \bar M(\tau)$ for $\tau \in (-\infty, \bar\tau)$ in the sense that

\[
(\supp\check\cM)_{\tau} \subset \Omega \setminus \{\bx + s \bar v(\bx,\tau)\nu_{\Sigma} : \bx \in \Sigma, s \in [0,1)\}.
\]

Recall that from Corollary <ref>  we have that $\bar v^m$ satisfies

\begin{equation}\label{eq:evolution-vm}
\partial_\tau \bar v^m = L\bar v^m + E
\end{equation}

where $E = \bar v^m E_1 + E_2(\nabla \bar v^m, \nabla \bar v^m)$ satisfy the estimates given in Corollary <ref>. Proposition <ref>, (5) then implies

$$ | \partial_\tau \bar v^m - L\bar v^m| \leq C e^{-2\mu\tau}\, .$$

This yields

$$\big| \tfrac{d}{d\tau} \langle \bar v^m, \varphi\rangle_W + \mu  \langle \bar v^m, \varphi\rangle_W \big| \leq Ce^{-2\mu\tau}\, ,$$

and thus for $\tau \in [\tau_m,\bar \tau]$

$$ \left| e^{\mu \tau} \langle \bar v^m(\cdot,\tau), \varphi\rangle_W - e^{\mu \tau_m}\langle \bar w^m, \varphi\rangle_W\right| \leq \int_{\tau_m}^\tau e^{-\mu s}\, ds \leq \int_{-\infty}^\tau e^{-\mu s}\, ds \leq C e^{-\mu \tau}\, .$$

By (<ref>), (<ref>) and (<ref>) we have

$$ \lim_{m\to\infty} e^{\mu \tau_m}\langle \bar w^m, \varphi\rangle_W = 1.$$

Let $\tilde v$ be any subsequential limit of the sequence $\bar v^m$. Note that $(-\infty, \bar\tau) \ni \tau \mapsto \tilde v(\cdot, \tau)$ is a entire graphical solution to rescaled mean curvature flow over $\Sigma$, lying below $\bar v$, which satisfies

$$ \left| e^{\mu\tau}  \langle \tilde v, \varphi\rangle_W  - 1 \right| \leq Ce^{-\mu\tau}$$

as well as

$$ \| \tilde v(\cdot\, \tau)\|_{C^3(\Sigma)} \leq C e^{-\mu\tau}\, .$$

Proposition <ref> thus implies that $\tilde v \equiv \bar v$. The assertion then follows by combining this with (4) in Proposition <ref>.

§.§ Global uniqueness
In this section we consider[If one is content with uniqueness in the ancient past, the condition $\Sigma\in\cS_n''$ suffices. The strictly stable end (as opposed to stable end) condition is only used to apply Proposition <ref> to upgrade uniqueness in the ancient past to uniqueness for all $t<0$. ]   $(\Sigma,\Omega)\in \cS_n'''$ and  $(\check \cM(\tau))_{\tau}$ an ancient (integral unit regular) rescaled Brakke flow with $(\supp \check \cM)_{\tau} \subset \overline \Omega$ for all $\tau < T$. Assume that $\check \cM(\tau)$ limits in $C^{\infty}_{\textrm{loc}}$ to $\Sigma$ with multiplicity one as $\tau\to -\infty$.

Let $\check a$ be given by Theorem <ref>.

We establish the following global uniqueness theorem.

(1)  If $\check a >0 $ then up to a shift in time, $\check \cM$ coincides with the rescaling of the flow constructed in Proposition  <ref> for all $t<0$.

(2)  If $\check a =0 $ then $\check \cM$ coincides with $\cM_\Sigma$ for all $t<0$.

We first consider case (1). By shifting time we can assume that (<ref>) holds. Thus by Proposition <ref> there exists $\bar \tau$, such that for $\tau \in (-\infty,\bar{\tau})$, $\check\cM(\tau)$ lies above[Here, we mean in the sense of Proposition <ref>.] $\bar M(\tau)$, where $\tau \mapsto \bar M(\tau)$ is the  smooth ancient rescaled mean curvature flow on one side of  $\Sigma$ constructed in Proposition <ref>, satisfying (<ref>). We denote with $\bar v:\Sigma \times (-\infty, \tau_0')\to \RR_+$ the function such that
$$\tau \mapsto \bar M(\tau)=\Graph_\Sigma(\bar v(\cdot,\tau))\, .$$

We can assume that $\tau_{0}'\leq \bar \tau$.

Our goal is to prove that there is some $\tau_{0}\in (-\infty,\tau_{0}']$ with $\check \cM(\tau) = \bar M(\tau)$ for all $\tau \in (-\infty,\tau_{0})$. Given this, the uniqueness for all (rescaled) time[and thus all negative non-rescaled time] follows from Proposition <ref>. Below, we will repeatedly (but only finitely many times) take $\tau_{0}$ more negative until we achieve this goal.

Fix $R=R_{1}$ as in Proposition <ref> . Note that Theorem <ref> together with (<ref>) and (<ref>) implies that for $\tau_0\ll 0$,

\begin{equation}\label{eq:first-decay}
\sup_{\Sigma \cap B_{5R}(\bOh)}|\check u(\cdot,\tau) - \bar v(\cdot, \tau)|\leq e^{\big(-\mu+\frac{\delta_0}{4}\big)\tau}
\end{equation}

for $\tau \leq \tau_0$. For $\tau\leq \tau_{0}\ll0$, we find

\begin{equation}\label{eq:ancient-past-defn-delta-tau}
\delta(\tau) := \sup_{\tau' \leq \tau} \|\check u(\cdot, \tau')- \bar v(\cdot, \tau') \|_{4,\alpha; \Sigma\cap B_{4R}(\bOh)} \leq \hat\delta\, ,
\end{equation}

where $\hat \delta$ is as in Proposition <ref>.
By Proposition <ref> we can assume that $\delta(\tau)>0$ for all $\tau \leq \tau_0$.

We now define $w = \check u - \bar v$ (on the domain of definition of $\check u$). Observe that the fact that $\check\cM(\tau)$ lies above $\bar M(\tau)$ implies that $w\geq 0$. We set

\[
W(\tau)^2 : = \sup_{s\leq \tau} \int_{\Sigma\cap B_{5R}(\bOh)} w(\cdot,s)^{2} e^{-\tfrac 14 |\bx|^{2}}.
\]

We define

\begin{equation}\label{eq:definiton-gamma}
\gamma:=\sup\{ \gamma'>0\, |\, W(\tau) \leq e^{(1+2\alpha)\gamma' \tau}\ \text{for}\ \tau\leq \tau_0\}\, .
\end{equation}

We assume (for contradiction) that $\gamma < \infty$ for all $\tau_{0}\ll0$. Note that if for some $\tau_{0}$ fixed, $\gamma=\infty$, then $W(\tau)=0$ for all $\tau\leq \tau_0$. Thus, (<ref>) together with Proposition <ref> implies that $\check \cM(\tau) = \bar{M}(\tau)$ for all $\tau \leq \tau_0$, proving the assertion.

To begin, we note that (<ref>) implies that

\begin{equation}\label{eq:first-decay-W}
W(\tau) \leq C e^{\big(-\mu+\frac{\delta_0}{4}\big)\tau}
\end{equation}

for $\tau \leq \tau_0$. This implies that

\begin{equation}\label{eq:first-estimate-gamma}
\gamma\geq \gamma_0:= (1+2\alpha)^{-1}\left( -\mu+\tfrac{\delta_0}{8}\right) > 0\,
\end{equation}

for $\tau_{0}\ll0$.

We now show derive pointwise estimates on $w$ from the given estimates on $W$. Interior parabolic Schauder estimates imply that there is a constant $C>0$,  such that

\begin{equation}\label{eq:control-delta}
\delta(\tau) \leq C W(\tau)
\end{equation}

for $\tau\leq \tau_0$. Recall the definition of the semi-global norm as in (<ref>)

\[ \Vert \cdot \Vert_{k,\alpha}(\rho)=   \Vert  \cdot \Vert_{k,\alpha;\, \text{cyl}}(\rho) + \Vert  \cdot \Vert_{k,\alpha;\Sigma_{\textnormal{con}}}^{(1)}\, .\]

Proposition <ref> implies that there is a constant $C_0\geq 2$ such that we can choose

\begin{equation}\label{eq:definition-graphical-radius}
\rho(\tau):= C_0^{-1} W(\tau)^{-\frac{1}{1+\alpha}}
\end{equation}

\[ \|w(\cdot, \tau)\|_{4,\alpha;\,\text{cyl}}(\rho(\tau)) \leq C W(\tau)^\theta\]

for any $\theta \in (0,\frac{1}{1+\alpha})$. In particular, by interpolating Lemma <ref> with the estimate that follows by using (<ref>) in Proposition <ref>,

\begin{equation}\label{eq:control-semi-global-decay}
\|w(\cdot, \tau)\|_{4,\alpha}(\rho(\tau)) \leq W(\tau)^{\frac{1}{1+2\alpha}}
\end{equation}

for $\tau\leq \tau_0 \ll0$. We note that

\begin{equation}\label{eq:lower-estimate-rho}
\rho(\tau)\geq 3 e^{-\gamma\tau}\,
\end{equation}

for $\tau_0$ sufficiently negative. This will be useful below.

In the following part of the argument, we will keep decreasing $\tau_0$, but will emphasize that all constants $C$ will be uniformly bounded independently of $\tau_0\to -\infty$ (and in particular will not depend on the value of $\gamma$). This will allow us to derive a contradiction to the assumption that $\gamma < \infty$ (as long as we take $\tau_{0}$ sufficiently negative).

We now show how to convert estimates on $W(\tau)$ into (potentially) optimal estimates on $\check u$. Propostion <ref> ensures that for $\tau\leq \tau_0$, we have

$$ \|\bar v\|_{C^5(\Sigma)}\leq C e^{-\mu\tau}\, ,$$

and combining this with (<ref>) we find

$$ \| \check u\|_{4,\alpha}(\rho(\tau))\leq C e^{\gamma_\mu\tau}\, ,$$

where we set

$$\gamma_\mu:= \min\{-\mu, \gamma\}\, .$$

Using these estimates in Corollary <ref>,  we find that $w$ satisfies

\begin{equation}\label{eq:evolution-w}
\partial_\tau w = L_\Sigma w + E^w
\end{equation}

$$ E^w(\bx) = w(\bx) F(\bx) + \nabla w(\bx) \cdot \bF(\bx) + \nabla^2 w(\bx) \cdot \mathcal{F}(\bx) \, ,$$

with the estimate

$$|F| + |\bF| + |\cF| + |\nabla \cF| \leq C e^{\gamma_\mu\tau}$$

on $\Sigma_\text{con}\cup(\Sigma_\text{cyl}\cap B_{\rho(\tau)}(\bOh))$ for $\tau\leq \tau_0$.

