We show that the mean curvature flow of a generic closed surface in $\RR^3$ avoids multiplicity one tangent flows that are not round spheres/cylinders. In particular, we show that any non-cylindrical self-shrinker with a cylindrical end cannot  arise generically.

§ INTRODUCTION

Mean curvature flow is the natural heat equation for submanifolds. A family of surfaces $M(t) \subset \RR^{3}$ flows by mean curvature flow if

\begin{equation} \label{eq:mean-curvature-flow}
\left(\tfrac{\partial}{\partial t} \bx \right)^{\perp} = \bH_{M(t)}(\bx),
\end{equation}

where $\bH_{M(t)}(\bx)$ denotes the mean curvature vector of $M(t)$ at $\bx$. When the initial surface $M(0)$ is compact, mean curvature flow is guaranteed to become singular in finite time. The simplest way to analyze such a singularity is to parabolically dilate around a singular point in space-time. Huisken's monotonicity formula [50] (cf. [56]) guarantees that a subsequential limit of such dilations will weakly limit to a tangent flow which will be a weak solution to (<ref>), evolving only by homothety. Ilmanen has showed that such a tangent flow will be supported on a smooth surface $\Sigma\subset \RR^3$ so that $(-\infty,0) \mapsto \sqrt{-t}\Sigma$ is a solution to (<ref>). Such a surface is called a self-shrinker (see Section <ref>).

A deeper understanding of such tangent flows is necessary in order to continue the flow past the onset of singularities (either by constructing a flow with surgery, or by showing that weak solutions to the flow have good partial regularity and well-posedness properties). The two main obstructions to doing so are the potential presence of multiplicity and of tangent flows supported on self-shrinkers whose scalar mean curvature changes sign. By a result of Huisken [50] (cf. [31]) a shrinker whose mean curvature does not change sign is either a plane or a round sphere or cylinder. Such singularities (when they occur with multiplicity one) are known to be well-behaved, thanks to regularity results of Colding–Minicozzi [34] (cf. [83, 29, 33]) and the resolution of the mean convex neighborhood conjecture by Choi–Haslhofer–Hershkovits [26].

In this paper, we show that the second obstruction does not occur for generic initial data $M(0)\subset \RR^3$. Our main result is roughly as follows (see Section <ref> for a more detailed overview and Theorem <ref> for the precise statement):

The mean curvature flow of a generic initial surface $M(0)\subset \RR^3$ encounters only spherical and cylindrical singularities at least until multiplicity occurs for some tangent flow. In particular, up to the first time multiplicity occurs, the flow is well-posed and is completely smooth for almost every time.

This confirms—up to the potential occurrence of multiplicity—a long-standing conjecture of Huisken <cit.>.

The well-known multiplicity-one conjecture posits that multiplicity never occurs, but this is widely open even for generic initial data (we note that considerable progress has been made under additional geometric assumptions on $M(0)$ or the flow; see [84, 71, 3, 62, 9, 63, 64, 7]). We review previous work related to generic mean curvature flow in Section <ref> below.

There is strong evidence that the generic hypothesis in the above result is necessary. Indeed, Ilmanen–White have indicated [85] a construction of a closed surface $M(0)\subset \RR^3$ whose mean curvature flow is not well-posed after the onset of singularities. (On the other hand, if $M(0)\subset \RR^3$ has genus zero, an important work of Brendle [10] shows that the statement above holds without “generic.”)

By combining our main result with a surgery construction by Daniels-Holgate [40], one can construct a mean curvature flow with surgery for a generic initial $M(0)\subset \RR^3$ until the first time multiplicity occurs for the weak flow above.

§.§ The no-cylinder conjecture

At the first time singularities appear (or more generally, the first time that some singularity other than multiplicity-one spherical/cylindrical singularities occur) we have seen that any tangent flow is supported on a smooth self-shrinker $\Sigma$ (possibly with multiplicity). A fundamental result of Wang [79] shows that if $\Sigma\subset\RR^3$ is a non-compact self-shrinker then any end of $\Sigma$ is smoothly asymptotic—with multiplicity one—to a smooth cone or a cylinder.

Ilmanen has asked if it is possible that a non-cylindrical shrinker has a cylindrical end (the no-cylinder conjecture <cit.>). See [80] for some partial progress towards non-existence. Moreover, we note that are by now many constructions of self-shrinkers (both numerical and rigorous) [4, 28, 60, 66, 59, 75, 8] but no example with a cylindrical end has been found.

The main new ingredient this paper can be stated as follows:

Even if a non-cylindrical shrinker with a cylindrical end exists, it does not arise generically (with multiplicity one at the first singular time).

This builds on our  previous work with Mantoulidis [22] where we proved that asymptotically conical and compact shrinkers do not arise generically, but left open the possibility of a shrinker with a cylindrical end. As we discuss below, the analysis in the cylindrical end case is significantly complicated by the a priori lack of graphicality of the corresponding ancient flow along the cylindrical end.

§.§ Generic initial data and one-sided ancient flows

In our previous paper with Mantoulidis [22], we developed a new approach to the study of the mean curvature flow of generic initial data by relating this to the classification of certain ancient solutions. (One may view this as the parabolic analogue of the Hardt–Simon foliation and corresponding generic regularity results for area minimizing hypersurfaces in $\RR^8$ [49, 69, 81], cf. [37].)

§.§.§ One-sided flows as local models for generic mean curvature flow
The basic strategy is as follows (see also <cit.> for a more detailed exposition of the method). Embed the initial surface $M_0(0)\subset \RR^3$ in a local foliation $M_s(0)$ and flow all of the surface simultaneously by mean curvature flow $M_s(t)$. Suppose that $M_0(t)$ encounters a singularity at time $T$ at point $\bx \in \RR^3$. (Let us assume that this is the first singular time, but in reality we can work with the first time with a singularity other a multiplicity one sphere/cylinder.)

As explained above, the parabolically dilated flows

\[
\lambda(M_0(T + \lambda^{-2}t  ) - \bx) : = M_0^\lambda(t)
\]

weakly limit subsequentially (as $\lambda\to\infty$) to the shrinking flow $t\mapsto \sqrt{-t}\Sigma$, $t<0$, associated to a smooth self-shrinker $\Sigma$. (Of course, this limit could also occur with some integer multiplicity $k\geq 2$; we will assume that $k=1$ below.)

The key observation in [22] is that if we simultaneously dilate the entire foliation

\[
\lambda(M_s(T + \lambda^{-2}t  ) - \bx) : = M_s^\lambda(t)
\]

then by choosing a subsequence of $s\to 0$ appropriately with $\lambda\to\infty$ we can pass $M_s^\lambda(t)$ to the limit to find $\check M(t)$ a (nontrivial) ancient mean curvature flow that's disjoint from $t\mapsto\sqrt{-t}\Sigma$ (we call such a flow one-sided). The main strategy is then to show that $\check M(t)$ has improved behavior as compared to $M_0(t)$ and $t\mapsto\sqrt{-t}\Sigma$. This is achieved by showing (see Section <ref>  for a more detailed discussion) that up to a parabolic dilation, there exists a unique ancient one-sided flow $\overline M(t)$ so that $\tfrac{1}{\sqrt{-t}}\overline M(t)$ is close (in a $C^\infty_\textrm{loc}$ sense, with multiplicity one) to $\Sigma$ as $t\to-\infty$ (existence is proven in Proposition <ref> and uniqueness is proven in Theorem <ref>) and moreover $\overline M(t)$ has the following improved properties (see Propositions <ref> and <ref> )

* $\overline M(t)$ is a shrinker mean-convex flow; geometrically this means that the space-time track is parabolically star-shaped[Compare with the “elliptic” star-shapedness of the Hardt–Simon foliation. ] around $(\bOh,0)$,

* as a consequence of the shrinker mean-convexity, all singularities of $\overline M(t)$ are spherical/cylindrical with multiplicity-one,

* for $t\ll0$, $\frac{1}{\sqrt{-t}}\overline M(t)$ is a small $C^\infty$ graph over $\Sigma$, and

* for $t<0$ sufficiently close to $0$, $\overline M(t)$ has genus zero.

Properties (1) and (2) can be used to show that for $s\sim 0$ appropriately chosen, the flow $M_s(t)$ will only have spherical/cylindrical singularities near $(\bx,T)$. Thus, if we assumed that the tangent flow to $M_0(t)$ at $(\bx,T)$ was supported on a more complicated shrinker, the flow $M_s(t)$ has improved locally. Of course, this does not rule out the possibility of $M_s(t)$ having other such singularities elsewhere.

To ensure that this improvement is actually reflected globally it is crucial that we only have to iterate this perturbation process finitely many times (if we perturb infinitely many times, smooth regions that we already controlled might become singular in the limit). This is guaranteed by combining (3) and (4) with the classification of genus zero self-shrinkers due to Brendle [10]. Indeed, at any non-spherical/cylindrical singularity, [10] and (3) implies that $\overline M(t)$ has non-trivial genus for $t \ll 0$. In particular, some handle of $M_s(t)$ “concentrates” near $(\bx,T)$. However, handle pinches off before time $T$, since by (4) it does so for the local model $\overline M(t)$. Along with a result of White [82] stating that genus is monotone non-increasing along mean curvature flow, this implies that we need only perturb finitely many times.

§.§.§ Existence/uniqueness of ancient flows
Fix a self-shrinker $\Sigma\subset \RR^3$. In the above sketch, we crucially used the existence of a one-sided ancient flow with properties (1)-(4) as well as the uniqueness of this flow. The basic mechanism involves the spectral properties of the $L$-operator

\begin{equation}\label{eq:intro-L-operator}
L u = \Delta_\Sigma u - \frac 12 \bx^T \cdot \nabla_\Sigma u + |A_\Sigma|^2 u + \frac 12 u
\end{equation}

obtained by linearizing $\bH + \tfrac 12 \bx^\perp$ at $\Sigma$ (with respect to normal variations). In their foundational work [31], Colding–Minicozzi introduced the $L$-operator and proved (among many other things) that the first eigenvalue $\mu$ of $L$ (with respect to the Gaussian weighted space) is negative.

We can thus give an indication at the linear level why the existence and uniqueness result should hold. After an appropriate rescaling, an ancient solution to (rescaled[Given $N(t)$ a solution to mean curvature flow, the rescaled solution around $(\bx,T)$ is the one-parameter family of surfaces $\tau\mapsto e^{\tau/2}(N(T-e^{-\tau}) -\bx)$. It is easily checked that this procedure (around $(\bOh,0)$) converts the shrinking self-shrinker $t\mapsto \sqrt{-t}\Sigma$ into a static $\tau\mapsto\Sigma$. In general, the rescaled family will solve $\left(\tfrac{\partial}{\partial t} \bx \right)^{\perp} = \bH(\bx) + \tfrac 12 \bx^\perp$ (compare with (<ref>)). ]) mean curvature flow that's close to $\Sigma$ can be modeled on an ancient solution to the parabolic equation

\begin{equation}\label{eq:par-L-eqn-intro}
\partial_\tau u = L u
\end{equation}

with $u\to 0$ as $\tau \to-\infty$. One-sidedness is reflected at the linear level by $u>0$ and shrinker mean convexity is reflected at the linear level by $\partial_\tau u \geq 0$ (shrinker mean convexity is equivalent to the rescaled mean curvature flow moving in one direction).

For “existence,” we have the solution $e^{-\mu\tau}\varphi$, for $\varphi$ the first eigenfunction. Since the first eigenfunction does not change sign, we see that this is a “one-sided” solution and since $\mu<0$ we see that this is a “shrinker mean convex” solution. Conversely, for “uniqueness” we note that by a straightforward linear analysis, any ancient solution to (<ref>) will be of the form

\begin{equation}\label{eq:intro-lin-analysis-stupid}
u(\bx,t) = c e^{-\mu\tau}\varphi +  \sum_{\mu < \lambda < 0} e^{-\lambda\tau} \varphi_\lambda(\bx)
\end{equation}

where the sum is taken over eigenvalues of $L$ and $\varphi_\lambda$ is a some corresponding eigenfunction (each of which must change sign, by the usual characterization of the first eigenfunction). Since $e^{-\lambda\tau} \gg e^{-\mu\tau}$ for $\tau\to-\infty$, we see that if $u$ is “one-sided” (i.e., $u>0$) then it holds that $u(\bx,t) = c e^{-\mu\tau}\varphi$, giving uniqueness in the linear setting.

Of course, in the actual problem one must account for the nonlinearity of the equation. To prove existence we rely on a new upper/lower barrier construction for the ancient flow (cf. Lemma <ref>). (This allows for a somewhat more direct proof that the one-sided flow is shrinker mean convex, cf. <cit.>.)

For uniqueness, if we were to assume a priori that the unknown one-sided rescaled flow is an entire graph (with good decay properties) over $\Sigma$ for very negative times, then a Merle–Zaag type analysis readily yields uniqueness, cf. Proposition <ref>. However, establishing this graphicality represents a major issue in this work. Indeed, in contrast to the conical ends (where graphicality follows automatically from pseudolocality, see <cit.>), an ancient one-sided rescaled flow is not a priori an entire graph over the cylindrical ends of $\Sigma$ for very negative times. Establishing this graphicality is the main technical ingredient needed to carry out the above linear analysis. We prove graphicality and the corresponding uniqueness statement in Section <ref>, see Section <ref> below for an outline of the proof.

§.§ Related work

The first major advance towards a theory of generic mean curvature flow was taken by Colding–Minicozzi who introduced the notion of entropy (cf. Section <ref>) and classified the sphere/cylinder as the unique linearly stable self-shrinkers in a precise sense [31]. In particular, their work shows that if $M(0) = \Sigma\subset \RR^3$ is a non-spherical/cylindrical closed self-shrinker, then there is $M'(0)$ a small $C^\infty$-graph over $M(0)$ so that the mean curvature flow $M'(t)$ does not have the shrinking flow associated to $\Sigma$ at any tangent flow.

It follows from Colding–Minicozzi's linear analysis that any shrinker becomes shrinker mean-convex when perturbed to one side. This crucial observation was first used by Colding–Ilmanen–Minicozzi–White [30] to classify the sphere as the closed self-shrinker of minimal entropy (see Section <ref>). This was later generalized in various ways by Bernstein–Wang [12, 13, 15] (cf. [89, 52]). Of particular relevance to this article, we note that Bernstein–Wang's proof [13] that the cylinder has the second-least entropy among non-flat shrinkers in $\RR^3$ proceeds by constructing a “pre-ancient” one-sided mean curvature flow (on one side of a hypothetical lower entropy asymptotically conical shrinker) and observing that it must lose genus at (non-rescaled) time $t=0$.

More recently, in a joint work with Mantoulidis [22] we introduced a new approach to the study of mean curvature flow with generic initial data. In particular, as explained above, we showed how the problem was related to the existence/uniqueness of ancient one-sided flows for a given shrinker $\Sigma$ and studied the case where $\Sigma$ is smoothly asymptotic to a cone. In this case, using pseudolocality (cf. [38]), such a flow is automatically an entire graph over $\Sigma$ in the ancient past and the techniques from work of the second-named author and Mantoulidis [36] could be used to prove existence/uniqueness. In particular, [22] proved that asymptotically conical self-shrinkers do not arise as multiplicity one tangent flows at the first (non-generic) singular time for a generic initial surface in $\RR^3$. (The current paper generalizes the techniques from [22] to account for cylindrical ends.)

We have also developed a related, but different, strategy to study generic mean curvature flows in a separate joint work with Mantoulidis [23]. This approach avoids the need for a classification of ancient flows and instead relies on a softer density drop argument. This argument is most effective in a low entropy setting and in particular proved that in $\RR^3$ a surface $M\subset \RR^3$ with entropy $\lambda(M) \leq 2$ can be perturbed so as to only encounter multiplicity one spherical/cylindrical singularities. However, the argument does not seem sensitive to the genus drop proven in this paper (and in [22]) which limits its applicability—at present—to the low entropy case. Similarly, the work [23] applied in $\RR^4$ to yield a similar result for hypersurfaces $M\subset \RR^4$ with entropy $\lambda(M) \leq \lambda(\SS^1\times \RR^2)$; combined with the surgery result of Daniels–Holgate [40] this proves that the Schoenflies conjecture holds under such a hypothesis (this improves on an influential work of Bernstein–Wang who established the Schoenflies conjecture under the assumption $\lambda(M) \leq \lambda(\SS^2\times \RR)$, cf.  [17, 16, 14, 20, 21, 19, 18]).

The existence and uniqueness result for ancient flows proven here in fact holds for self-shrinking hypersurfaces $\Sigma\subset \RR^{n+1}$ ($n+1 = 3,\dots,7$) with smooth conical and cylindrical ends. In fact, we can even allow for $\Sigma$ to have ends modeled on cylinders of the form $\hat\Sigma\times\RR$, for $\hat\Sigma\subset \RR^n$ an arbitrary closed shrinker. However, in the case of ends modeled on $\hat\Sigma\times \RR$, we have to impose an extra condition on $\Sigma$, namely we require that the first eigenfunction of the $L$ operator is $o(1)$ at spatial infinity (we call this $\Sigma$ having strictly stable ends). (This holds automatically for $\hat\Sigma$ a round sphere, by work of Colding–Minicozzi [31]; cf. Proposition <ref> and Lemma <ref>). This generalization has some consequences for perturbing away such singularities in higher dimensions (in particular, yielding precise local information in contrast with the local perturbation result described in  <cit.>).

Some time after our work with Mantoulidis [22, 23] appeared, a more analytic approach to the mean curvature flow of generic initial data has been studied by Sun–Xue [77, 76, 78] (modeled on an earlier program introduced by Colding–Minicozzi [35]). They have proven that tangent flows modeled on (multiplicity one) compact or asymptotically conical tangent flows (other than spheres/cylinders) at the first singular time can be locally perturbed away by adjusting the initial data. It is yet unclear if their methods can be applied to shrinkers with cylindrical ends (as we do in this paper) or if their methods can recover the crucial genus drop property used here to prove global results.

§.§ Outline of existence/uniqueness for the one-sided ancient flow

In this section we give a more detailed description of the existence and uniqueness of the one-sided ancient flow. Consider $\Sigma\subset \RR^{n+1}$ (for[This is a technical assumption related to certain difficulties in White's theory of mean convex flows in dimensions admitting non-flat stable minimal cones; we expect that it could be removed, but have not pursued this further. Of course, this  restriction does not prevent applications in the most relevant dimensions $n=2,3$.] $n\leq 6$) a smoothly embedded self-shrinker. We assume that any end of $\Sigma$ is either smoothly asymptotic to a smooth cone (with multiplicity one) or smoothly asymptotic to a cylinder $\hat\Sigma\times \RR$ for $\hat\Sigma\subset \RR^n$ a closed smooth self-shrinker. We call this the nice ends hypothesis (cf. Definition <ref>).

It follows from standard linear theory for the Gaussian weighted $L^2$-space that the $L$ operator (<ref>) admits a positive first eigenfunction $\varphi$. We will assume that $\Sigma$ has strictly stable ends  (cf. Definition <ref>) meaning that $\varphi(\bx) = o(1)$ as $\bx\to \infty$. By work of Colding–Minicozzi and the barrier argument given in Lemma <ref> this automatically holds if all cylindrical ends of $\Sigma$ are asymptotic to round cylinders but we have been unable to determine if it holds in general. (Note that for $\Sigma\subset \RR^3$, work of Wang and Colding–Minicozzi guarantee that $\Sigma$ has nice, stable ends, see Proposition <ref>.)

The stable ends hypothesis does not seem to have been considered elsewhere. If there exists $\Sigma\subset \RR^{n+1}$ with nice but unstable ends, we conjecture that there exists an ancient one-sided flow coming out of $\Sigma$, but the flow is not graphical in the ancient past. Indeed, one may expect that the ancient flow along a cylindrical end $\hat\Sigma\times \RR$ is modeled on the elliptic regularization translator solution that limits to the one-sided flow coming out of $\hat\Sigma$ (in one dimension lower).
To prove existence and uniqueness in this setting one would need to analyze the “tips" of the cylindrical ends by generalizing the arguments from [2] to apply in this setting. Since this is not needed for the main result in $\RR^3$, we leave such investigations for future work.

We now state (somewhat less informally) our existence and uniqueness results for the ancient one-sided flow.

[Existence of ancient one-sided flow, informal]
Consider $\Sigma\subset \RR^{n+1}$, $n\leq 6$, a self-shrinker with nice, strictly stable ends. Fix one of the two open sets $\Omega$ with $\Sigma = \partial\Omega$ and choose the inwards pointing unit normal for $\Sigma$. There exists a weak[precisely, $\bar M(t)$ is an integral cyclic unit regular Brakke flow with corresponding weak set flow $\cK$ so that $\bar M(t) = \cH^n\lfloor \partial\cK(t)$] rescaled mean curvature flow $(\bar M(\tau))_{\tau \in \RR}$, with the following properties:

* the flow is one-sided: $\bar M(\tau) \subset \Omega$

* the flow is shrinker mean convex: the speed of $\tau \mapsto \bar M(\tau)$ is everywhere positive

* the flow has good regularity properties: $\bar M(\tau)$ has only spherical/cylindrical multiplicity one singularities

* the flow is graphical in the ancient past: for $\tau \ll0$, there is $v(\cdot,\tau) > 0 \in C^\infty(\Sigma)$ so that $\bar M(\tau)$ is the normal graph of $v(\cdot,\tau)$,

* convergence in the ancient past: the function $v(\cdot,\tau)\to0$ in $C^k(\Sigma)$ as $\tau \to -\infty$, and

* the graph is first eigenfunction dominated: the function $v(\cdot,\tau)$ satisfies $\tfrac 12 e^{-\mu\tau} \varphi(\cdot) \leq v(\cdot,\tau) \leq 2 e^{-\mu\tau} \varphi(\cdot)$, for $\mu$ and $\varphi$ the first eigenvalue and eigenfunction of the $L$-operator.