We consider a smooth cut-off function $\eta\geq0$ satisfying $\eta(r) =1 $ for $|r|\leq 1, \eta(r) =0$ for $|r|\geq 2, |\eta'|\leq 2$, and $|\eta''| \leq 2$. Using (<ref>), we can define on $\Sigma$

\begin{equation}\label{eq:defn.hat.w}
\hat w(\bx,\tau) = w(\bx, \tau) \,\hat\eta(\bx,\tau)\, ,
\end{equation}

where $\hat \eta =1$ on $\Sigma \setminus \Sigma_\textnormal{cyl}$, and $\hat \eta(\bx,\tau) = \eta(10^{-1} e^{\gamma \tau} |\bx|)$ on $\Sigma_\textnormal{cyl}$. We introduce

\begin{equation}\label{eq:def-hat-w}
\hat W(\tau) := \sup_{s\leq \tau} \|\hat w(\cdot, s)\|_W
\end{equation}

and note that for $\tau_0$ sufficiently negative Proposition <ref> (as in the proof of Proposition <ref>) combined with (<ref>) implies that

\begin{equation}\label{eq:hat-W-first-decay}
\hat W(\tau) \leq C e^{(1+2\alpha)\gamma\tau}\leq C e^{\big(-\mu +\frac{\delta_0}{8}\big)\tau}
\end{equation}

and we have

$$W(\tau) \leq \hat W(\tau)\ .$$

The evolution of $\hat w$ is given by

\begin{equation}\label{eq:evolution-hat-w}
\partial_\tau \hat w = L_\Sigma \hat w +  \hat E^w =  L_\Sigma \hat w + \hat w F +  \nabla \hat w \cdot \bF + \nabla^2 \hat w \cdot \mathcal{F} + G
\end{equation}

\begin{equation*}\label{eq:error-hat-w}
\begin{split}
G &= - 2 \nabla \hat \eta \cdot \nabla w - w\, \Delta \hat \eta + \tfrac{1}{2} w \, (\bx^T\cdot \nabla \hat \eta) - w \nabla \hat \eta \cdot \bF\,  - (\nabla w \otimes \nabla \hat \eta + \nabla \hat \eta \otimes \nabla w )\cdot \mathcal{F}\\
&\quad - w  \nabla^2 \hat \eta \cdot \mathcal{F} + w\, \partial_\tau \hat \eta\, .
\end{split}
\end{equation*}

Note that $|\nabla \hat \eta|, |\nabla^2\hat\eta| \leq C \chi_A e^{\gamma \tau}$ where $\chi_A$ is the characteristic function of the annulus $A=\{ \bx\, |\,  e^{-\gamma \tau} \leq |\bx| \leq 2 e^{-\gamma \tau}\}$. Note that by (<ref>), $2e^{-2\gamma\tau} < \rho(\tau)$ for $\tau_0$ sufficiently negative. Thus we can estimate using (<ref>) and (<ref>)

$$|G| \leq C \chi_A e^{\gamma \tau} \big((|w| + |\nabla w|) + (1 + \gamma) |\bx| |w|)\big)\leq C  \chi_A .$$

We emphasize that the final constant $C$ is independent of $\gamma$ (which depends on $\tau_0$). Indeed,

\[
\chi_A  e^{\gamma \tau}\gamma  |\bx| |w| = \chi_A ((|\bx| e^{\gamma \tau})(\gamma |w|))\leq \chi_A (2\gamma e^{\gamma \tau})
\]

Since $\tau_0 < 0$, $\gamma e^{\gamma \tau}$ is bounded independently of $\gamma>0$, so $C$ can be chosen as claimed.

Because $e^{-\frac{|\bx|^{2}}{4}}\leq \exp\big(-\tfrac{1}{4}e^{-2\gamma \tau}\big)$ on $A$, we can derive the following coarse estimates

\begin{align}
|\langle \hat w, G \rangle_W| & \leq e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau} \label{eq:estimate-error.1} \\
\|G\|_W^2 & \leq e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau}\,,\label{eq:estimate-error.2}
\end{align}

valid for $\tau\leq \tau_{0}\ll0$.

Combining (<ref>), (<ref>), (<ref>) with Corollary <ref> and with integration by parts (off of the $\nabla^{2}\hat w$ term),  we see that[To handle the error terms arising after integrating by parts, we can use the estimate for $\nabla \cF$ as well as (<ref>) in Corollary <ref> (to control the terms when the derivative hits the Gaussian weight).  ]

\begin{equation}\label{eq:estimate-error.3}
|\langle \hat w, \hat E^w \rangle_W| \leq C e^{\gamma_\mu\tau} \|\hat w\|^2_{W,1} + e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau}\, .
\end{equation}

Thus we can compute

\begin{equation}\label{eq:estimate-evol-hat-w.0}
\tfrac{1}{2}\tfrac{d}{d\tau}\|\hat w\|^2_W = \langle \hat w, L_\Sigma \hat w  + \hat E^w\rangle_W \leq -\mu \|\hat w\|^2_W +  C e^{\gamma_\mu\tau} \|\hat w\|^2_{W,1} + e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau}\, ,
\end{equation}

and therefore

\begin{equation}\label{eq:estimate-evol-hat-w.1}
\tfrac{1}{2}\tfrac{d}{d\tau}\big(e^{2\mu\tau}\|\hat w \|^2_W\big)  \leq   C e^{(2\mu + \gamma_\mu)\tau} \|\hat w\|^2_{W,1} + e^{(2\mu + 2(1+2\alpha)\gamma + \gamma_{\mu})\tau}\, .
\end{equation}

We recall from <cit.> the identity

\begin{multline}\label{eq:integration-by-parts-hessian-laplacian}
\int_\Sigma (\Delta_\Sigma \hat w -\tfrac{1}{2} \bx \cdot \nabla_\Sigma \hat w)^2 e^{-\tfrac 14 |\bx|^{2}} d\cH^n \\ = \int_\Sigma (|\nabla^2_\Sigma \hat w|^2 - A^2(\nabla_\Sigma \hat w, \nabla_\Sigma \hat w) + \tfrac{1}{2}|\nabla_\Sigma\hat w|^2)\, e^{-\tfrac 14 |\bx|^{2}} d\cH^n
\end{multline}

and the identity following on that, which in the present situation can be written as

\begin{equation*}
\begin{split}
\tfrac{1}{2}\tfrac{d}{d\tau}\|\nabla_\Sigma \hat w\|_W^2 &=  - \Vert \Delta_\Sigma \hat w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma \hat w \Vert_W^2 + \Vert (\tfrac12 + |A_\Sigma|^2)^{\frac12} \nabla_\Sigma \hat w \Vert_W^2\\ & \qquad  + \langle \nabla_\Sigma \hat w, \hat w \, \nabla_\Sigma |A_\Sigma|^2 \rangle_W
- \langle \Delta_\Sigma \hat w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma \hat w, \hat E^w \rangle_W.
\end{split}
\end{equation*}

We can thus estimate, using (<ref>), (<ref>), (<ref>), as follows

\begin{equation}\label{eq:estimate-evol-hat-w.2}
\begin{split}
\tfrac{1}{2}\tfrac{d}{d\tau}\|\nabla_\Sigma \hat w\|_W^2 &\leq  - \tfrac{1}{2} \Vert \Delta_\Sigma \hat w - \tfrac12 \mathbf{x} \cdot \nabla_\Sigma \hat w \Vert_W^2 + \tfrac{1}{2}\|\hat E^w\|_W^2 + C \|w\|_{W,1}^2\\
&\leq  -\tfrac{1}{2} \Vert \nabla^2_\Sigma \hat w\Vert_W^2 + 2\| \nabla^2 \hat w \cdot \mathcal{F}\|_W^2 + C \|w\|_{W,1}^2 + e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau}\\
&\leq C \|w\|_{W,1}^2 + e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau}
\end{split}
\end{equation}

Alternatively, we can integrate by parts rather than using the spectral estimate in (<ref>) to find (decreasing $\tau_0$ if necessary and using (<ref>))

\begin{equation}\label{eq:estimate-evol-hat-w.3}
\begin{split}
\tfrac{1}{2}\tfrac{d}{d\tau}\|\hat w\|^2_W &= - \Vert \nabla_\Sigma \hat w \Vert_W^2 + \langle \hat w, (\tfrac12 + |A_\Sigma|^2)\hat w + \hat E^w \rangle_W \\
& \leq -  \Vert \nabla_\Sigma \hat w \Vert_W^2 + C \Vert \hat w \Vert_W^2 + C e^{\gamma_\mu\tau} \|\hat w\|^2_{W,1} + e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau}\\
&\leq - \tfrac12 \Vert \nabla_\Sigma \hat w \Vert_W^2 + C \Vert \hat w \Vert_W^2 + e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau}\, .
\end{split}
\end{equation}

Combining (<ref>) and (<ref>) yields

\begin{equation}\label{eq:final-unique-W1-ode}
\tfrac{d}{d\tau} (\|\nabla_\Sigma \hat w\|_W^2 + C \|\hat w\|^2_W)\leq C \Vert \hat w \Vert_W^2 + C e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau}\leq C \hat W(\tau)^2 + C e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau}
\end{equation}

By (<ref>) find that

\[ \hat W(\tau)^2 \leq C e^{2(1+2\alpha) \gamma \tau} \, ,\]

for $\tau\leq \tau_0 \ll0$, so by integrating (<ref>), we find

\[ \|\hat w(\cdot, \tau)\|^2_{W,1} \leq C \gamma^{-1} e^{2(1+2\alpha)\gamma\tau} + C \gamma^{-1} e^{(2(1+2\alpha)\gamma + \gamma_{\mu})\tau} \leq C e^{2(1+2\alpha)\gamma\tau} \, \]

for $\tau\leq \tau_{0}$. In the final inequality we used $\gamma_{\mu}>0$ and $\gamma^{-1} \leq \gamma_{0}^{-1}<\infty$ (where we emphasize that $\gamma_{0}$ is fixed in (<ref>) and thus is independent of $\tau_{0}$).

We can now return to (<ref>) to find

\[
\tfrac{1}{2}\tfrac{d}{d\tau}\big(e^{2\mu\tau}\|\hat w \|^2_W\big)  \leq C e^{(2\mu + 2(1+2\alpha)\gamma + \gamma_{\mu})\tau}\, .
\]

Note that $2\mu + 2(1+2\alpha)\gamma \geq \tfrac{\delta_0}{4} > 0$, so we can integrate this to find

\[ \|\hat w(\cdot, \tau)\|^2_{W} \leq  e^{\big(2(1+2\alpha)\gamma +\frac{\gamma_\mu}{2}\big)\tau} \]

for $\tau \leq \tau_{0}\ll0$. Thus,

\[W(\tau) \leq   e^{\big((1+2\alpha)\gamma +\frac{\gamma_\mu}{4}\big)\tau}\]

for $\tau \leq \tau_0$. But this contradicts the extremality of $\gamma$ in (<ref>) (for $\tau_{0}$ fixed sufficiently negative so that each step above is justified) and thus $\gamma = \infty$ and $W(\tau) \equiv 0$ for $\tau \leq \tau_0$.

In case (2) we can argue as above, replacing $\tau \mapsto \bar{M}(\tau)$ by the static flow $\tau \mapsto \Sigma$.

Application to mean curvature flow in $\RR^3$

§ GENUS DROP FOR ANCIENT ONE-SIDED FLOWS IN $\RR^{3}$

In this section we consider $(\Sigma,\Omega) \in \cS_{2}^*\setminus \cS^\textrm{gen}_2$. In other words, we consider $\Sigma$ a non-flat self-shrinker in $\RR^3$ that is not a sphere or cylinder, and $\Omega$ a chosen side of $\Sigma$. Recall that by Proposition <ref> it holds that $\Sigma \in \cS_{2}'''$.

Consider the (one-sided) ancient weak set flow $\cK$ and associated Brakke flow $\cM$ as considered in Proposition <ref> (cf. Corollary <ref>). The next lemma follows by combining <cit.> with Proposition <ref>.

$\partial\cK(t)$ is smooth for a.e., $t<0$.

In this section we will show that $\partial\cK(t)$ loses genus as $t\to 0$ (recall that Brendle has shown [10] that $\Sigma$ has positive genus). We will later use this in the global perturbation argument to ensure that we only need to perturb the initial conditions finitely many times.