Precise references are as follows: for (1)-(3) see Propositions <ref>  and <ref>, for (4)-(6) see Proposition <ref>.

[Uniqueness of ancient one-sided flow, informal]
For $\Sigma,\Omega,\bar M(\tau)$ as in the existence result, consider $\check M(\tau)$ an ancient weak[integral unit regular Brakke flow] rescaled mean curvature flow with the following properties:

* one-sided: $\check M(\tau) \subset \Omega$

* smoothly limiting in the ancient past: $\check M(\tau)$ limits to $\Sigma$ in $C^\infty_\textnormal{loc}(\Sigma)$ with multiplicity one as $\tau\to-\infty$.

Then there is $\tau_0 \in \RR$ so that $\check M(\tau)  = \bar M(\tau + \tau_0)$.

The precise reference is Theorem <ref>.

§.§.§ Outline of the proof of existence

For $\mu,\varphi$ the first eigenvalue/eigenfunction for the $L$-operator, we prove in Lemma <ref> that the graphs of

\[
v_\pm(\bx,\tau) : =  (e^{-\mu \tau}\pm M e^{-2\mu \tau}) \varphi(\bx),
\]

define a supersolution (for $v_+$) and subsolution (for $v_-$) for rescaled mean curvature flow, for $\tau \ll0$ and $M\gg0$ (see Lemma <ref>).[This is a key innovation in the existence part of the present paper, allowing us to simplify several arguments from [22] even though the cylindrical ends seem like they should complicate certain steps.] The key feature is that to highest order $v_\pm \sim e^{-\mu\tau}\varphi$ decays at exactly the expected rate. Using these solutions as barriers, we see that the rescaled mean curvature flow with initial condition the graph of $\eps\varphi$ starting at time $\tau = |\mu|^{-1} \log \eps$—where $\eps>0$ is sufficiently small—exists until some fixed time $\tau_0$ independent of $\eps>0$ (this follows from the bootstrap arguments in Section <ref>, see Proposition <ref>).

Because the flows exist for a uniform amount of time as $\eps\to 0$, we can pass to the $\eps\to 0$ limit to obtain a graphical ancient one-sided rescaled mean curvature flow $\bar M(\tau)$ defined for $\tau \in (-\infty,\tau_0]$. In fact, using standard elliptic regularization methods we can continue $\bar M(\tau)$ to all $\tau \in\RR$ as a weak flow. This yields (1). Properties (4)-(6) follow from the barriers.

Shrinker mean convexity for all $\tau\in\RR$ follows from shrinker mean convexity in the ancient past and spatial infinity. This proves (2) and allows us (as in [22]) to apply White's regularity theory to deduce (3).

§.§.§ Outline of the proof of uniqueness

As mentioned above, establishing uniqueness of the ancient one-sided flow represents the central difficulty in this paper.

For a shrinker $\Sigma = \partial\Omega$ with nice, stable ends consider an ancient weak rescaled mean curvature flow $\check M(\tau)$ that converges to $\Sigma$ in $C^\infty_\textrm{loc}$ with multiplicity-one as $\tau\to-\infty$. We claim that $\check M (\tau) = \bar M(\tau + \tau_0)$ for some $\tau_0\in\RR$, where $\bar M(\cdot)$ is the ancient one-sided flow constructed above.

For simplicity, we assume in this outline that $\Sigma$ has exactly one end and is asymptotically cylindrical, modeled on $\hat\Sigma\times \RR$. Multiple cylindrical ends do not interact in a significant way and conical ends can be handled by the arguments from [22], so this is not a significant loss of generality. We will choose coordinates $(\bomega,z) \in \RR^n\times \RR$ so that the $\RR$-factor of the asymptotic cylinder  points in the $\be = \partial_z$ direction.

The steps to prove uniqueness are as follows:

* We prove a nearly sharp upper bound for the graphical function $\check u$ in a fixed compact region $\Sigma_0\Subset \Sigma$ (with smooth boundary) as $\tau\to-\infty$. Namely, for any $\mu_0$ the first eigenvalue of $L$ on $\Sigma_0$ (with Dirichlet boundary conditions) we show (see Theorem <ref>) that

\begin{equation}\label{eq:nearly-sharp-decay-intro-sketch}
\sup_{\Sigma_0} \check u \leq C e^{-\mu_0\tau}
\end{equation}

as $\tau\to-\infty$. (By taking $\Sigma_0$ larger we can take $\mu_0$ as close to $\mu$ as we want.)

The basic idea is to consider $\varphi_0$ the first eigenfunction for $L$ on $\Sigma_0$ with Dirichlet boundary and to form a subsolution[In reality, this expression fails to be a subsolution near $\partial\Sigma_0$. We use a different argument to prevent contact there by contrasting the Hopf boundary point lemma with a differential Harnack inequality proven for $\check u$. ] of the form

\[
a e^{-\mu_0\tau}(1-aM e^{-\mu_0\tau})\varphi_0
\]

(this is similar to the ansatz used in the proof of existence). If the desired decay was false then by using the (parabolic) Harnack inequality we would be able to fit the subsolution below $\check M(\tau)$ for larger and larger values of $a$ (as $\tau\to-\infty$).

This part of the argument is very general and we expect it will be useful in other contexts.

* Fixing $\delta(\tau) : = \Vert \check u(\cdot,\tau)\Vert_{C^4(\Sigma\cap B_{4R}(\bOh))}$ for $R\gg0$ fixed, step (1) (along with Schauder theory) imply that $\delta(\tau)$ has nearly sharp exponential decay as $\tau \to-\infty$. We combine this with a barrier argument to extend this to graphicality/decay in the $z$-direction (i.e., along the cylindrical end). In particular, we prove that there is some $\alpha \gg0$ so that $\check M(\tau)$ is a small graph over the exponentially growing region $\Sigma \cap B_{\tfrac 12 \delta(\tau)^{-\frac{1}{1+\alpha}}}(\bOh)$ (see Proposition <ref>). The barrier argument proceeds in the following manner:

* Construction of the barriers: In the case that the cylindrical end was modeled on a round cylinder, we could try to use the foliation of the complement of the end of the cylinder by self-shrinkers as introduced by Angenent–Daskalopoulos–Šešum [1] (cf. [61]).

However, since the actual end of $\Sigma$ is likely to cross in and outside of the cylinder, these shrinkers will not fit perfectly inside/outside of $\Sigma$. To solve this problem, we observe that we can construct a “long” barrier by first writing the end of $\Sigma$ as a graph over $\hat\Sigma\times \RR$ and then perturbing this graph by $\eta z^\alpha \hat\varphi(\bomega)$ for $\hat \varphi$ the first eigenfunction of $L_{\hat\Sigma}$ (see Proposition <ref>). When $\eta>0$ is sufficiently small, this property holds for $\alpha \leq z \leq \eta^{-\frac 1\alpha}$. When $z = \eta^{-\frac 1\alpha}$, the long barrier has peeled away from $\Sigma$ a sufficient amount so that it can be “welded”  (cf. Proposition <ref>) to the $z\geq \eta^{-\frac 1\alpha}$ portion of a ADS shrinker that fits within the end of $\Sigma$.

(When $\hat\Sigma$ is some arbitrary closed shrinker rather than the round sphere, we construct a replacement for the ADS shrinkers formed from the translator approximations to the rescaled (ancient) mean curvature flow coming out of $\hat\Sigma$; see Section <ref>. We note also that Sun–Wang have very recently [74] constructed complete translators asymptotic to $\hat\Sigma\times \RR$ which could alternatively be used here.)

* The flow fits under the barrier with $\eta \sim \delta(\tau)$: We then need to show that the flow $\check M(\tau)$ can be controlled along the end of $\Sigma$ by using the barrier described in (a). To do so, we prove that the barriers from (a) either have bounded length or are asymptotically conical (see Lemma <ref>) and that $\check M(\tau)$ decays to $\Sigma$ in a sublinear manner (see Proposition <ref>). Combining these facts (and the decay from (1) above, we see that the flow $\check \cM(\tau)$ is pinched between $\Sigma$ and the barrier.

* Upgrading the barrier argument to graphicality:  A priori it could hold that the flow bends into multiple sheets or develops a singularity in the region where it is pinched between the barrier and $\Sigma$. We know that this does not occur on fixed compact sets. Using a geometric continuity/bootstrap argument we can then extend this to the full barrier region (see Proposition <ref>).

* At this point the barrier argument from (b) has given that $\check M(\tau)$ is graphical on a exponentially growing region (with nearly sharp pointwise estimates). We can use this, along with a Merle–Zaag type spectral dynamic analysis to prove that the graphical function $\check u$ is equal to $\check a e^{-\mu \tau} \varphi + O(e^{(-\mu + \tfrac {\delta_0} {4}) \tau})$ on compact sets, i.e., $\check u$ behaves like one would expect based on a naive linear analysis as in (<ref>) above (see Theorem <ref>). Below we will assume that $\check a = 1$ (this can be arranged by a translation in time unless $\check a = 0$; we ignore this possibility in this sketch, as it is easily handled by a modification of the $\check a = 1$ argument). We will also assume that $\bar M(\tau)$ (the graphical flow constructed above) has the same top order spectral behavior, i.e., $\bar a=1$.

* We now prove that $\bar M(\tau)$ lies (weakly) below $\check M(\tau)$. (Later we will show they agree.) The main idea is for $\tau_m\to-\infty$ appropriately chosen, we can cut off $(1-o(1))e^{-\mu\tau_m}\varphi(\cdot)$ near radius $r_m\to\infty$ to form an initial condition for the rescaled mean curvature flow that lies below both solutions $\check M(\tau_m),\bar M(\tau_m)$. Since $\bar M(\tau_m)$ remains graphical (with good exponential growth estimates) until a fixed time (independent of $m$), a bootstrap argument implies that the rescaled mean curvature flow of this initial condition also exists until a fixed time (independent of $m$) and remains below both solutions $\check M(\tau),\bar M(\tau)$.

As $m\to\infty$ these new solutions limit to ancient one-sided solutions that satisfy similar graphicality/decay estimates to the constructed graphical solution $\bar M(\tau)$ (see Proposition <ref>). Because the initial condition is formed by cutting off $(1-o(1))e^{-\mu\tau_m}\varphi(\cdot)$, a simple spectral dynamics and uniqueness argument implies that in fact this new solution agrees with $\bar M(\tau)$. Thus, $\bar M(\tau)$ lies below $\check M(\tau)$ (see Proposition <ref>).

* It thus remains to estimate $\check M(\tau)$ “from above” (in reality we do not yet know that $\check M(\tau)$ is an entire graph over $\Sigma$). As a first step, we observe that we could repeat the barrier argument used in (2) to pinch $\check M(\tau)$ between barriers and $\bar M(\tau)$ (instead of pinching it between barriers and $\Sigma$). This involves observing that the “long barrier” can be also considered as a graph over $\bar M(\tau)$ instead of $\Sigma$ and one can construct suitable barriers in this setting as well. This allows us to bootstrap estimates for the quantity

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

Indeed, when $W(\tau)$ is very small, we can fit very close barriers (upgrading $L^2$-smallness to $L^\infty$-smallness using Schauder estimates) to obtain graphicality on a very large scale $\sim W(\tau)^{-\frac{1}{1+\alpha}}$. Then, this implies that the cutoff error estimates in a spectral dynamics argument for $\check u(\cdot,s)-\bar u(\cdot,s)$ are very small, leading to further improved estimates for $W(\tau)$. Eventually (see Theorem <ref>), this implies that $W(\tau) \equiv 0$, finishing the proof.

§.§ Outline of application to mean curvature flow in $\RR^3$

Consider $M(0)\subset \RR^3$ a closed embedded surface. Assume that some weak[integral unit regular Brakke flow] mean curvature flow $M(t)$ starting at $M(0)$ has only multiplicity one spherical and cylindrical tangent flows until time $t=T$[By the resolution of the mean convex neighborhood conjecture [26] any two integral unit regular Brakke flows with the given initial condition agree at least until $t=T$. ]. Assume that  all singularities at time $t=T$ have multiplicity one. For simplicity, we assume that there is exactly one singular point at $t=T$, at $\bx \in \RR^3$.

We now consider a local foliation $M_s(0)$ around $M(0)$ and flow each leaf simultaneously into $M_s(t)$. Fixing $s_j\to 0$ so that the level set flow of $M_{s_j}(0)$ does not fatten, we parabolically dilate $M_{s_j}(t)$ and $M_0(t)$ around $(\bx,T)$ by the parabolic distance $d_j$ of the (space-time) support of the flow $t\mapsto M_{s_j}(t)$ to $(\bx,T)$. Passing both flows to the limit, we obtain an ancient flow $\check M(t)$ on one side of some[Uniqueness of tangent flows has not been proven in the case of shrinkers with cylindrical ends, cf. [67, 33, 38]. ] tangent flow to $t\mapsto M_0(t)$ at $(\bx,T)$. By a standard argument based on Huisken's monotonicity formula (cf. the proof of Lemma <ref>) we have $\lambda(\check M(t)) \leq F(\Sigma)$, so the tangent flow to $t\mapsto \check M(t)$ is $t\mapsto \cH^2\lfloor \sqrt{-t}\Sigma$ (i.e., the multiplicity is one).  By the choice of rescaling, $\check M(t)$ is not $t\mapsto \cH^2\lfloor \sqrt{-t}\Sigma$, so by the uniqueness result discussed above, it must be a parabolic dilation of the ancient one-sided flow $\bar M(t)$. We have that $t\mapsto \bar M(t)$ has only multiplicity one spherical/cylindrical singularities, at least for $t<0$[For the arguments developed here, we only need to analyze the flow for $t<0$. This is a strong contrast to [22] which considers the properties of the one-sided flow for all $t \in \RR$ (see, however <cit.>). ].

In particular, returning to the original scale, we find that $t\mapsto M_{s_j}(t)$ has only multiplicity one spherical/cylindrical singularities, at least until time $ t= T - o(d_j^2)$ as $j\to\infty$. We claim that the genus of $M_{s_j}(T - o(d_j^2))$ has strictly dropped in comparison to the genus of $M_0(t)$ right before $t=T$. Recalling that the genus is non-increasing along mean curvature flow by work of White [82], we thus conclude the following result after $\leq \genus(M(0))$ such perturbations:
[Mean curvature flow of generic initial data in $\RR^3$, informal]
For $M(0)\subset \RR^3$ closed embedded surface, there is $M'(0)\subset\RR^3$ an arbitrarily small $C^\infty$ normal graph over $M(0)$ so that the mean curvature flow $t\mapsto M'(t)$ satisfies one of the following conditions:

* all tangent flows at singular points are multiplicity one shrinking spheres or cylinders, or

* at the first time this fails, some tangent flow has multiplicity $\geq 2$.

See Theorem <ref> for the precise statement.

As such, the remaining ingredient we need to discuss is the genus drop. It suffices to show that the genus of the one-sided ancient rescaled flow $\tau \mapsto \bar M(\tau)$ loses genus at some (rescaled) time $\tau<\infty$. Indeed, by an important result of Brendle [10], $\Sigma$ has non-zero genus (if it is not a sphere/cylinder) and thus $\bar M(\tau)$ does as well, for $\tau\ll0$, since it is graphical at such times. In [22] we established the genus drop (when $\Sigma$ is asymptotically conical or compact) by generalizing an observation of Bernstein–Wang [13] who noted that shrinker mean-convexity implies that the flow is star-shaped (and thus genus zero) at time $t=0$.

Here, we use a softer argument (cf. Proposition <ref>), still using shrinker mean convexity, but only in the sense that the rescaled flow $\tau\mapsto \bar M(\tau)$ flows in one direction. If $M(\tau)$ has positive genus for $\tau \gg0$ we can find a non-trivial loop in the component of $\RR^3\setminus M(\tau)$ that $M(\tau)$ is flowing into. (Some care must be taken to avoid a loop $\gamma$ that just goes around a cylindrical end.) The avoidance principle for mean curvature flows rescales to imply that $M(\tau)$ diverges as $\tau \to\infty$. Thus the loop $\gamma$ can be chosen outside of any fixed compact set. In particular, with respect to $M(\tau)$, $\tau \ll0$ (so $M(\tau)$ is $C^\infty$ close to $\Sigma$), the loop $\gamma$ lies near infinity and thus bounds a disk (near infinity $\Sigma$ only consists of cylindrical/conical ends and we can choose $\gamma$ not going around any of these ends). By work of White [82], a loop cannot bound a disk in the complement of a mean curvature flow at an earlier time but not a later time. This is a contradiction, completing the proof.

§.§ Organization of the paper
We collect several fundamental definitions and conventions in Section <ref>. In Section <ref> we discuss self-shrinkers including their ends, and linear theory for the $L$ operator. Section <ref> contains some techniques for constructing graphical rescaled mean curvature flows over shrinkers. We prove existence of the ancient one-sided flow $\tau \mapsto \bar M(\tau)$ in Section <ref>. In Section <ref> we construct barriers along the cylindrical end and then use these (along with other arguments) in Section to prove uniqueness of ancient one-sided flows in Section <ref>. We prove the genus drop for ancient one-sided flows in $\RR^3$ in Section <ref> and combine everything to prove the main generic flow result in Section <ref>.

There are also several appendices. Appendix <ref> contains some important computations concerning the specific form of the quadratic error term in the linearization of the rescaled mean curvature flow equation over a shrinker. Appendix <ref> continues this investigation with the additional assumption that the graph of a positive function, proving a Li–Yau estimate used in the proof of Theorem <ref>. Appendix <ref> recalls a definition of Brakke flow with additional forces while Appendix <ref> recalls Ilmanen's avoidance principle. Appendix <ref> contains a result (possibly of independent interest) concerning the avoidance of a Brakke flow and a non-compact flow with uniformly bounded curvature.  Appendix <ref> contains some results about the topology of regions in $\RR^3$ and Appendix <ref> recalls some concepts from [22] concerning localized topological monotonicity for  mean curvature flows. Finally, Appendix <ref> contains some results about weak versions of uniqueness of tangent flows in $\RR^3$ (also possibly of independent interest).

§.§ Acknowledgements

We are grateful to Christos Mantoulidis for countless discussions about generic mean curvature flow and related topics. We would also like to thank Steve Kerckhoff and Lu Wang for answering some questions.  O.C. was supported by an NSF grant (DMS-2016403), a Terman Fellowship, and a Sloan Fellowship.  K.C. was supported by KIAS Individual Grant MG078901 and POSCO Science Fellowship. F.S. was partially supported by a Leverhulme Trust Research
Project Grant RPG-2016-174.

§ PRELIMINARIES

In this section we collect some preliminary definitions, conventions, and results.

§.§ Spacetime and the level set flow

We define the time map $\ft : \RR^{n+1}\times \RR\to \RR$ to be the projection $\ft(\bx,t) = t$. For $E\subset \RR^{n+1}\times \RR$ we will write $E(t) : = E\cap\ft^{-1}(t)$. The knowledge of $E(t)$ for all $t$ is the same thing as knowing $E$, so we will often ignore the distinction.

For a compact $n$-manifold $M$ (possibly with boundary), we consider $f: M\times [a,b] \to \RR^{n+1}$ so that (i) $f$ is continuous (ii) $f$ is smooth on $(M\setminus\partial M)\times (a,b]$ (iii) $f|_{M\times \{t\}}$ is injective for each $t \in [a,b]$ and (iii) $t\mapsto f(M\setminus\partial M,t)$ is flowing by mean curvature flow. In this case we call

\[
\cM : =\cup_{t\in [a,b]} f(M,t)\times \{t\} \subset \RR^{n+1}\times \RR
\]

a classical mean curvature flow and define the heat boundary of $\cM$ by

\[
\partial\cM : = f(M,a) \cup f(\partial M,[a,b]).
\]

Classical flows that intersect must intersect in a point that belongs to at least one of their heat boundaries (cf. <cit.>).

For $\Gamma \subset \RR^{n+1}\times [0,\infty)$, $\cM \subset \RR^{n+1}\times \RR$ is a weak set flow (generated by $\Gamma$) if $\cM(0) =\Gamma(0)$ and if $\cM'$ is a classical flow with $\partial \cM'$ disjoint from $\cM$ and $\cM'$ disjoint from $\Gamma$ then $\cM'$ is disjoint from $\cM$. There may be more than one weak set flow generated by $\Gamma$.

The biggest such flow is called the level set flow, which can be constructed as follows: For $\Gamma \subset \RR^{n+1}\times [0,\infty)$ as above, we set

\[
W_0 : = \{(\bx,0) : (\bx,0) \not \in \Gamma\}
\]

and then let $W_{k+1}$ denote the union of all classical flows $\cM'$ with $\cM'$ disjoint from $\Gamma$ and $\partial\cM\subset W_k$. The level set flow generated by $\Gamma$ is then defined by

\[
\cM : = (\RR^{n+1}\times [0,\infty))\setminus \cup_k W_k \subset \RR^{n+1}\times [0,\infty).
\]

See [43, 55, 82]. If $\Gamma\subset \RR^{n+1}\times \{0\}$, we will write $F_t(\Gamma) : = \cM(t)$ for the time $t$ slice of the corresponding level set flow.

Fix $\Gamma \subset \RR^{n+1}$ compact. We say that the level set flow of $\Gamma$ is non-fattening if $F_t(\Gamma)$ has no interior for each $t \geq 0$. This condition holds generically, namely if $u_0$ is a continuous function with compact level sets $u_0^{-1}(s)$ then the level set flow of $u_0^{-1}(s)$ fattens for at most countably many values of $s$ <cit.>.

§.§ Integral Brakke flows
An ($n$-dimensional[Of course one can consider $k$-dimensional flows in $\RR^{n+1}$ but we will never do so in this paper, so we will often omit the “$n$-dimensionality” and implicitly assume that all Brakke flows are flows of “hypersurfaces.”]) integral Brakke flow in $\RR^{n+1}$ is a $1$-parameter family of Radon measures $(\mu(t))_{t\in I}$ so that

* For almost every $t\in I$ there is an integral $n$-dimensional varifold $V(t)$ with $\mu(t) = \mu_{V(t)}$ and so that $V(t)$ has locally bounded first variation and mean curvature $\bH$ orthogonal to $\textrm{Tan}(V(t),\cdot)$ almost everywhere.