We recall the following definitions:

If $M$ is an oriented compact connected surface without boundary, decomposing $M = \cup_{\alpha} M _{\alpha}$ into connected components, then we set

\[
\genus(M) : = \sum_{\alpha} \genus(M_{\alpha}) = \tfrac 12 b_{1}(M).
\]

If $M^2$ is an oriented compact surface with boundary, the genus of $M$ is defined to be the genus of the oriented surface $\tilde M$ formed by capping off each  circle in $\partial M$ by a disk.

For $U\subset \RR^3$ an open set with smooth boundary, a simple flow in $U\times I$ is a closed subset of space-time $\cM \subset \RR^{3}\times \RR$ so that there is a compact $2$-manifold with boundary and a continuous map $f: M \times I \to \RR^3$ so that

* $\cM(t) \cap \overline U= f(M\times \{t\})$ (where $\cM(t) = \{\bx \in \RR^3 : (\bx,t) \in \cM\}$)

* $f$ is smooth on $M^\circ \times I$ where $M^\circ = M\setminus\partial M$

* $f(\cdot,t)$, $t\in I$ is an embedding of $M^\circ$ into $U$

* $t\mapsto f(M^\circ\times \{t\})$ is a smooth mean curvature flow

* $f|_{\partial M\times I}$ is a smooth family of embeddings of $\partial M$ into $\partial U$.

We can now state the main result.

Let $(\Sigma,\Omega)$, $\cK$ be as above. There is $t_{0}<t_{1}<0$, $\kappa_{0} \in (0,t_{1}-t_{0})$, $\kappa_{1} \in (0,-t_{1})$ and $R>0$ with the following properties.

* for any $t \in [t_{0},t_{0}+\kappa_{0}]$ so that $\partial\cK(t)$ is smooth, $\genus(\partial\cK(t) \cap B_{R}(\bOh)) > 0$,

* for any $t \in [t_{1},t_{1}+\kappa_{1}]$ so that $\partial\cK(t)$ is smooth, $\genus(\partial\cK(t) \cap B_{R}(\bOh)) = 0$,

* $t\mapsto \partial\cK(t)$ is a simple flow in $\{R/2 <  |\bx| < 3R\} \times [t_0,t_1+\kappa_{1}]$,

* and
for $(\bx,t) \in \partial\cK(t) \cap \{R/2 < |\bx| < 3R\}\times  [t_0,t_1+\kappa_{1}]$, we have $|\bx^T| \geq \tfrac 12 |\bx|$, for $\bx^T$ the projection of $\bx$ to $T_\bx\partial\cK(t)$.

We first sketch the idea of the proof. For simplicity let us assume that $\Sigma$ is compact (in reality one must be somewhat careful with the cylindrical ends, but the main idea  is the same). Write $\Omega(\tau)$ for the inside of the rescaled one-sided flow (so $\Omega(\tau) \subset \Omega(\tau') \subset \Omega$ for $\tau > \tau'$). Suppose that $\partial\Omega(\tau)$ has non-zero genus for all $\tau$ large. Note that $d(\Sigma, \Omega(\tau))$ is exponentially increasing as $\tau\to\infty$ (since the distance between the non-rescaled flows is non-decreasing).

Thus, the non-trivial situation is when $\Omega$ is the unbounded component of $\RR^3\setminus\Sigma$. If $\partial\Omega(\tau)$ has positive genus, then by a result in algebraic topology (cf. Lemma <ref> below), there is a non-zero element $[\sigma] \in H_1(\Omega(\tau))$. Because the flows are shrinker mean convex, we find $\sigma \subset \Omega$ and $d(\sigma,\Sigma)$ is arbitrarily large. However, it is easy to see that when $d(\sigma,\Sigma)$ is sufficiently large, then $[\sigma] = 0 \in H_1(\Omega)$. This contradicts White's topological monotonicity result <cit.>, which says that a $1$-cycle in the complement of a ($2$-dimensional) weak set flow cannot suddenly cease to bound a $2$-chain.

The argument used here avoids the need for a potentially delicate analysis of the star-shaped properties of the flow (along cylindrical ends) near $t=0$ (cf. <cit.>).

The following is a direct consequence of Lemma <ref>.

We can take $R_0$ sufficiently large so that for $R\geq R_0$, $\Sigma$ intersects $\partial B_R(\bOh)$ transversely and $\Sigma \setminus B_R(\bOh)$ (resp. $\Omega \setminus B_R(\bOh)$) is diffeomorphic to $\Sigma \cap \partial B_R(\bOh) \times [R,\infty)$ (resp. $\Omega \cap \partial B_R(\bOh)\times [R,\infty)$).

Note that $\Sigma \cap \partial B_R(\bOh)$ is a finite union of pairwise disjoint embedded loops in $\partial B_R(\bOh) \approx \SS^2$ and $\Omega\cap\partial B_R(\bOh)$ is a finite union of regions in $\partial B_R(\bOh)$ bounded by said loops.

There is $r >0$ and $s_{1}<0$ so that $\cK(t) \cap B_{r}(\bOh) = \emptyset$ for $t \in [s_{1},0)$.

This follows from (1) in Proposition <ref>.

As such, taking $t_1 \in [s_{1},0)$ sufficiently close to $0$, we find that for all $t \in [t_{1},0)$,

\begin{equation}\label{eq:genus.drop.defn.t1}
\tfrac{1}{\sqrt{-t}}\cK(t) \cap B_{R_0}(\bOh) = \emptyset,
\end{equation}

wheres $R_0$ is fixed in Lemma <ref> above. This fixes $t_{1}$. Now we fix an arbitrary $\kappa_{1} \in (t_{1},0)$.

There is $t_{0}<t_{1}$ sufficiently negative, $\kappa_{0} \in (0,t_{1}-t_{0})$ and $R_{1}\geq R_{0}$ so that for any $R\geq R_{1}$ and $t \in [t_{0},t_{0}+\kappa_{0})$, $\tfrac{1}{\sqrt{-t}}\partial\cK(t)$ is a smooth surface meeting $\partial B_{R}(\bOh)$ transversally with $\genus(\tfrac{1}{\sqrt{-t}} \partial\cK(t) \cap B_{R}(\bOh)) > 0$ and $\tfrac{1}{\sqrt{-t}}\partial\cK(t) \setminus B_{R}(\bOh)$ is diffeomorphic to $(\tfrac{1}{\sqrt{-t}}\partial\cK(t) \cap \partial B_{R}(\bOh))\times [R,\infty)$

Combine Lemmas <ref> and <ref>, property (4) in Proposition <ref>, and <cit.>.

There is $R_{2}\geq R_{1}$ sufficiently large so that:

* For $R \geq R_{2}$ and $t \in [t_{0},t_{1}+\kappa_{1}]$ we have that $\tfrac{1}{\sqrt{-t}} \partial\cK(t)$ is smooth in a neighborhood of $\partial B_{R}(\bOh)$ and these surfaces have transverse intersection.

* For any $t > t_{0}$, each component of $\overline{(\tfrac{1}{\sqrt{-t}}\cK(t) \setminus \tfrac{1}{\sqrt{-t_0}}\cK(t_{0}))} \cap \partial B_{R}(\bOh)$
is an annulus.

By (3) in Proposition <ref> there is $R(t)$ monotonically increasing on $(-\infty,0)$ so that

\[
\partial \cK(t) \setminus B_{R(t)}(\bOh)
\]

is a smooth strictly shrinker mean convex flow that can be written as a small $C^{3}$-graph over a subset of $\Sigma$. Take $R_{2} > \max_{t\in[t_{0},t_{1}+\kappa_{1}]}\tfrac{1}{\sqrt{-t}}R(t) = \tfrac{1}{\sqrt{-(t_{1}+\kappa_{1})}} R(t_{1}+\kappa_{1})$. This yields (1). To prove (2), we combine the above reasoning with the geometric reformulation of shrinker mean convexity: $\tfrac{1}{\sqrt{-t}} \cK(t) \subset \tfrac{1}{\sqrt{-s}} \cK(s)$ for $t>s$.

Loosely speaking, the next lemma shows that if for a time $t$ close to zero $(-t)^{-1/2} \partial\cK(t)$ still has genus in a large ball, then this can be detected by a homologically nontrivial  loop in $(-t)^{-1/2} \cK(t)$ (relative to the homology created by the ends of the shrinker) that was trivial in the ancient past. Note that we must take care to avoid loops that are homologically nontrivial only because they wrap around the ends of the shrinker/flow.

Suppose  that for some $t\in [t_{1},t_{1}+\kappa_{1}]$ and $R\geq R_{2}$ it holds that $\partial \cK(t)$ is smooth and $\genus(\tfrac{1}{\sqrt{-t}} \partial\cK(t)\cap B_{R}(\bOh)) > 0$. Then, there is a loop $\sigma \subset \tfrac{1}{\sqrt{-t}} \cK(t)\cap B_{R}(\bOh)$ so that

* for any $1$-cycle  $\beta \subset \partial B_{R}(\bOh)\cap \tfrac{1}{\sqrt{-t}}\cK(t)$, it holds that  $[\sigma] \neq [\beta]$ in $H_{1}(\tfrac{1}{\sqrt{-t}} \cK(t)\cap \overline{B_{R}(\bOh)};\ZZ)$, but

* there is a $1$-cycle $\beta \subset \partial B_{R}(\bOh)\cap \tfrac{1}{\sqrt{-t_{0}}}\cK(t_{0})$ so that $[\sigma] = [\beta]$ in $H_{1}(\tfrac{1}{\sqrt{-t_{0}}}\cK(t_{0})\cap \overline{B_{R}(\bOh)};\ZZ)$.

Applying Lemma <ref> to the set $\tilde \Omega$ formed by smoothing out the corners of $\tfrac{1}{\sqrt{-t}} \cK(t) \cap B_{R+\eps}(\bOh)$ and ball $B=B_{R}(\bOh)$ yields $\sigma \subset  \tfrac{1}{\sqrt{-t}} \cK(t)\cap B_{R}(\bOh)$ satisfying the first bullet point (since
\genus(\partial\tilde\Omega \cap B_{R}(\bOh)) > 0 $
by assumption).

Now, we observe that by shrinker mean convexity and (<ref>)

\[
\sigma \subset \tfrac{1}{\sqrt{-t_{0}}} \cK(t_{0}) \cap (B_{R}(\bOh)\setminus B_{R_{0}}(\bOh)).
\]

By Lemma <ref> we have

\[
\tfrac{1}{\sqrt{-t_{0}}} \cK(t_{0}) \cap (\overline{B_{R}(\bOh)}\setminus B_{R_{0}}(\bOh))
\]

deformation retracts to

\[
\tfrac{1}{\sqrt{-t_{0}}} \cK(t_{0})\cap \partial B_{R}(\bOh)
\]

so the second bullet point follows.

Recall that $\partial\Omega = \Sigma$. For some $R\geq R_{2}$ fixed, suppose that there is a loop $\sigma \subset \Omega \cap B_{R}(\bOh)$ and a $1$-cycle $\beta \subset  \Omega \cap \partial B_{R}(\bOh)$ so that $[\sigma] = [\beta]$ in $H_{1}(\Omega \cap \overline{B_{R}(\bOh)};\ZZ)$. Then, there is a connected smooth embedded self-shrinker $\Gamma\subset \Omega$ with $\partial \Gamma = \sigma$. Moreover,

\[
\sqrt{0} \Gamma \cap \partial B_{1}(\bOh) \qquad \textrm{and} \qquad \sqrt{0} \Sigma \cap \partial B_{1}(\bOh)
\]

are disjoint.

As in <cit.>,  we can choose a sequence of Riemannian metrics $g_{k}$ on $\RR^{n+1}$ converging in $C^{\infty}_{\textrm{loc}}(\RR^{n+1})$ to $e^{-\frac{|\bx|^{2}}{2n}}g_{\RR^{n+1}}$ so that $\Omega \cap B_{2k}(\bOh)$ is (weakly) shrinker mean convex in the sense of Meeks–Yau <cit.>. By Lemma <ref>, we can choose $\beta_{k} \subset \Omega \cap \partial B_{2k}(\bOh)$ homologous to $\beta$ in $\Omega$.

Let $\Gamma_{k}$ minimize $g_{k}$-area among oriented embedded surfaces $\Gamma' \subset \Omega$ with $\partial\Gamma' = \sigma - \beta_{k}$. Using standard interior regularity for minimizers (and [48]) we can pass a subsequence and obtain a connected smooth embedded self-shrinker with $\Gamma\subset \Omega$ (thanks to the strong maximum principle) and $\partial\Gamma = \sigma$. The statement about the asymptotic cones follows from Ilmanen's avoidance principle (Theorem <ref>) as in Lemma <ref>. Indeed, for $t<0$, $t\mapsto \sqrt{-t}\Gamma,t\mapsto \sqrt{-t}\Sigma$ are disjoint mean curvature flows in any small ball around any point in $\partial B_{1}(\bOh)$. Applying Ilmanen's avoidance principle, we find that they remain a definite distance apart at $t=0$ (in a smaller ball). This completes the proof.

Fix $\sigma,\Gamma$ as in Lemma <ref>. Assume that $\sigma \subset\frac{1}{\sqrt{-t'}} \cK(t')$ for some $t'<0$. Then $\Gamma \subset\frac{1}{\sqrt{-t'}} \cK(t')$

Combining Lemma <ref> with the sublinear growth estimates in Proposition <ref> applied to $\cM(t)$ (recall that $\cM(t)$ is the one-sided Brakke flow with $\supp\cM(t) = \partial\cK(t)$), we have that

\[
d((\tfrac{1}{\sqrt{-t}} \partial\cK(t)) \setminus B_{R}(\bOh), \Gamma \setminus B_{R}(\bOh)) \to \infty
\]

as $R\to\infty$, uniformly for $t\leq t'$. Using this,   together with the fact that $\frac{1}{\sqrt{-t}} \cK(t)$ exhausts $\Omega$ as $t\to -\infty$ and the avoidance principle, we find that $(\tfrac{1}{\sqrt{-t}} \partial\cK(t)) \cap \Gamma = \emptyset$ for all $t\leq t'$. This proves the assertion (because $\Gamma$ is connected, if there was a point in $\Gamma \setminus \tfrac{1}{\sqrt{-t}}\cK(t)$ then there would be a point in $\Gamma \cap \tfrac{1}{\sqrt{-t}}\partial \cK(t)$).

With these preparations we can now prove the genus drop result.

Fix $t_{1},\kappa_{1}$ as in (<ref>) and $t_{0},\kappa_{0}$ as in Lemma <ref>. Then, as long as we fix $R\geq \tfrac{1}{\sqrt{-(t_{1}+\kappa_{1})}}R_{2}$ sufficiently large, we can ensure that (3) and (4) hold by using stability of cylinders (Lemma <ref>) and pseudolocality (along the conical ends).

Property (1) follows from Lemma <ref> (with $\sqrt{-t}R \geq R_{1}$ in place of $R$). To prove property (2), we assume that for some $t \in [t_{1},t_{1}+\kappa_{1}]$, $\partial\cK(t)$ is smooth and $\genus(\partial\cK(t)\cap B_{R}(\bOh))>0$. Hence,

\[
\genus(\tfrac{1}{\sqrt{-t}} \partial\cK(t)\cap B_{R'}(\bOh)) > 0
\]

for $R' = \tfrac{1}{\sqrt{-t}} R \geq R_{2}$. Fix $\sigma$ as in Lemma <ref> (with $R'$ in place of $R$) and then $\Gamma$ as in Lemma <ref>. By Corollary <ref> (with $t=t'$) we find that

\[
\Gamma \subset \tfrac{1}{\sqrt{-t}} \cK(t).
\]

We can perturb $\Gamma$ slightly to find a surface with $\Gamma \subset \tfrac{1}{\sqrt{-t}} \cK(t)$ and $\Gamma$  transversal to $\partial B_{R'}(\bOh)$. This yields

\[
\sigma = \partial (\Gamma \cap B_{R}(\bOh)) - \beta
\]

for some $\beta \subset (\tfrac{1}{\sqrt{-t}} \cK(t)) \cap \partial B_{R'}(\bOh)$. This contradicts the first bullet point in Lemma <ref>, completing the proof.

§ THE FIRST NON-GENERIC TIME FOR FLOWS IN $\RR^3$

Consider $\cM$ a cyclic integral unit-regular Brakke flow of surfaces in $\RR^3$.

* We define $\sing_\textnormal{gen}\cM\subset \sing\cM$ to be the set of singular points for $\cM$ so that one tangent flow (and thus all [33])  is a multiplicity-one shrinking sphere or cylinder.

* Then, we define $T_\textrm{gen}(\cM) : = \inf \ft(\sing\cM \setminus \sing_\textrm{gen}\cM)$, i.e., the first time that there is a “non-generic” singularity.

* Finally, we set $\fS_\textrm{first}(\cM) : = (\sing \cM \setminus \sing_\textrm{gen}\cM) \cap \{t=T_\textrm{gen}(\cM)\}$, i.e., this is the set of “non-generic” singularities at the first time that they exist.

The following is our main result.

Suppose that $M\subset \RR^3$ is a closed embedded surface. Then, there exist arbitrarily small $C^\infty$ graphs $M'$ over $M$ and corresponding cyclic integral unit-regular Brakke flows $\cM'$ with $\cM'(0) = \cH^2\lfloor M'$ so that either:

* $\cM'$ has only generic singularities: $\sing\cM' = \sing_\textnormal{gen}\cM'$, or

* multiplicity occurs: there is $\bx \in \RR^3$ so that any tangent flow to $\cM'$ at the space-time point $(\bx,T_\textnormal{gen}(\cM'))$ is $k\cH^2\lfloor \sqrt{-t}\Sigma$ for $k\geq 2$ and $\Sigma \in \cS_2$ a smooth shrinker.

Compare with <cit.> which also left open the possibility that a non-cylindrical shrinker with a cylindrical end could occur. The proof here follows a similar strategy (granted the stronger one-sided uniqueness result Theorem <ref>)  but differs in some minor, but technically important ways.

In this section we will establish various preliminary results that lead up to the proof of Theorem <ref>.

If $\cM,\cM'$ are cyclic integral unit-regular Brakke flows in $\RR^3$ with $\cM(0) = \cM'(0) = \cH^2\lfloor M$ for a closed embedded surface $M\subset \RR^3$ then $T_\textnormal{gen}(\cM) =T_\textnormal{gen}(\cM')$, $\cM(t) = \cM'(t)$ for $t< T_\textnormal{gen}(\cM)$, and $\fS_\textnormal{first}(\cM) = \fS_\textnormal{first}(\cM')$.

By definition of $T_\textnormal{gen}$ and <cit.>, we find that $\reg \cM \cap \{t < T_\textrm{gen}(\cM)\}$ is connected. Thus, the assertion follows from unit-regularity and the resolution of the mean convex neighborhood conjecture <cit.> (cf. [53]).

Consider  $M \subset \RR^3$ a closed embedded surface. Let $U$ denote the unique compact region bounded by $M$. Write $M(t)$ for the level set flow of $M$ and $U(t)$ for the level set flow of $U$.

Consider a cyclic integral unit-regular Brakke flow $\cM$ in $\RR^3$ with $\cM(0) = \cH^2\lfloor M$. We say
the set of times $\cT_\textrm{reg}(\cM)$ is the complement of the set of singular times $\ft(\sing\cM)$. Note that by upper semi-continuity of density $\cT_\textrm{reg}(\cM)$ is open. Furthermore, for $t\in \cT_\textrm{reg}(\cM)$ the level set flow $M(t)$ of $M$ at time $t$ is a smooth embedded surface $M(t)$ with $M(t) = \partial U(t)$ and $\cM(t) = \cH^2\lfloor M(t)$.

The next result follows from combining several important results in the literature (cf. <cit.>).

Consider a cyclic integral unit-regular Brakke flow $\cM$ in $\RR^3$ with $\cM(0) = \cH^2\lfloor M$ for a closed embedded surface $M \subset \RR^3$. Write $U$ for the unique compact set bounded by $M$ and $U(t)$ for the level set flow of $U$. Then:

* the level set flow $M(t)$ of $M$ does not fatten before $T_\textnormal{gen}(\cM)$,

* almost every $t\in [0,T_\textnormal{gen}(\cM)]$ is a regular time for the flow, and

* $t\mapsto \genus(M(t))$ is non-increasing for $t \in \cT_\textnormal{reg}$.

Now, assume that $T_\textnormal{gen}(\cM) <\infty$. Then:

* Every tangent flow at $(\bx,T_\textnormal{gen}(\cM)) \in \sing\cM$ is of the following form for $t<0$

\[
t\mapsto k \cH^2\lfloor\sqrt{-t}\Sigma
\]

where $k \in \NN$ is the multiplicity and $\Sigma$ a smooth embedded self-shrinker and

* the self-shrinker $\Sigma$ in the previous bullet point satisfies

\[
\genus(\Sigma) \leq \lim_{\cT_\textrm{reg} \ni t \nearrow T_\textnormal{gen}(\cM)}\genus(M(t)) .
\]

and $\Sigma \in \cS_2''$ has nice stable ends.

Finally, assume that for some point $(\bx,T_\textnormal{gen}(\cM)) \in \fS_\textnormal{first}(\cM)$ there is some tangent flow of the form $t\mapsto \cH^2\lfloor \sqrt{-t}\Sigma$, i.e., the tangent flow is multiplicity one. Then:

* all tangent flows at $(\bx,T_\textnormal{gen}(\cM))$ occur with multiplicity-one—possibly with a different $\Sigma$,

* $\genus(\Sigma) \geq 1$, and

* $F(\Sigma) \geq \lambda_1 + \delta_0$ for some numerical constant $\delta_0$.

By <cit.> the level set flow of $M$ does not fatten before $T_\textrm{gen}(\cM)$. Hence, by <cit.> (and <cit.>), we have that $M(t)$ is smooth and $\cM(t) = \cH^2\lfloor M(t)$ for almost every $t \in [0,T_\textrm{gen}(\cM)]$. Combined with <cit.>, this implies that almost every $t \in [0,T_\textrm{gen}(\cM)]$ is a regular time for the flow. Now, by <cit.>, we have that $t\mapsto \genus(M(t))$ is non-increasing for $t \in \cT_\textrm{reg}$. This verifies the first set of bullet points.

The second set of bullet points follows from a straightforward adaptation of [56] to avoid the singular time-slices (for $\Sigma \in\cS_2''$, see Proposition <ref>). Finally, for the third set of bullet points, the first one is proven in Proposition <ref>. The second follows from <cit.> and the third from <cit.>. This completes the proof.

It is convenient to define the quantity

\[
\genus_{T_\textnormal{gen}}(\cM) : = \lim_{\cT_\textrm{reg} \ni t \nearrow T_\textnormal{gen}(\cM)}\genus(M(t)) .
\]

We will refer to this below. Note that by Proposition <ref> we have $\genus_{T_\textnormal{gen}}(\cM) \leq \genus M$. Moreover, by Lemma <ref> and Proposition <ref> we find that the quantities $T_\textrm{gen}(\cM)$ and $\genus_{T_\textnormal{gen}}(\cM)$ depend only on the initial surface $M$, not the chosen Brakke flow $\cM$ (by <cit.> a flow $\cM$ as above always exists). As such, we will sometimes write $T_\textrm{gen}(M)$ and $\genus_{T_\textnormal{gen}}(M)$.