* For a bounded interval $[t_1,t_2]\in I$ and $K\subset \RR^{n+1}$ compact, we have

\[
\int_{t_1}^{t_2} \int_K (1+|\bH|^2) d\mu(t) dt < \infty.
\]

* If $[t_1,t_2]\subset I$ and $f\in C^1_c(\RR^{n+1}\times [t_1,t_2])$ has $f\geq 0$ then

\[
\int f(\cdot,t_2) d\mu(t_2) - \int f(\cdot,t_1) d\mu(t_1) \leq \int_{t_1}^{t_2} \int (-|\bH|^2 f + \bH \cdot\nabla f + \tfrac{\partial f}{\partial t}) d\mu(t) dt.
\]

We will sometimes write $\cM$ to represent a Brakke flow.

We define the support of $\cM = (\mu(t))_t$ to be $\overline{\cup_t \supp \mu(t) \times \{t\}} \subset \RR^{n+1}\times \RR$.  It is useful to recall that the support a Brakke flow (with $t \in [0,\infty)$) is a weak set flow (generated by $\supp \mu(0)$) <cit.>.

We say that a sequence of integral Brakke flows $\cM_i$ converges to another integral Brakke flow $\cM$ (written $\cM_i \rightharpoonup \cM$) if $\mu_i(t)$ weakly converges to $\mu(t)$ for all $t$ and for almost every $t$, after passing to a further subsequence depending on $t$, the associated integral varifolds converge $V_i(t) \to V(t)$. (Recall that if $\cM_i$ is a sequence of integral Brakke flows with uniform local mass bounds then a subsequence converges to an integral Brakke flow <cit.>.)

We denote the set of points so that an integral Brakke flow $\cM$ is locally (in a backwards-forwards parabolic neighborhood) a smooth multiplicity one flow as $\reg \cM$ and if $\cM$ is defined on $[T_0,\infty)$, we set $\sing \cM : = \supp\cM \setminus (\reg\cM \cup \RR^{n+1}\times \{T_0\})$.

§.§ Density, Huisken's monotonicity, and entropy
For $X_0 = (\bx_0,t_0)\in\RR^{n+1}\times \RR$ we consider the ($n$-dimensional) backwards heat kernel based at $X_0$:

\begin{equation}\label{eq:gauss.dens.defn}
\rho_{X_0} (\bx,t) : = (4\pi(t_0-t))^{-\frac n2} \exp\left( -\frac{|\bx-\bx_0|^2}{4(t_0-t)}\right)
\end{equation}

for $\bx\in\RR^{n+1},t<t_0$. For $\cM$ a Brakke flow defined on $[T_0,\infty)$, $t_0>T_0$ and $0<r\leq\sqrt{T_0-t_0}$, we set

\[
\Theta_\cM(X_0,r) : = \int \rho_{X_0}(\bx,t_0-r^2) d\mu(t_0-r^2).
\]

Huisken's monotonicity formula [50, 56] implies that $r\mapsto \Theta_{\cM}(X_0,r)$ is non-decreasing (and constant only for a shrinking self-shrinker centered at $X_0$, cf. Section <ref>). In particular we can define the density of $\cM$ at such $X_0$ by

\[
\Theta_\cM(X_0) : = \lim_{r\searrow 0} \Theta_\cM(X_0,r).
\]

We call an integral Brakke flow $\cM$ unit-regular if $\cM$ is smooth in a forwards-backwards space-time neighborhood of any space-time point $X$ with $\Theta_\cM(X) = 1$. Note that we can then write $\sing\cM = \{X \in\RR^{n+1}\times \RR : \Theta_\cM(X) > 1\}$.  Note that by <cit.> the class of unit-regular integral Brakke flows is closed under the convergence of Brakke flows. Furthermore, combining  <cit.> and [86] it follows that there is $\varepsilon_0>0$, depending only on dimension, such that every point $X\in \sing\cM$ has $\Theta_\cM(X) \geq 1 + \varepsilon_0$. Upper semi-continuity of density then implies that $\sing\cM$ is closed.

Following [31] for a Radon measure $\mu$ on $\RR^{n+1}$ we define the entropy of $\mu$ by

\[
\lambda(\mu) : = \sup_{\bx_0\in\RR^{n+1},t_0>0} \int \rho_{(\bx_0,t_0)}(\bx,0) d\mu
\]

where $\rho$ is the Gaussian as in (<ref>). When $\mu = \cH^n \lfloor M$ for $M\subset \RR^{n+1}$ a properly embedded submanifold we will write $\lambda(M)$ instead. We define $\lambda_j : = \lambda(\SS^j \subset \RR^{j+1})$ and note that $\lambda_j = \lambda(\SS^j \times \RR^{n-j})$.

§.§ Cyclic Brakke flows

We call an integral Brakke flow $\cM = (\mu(t))_{t\in I}$ cyclic if for a.e. $t\in I$ it holds that $\mu(t) = \mu_{V(t)}$ for an integral varifold $V(t)$ whose uniquely associated rectifiable mod-$2$ flat chain $[V(t)]$ has $\partial[V(t)]=0$. Loosely speaking, the cyclic condition can be used to rule out triple junctions (and generalizations thereof). By <cit.> the cyclic property is preserved under weak limits of integral Brakke flows.

Existence and uniqueness of one sided ancient flows

§ SELF-SHRINKERS

§.§ Self-shrinkers and curvature conventions
We recall that a self-shrinker is a hypersurface $\Sigma^n \subset \RR^{n+1}$ satisfying

\begin{equation}\label{eq:shrinker-def}
\bH +\frac{\bx^\perp}{2} = 0\,
\end{equation}

where $\bH$ is its mean curvature vector, $\bx$ the position vector and ${}^\perp$ the projection to the normal space.

Recall that (<ref>) is equivalent to any of the following conditions:

* $t\mapsto \sqrt{-t}\Sigma$ is a mean curvature flow for $t<0$,

* $\Sigma$ is a minimal hypersurface with respect to $e^{-\frac{1}{2n}|\bx|^2} g_{\RR^{n+1}}$, or

* $\Sigma$ is a critical point of the $F$-functional

\[
F(\Sigma) : = (4\pi)^{-\frac n2} \int_\Sigma e^{-\frac 14 |\bx|^2}
\]

among compactly supported deformations and translations/dilations.

See <cit.>.

We denote with $\cS_n$ the set of smooth properly embedded self shrinkers $\Sigma^n \subset \RR^{n+1}$ with finitely many ends and let $\cS_n^*$ denote the subset of non-flat ones. Moreover (cf. [15]) set

\[
\cS_n(\Lambda) = \{\Sigma \in \cS_n : \lambda(\Sigma) < \Lambda\}.
\]

Furthermore, we consider tuples $(\Sigma, \Omega)$ where $\Sigma \in \cS_n$ and $\Omega$ is one of the two unique choices of open subsets $\Omega \subset \RR^{n+1}$ such that $\partial \Omega = \Sigma$. By an abuse of notation we will write $(\Sigma,\Omega) \in \cS_n$.

We also define

\[
\cS^\textrm{gen}_n : = \left\{O(\SS^{j}(\sqrt{2j}) \times \RR^{n-j}) \in \cS_n : j = 1,\dots,k, \, O \in O(n+1) \right\}
\]

to be the set of (round) self-shrinking spheres and cylinders in $\RR^{n+1}$.

We denote with $\nu_\Sigma$ the choice of unit normal vector field to $\Sigma$ pointing into $\Omega$. We define its shape operator (or Weingarten map)

\begin{equation}\label{eq:convent.shape.oper}
S_p: T_p\Sigma \to T_p\Sigma, \qquad \xi \mapsto - D_\xi\nu_\Sigma
\end{equation}

and its second fundamental form

\begin{equation}\label{eq:convent.sff}
A: T_p\Sigma \times T_p\Sigma \to \RR,\qquad (\xi,\zeta) \mapsto A(\xi,\zeta) = S_p(\xi) \cdot \zeta\, .
\end{equation}

We fix the sign of the scalar mean curvature $H$ as follows

$$ \bH = H\, \nu_\Sigma\, ,$$

and thus $H = \tr S  = \tr A $, with the principal curvatures of $\Sigma$ being the eigenvalues of $S$. Observe that the shrinker mean curvature can be written as

\[
\bH + \frac{\bx^\perp}{2} = \left(H + \frac 12  \bx \cdot \nu_\Sigma \right) \nu_\Sigma
\]

Note that with these conventions, a sphere bounding a ball has normal vector pointing to the inside, positive mean curvature and positive principal curvatures. Similarly, a sphere bounding the complement of a closed ball has normal vector pointing to the outside, negative mean curvature and negative principal curvatures.

For a Riemannian manifold $(M,g)$ with Levi–Civita connection $\nabla$, we define the Riemannian curvature tensor by

\begin{align}
R(\bX,\bY)\bZ & = \nabla^2_{\bX,\bY} \bZ - \nabla^2_{\bY,\bX} \bZ \label{eq:curv.con1}\\
R(\bX,\bY,\bZ,\bW) & = g(R(\bX,\bY)\bW,\bZ)\label{eq:curv.con2}
\end{align}

With this convention, if $\be_1,\be_2\in T_p M$ are orthonormal, the sectional curvature of $\Pi = \Span\{\be_1,\be_2\}$ is given by $K(\Pi) = R(\be_1,\be_2,\be_1,\be_2)$. Furthermore, if $M \subset \RR^{n+1}$ is a hypersurface then the Gauss equations become

\begin{equation}\label{eq:Gauss.convention}
R(\bX,\bY,\bZ,\bW) = A(\bX,\bZ)A(\bY,\bW) - A(\bX,\bW)A(\bY,\bZ).
\end{equation}

In particular, we have

\begin{equation}\label{eq:Gauss.convention.traced}
\Ric(\bX,\bY) = H A(\bX,\bY) - A^2(\bX,\bY),
\end{equation}

where $A^2(\bX,\bY) = \sum_{i=1}^n A(\bX,\be_i)A(\bY,\be_i)$ for an orthonormal basis $\be_i$.

§.§ Ends of shrinkers
For any $\Sigma \in \cS_n$, define

\[
\sqrt{0}\Sigma : = \lim_{t\nearrow 0} \sqrt{-t}\Sigma
\]

where the limit is taken in the Hausdorff sense. For $\bx \in \sqrt{0}\Sigma$, define $\cT_\Sigma(\bx)$ to be the set of tangent flows to the flow $\{\sqrt{-t}\Sigma\}_{\{t<0\}}$ at $(\bx,0)$. Note that if $\bx \neq \bOh$, any tangent flow in $\cT_\Sigma(\bx)$ (weakly) splits a line in the $\bx$ direction.

We define the class of shrinkers that we consider in this paper as follows.