The following lemma will be important later.

Consider $\cM$ a cyclic integral unit-regular Brakke flow in $\RR^3$ with $\cM(0)  = \cH^2\lfloor M$ a closed embedded surface $M\subset \RR^3$. Assume that $T_\textnormal{gen}(\cM) < \infty$ and at any $(\bx,T_{\textnormal{gen}}(\cM)) \in \fS_\textnormal{first}(\cM)$, all tangent flows occur with multiplicity one. Then $\fS_\textnormal{first}(\cM)$ is a non-zero finite set.

It is clear that $\fS_\textnormal{first}(\cM)$ is non-zero by definition of the first non-generic time. We claim that $\fS_\textnormal{first}(\cM)$ is closed. Indeed, if a sequence $(\bx_j,T_\textrm{gen}(\cM)) \in \fS_\textnormal{first}(\cM)$ has $(\bx_j,T_\textrm{gen}(\cM)) \to (\bx_\infty,T_\textrm{gen}(\cM))$ we note that

\[
\Theta_{\cM}(\bx_\infty,T_\textrm{gen}(\cM)) \geq \limsup_{j\to\infty} \Theta_{\cM}(\bx_\infty,T_\textrm{gen}(\cM)) \geq \lambda_1 + \delta_0
\]

by upper-semicontinuity of density and the final bullet point in Proposition <ref>. This implies that $(\bx_\infty,T_\textrm{gen}(\cM)) \in \fS_\textnormal{first}(\cM)$.

We now claim that $\fS_\textrm{first}(\cM)$ is discrete. (This will finish the proof.) Indeed, if $(\bx_j,T_\textrm{gen}(\cM)) \to (\bx_\infty,T_\textrm{gen}(\cM))$ are distinct elements of $\fS_\textrm{gen}(\cM)$ we can rescale $\cM$ around $(\bx_\infty,T_\textrm{gen}(\cM))$ by $|\bx_j - \bx_\infty|^{-1}$ to yield $\tilde \cM_j$. Note that up to a subsequence, $\tilde\cM_j$ converges to a tangent flow $\tilde\cM$ to $\cM$ at $(\bx_\infty,T_\textrm{gen}(\cM))$. By Proposition <ref> and the multiplicity one assumption, $\tilde\cM = \cH^2\lfloor \Sigma$ for some $\Sigma\in \cS_2''$. Rescaling the points $\bx_j$ and passing to the limit we find $\tilde \bx \in \RR^3$ with $|\tilde \bx| = 1$ and $\Theta_{\tilde\cM}(\bx,0) \geq \lambda_1 + \delta_0$ (by upper-semicontinuity of density and the final bullet point in Proposition <ref>). This contradicts the fact that all tangent flows to $\tilde\cM$ at $(\bx,0) \neq (\bOh,0)$ are multiplicity one cylinders or planes. This completes the proof.

Consider a cyclic integral unit-regular Brakke flow $\cM$ in $\RR^3$ with $\cM(0) = \cH^2\lfloor M$ for a closed embedded surface $M \subset \RR^3$. Write $U$ for the unique compact region bounded by $M$ and choose the inwards pointing unit normal. Assume that $T_\textnormal{gen}(\cM) < \infty$ and at any $(\bx,T_{\textnormal{gen}}(\cM)) \in \fS_\textnormal{first}(\cM)$, all tangent flows occur with multiplicity one. If $M_j$ is a sequence of normal graphs of functions $0<u_j$ over $M$ with $u_j\to 0$ in $C^\infty(M)$, then

\[
\limsup_{j\to\infty} \genus_{T_\textnormal{gen}}(M_j) \leq  \genus_{T_\textnormal{gen}}(M) - 1.
\]

This result would also hold (with the same proof) if we assumed that $0>u_j$.

Before proving Proposition <ref>, we observe that it immediately implies Theorem <ref> after finitely many iterations.

For $M\subset \RR^3$ a closed embedded surface, fix (using <cit.>) a cyclic integral unit-regular Brakke flow $\cM$ with $\cM(0) = \cH^2\lfloor M$.  If $T_{\textrm{gen}}(\cM) = \infty$ then $\sing \cM = \sing_\textrm{gen}\cM$ so we're in case (1) of Theorem <ref> with $M'=M$ (i.e. no perturbation is necessary).  If $T_{\textrm{gen}}(\cM) < \infty$, then either it holds that all $(\bx,T_{\textnormal{gen}}(\cM)) \in \cS_\textnormal{first}(\cM)$ have some tangent flow with multiplicity $k=1$ or some $(\bx,T_{\textnormal{gen}}(\cM)) \in \cS_\textnormal{first}(\cM)$ has only tangent flows with multiplicity $k\geq 2$. In the latter case, we're in case (2) of Theorem <ref> (again with $M'=M$).

It thus remains to consider the case where $T_{\textrm{gen}}(\cM) < \infty$ and at all $(\bx,T_{\textnormal{gen}}(\cM)) \in\fS_\textnormal{first}(\cM)$, some tangent flows occur with multiplicity-one. By Proposition <ref> (the “...Finally,” bullet points) it holds that at all $(\bx,T_{\textnormal{gen}}(\cM)) \in\fS_\textnormal{first}(\cM)$, all tangent flows occur with multiplicity-one.

Note also that by Proposition <ref> we see that $\genus_{T_\textrm{gen}}(M) \geq 1$. Using Proposition <ref> we can thus find $M'$ a (small) normal graph over $M$ with

\[
\genus_{T_\textrm{gen}}(M') \leq \genus_{T_\textrm{gen}}(M) - 1.
\]

Now, by the argument in the previous paragraph if $M'$ fails to satisfy the desired properties in Theorem <ref> for some (and thus any) cyclic unit-regular integral Brakke flow $\cM'$ with $\cM'(0) = \cH^2\lfloor M'$ it holds that $T_{\textrm{gen}}(\cM') < \infty$ and at all $(\bx,T_{\textnormal{gen}}(\cM')) \in \fS_\textnormal{first}(\cM)$, all tangent flows occur with multiplicity one. We can thus apply  Proposition <ref> with $M'$ in place of $M$. Note that at each step the genus at the non-generic time must strictly decrease. As such, the perturbations must eventually satisfy the asserted properties in Theorem <ref>. (By choosing all perturbations sufficiently small, we can arrange that the final surface is a small graph over $M$.) This completes the proof.

The remainder of this section is thus devoted to the proof of Proposition <ref>. Below we fix a cyclic integral unit-regular Brakke flow $\cM$ in $\RR^3$ with $\cM(0) = \cH^2\lfloor M$ for a closed embedded surface $M \subset \RR^3$.  Assume that $T_\textnormal{gen}(\cM) < \infty$ and at any $(\bx,T_{\textnormal{gen}}(\cM)) \in \fS_\textnormal{first}(\cM)$, all tangent flows occur with multiplicity one.

Consider $M_j$ is a sequence of normal graphs of functions $0<u_j$ over $M$ with $u_j\to 0$ in $C^\infty(M)$ and fix cyclic integral unit-regular Brakke flows $\cM_j$ with $\cM_j(0) = \cH^2\lfloor M_j$. Using  Lemma <ref> it suffices to assume (after passing to a subsequence and replacing $\cM$ by the limiting Brakke flow) that $\cM_j \rightharpoonup \cM$.

\begin{equation}\label{eq:R3-defn-dj}
d_j : = \max_{(\bx,T_{\textrm{gen}}(\cM)) \in \fS_\textrm{first}(\cM)} d((\bx,T_{\textrm{gen}}(\cM)), \supp\cM_j)
\end{equation}

(using the parabolic distance). Note that $0<d_j =o(1)$. (Lemma <ref> implies that $\fS_\textrm{first}(\cM)$ is a finite set.)

Before proving Proposition <ref> we first show that the first non-generic time does not significantly decrease under the perturbation.

For $M,M_j,\cM,\cM_j$ as above we have $T_\textnormal{gen}(\cM_j) \geq T_\textnormal{gen}(M) - o(d_j^2)$ as $j\to\infty$.

Passing to a subsequence, we can assume for contradiction that there is some $\eta>0$ so that

\[
T_\textnormal{gen}(\cM_j) \leq T_\textnormal{gen}(M) - \eta d_j^2
\]

By stability of cylindrical (generic) singularities (this follows from upper-semicontinuity of density and the final bullet point in Proposition <ref>) we see that in fact

\begin{equation}\label{eq:gen-time-not-sig-decrease-contr}
T_\textnormal{gen}(M) - o(1) \leq T_\textnormal{gen}(\cM_j) \leq T_\textnormal{gen}(M) - \eta d_j^2
\end{equation}

as $j\to\infty$. By translating in space-time and using (<ref>) we can assume that $T_\textrm{gen}(\cM) = 0$ and that there is $(\bx_j,T_j) \in \fS_\textrm{first}(\cM_j)$ with $T_j = T_\textrm{gen}(\cM_j)$ so that $(\bx_j,T_j)\to (\bOh,0) \in \fS_\textrm{first}(\cM)$ but

\begin{equation}\label{eq:gen-time-not-sig-decrease-contr-2}
T_j \leq - \eta d_j^2.
\end{equation}

Now, rescale $\cM,\cM_j,(\bx_j,T_j)$ around the space-time origin to $\tilde \cM_j,\check \cM_j,(\check\bx_j,\check T_j)$ so that $|(\check \bx_j,\check T_j)| = 1$. Passing to a further subsequence, we assume that $\tilde \cM_j \rightharpoonup \tilde\cM$, $\check \cM_j \rightharpoonup \check \cM$, $(\check\bx_j,\check T_j) \to (\check\bx_0,\check T_0)$. Note that $\check T_0 \leq 0$ and $\tilde\cM(t) = \cH^2\lfloor\sqrt{-t}\Sigma$ for some $\Sigma\in \cS_2''$.

We claim that is $\check \cM$ either $\check\cM(t) = \cH^2\lfloor \sqrt{-t}\Sigma$ for all $t<0$ or $\check \cM$ is (up to a parabolic dilation around the origin) the ancient flow constructed in Proposition  <ref>. We will verify this claim below but first will show how it completes the proof. By upper-semicontinuity of density and the final bullet point in Proposition <ref> we see that $\Theta_{\check\cM}(\check \bx_0,\check T_0) \geq \lambda_1 + \delta_0$.  Thus, the first case ($\check\cM(t) = \cH^2\lfloor \sqrt{-t}\Sigma$) cannot occur, since $\check T_0 \leq 0$, so we find that $\check\cM$ is a parabolic dilation of the ancient flow constructed  in Proposition  <ref>. By (4) in Proposition <ref> we see that $\check T_0 = 0$. By the scaling invariance of (<ref>), $\check T_j \to 0$ and the definition of $d_j$ we see that

$$ d((\bOh,0),\supp \check \cM_j) \to 0$$

and thus $d((\bOh,0),\supp\check \cM) = 0$. This contradicts (1) in Proposition <ref>.