We say that $\Sigma \in \cS_n$ has nice ends if for $\bx \in \sqrt{0}\Sigma\setminus\{\bOh\}$, the set $\cT_\Sigma(\bx)$ has precisely one element $(-\infty,0) \ni t\mapsto \cM(t)$ and moreover, one of the following holds:

* $\cM(t)$ is a multiplicity one plane, or

* $\cM(t)$ is the the multiplicity one flow associated to the smooth flow $\sqrt{-t} \hat\Sigma \times \RR_\bx$, for $\hat\Sigma$ a compact smooth shrinker in $\RR^{n}_{\bx^\perp} : = \{\by \in \RR^{n+1} : \bangle{\by,\bx} = 0\}$.

If $\Sigma\in\cS_n$ has nice ends, we write $\Sigma\in \cS'_n$. We define $\cS_n'(\Lambda) = \cS_n'\cap\cS_n(\Lambda)$.

The following lemma follows immediately from the definition of $\cS_n'$ combined with Brakke regularity.

For $\Sigma \in \cS_n'$, there is $R=R(\Sigma)$ sufficiently large so that

\[
\Sigma \setminus B_R(\bOh) = \Sigma_\textnormal{cyl} \cup \Sigma_\textnormal{con}
\]

so that $\lambda \Sigma_\textnormal{con}$ converges in $C^\infty_\textnormal{loc}(\RR^{n+1}\setminus\{0\})$ with multiplicity one to $\sqrt{0}\Sigma_\textnormal{con}$ as $\lambda \to 0$, and for any connected component $\Sigma ' \subset \Sigma_\textnormal{cyl}$ there exists $\bx \in \SS^n \subset \RR^{n+1}$ and a compact smooth shrinker $\hat \Sigma \subset \RR^n_{\bx^\perp}$ so that $\Sigma' - s\bx$ converges in $C^\infty_\textnormal{loc}(\RR^{n+1})$ with multiplicity one to $\hat \Sigma\times \RR_\bx$ as $s\to \infty$.

§.§ Function spaces
For ease of reference, we collect below various norms to be used in later sections. We will use the notation $L^{2}_{W}$ for the Gaussian weighted $L^{2}$-space on a shrinker $\Sigma$, i.e.,

\[
\bangle{u,v}_{W} = (4\pi)^{-\frac n2} \int_{\Sigma} vw e^{-\tfrac 14 |\bx|^{2}}
\]

The following weighted Hölder spaces will be used along the conical ends.

\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{con}}} \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{con}}} + [\nabla^{k}_{\Sigma}f]^{(d-k)}_{\alpha;\Sigma_{\textrm{con}}}.
\end{align*}

When working with flows that are not known to be graphical over the entire cylindrical end, we will use the following semi-global norms

\[
\Vert \cdot \Vert_{k,\alpha; \text{cyl}}(\rho)=   \Vert  \cdot \Vert_{k,\alpha;\Sigma \cap B_{2R_0}(\bOh)} + \Vert \cdot \Vert_{k,\alpha;\Sigma_\textnormal{cyl} \cap B_{\rho}(\bOh)}
\]

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

where the first norm only considers the compact part along with a finite part of the  cylindrical piece, while the second normal also controls the conical ends (with an appropriate weight). These norms are used in the proof of uniqueness in the ancient past in Section <ref>.

§.§ Linear theory on shrinkers with stable ends
Consider a shrinker with nice ends $\Sigma \in \cS'_n$. By Lemma <ref> the ends of $\Sigma$  decompose as the disjoint union $\Sigma_\textnormal{cyl} \cup \Sigma_\textnormal{con}$ of cylindrical and smoothly asymptotically conical ends. We denote the asymptotically conical ends by $\Sigma'_{\textnormal{con}, i}$ for $i =1, \dots, N_\textnormal{con}$ with asymptotic cones $\cC_i$ and the asymptotically cylindrical ends $\Sigma'_{\textnormal{cyl}, j}$ for $j =1, \dots, N_\textnormal{cyl}$ with asymptotic directions $\bx_j \in \SS^n$ and asymptotic compact smooth shrinkers $\hat{\Sigma}_j\subset \RR^n_{\bx_j^\perp}$.

We denote by $\mu$ the smallest eigenvalue of the following operator[Recall that we say that $\mu$ is an eigenvalue of $L$ with eigenfunction $\varphi$ if $L\varphi = - \mu \varphi$. We will write $L_\Sigma$, $\Delta_\Sigma$, etc., when the dependence on $\Sigma$ is not clear.] acting on functions $u \in C^2_\textrm{loc}(\Sigma)$

$$Lu=\Delta u -\frac{1}{2} \bx^T \cdot \nabla u +\tfrac{1}{2}u +|A|^2 u$$

(where $A$ is the second fundamental form of $\Sigma$) and will write $0<\varphi \in H^1_W(\Sigma)$ for the corresponding first eigenfunction (unique up to scaling). We normalize $\Vert \varphi \Vert_{W} = 1$. The existence and uniqueness of $\varphi$ follows as in the proof of <cit.>, using the results in [31]. We denote the corresponding eigenvalues and first eigenfunctions of the asymptotic shrinkers $\hat{\Sigma}_j$ by $\hat{\mu}_{j}$ and $\hat{\varphi}_{j}$ respectively. Recall that by Colding–Minicozzi's classification of entropy stable shrinkers, <cit.>, all non flat shrinkers which are not a sphere or a (generalised) round cylinder satisfy $\mu < -1$.

We say a shrinker $\Sigma \in \cS_n$ has stable ends if the first eigenfunction $\varphi$ satisfies $\varphi \in L^{\infty}(\Sigma)$.
If $\Sigma\in\cS'_n$ has nice stable ends, we write $\Sigma\in \cS''_n$. We define $\cS_n''(\Lambda) = \cS_n''\cap\cS_n(\Lambda)$.

We say a shrinker $\Sigma \in \cS_n$ has strictly stable ends if the first eigenfunction $\varphi$ satisfies $\varphi(\bx) = o(1)$ for $\Sigma\ni \bx \to\infty$. If $\Sigma\in\cS'_n$ has nice strictly stable ends, we write $\Sigma\in \cS'''_n$. We define $\cS_n'''(\Lambda) = \cS_n'''\cap\cS_n(\Lambda)$.

Clearly if $\Sigma = \hat \Sigma\times \RR$ for $\hat\Sigma \in \cS_{n-1}$ a smooth compact shrinker, it holds that $\Sigma$ has nice stable ends $\Sigma \in \cS_{n}''$ (but $\Sigma \not \in \cS_n'''$). For a non-cylindrical shrinker with nice ends, we show in part (2) of Proposition <ref> below that if $\hat{\mu}_{j}$ for $j = 1, \dots, N_\textnormal{cyl}$ are the first eigenvalues of the cylindrical ends of $\Sigma \in \cS_{n}'$, then a sufficient condition for $\Sigma$ to have strictly stable ends is $\mu< \hat{\mu}_{j}$ for $j = 1, \dots, N_\textnormal{cyl}$.

We note that by the result of Wang [79], for $\Sigma \in \cS_2$ (i.e., an embedded shrinker in $\RR^3$), all ends are either cylindrical, with $\hat \Sigma$ being the shrinking circle, or asymptotically conical. Thus, $\Sigma \in \cS_{2}'$. Combining this  with Colding–Minicozzi's estimate $\mu<-1$  and  properties (1) and (2) Proposition <ref> below we see that any non-cylindrical shrinker in $\cS_2$ has nice, strictly stable ends.

$\cS_2 = \cS_2' = \cS_2''$ and $\cS_2 \setminus \cS_2^\textnormal{gen} = \cS_2'''$.

It is an interesting question to what extent the inclusions $\cS_3'' \subset \cS_3'\subset \cS_3$, $\cS_3'''\subset \cS_3' \setminus \cS_3^\textrm{gen}$ are strict.[For example, if we fix a smooth closed embedded shrinker $\hat\Sigma\subset\RR^3$ (e.g., the round sphere, or the Angenent torus), we do not know if there exists $\Sigma \in \cS_3'$ with an end asymptotic to $\hat\Sigma\times \RR$ (other than $\Sigma = \hat \Sigma\times \RR$) and if such $\Sigma$ exists, we do not know if it must have (strictly) stable ends when $\Sigma$ is not a round sphere. These questions are likely to be relevant to the understanding of (generic) mean curvature flow of hypersurfaces in $\RR^4$.]

We extend here the analysis in <cit.> of the first eigenfunction of $L_\Sigma$ from asymptotically conical shrinkers to shrinkers with nice ends.

Assume $(\Sigma, \Omega) \in \cS'_n$ is a smooth shrinker with nice ends. Write $\mu$ for the first eigenvalue of $L_{\Sigma}$ on $\Sigma$.

* Fix $\beta>0$. Along $\Sigma_\textnormal{con}$ we have

\begin{align*}
(1+|\mathbf{x}|^2)^{\frac 12 + \mu - \beta} \lesssim \varphi(\mathbf{x}) & \lesssim (1+|\mathbf{x}|^2)^{\frac 12 + \mu + \beta},\\
|\nabla^m \varphi(\mathbf{x})| & \lesssim (1+|\mathbf{x}|^2)^{\frac 12 + \mu + \beta - \frac m2}.
\end{align*}

* Write $\mu_{j}$ for the first eigenvalue of $L_{\hat\Sigma_{j}\times\RR}$ for the exact cylinder associated to each cylindrical end $\Sigma_{\textnormal{cyl},j}'$, $j=1,\dots,N_{\textnormal{cyl}}$. Assume that

\begin{equation}\label{eq:eig-cond-impl-stable-ends}
\mu<\mu_{j} \textrm{ for } j=1,\dots,N_{\textnormal{cyl}}.
\end{equation}

Then, along $\Sigma'_{\textnormal{cyl}, j} \subset \Sigma_\textnormal{cyl}$, we have

\begin{align}
(1+|\mathbf{x}|^2)^{\mu - \hat{\mu}_{j} - \beta} \lesssim \varphi (\mathbf{x}) & \lesssim (1+|\mathbf{x}|^2)^{\mu - \hat{\mu}_{j}+ \beta} \label{eq:C0_estimate_cylindrical_end},\\
|\nabla^m \varphi (\mathbf{x})| & \lesssim (1+|\mathbf{x}|^2)^{\mu - \hat{\mu}_{j}+ \beta}. \label{eq:Ck_estimate_cylindrical_end}
\end{align}

In particular, (<ref>) implies that $\Sigma \in \cS_{n}'''$.

* Assume that $\Sigma\in\cS_{n}''$, i.e., $\Sigma$ has stable ends (but do not necessarily assume the $\mu<\mu_{j}$ condition in (<ref>)). Then, $|\nabla^{m}\varphi|\in L^{\infty}(\Sigma)$ and $\varphi^{-1}|\nabla \varphi|^{2} \in L^{\infty}(\Sigma)$.

Moreover, there is $\eps_0 = \eps_0(\Sigma)>0$ with the following properties. For $\eps \in (0,\eps_0)$, the normal graph of $\eps\varphi$ is a smooth surface $\Sigma_\eps \subset \Omega$. Denote $\Omega_\eps \subset \Omega$ by the open set with $\partial\Omega_\eps = \Sigma_\eps$. The surface $\Sigma_\eps$ is strictly shrinker mean convex to the interior of $\Omega_\eps$ in the sense that

\begin{equation}\label{eq:shrinker_mean_convexity}
2H_{\Sigma_\eps} + \mathbf{x}\cdot \nu_{\Sigma_\eps} \geq  \eps (2 |\mu| - C\eps) \varphi(\bx)
\end{equation}

for $C=C(\Sigma)$. Furthermore,  the entropy of $\Sigma_\eps$ satisfies the bound $\lambda(\Sigma_\eps) \leq F(\Sigma) + o(1)$ as $\eps \to 0$.

Recall from the proof of <cit.> that for every $\beta>0$ there exists $R = R(\Sigma, \beta)$ such along each asymptotically conical end $\Sigma'_{\textnormal{con}, i}$ we have barriers

\begin{equation}\label{eq: barrier asym.1}
\overline{g}(\bx) = |x|^{1+2\mu+2\beta} \text{ and } \underline{g}(\bx) = |x|^{1+2\mu-2\beta}\, ,
\end{equation}

which satisfy

\begin{equation}\label{eq: barrier asym.2}
L\overline{g} + \mu \overline{g} < 0 \text{ and } L\underline{g} + \mu \underline{g} > 0
\end{equation}

on $\Sigma'_{\textnormal{con}, i} \setminus \overline{B}_R$.

Similarly, for an asymptotically cylindrical end $\Sigma'_{\textnormal{cyl}, j}$, so that $\mu_{j}<\mu$,
let $\be:=\bx_j$ and denote the coordinate in direction $\be$ by $z$. Let $\bar{\varphi} \in C^\infty_c(\RR^n_{\be^\perp})$ be an extension of the first eigenfunction of $L_{\hat \Sigma_i}$, $\hat{\varphi}_{j}$, to a neighborhood of $\hat{\Sigma}\subset \RR^n_{\be^\perp}$. We make the ansatz for a barrier as the ambient function

$$g := z^{-\alpha} \bar{\varphi}$$
for some $\alpha>0$, where we extend $\bar{\varphi}$ constant in $\be$-direction. Note that

\begin{align}
\nabla_{\RR^{n+1}} g &= - \alpha z^{-\alpha-1} \bar{\varphi} \be + z^{-\alpha} \nabla_{\RR^{n+1}} \bar{\varphi} \notag \\
\nabla^2_{\RR^{n+1}} g &= \alpha (\alpha + 1) z^{-\alpha-2} \bar{\varphi}\, dz \otimes dz - \alpha z^{-\alpha -1} \left( dz \otimes d\bar{\varphi} + d\bar{\varphi} \otimes dz \right) + z^{-\alpha} \nabla^2_{\RR^{n+1}} \bar{\varphi} \label{eq: hess psi}
\end{align}

Since $g$ is an ambient function we compute (compare the proof of <cit.>)

$$L g = \Delta_{\RR^{n+1}}g - \nabla^2_{\RR^{n+1}} g(\nu, \nu) - \tfrac{1}{2} \bx \cdot \nabla_{\RR^{n+1}} g + \tfrac{1}{2} g + |A|^2 g\, .$$

Note that

$$ - \tfrac{1}{2} \bx \cdot \nabla_{\RR^{n+1}}g = \tfrac{\alpha}{2} g - \tfrac{1}{2} z^{-\alpha} \bx \cdot \nabla_{\RR^{n}_{\be^{\perp}}} \bar{\varphi} \, .$$

Furthermore, since $\hat{\varphi}_{j} > 0$, using (<ref>), we can estimate as $z \to \infty$ that

\begin{align*}
\nabla^2_{\RR^{n+1}} g(\nu_\Sigma,\nu_\Sigma) &= z^{-\alpha} \nabla^2_{\RR^{n}_{\be^{\perp}}} \bar{\varphi}(\nu_{\hat{\Sigma}_j}, \nu_{\hat{\Sigma}_j}) + o(1) g \\
\Delta_{\RR^{n+1}}g &=  z^{-\alpha} \Delta_{\RR^n_{\be^\perp}} \bar{\varphi} + o(1) g\\
|A_\Sigma|^2 g &=  z^{-\alpha} |A_{\hat{\Sigma}_j}|^2\bar{\varphi} + o(1) g\, ,
\end{align*}

where have extended $\nu_{\hat{\Sigma}_j}$ and $|A_{\hat{\Sigma}_j}|^2$ to a neighborhood of $\hat{\Sigma}\subset \RR^n_{\be^\perp}$. We thus can estimate

\begin{equation*}\begin{split}
L g &= z^{-\alpha} \left( \Delta_{\RR^n_{\be^\perp}} \bar{\varphi} - \nabla^2_{\RR^n_{\be^\perp}} \bar{\varphi}(\nu_{\hat{\Sigma}_j}, \nu_{\hat{\Sigma}_j}) - \tfrac{1}{2} \bx \cdot \nabla_{\RR^{n}_{\be^{\perp}}} \bar{\varphi} + \tfrac{1}{2} \bar{\varphi} + |A_{\hat{\Sigma}_j}|^2\bar{\varphi} \right) + \tfrac{\alpha}{2} g + o(1) g \\
&= \left( \tfrac{\alpha}{2} - \hat{\mu}_{j} + o(1)\right) g\, .
\end{split}\end{equation*}

Thus for every $\beta > 0$ there is $R = R(\Sigma'_j, \beta)$ such that choosing $\alpha =  2(\hat{\mu}_{j} - \mu - \beta)$ we have an upper barrier $\overline{g}$ and choosing $\alpha =  2(\hat{\mu}_{j} - \mu + \beta)$ we have a lower barrier $\underline{g}$ satisfying

\begin{equation}\label{eq: barrier asym.3}
L\overline{g} + \mu \overline{g} < 0 \text{ and } L\underline{g} + \mu \underline{g} > 0
\end{equation}

on $\Sigma'_j \setminus B_R$.
In both the asymptotically conical and the asymptotically cylindrical case we can choose $C>1$ such that

$$ C^{-1} \underline{g} < \varphi < C \overline{g} $$

on $\partial(\Sigma' \setminus B_R)$.  As such, $v = \varphi - C \overline{g}$ satisfies $L v + \mu v > 0$ and $v < 0$ in a neighborhood of $\partial(\Sigma' \setminus B_R)$. We now argue that $v \leq 0 $ on $\Sigma' \setminus B_R$. Since $v^+ \in H^1_W$, we find that

\begin{equation*}\begin{split}
-\mu \int_{\Sigma \setminus B_R} (v^+)^2 \rho\, d\cH^n &\leq \int_{\Sigma \setminus B_R} v^+ L v \rho\, d\cH^n \\
&= \int_{\Sigma \setminus B_R} \left(-|\nabla v^+|^2 + \left(\frac{1}{2} + |A_\Sigma|^2 \right)(v^+)^2\right) \rho\, d\cH^n\, .
\end{split}
\end{equation*}

Thus, using Ecker's Sobolev inequality [41] (cf. <cit.>) and that $|A_\Sigma|$ is bounded, we find that

$$ R^2 \int_{\Sigma \setminus B_R}  (v^+)^2 \rho\, d\cH^n \leq C \int_{\Sigma \setminus B_R}  (v^+)^2 \rho\, d\cH^n\, .$$

Thus choosing $R$ sufficiently large, if necessary, we thus see that $v^+ \equiv 0$ on $\Sigma'\setminus B_R(0)$.

Thus $\varphi \leq C \overline{g}$ on $\Sigma'\setminus B_R(0)$. The same reasoning gives $\varphi \geq C^{-1} \underline{g}$ on $\Sigma'\setminus B_R(0)$. This proves the $C^{0}$-estimate in properties (1) and (2). In particular, this proves that (<ref>) implies $\Sigma \in \cS_n'''$.

We now explain the derivative estimates in (1) and (2). In the asymptotically conical case, the decay estimates for the higher derivatives of $\varphi$ follow as in the proof of <cit.>.
For the asymptotically cylindrical ends, consider for $t<0$ and $\mathbf{q} \in \sqrt{-t} \Sigma : = \Sigma_t$ the function

$$ \tilde{\varphi}(\mathbf{q},t) = \varphi\left(\frac{\mathbf{q}}{\sqrt{-t}}\right)\, .$$

Then since $L\varphi = -\mu \varphi$ we have that

$$ \left(\frac{d}{dt} - \Delta_{\Sigma_t}\right) \tilde{\varphi} = \left(|A_{\Sigma_t}|^2 + \frac{1}{2} + \mu\right) \tilde{\varphi}\, .$$

Note that (<ref>) implies that for $t \in (-3,-1]$ along $(\Sigma'_j)_t$

$$ \tilde{\varphi}(\bx) \leq C (1+ |\bx|^2)^{\mu-\hat{\mu}_j + \beta}$$

which is decaying for $\beta < \hat{\mu}_j - \mu$ and thus bounded. Standard interior estimates imply that all higher derivatives of $\tilde\varphi$ are uniformly bounded along $(\Sigma'_j)_t$ for $t \in [-2,-1]$ . Then (<ref>) follows from (<ref>) via interpolation.

Finally, we consider (3). Assume that $\Sigma\in \cS_{n}''$. We first note that the bounds for higher derivatives of the first eigenfunction (with no decay) follows from interior estimates (on $t\mapsto\sqrt{-t}\Sigma$) exactly as in the above argument. We now estimate $\varphi^{-1}|\nabla \varphi|^{2}$. Using the fact that $\Sigma$ has  bounded geometry and the fact that $\varphi$ has bounded $C^{2}$ norm, there is $r>0$ so that normal coordinates exist on a ball of radius $r$ centered at each $\bx \in \Sigma$ and for $\by \in B_{r}(\bx)$ we have the following Taylor expansion

\[
0 \leq \varphi(\by) \leq \varphi(\bx) + \partial_{i}\varphi|_{\bx} (\by^{i}-\bx^{i}) + C|\by-\bx|^{2}.
\]

in the normal coordinates (with constant $C$ uniformly bounded for all such $\bx,\by$). Since $\nabla \varphi \in L^{\infty}(\Sigma)$, we can choose $\by^{i} := - s \partial_{i}\varphi|_{\bx} + \bx^{i} \in B_{r}(\bx)$, for $s>0$ sufficiently small (independent of $\bx$). Thus, we find

\[
0 \leq \varphi(\bx) - s |\nabla \varphi(\bx)|^{2} + C s^{2} |\nabla \varphi(\bx)|^{2}.
\]

Taking $s>0$ even smaller, we conclude that $\varphi^{-1}|\nabla \varphi|^{2}\in L^{\infty}(\Sigma)$.[Note that this estimate could alternatively have been proven by appropriately modifying the argument in <cit.>.]