We now prove the claim about $\check \cM$. We can choose an open set $\Omega$ so that $\partial\Omega = \Sigma$ and $\supp\check\cM(t) \subset \sqrt{-t} \overline\Omega$ (to see that $\check \cM$ is contained to one side of $\sqrt{-t}\Sigma$ we can use the fact that the level set flow of the inside $U_j$ of $M_j$ contains the level set flow $U$, the inside of $M$). We claim that
\begin{equation}\label{eq:R3-sing-time-lemm-ent-bd}
\lambda(\check \cM) \leq F(\Sigma).
\end{equation}

Indeed, we can choose $(\by_j,s_j) \in \RR^3\times \RR$ with $(\by_j,s_j) \to (\bOh,0)$ and $r_j\to 0$ and

$$\Theta_{\cM_j}((\by_j,s_j),r_j) = \lambda(\check\cM) + o(1)\, .$$

For $r>0$ fixed, we can take $j$ large and apply Huisken's monotonicity to yield

\[
\Theta_{\cM_j}((\by_j,s_j),r) \geq \Theta_{\cM_j}((\by_j,s_j),r_j) = \lambda(\check\cM) + o(1).
\]

Sending $j\to\infty$, we find

\[
\Theta_\cM((\bOh,0),r) \geq \lambda(\check\cM).
\]

Then, sending $r\to 0$, we obtain (<ref>).

Since $\supp\check\cM(t) \subset \sqrt{-t} \overline\Omega$ we find that any tangent flow to $\check\cM$ at $-\infty$ is associated to an integral unit-regular varifold shrinker ($F$-stationary varifold) $V$ with $\supp V\subset \overline \Omega$. The Frankel property for shrinkers (Corollary <ref>) implies that $\supp V \cap \Sigma \not = \emptyset$. Then the strong maximum principle for stationary varifolds [70, 57] (either applies since $\Sigma$ is smooth) gives $V = m \cH^2\lfloor \Sigma$ for some $m \in \NN$. The entropy bound (<ref>) then implies that $m=1$.

We can thus apply the global uniqueness result Theorem <ref> to see that either $\check\cM(t) = \cH^2\lfloor \sqrt{-t}\Sigma$ for all $t<0$ or $\check \cM$ is (up to a parabolic dilation around the origin) the ancient flow constructed in Proposition  <ref>. This proves the claim and thus completes the proof.

We now complete the proof of Theorem <ref> by establishing Proposition <ref>.

Recall that $M\subset \RR^3$ is a closed embedded surface and $M_j$ are the normal graphs of functions $0<u_j$ with $u_j\to0$ in $C^\infty(M)$. Write $U_j$ for the unique compact region bounded by $M$ and let $U_j(t)$ denote the level set flow of $U_j$. Write $\cU_j$ for the space-time track of $U_j(t)$.

We have fixed cyclic integral unit-regular Brakke flows $\cM_j$ with $\cM_j(0) = \cH^2\lfloor M_j$ and so that $\cM_j\rightharpoonup \cM$ a cyclic integral unit-regular Brakke flow with $\cM(0) = \cH^2\lfloor M$ and so that all tangent flows to $\cM$ at time $T_\textrm{gen}(\cM)$ occur with multiplicity one.

By Lemma <ref>, the set $\fS_\textrm{first}(\cM)$ is finite. Thus, by passing to a subsequence and recalling the definition of $d_j$ in (<ref>) we can translate $\cM,\cM_j$ by the same fixed space-time vector to assume that $T_\textrm{gen}(\cM) = 0$ and

\[
d_j = d((\bOh,0),\supp\cM_j)
\]

(i.e., that the maximum in $d_j$ is achieved at the space-time origin). Define $\tilde \cM_j,\check\cM_j$ by parabolically dilating around the space-time origin by $d_j^{-1}$. Exactly as in the proof of Lemma <ref> we find that (after passing to a subsequence) $\tilde\cM_j$ limits to $\cH^2\lfloor\sqrt{-t}\Sigma$ and $\check \cM_j \rightharpoonup \check\cM$ a parabolic dilation of the ancient flow constructed in Proposition  <ref>. (Note that the other possibility considered in the proof of Lemma <ref>, namely $\check\cM = \cH^2 \lfloor \sqrt{-t}\Sigma$ cannot occur since $d((\bOh,0),\supp\check\cM_j) = 1$ by the choice of rescaling.)

Because the ancient flow $\check \cM = \cH^n \lfloor \partial\cK(t)$ is a parabolic dilation of the flow considered in Section <ref>, we can apply Proposition <ref> to find $t_0 < t_1 < 0$ and $\kappa_0 \in (0,t_1-t_0),\kappa_1 \in (0,-t_1)$ and $R>0$ so that

* for any $t \in [t_{0},t_{0}+\kappa_{0}]$ so that $\partial\cK(t)$ is smooth, $\genus(\partial\cK(t) \cap B_{R}(\bOh)) > 0$,

* for any $t \in [t_{1},t_{1}+\kappa_{1}]$ so that $\partial\cK(t)$ is smooth, $\genus(\partial\cK(t) \cap B_{R}(\bOh)) = 0$,

* $t\mapsto \partial\cK(t)$ is a simple flow in $\{R/2 <  |\bx| < 3R\} \times [t_0,t_1+\kappa_{1}]$,

* and
for $(\bx,t) \in \partial\cK(t) \cap \{R/2 < |\bx| < 3R\}\times  [t_0,t_1+\kappa_{1}]$, we have $|\bx^T| \geq \tfrac 12 |\bx|$, for $\bx^T$ the projection of $\bx$ to $T_\bx\partial\cK(t)$.

(Note that the various constants will be changed by the parabolic dilation, but once this is taken into account they will be fixed with the above properties and will not depend on $j$. This is all we will need later.) Note that by Lemma <ref> and choice of translation/rescaling we have $T_\textrm{gen}(\check\cM_j) \geq -o(1)$. In particular, for $j$ sufficiently large, $T_\textrm{gen}(\check\cM_j) \geq t_1 + \kappa_1$.

By unit-regularity and (3) and (4) above, we conclude that for $j$ sufficiently large,

\begin{equation}\label{eq:checkM_j-simple-flow}
\textrm{$\check\cM_j$ is a simple flow in $\{3R/4<|\bx| < 2R\}\times [t_0 + \kappa_0/2,t_1+\kappa_1/2]$.}
\end{equation}

Furthermore, by (1) (and (6) in Proposition <ref> combined with unit-regularity)  there is $t_{0,j} \in [t_0 + \kappa_0/2,t_0+3\kappa_0/4] \cap \cT_\textrm{reg}(\check\cM_j)$ with

\begin{equation}\label{eq:R3genusdrop.genus.inside}
\genus(\check M_j(t_{0,j}) \cap B_R(\bOh)) > 0.
\end{equation}

Similarly, by (2) there is $t_{1,j} \in [t_1 + \kappa_1/2,t_1+3\kappa_1/4] \cap \cT_\textrm{reg}(\check\cM_j)$ with

\begin{equation}\label{eq:R3genusdrop.genus.inside.dies}
\genus(\check M_j(t_{1,j}) \cap B_R(\bOh)) = 0.
\end{equation}

Discarding finitely many terms we can ensure (thanks to $\cM_j\rightharpoonup\cM$ and the monotonicity of genus) that for all $j$, it holds that

\[
g_{0,j} : = \genus({\check M_j(t_{0,j})}) \leq \genus_{T_\textrm{gen}}(M).
\]

Defining $g_{1,j} : = \genus(\check M_j(t_{1,j}))$, we claim that

\begin{equation}\label{eq:genus-ineq-R3-in-proof}
g_{1,j} \leq g_{0,j} - 1
\end{equation}

for $j$ sufficiently large. Thanks to (1)-(4) above, the proof of (<ref>) is now completed exactly as in <cit.> by using the localized topological monotonicity results proven in <cit.>.

For the reader's convenience we recall the proof here. If (<ref>) fails, we can take $j$ sufficiently large so that (<ref>) and (<ref>) hold but

\[
g_{0,j} = g_{1,j} : = g
\]

For this $j$ fixed, we write $W_j[t]:=(\check U_j(t))^c$ and $W_j[a,b] : = \check \cU_j^c \cap \ft^{-1}([a,b])$ (where $\check \cU_j$ is the space-time track of the level set flow of $U_j$ translated in space-time and then parabolically rescaled by $d_j^{-1}$ as above and $\check U_j(t)$ is the time $t$ slice.)

Fix $B=B_R(\bOh)$ and recall (<ref>). By Lemmas <ref> and <ref> and (<ref>) we can choose loops

\[
\gamma_1^{(1)},\dots, \gamma_g^{(1)} \subset W_j[t_{1,j}]
\]

so that

\[
\{[\gamma_1^{(1)}],\dots, [\gamma_g^{(1)}]\} \subset H_1(W_j[t_{1,j}],\ZZ)
\]

is linearly independent and each $\gamma_j^{(1)}$ satisfies one of the following:

* $\gamma_j^{(1)}$ is contained in $\overline B^c$ , or

* there is some component $\cB_j^{(1)}$ of $\partial B\cap W_j[t_{1,j}]$ that has non-zero mod-2 intersection with $\gamma_j^{(1)}$ and zero mod-2 intersection with each previous $\gamma_i^{(1)}$, $i<j$.

(No $\gamma_j^{(1)}$ is contained in $B$, thanks to (<ref>).) By <cit.> we have that

\[
\{[\gamma_1^{(1)}],\dots, [\gamma_g^{(1)}]\} \subset H_1(W_j[t_{0,j},t_{1,j}],\ZZ)
\]

is linearly independent. We now choose loops $\gamma_1^{(0)},\dots,\gamma_g^{(0)} \subset W_j[t_{0,j}]$ so that

* Each $\gamma_j^{(1)}$ is homotopic to $\gamma_j^{(0)}$ in $W_j[t_{0,j},t_{1,j}]$; see <cit.>.

* If $\gamma^{(1)}_j$ is contained in $\overline B^c$ then so is $\gamma^{(0)}_j$ (in fact, so is the entire homotopy between them); see Theorem <ref>.

* If $\gamma^{(1)}_j$ is not entirely contained in $\overline B^c$ then there is some component $\cB^{(0)}_j$ of $\partial B\cap W_j[t_{0,j}]$ that has non-zero mod-2 intersection with $\gamma_j^{(0)}$ and zero mod-2 intersection with each previous $\gamma_i^{(0)}$, $i<j$; this follows from simplicity of the flow near $\partial B$ and the fact that mod-2 intersection is preserved under homotopy.

By Lemma <ref> we have that

\[
\genus(\check M_j(t_{0,j})\cap B_R(\bOh)) = 0.
\]

This contradicts (<ref>), completing the proof of (<ref>) and thus the proposition.

§ GRAPHS OVER SHRINKERS

Recall the conventions for the second fundamental form and mean curvature as described in Section <ref>. Suppose that $\Sigma^n\subset \RR^{n+1}$ is a smooth shrinker (the estimates here are local so $\Sigma$ could just be a small piece of a shrinker) and consider for $I \subset \RR$ an interval of time $u : \Sigma \times I \to \RR$ so that

\[
|u||A| < \eta < 1
\]

along $\Sigma\times I$, where $A$ is the second fundamental form of $\Sigma$. This allows us to define the graph

\[
\Gamma_\tau  : =\{ \bx + u(\bx,\tau) \nu_\Sigma(\bx): \bx \in \Sigma\}.
\]

We compute here various geometric quantities associated to $\Gamma_\tau$. Define

\begin{equation}\label{app:graphical-eqns-defn-v}
v(\bx,\tau) = (1+|(\Id - u S_\Sigma)^{-1}(\nabla_\Sigma u)|^2)^{\frac 12}.
\end{equation}

The upwards pointing normal along $\Gamma_\tau$ is

\begin{equation}\label{eq:app-unit-normal-graph}
\nu_\Gamma = v^{-1}(-(\Id - u S_\Sigma)^{-1}\nabla_\Sigma u + \nu_\Sigma).
\end{equation}

In particular

\begin{equation}\label{eq:app-defi-vdotv-v}
v = (\nu_\Sigma \cdot \nu_\Gamma)^{-1}.
\end{equation}

Note that $\bZ \in T_{\bx} \Sigma$ if and only if

\[
(\Id - uS_\Sigma) \bZ  + du(\bZ) \nu_\Sigma( \bx)  \in T_{\bx + u(\bx,\tau) \nu_\Sigma(\bx)}\Gamma_\tau.
\]

Since the shape operator is self-adjoint, we thus find that the expression for $\nu_\Gamma$ is orthogonal to $\Gamma_\tau$. The assertion thus follows from the definition of $v$.

For $\ell \in \{0,1\}$ set

\begin{equation}\label{eq:defn-sigma-x-app}
\sigma_{\ell}(\bx,\tau) : = \sum_{j=0}^\ell \sum_{k=0}^{4-2j} |\partial_\tau^j\nabla^k u(\bx,\tau)|.
\end{equation}

We now study the form of the error terms in the linearization of the rescaled mean curvature flow equation.

The mean curvature of $\Gamma_\tau$ at $\bx + u(\bx,\tau) \nu_\Sigma(\bx)$ satisfies

\begin{equation}
v(\bx,\tau) H_\Gamma(\bx + u(\bx,\tau) \nu_\Sigma(\bx),\tau) = H_\Sigma(\bx) +  (\Delta_\Sigma u + |A_\Sigma |^2 u)(\bx) + E^H
\end{equation}

where the error $E^H$ can be decomposed into terms of the form

\[
E^H = u E^H_1  + E^H_2(\nabla_\Sigma u,\nabla_\Sigma u)
\]

where $E^H_1 \in C^\infty(\Sigma)$ and $E^H_2 \in C^\infty(\Sigma ; T^* \Sigma \otimes T^*\Sigma)$ satisfy the following estimates:

\[
|\partial_\tau  E_1^H(\bx,\tau)| \leq C_1^H\sigma_1(\bx,\tau),\qquad  \sum_{k=0}^2 |\nabla^k_\Sigma E_1^H(\bx,\tau)|  \leq C_1^H \sigma_0(\bx,\tau)
\]

and[recall that $E_2$ is a section of $T^*\Sigma\otimes T^*\Sigma$ so e.g., $\nabla_\Sigma E_2$ is a section of $T^*\Sigma\otimes T^*\Sigma \otimes T^*\Sigma$]

\[
|\partial_\tau  E_2^H(\bx,t)| \leq C_2^H (1+\sigma_1(\bx,\tau)),\qquad \sum_{k=0}^2 |\nabla^k E_2^H(\bx,t)|  \leq C_2^H (1+ \sigma_0(\bx,\tau))
\]

where $C_1^H,C_2^H$ depend only on $\eta$ and an upper bound for $\sum_{k=0}^3 |\nabla^k A|(\bx)$.

The area formula <cit.> implies that for $\tau$ fixed, $\Psi(\bx) = \bx + u(\bx,\tau)\nu_\Sigma(\bx)$ satisfies

\[
\Psi^* d\mu_\Gamma = \sqrt{\det \cG } \, d\mu_\Sigma
\]

where $\cG: T_{\bx}\Sigma \to T_{\bx}\Sigma$ is the linear map

\[
\cG = (\Id - u S)^2 + \nabla_\Sigma u \otimes du.
\]

It is straightforward to check that $\cG$ is invertible as long as $|u| |A|< 1$.

Set $u_s = u+s\varphi$ for $\varphi\in C^\infty_c(\Sigma)$ arbitrary. Then (writing $\cG_s$ for $\cG$ with $u$ replaced by $u+s\varphi$) the first variation formula yields

\[
\frac{d}{ds}\Big|_{s=0} \int_\Sigma \sqrt{\det \cG_s } \, d\mu_\Sigma = - \int_\Sigma H_\Gamma (\nu_\Sigma \cdot \nu_\Gamma) \varphi \sqrt{\det \cG} \, d\mu_\Sigma.
\]

On the other hand, we compute

\[
\frac{d}{ds}\Big|_{s=0} \cG_s = -2  (\Id - u S)  S \varphi+ \nabla \varphi \otimes du + \nabla u \otimes d\varphi,
\]

\begin{align*}
\frac{d}{ds}\Big|_{s=0} \sqrt{\det \cG_s} & = \tfrac 12 \tr \left( \cG^{-1} \tfrac{d}{ds}\Big|_{s=0} \cG_s \right)  \sqrt{\det \cG}\\
& = \tr \left( \cG^{-1} \left( \nabla u \otimes d\varphi - (S - u S^2)  \varphi \right)\right)  \sqrt{\det \cG}.
\end{align*}

Integrating by parts, we thus find

\begin{equation}\label{eq:mean-curv-graph-div-form}
v H_\Gamma   = \frac{v^2}{\sqrt{\det\cG}} \Div (\sqrt{\det\cG} \, \cG^{-1} \nabla u) + v^2 ( \tr \cG^{-1} S -   \tr \cG^{-1} S^2 u ).
\end{equation}

where we emphasize that all quantities (on the right hand side) are taken along $\Sigma$.

We now simplify this expression. For notational simplicity, below we will write $f \approx g$ if $f=g + u e_1 + e_2(\nabla u,\nabla u)$ for $e_1,e_2$ satisfying the estimates for $E_1^H,E_2^H$ in the statement of the lemma (i.e., terms we will absorb in to the error terms). Note that $v\approx 1$ and if we abuse the notation slightly to allow for tensor valued expressions, $\cG \approx \Id -2uS$ so $\cG^{-1}\approx \Id + 2uS$. We compute

\begin{align*}
\tr \cG^{-1} S -   \tr \cG^{-1} S^2 u & =  \tr  \left((1+2u S) S \right) - \tr \left( S^2 u\right) \\
& +  \tr  \left( (\cG^{-1} - \Id - 2u S) S  -  (\cG^{-1} - \Id) S^2 u \right)\\
& =  H + |A|^2 u\\
& +  \tr  \left( (\cG^{-1} - \Id - 2u S_\Sigma) S  -  (\cG^{-1} - \Id) S^2 u \right),
\end{align*}

which implies that

\begin{align*}
v^2 \left( \tr \cG^{-1} S -   \tr \cG^{-1} S^2 u\right) & \approx  H + |A|^2 u.
\end{align*}

We now consider the divergence term in (<ref>). Write

\[
\frac{v^2}{\sqrt{\det\cG}} \Div (\sqrt{\det\cG} \, \cG^{-1} \nabla u) \approx \Div(\cG^{-1}\nabla u) + \frac 12 \nabla \log \det \cG \cdot \nabla u
\]

Since $\nabla u$ enters $\cG$ quadratically, we easily see that $ \nabla \log \det \cG \cdot \nabla u \approx 0$. Finally, we have

\begin{align*}
\Div(\cG^{-1}\nabla u) & \approx \Div \left( (\Id + \nabla u \otimes du)^{-1} \nabla u\right) \\
& = \Div \left( \left( (\Id - \frac{ \nabla u \otimes du}{1+|\nabla u|^2}  \right) \nabla u\right)\\
& = \Div \left( \frac{\nabla u }{1+|\nabla u|^2} \right) \approx \Delta u .
\end{align*}

This completes the proof.

The support function along $\Gamma_\tau$ satisfies

\begin{equation}
v(\bx,\tau) \left( \bx_\Gamma \cdot \nu_\Gamma \right) = \bx \cdot \nu_\Sigma + u(\bx,\tau) - \bx^T \cdot \nabla u + u E^{\bx\cdot \nu}
\end{equation}

at $ \bx_\Gamma = \bx + u(\bx,\tau)  \nu_\Sigma(\bx)$, where $E^{\bx\cdot \nu}$ satisfies

\[
|\partial_\tau  E^{\bx\cdot\nu}(\bx,\tau)| \leq C \sigma_1(\bx,\tau) , \qquad  \sum_{k=0}^2 |\nabla^k E^{\bx\cdot\nu}(\bx,\tau)|  \leq C \sigma_0(\bx,\tau)
\]

for $C$ depending only on $\eta$ and an upper bound[in particular $C$ does not depend on $|\bx|$] for $\sum_{k=0}^3 |\nabla^k A|(\bx)$.

Differentiating the shrinker equation $H + \tfrac 12 \bx \cdot \nu_\Sigma = 0$ we find

\[
S(\bx^T) =  2 \nabla H .
\]

Hence, using (<ref>) we compute

\begin{align*}
v(\bx,\tau) \left( \bx_\Gamma \cdot \nu_\Gamma \right) & = \bx \cdot \nu_\Sigma  + u - \bx \cdot (\Id - u S)^{-1} \nabla u\\
&  = \bx \cdot \nu_\Sigma  + u - \bx\cdot \nabla u - u \sum_{k=1}^\infty u^{k-1} \bx \cdot S^k(\nabla u)\\
&  = \bx \cdot \nu_\Sigma  + u - \bx\cdot \nabla u - u \sum_{k=0}^\infty u^{k} S(\bx^T) \cdot S^k(\nabla u)\\
&  = \bx \cdot \nu_\Sigma  + u - \bx^T \cdot \nabla u - 2 u \sum_{k=0}^\infty u^{k} \nabla H \cdot S^k(\nabla u).
\end{align*}

This completes the proof.

We have

\begin{align*}
v(\bx,\tau) (\partial_{\tau}\bx_{\Gamma} - \bH -\tfrac 12 \bx_{\Gamma}) \cdot \nu_{\Gamma} & = \partial_{\tau}u  - {\big (\underbrace{\Delta u - \tfrac 12 \bx^T \cdot \nabla u + ( |A|^2 + \tfrac 12) u}_{=L u} \big) }+ E
\end{align*}

at $\bx_{\Gamma} = \bx + u(\bx,\tau)\nu_{\Gamma}(\bx)$ for  $E=uE_1 + E_2(\nabla u,\nabla u)$ for $E_1,E_2$ satisfying

\[
|\partial_\tau  E_1(\bx,\tau)| \leq C_1\sigma_1(\bx,\tau), \qquad \sum_{k=0}^2 |\nabla^k E_1(\bx,\tau)|  \leq C_1\sigma_0(\bx,\tau)
\]

\[
|\partial_\tau  E_2(\bx,\tau)| \leq C_2(1+\sigma_1(\bx,\tau)),\qquad \sum_{k=0}^2 |\nabla^k E_2(\bx,\tau)|  \leq C_2 (1+ \sigma_0(\bx,\tau))
\]

where $C_1,C_2$ depend only on $\eta$ and an upper bound for $\sum_{k=0}^3 |\nabla^k_\Sigma A|(\bx)$.

In particular, if $\Gamma_\tau$ is a solution to rescaled mean curvature flow, i.e.,

\[
(\partial_\tau \bx)^\perp = \bH + \tfrac 12\bx^\perp
\]

\begin{equation}\label{eq:error-term-linearize}
\partial_\tau u = Lu + E
\end{equation}

for $E$ as above.

Recall that our convention is that $\bH = H \nu_{\Gamma} $. Thus, when combined with Lemmas <ref> and <ref>, we compute

\begin{align*}
& (\partial_{\tau}\bx_{\Gamma} - \bH - \tfrac 12 \bx_{\Gamma}) \cdot \nu_{\Gamma} \\
& = (\nu_{\Sigma}\cdot \nu_{\Gamma}) \partial_{\tau}u - H - \tfrac 12 \bx_{\Gamma}\cdot\nu_{\Gamma} \\
& =  - \underbrace{( H_{\Sigma}(\bx) + \tfrac 12 \bx\cdot \nu_{\Sigma})}_{=0}  + \partial_{\tau}u - (\Delta_{\Sigma} u + |A_{\Sigma}|^{2}u - \tfrac 12 \bx^{T}\cdot \nabla u + \tfrac 12 u  ) + E^{\bx\cdot\nu} - E^{H} .
\end{align*}

Combined with the estimates from Lemmas <ref> and <ref>, this completes the proof.

Finally, we recall the improved estimates that hold along an asymptotically conical end $\Sigma_\textrm{con}$, proven in <cit.> .

Along an asymptotically conical end $\Sigma = \Sigma_\textnormal{con}$ there is $\eta,C$ depending on $\Sigma$ so that if

\[
|\bx|^{-1} |u(\bx)| + |\nabla u(\bx)| + |\bx| |\nabla^2u| \leq \eta
\]

then the error term $E$ in (<ref>) decomposes as

\begin{multline} \label{eq:linearized.equation.error.decomposition.con}
E(u)(\mathbf{x}) = u(\mathbf{x}) E_1(\mathbf{x}, u(\mathbf{x}), \nabla u(\mathbf{x}), \nabla^2 u(\mathbf{x})) \\
+  \nabla u(\mathbf{x}) \cdot \mathbf{E}_2(\mathbf{x}, u(\mathbf{x}), \nabla u(\mathbf{x}), \nabla^2 u(\mathbf{x})),
\end{multline}

where $E_1$, $\mathbf{E}_2$ are smooth functions on the following domains

\begin{align*}
&E_1(\mathbf{x}, \cdot, \cdot, \cdot) : \RR \times T_{\mathbf{x}} \Sigma \times \operatorname{Sym}(T_{\mathbf{x}} \Sigma \otimes T_{\mathbf{x}} \Sigma) \to \RR,\\
&\mathbf{E}_2(\mathbf{x}, \cdot, \cdot, \cdot) : \RR \times T_{\mathbf{x}} \Sigma \times \operatorname{Sym}(T_{\mathbf{x}} \Sigma \otimes T_{\mathbf{x}} \Sigma) \to T_{\mathbf{x}} \Sigma.
\end{align*}

Moreover, we can estimate:

\begin{align*}
& r(\mathbf{x}) ^{2+j-\ell} |\nabla_{\mathbf{x}}^i \nabla_z^j \nabla_{\mathbf{q}}^k \nabla_{\mathbf{A}}^\ell E_1(\mathbf{x}, z, \mathbf{q}, \mathbf{A})| \nonumber \\
& \qquad \qquad \qquad \leq C( r(\mathbf{x})^{-1} |z| + |\mathbf{q}| + r(\mathbf{x}) |\mathbf{A}|)^{\max\{0, 1-j-k-\ell\}}, \\
& r(\mathbf{x}) ^{1+j-\ell} |\nabla_{\mathbf{x}}^i \nabla_z^j \nabla_{\mathbf{q}}^k \nabla_{\mathbf{A}}^\ell \mathbf{E}_2(\mathbf{x}, z, \mathbf{q}, \mathbf{A})| \nonumber \\
& \qquad \qquad \qquad \leq C( r(\mathbf{x})^{-1} |z| + |\mathbf{q}| + r(\mathbf{x}) |\mathbf{A}|)^{ \max\{0, 1-j-k-\ell\}} .
\end{align*}

In the above, $C = C(\Sigma,i,j,k,\ell)$, and $i,j,k,\ell \geq 0$.

Along an asymptotically cylindrical end the estimates can be extended as follows.

Along the core $\Sigma \cap B_R(\bOh)$ and every asymptotically cylindrical end $\Sigma = \Sigma_\textnormal{cyl}$ there is $\eta,C$ depending on $\Sigma$ so that if $ \|u\|_{C^2} \leq \eta$ as well as $|u| |A| \leq \eta$ then the error term $E$ in (<ref>) decomposes as

\begin{multline} \label{eq:linearized.equation.error.decomposition.cyl}
E(u)(\mathbf{x}) = u(\mathbf{x}) E_1(\mathbf{x}, u(\mathbf{x}), \nabla u(\mathbf{x}), \nabla^2 u(\mathbf{x})) \\
+  \nabla u(\mathbf{x}) \cdot \mathbf{E}_2(\mathbf{x}, u(\mathbf{x}), \nabla u(\mathbf{x}), \nabla^2 u(\mathbf{x})),
\end{multline}

where $E_1$, $\mathbf{E}_2$ are smooth functions as in Lemma <ref>, where we can estimate:

\begin{align*}
&  |\nabla_{\mathbf{x}}^i \nabla_z^j \nabla_{\mathbf{q}}^k \nabla_{\mathbf{A}}^\ell E_1(\mathbf{x}, z, \mathbf{q}, \mathbf{A})|  \leq C( |z| + |\mathbf{q}| + |\mathbf{A}|)^{\max\{0, 1-j-k-\ell\}}, \\
& |\nabla_{\mathbf{x}}^i \nabla_z^j \nabla_{\mathbf{q}}^k \nabla_{\mathbf{A}}^\ell \mathbf{E}_2(\mathbf{x}, z, \mathbf{q}, \mathbf{A})|  \leq C(  |z| + |\mathbf{q}| + |\mathbf{A}|)^{ \max\{0, 1-j-k-\ell\}} .
\end{align*}

In the above, $C = C(\Sigma,i,j,k,\ell)$, and $i,j,k,\ell \geq 0$.

As in the proof of <cit.> one can split the error into $E^H(u)$ and $E^{\bx \cdot \nu}(u)$. The estimate for the $E^H(u)$ term is as in <cit.>, whereas for the $E^{\bx \cdot \nu}(u)$ term we can argue as in the proof of Lemma <ref>.

We also need to understand the linearization of the shrinker mean curvature of two graphs over $\Sigma$, relative to each other and relative to the base $\Sigma$.

For $\delta < \tfrac 12 (\sup_\Omega |A_\Sigma|)^{-1}$ and  $u_i \in C^\infty(\Sigma), i = 0,1$ with

\begin{equation}\label{eq:C2-bound}
|u_i(\bx)|+ |\nabla_\Sigma u_i(\bx)| + |\nabla^2_\Sigma u_i(\bx) | + |\nabla^3_\Sigma u_i(\bx) | \leq \delta
\end{equation}

for all $\bx \in \Sigma$. Letting

$$\Gamma_i:= \{ \bx + u_i(\bx) \nu_\Sigma(\bx): \bx \in \Sigma\}$$

then denoting $w =u_1-u_0$ and $v_i:= (\nu_{\Gamma_i} \cdot \nu_\Sigma)^{-1}$ there exists $C=C(\sup_{\Sigma} |A_{\Sigma}| + |\nabla A_{\Sigma}|+ |\nabla^2 A_{\Sigma}|)$ such that

\begin{equation*}
v_1\Big(H_{\Sigma_1} + \tfrac 12 \bx_{\Sigma_1} \cdot \nu_{\Sigma_1}\Big) - v_0\Big(H_{\Sigma_0}+ \tfrac 12\bx_{\Sigma_0}\cdot \nu_{\Sigma_0}\Big)
= L_\Sigma w + E^w
\end{equation*}

where $\bx_{\Sigma_i} = \bx + u_i(\bx)\nu_\Sigma(\bx)$ and the error term $E$ satisfies

$$ E^w(\bx) = w(\bx) F(\bx) + \nabla w(\bx) \cdot \bF(\bx) + \nabla^2 w(\bx) \cdot \mathcal{F}(\bx) \, ,$$

with the estimate

$$|F| + |\bF| + |\cF| + |\nabla \cF| \leq C\delta$$

for all $\bx \in \Sigma$. Furthermore, under the assumption that $\Sigma$ is a properly embedded shrinker, it holds

\begin{equation} \label{eq:relative-estimate-ibp-hits-gaussian}\left|\int_\Sigma w (\bx^T\otimes \nabla_\Sigma w) \cdot \cF\, \rho\, d\cH^n\right| \leq C \delta \|w\|^2_{W,1}\, .\end{equation}

In the case $\Sigma_\textnormal{con} \subset \Sigma$, all the above statements remain the same if along the conical ends $\Sigma_\textnormal{con}$ one replaces (<ref>) with the weaker assumption

$$ \|u_i\|_{3,\alpha;\Sigma_\textnormal{con}}^{(1)} \leq \delta$$

for $i=1,2$.

The proof follows similarly as the proof of <cit.>. Denoting

$$w: =u_1-u_2,\  E^w:= E(u_1)-E(u_2)$$

we have

$$ \big(\tfrac{\partial}{\partial \tau} - L \big) w = E^w\, .$$

Using Lemma <ref> and the fundamental theorem of calculus, we obtain as in <cit.>

\begin{equation} \label{eq:uniqueness.w.error}
\begin{split}
E^w 	& = w E_1(\cdot, u_2, \nabla_\Sigma u_2, \nabla_\Sigma^2 u_2) + \nabla_\Sigma w \cdot \mathbf{E}_2(\cdot, u_2, \nabla_\Sigma u_2, \nabla_\Sigma^2 u_2)  \\
& \quad + \Big[ u_1 \int_0^1 D_z E_1(\cdots) \, dt \Big] w  + \Big[ u_1 \int_0^1 D_{\mathbf{q}} E_1(\cdots) \, dt \Big] \cdot \nabla_\Sigma w  \\
& \quad + \Big[ u_1 \int_0^1 D_{\mathbf{A}} E_1(\cdots) \, dt \Big] \cdot \nabla^2_\Sigma w + \Big[ \nabla_\Sigma u_1 \cdot \int_0^1 D_z \mathbf{E}_2(\cdots) \, dt \Big] w  \\
& \quad + \Big[ \nabla_\Sigma u_1 \cdot \int_0^1 D_{\mathbf{q}} \mathbf{E}_2(\cdots) \, dt \Big] \cdot \nabla_\Sigma w\\
&\quad  + \Big[ \nabla_\Sigma u_1 \cdot \int_0^1 D_{\mathbf{A}} \mathbf{E}_2(\cdots) \, dt \Big] \cdot \nabla^2_\Sigma w,
\end{split}
\end{equation}

where, in all six instances, $\cdots$ stands for $(\cdot, u_2 + tw, \nabla_\Sigma u_2 + t \nabla_\Sigma w, \nabla_\Sigma^2 u_2 + t \nabla^2_\Sigma w)$. We note that we can thus write

$$ E^w= w F + \nabla_\Sigma w \cdot \mathbf{F} + \nabla^2_\Sigma w \cdot \mathcal{F}\, .$$

The estimates for $F,\mathbf{F},\mathcal{F},\nabla\mathcal{F}$ then follow from Lemma <ref>.

In case $\Sigma$ is a properly embedded shrinker, recall that Ecker's Sobolev inequality [41] (cf. <cit.>) implies that for $f \in H^1_W(\Sigma)$

$$ \| |\bx| f  \|^2_W \leq 4n \| f\|_{W,1}^2\, ,$$

and we can thus estimate

$$ \left| \int_\Sigma w\, (\bx^T \otimes \nabla_\Sigma w) \cdot \mathcal{F} \,e^{-\tfrac14|\bx|^{2}} d\mathcal{H}^n \right| \leq C \delta (\| |\bx|w\|_W \|\nabla_\Sigma w\|_W) \leq C \delta \| w\|_{W,1}^2 \, .$$

The claim for the conical ends $ \|u_i\|_{2,\alpha;\Sigma_\text{con}}^{(1)} \leq \delta$ for $i=1,2$, follow similarly, (compare
<cit.>), using Lemma <ref> instead of  Lemma <ref>.

§ POSITIVE GRAPHS OVER SHRINKERS

We continue the analysis in Appendix <ref> and consider $u:\Sigma\times I \to \RR$ (where we continue to allow for $\Sigma$ to be possibly incomplete) so that the graph $\Gamma_\tau$ is flowing by rescaled mean curvature flow. We continue to require that

\[
|u| |A| \leq \eta < 1
\]