Because $\Sigma$ has a uniform tubular neighborhood, for $\eps>0$, the normal graph of $\eps\varphi$, $\Sigma_{\eps}$, is a smooth surface. Moreover, by Corollary <ref> (and recalling that $v = (\nu_{\Gamma}\cdot\nu_{\Sigma})^{-1} = 1+o(1)$ as $\eps\to 0$)  we have

\[
v(\bx) (H + \tfrac 12 \bx_{\Gamma}\cdot\nu_{\Gamma}) = \eps L \varphi + E = - \eps \mu \varphi + E
\]

\[
E= \eps \varphi E_{1} + \eps^{2} E_{2}(\nabla \varphi,\nabla \varphi)
\]

with $|E_{1}|= O(\eps),|E_{2}| = O(1)$ as $\eps\to 0$, uniformly on $\Sigma$. Using the $\varphi^{-1}|\nabla \varphi|^{2} \in L^{\infty}(\Sigma)$ estimate from above, we find

\[
v(\bx) (H + \tfrac 12 \bx_{\Gamma}\cdot\nu_{\Gamma}) = \eps L \varphi + E = - \eps \mu \varphi + O(\eps^{2} \varphi)
\]

This proves the strictly shrinker mean convex assertion.

The final statement on the entropy follows directly from the smooth convergence of $\Sigma_\eps \to \Sigma$ as $\eps \to 0$.

§ GEOMETRIC ESTIMATES FOR GRAPHICAL FLOWS

This section contains some well-known considerations relating graphs over shrinkers to rescaled mean curvature flows, including a long-time existence for rescaled flows with small $L^\infty$-norm.

We begin by using pseudolocality and interior estimates to prove higher derivative estimates for a graphical rescaled mean curvature flow over a shrinker.

Fix a shrinker $\Sigma \in \cS_n'$. There are constants $\psi_0= \psi_0(\Sigma), \psi_1= \psi_1(\Sigma)$ with the following properties. For $T > 1$, suppose that $v \in C^2_\textnormal{loc}(\Sigma \times [0,T))$ has

\begin{equation}\label{eq:restart.graph.flow}
\sup_\Sigma |v(\cdot,\tau)| \leq \psi_0, \qquad \sup_\Sigma |\nabla_\Sigma v(\cdot,\tau)|\leq \psi_1
\end{equation}

for all $\tau \in [0,T)$ and $\tau \mapsto \Gamma(\tau) : = \Graph_\Sigma v(\cdot,\tau)$ is a rescaled mean curvature flow.

Then, for $m\in \NN$, it holds that

\sup_{\Sigma\times [1,T)} |\nabla^m_\Sigma v| \leq C= C(m,\Sigma)

Furthermore, if

\limsup_{\tau \to T} \sup_\Sigma |v(\cdot,T)| < \psi_0

then we can extend the flow to $v' \in (\Sigma\times [0,T'])$ for some $T'>T$ with $v'$ still satisfying (<ref>) for all $\tau \in [0,T']$.

Choose $\psi_1=\psi_1(\Sigma)$ sufficiently small so that if $w \in C^2_\textnormal{loc}(\Sigma)$ has

\[
\sup_\Sigma |w(\cdot)| , \sup_\Sigma |\nabla_\Sigma w(\cdot)|\leq \psi_1
\]

then $\Graph_\Sigma w(\cdot)$ is a uniformly small Lipschitz graph in balls of uniform size (over tangent planes to $\Sigma$) and $F(\Graph_\Sigma w(\cdot)) \leq (2-\delta) F(\Sigma)$ for some $\delta\in(0,1)$ fixed.

Now, consider $T > 1$ and $v \in C^2_\textnormal{loc}(\Sigma \times [0,T))$ with

\begin{equation}\label{eq:restart.graph.flow.1}
\sup_\Sigma |v(\cdot,\tau)| \leq \psi_1, \qquad \sup_\Sigma |\nabla_\Sigma v(\cdot,\tau)|\leq \psi_1
\end{equation}

for all $\tau \in [0,T)$. Assume that $\tau\mapsto\Gamma(\tau)$ is a rescaled mean curvature flow. Due to the choice of $\psi_1$ above, we can locally write the flow,

\[
[-e,-1] \ni t\mapsto \sqrt{-t} \left( \Graph_\Sigma v(\cdot,\bar \tau - \log(-t))\right)
\]

as a uniformly bounded Lipschitz graphs over balls of uniform size. Interior estimates <cit.> for mean curvature flow applied at $t=-1$ thus imply that

\[
\sup_{\Gamma(\bar\tau)}|\nabla^m A_{\Gamma(\bar\tau)}| \leq C'(m,\Sigma)
\]

This yields the asserted $C^m$-estimates as long as $\psi_0 \leq \psi_1$.

It remains to choose $\psi_0=\psi_0(\Sigma) \leq \psi_1$ sufficiently small so that the second property holds. Before doing so, we observe that the curvature estimates just derived show that $\Gamma(\tau)$ smoothly limits to some $\Gamma(T)$ and combined with <cit.>, we find that there is $T''>T$ (independent of $\psi_0$) so that the rescaled mean curvature flow $\tau \mapsto \Gamma(\tau)$ exists for $\tau \in [0,T'']$. The same argument used above implies that

\begin{equation}\label{eq:curv.est.boostrap}
\sup_{\Gamma(\bar\tau)}|\nabla^m A_{\Gamma(\bar\tau)}| \leq C'(m,\Sigma)
\end{equation}

holds for $\tau \in [T,T'']$ (after shrinking $T''$ if necessary).

We now fix $\psi_0=\psi_0(\Sigma,\psi_1) \leq \psi_1$ so that if $\Gamma \subset \RR^{n+1}$ is a properly embedded hypersurface with $\Gamma \subset U_{\psi_0}(\Sigma)$, $F(\Gamma) \leq (2-\delta) F(\Sigma)$ and $|A_\Gamma| \leq C'(1,\Sigma)$ (the constant in (<ref>) with $m=1$), then

\[
\Gamma = \Graph_\Sigma w,
\]

where $w\in C^2_\textrm{loc}(\Sigma)$ satisfies

\[
\sup_\Sigma |w(\cdot)|\leq \psi_0,\qquad \sup_\Sigma |\nabla_\Sigma w(\cdot)|\leq \psi_1.
\]

(To see that $\psi_{0}$ exists, note that as $\psi_{0}\to 0$, $\Gamma$ will converge to $\Sigma$ in $C^{1,\alpha}$ due to the entropy and second fundamental form bounds. In this limit, we have the uniform decay $\nabla_\Sigma w = o(1)$, while $|w|\leq \psi_0$ follows from $\Gamma \subset U_{\psi_0}(\Sigma)$.)

We now assume that

\[
\limsup_{\tau \to T} \sup_\Sigma |v(\cdot,T)| < \psi_0
\]

where we have chosen $\psi_0$ above. Using small spheres as barriers we can conclude that there is $T'\in[T,T'']$ so that $\Gamma(\tau) \subset U_{\psi_0}(\Sigma)$ for $\tau \in [T,T']$. Moreover, $F(\Gamma(\tau)) \leq (2-\delta)F(\Sigma)$ by monotonicity of $F$ under the rescaled flow. Finally, (<ref>) implies that $|A_{\Gamma(\tau)}| \leq C'(1,\Sigma)$ for $\tau \in [T,T']$. Thus, by the choice of $\psi_0$, we see that $\Gamma(\tau) = \Graph_\Sigma v(\cdot,\tau)$ with

\[
\sup_\Sigma |v(\cdot,\tau)|\leq \psi_0,\qquad \sup_\Sigma |\nabla_\Sigma v(\cdot,\tau)|\leq \psi_1.
\]

for all $\tau \in [T,T']$. This completes the proof.

Fix a shrinker $\Sigma \in \cS_n'$. There is a constant $\psi^* = \psi^*(\Sigma)$ so that if $v_0 \in C^2(\Sigma)$ has $\Vert v_0\Vert_{C^2(\Sigma)} \leq \psi^*$ then there is $T\in [1,\infty]$ and $v \in C^2_\textnormal{loc}(\Sigma\times [0,T])$ so that:

* $v(\cdot,0) = v_0(\cdot)$,

* $\Gamma(\tau) : = \Graph_\Sigma v(\cdot,\tau)$ is a rescaled mean curvature flow,

* we have

\[
\sup_\Sigma |\nabla^m v(\cdot,\tau)| \leq C(m,\Sigma)
\]

for all $\tau \in [1,T]$ and $m \in\NN$, and

* if $T<\infty$, then it must hold that

\[
\sup_\Sigma |v(\cdot,T)| = \psi_0,
\]

for $\psi_0=\psi_0(\Sigma)$ fixed in Proposition <ref>.

Taking $\psi^*=\psi^*(\Sigma)$ sufficiently small it is easy to see (e.g. using <cit.>) that the graphical rescaled mean curvature flow starting at $v_0$ exists and satisfies

\begin{equation}\label{eq:bootstrap.graphical.conditions}
\sup_\Sigma |v(\cdot,\tau)|\leq \psi_0, \qquad \sup_\Sigma |\nabla_\Sigma v(\cdot,\tau)| \leq \psi_1,
\end{equation}

at least for $\tau \in [0,2]$. Choose $T \in (1,\infty]$ maximal so that the flow exists and satisfies (<ref>) for $\tau \in [0,T)$. Property (3) for $\tau < T$ follows immediately from Proposition <ref>. This verifies (1) and (2) (since if $T<\infty$ then (3) guarantees the flow extends to $\tau =T$.)

In the case that $T=\infty$, condition (4) is vacuous so the proof is completed. Thus, we must verify (4) when $T<\infty$. If (4) fails, then

\[
\sup_\Sigma |v(\cdot,T)| < \psi_0.
\]

However, due to Proposition <ref>, this would would contradict the maximality of $T$, so this case cannot occur. This completes the proof.

Fix a shrinker $(\Sigma,\Omega) \in \cS_{n}'$ and the corresponding constant $\psi^*$ from Corollary <ref>. For $v_0 \in C^2(\Sigma) \setminus\{0\}$ with $v_0 \geq 0$ and $\Vert v_0\Vert_{C^2(\Sigma)} \leq \psi^{*}$, let $[0,T]\ni \tau \mapsto v(\cdot,\tau)$ denote the function whose graph is a rescaled mean curvature flow as in Corollary <ref>. Then $1\leq T< \infty$.

In particular,

\[
\sup_\Sigma |v(\cdot,T)| = \psi_0.
\]

Moreover, if $[0,T]\ni \tau \mapsto \cM(\tau)$ is a rescaled Brakke flow with $\supp \cM(\tau) \subset\Omega$ for all $\tau \in [0,T]$ and

\[
\supp\cM(0) \subset \Omega\setminus \{\bx + s v_0(\bx)\nu_\Sigma(\bx) : \bx \in\Sigma,s\in [0,1]\}
\]

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

Assume that $T = \infty$. First note that $v(\cdot,\tau) >0 $ for all $\tau \in [0,\infty)$ by the Ecker–Huisken maximum principle <cit.>. Then, Ilmanen's localized avoidance principle (Theorem <ref>) applied to $t\mapsto \sqrt{-t}\Sigma$ and $t\mapsto \Graph_\Sigma v(\cdot,-\log(-t))$ implies that for any $R>0$, there is $t_0=t_0(R) < 0$ so that $\Graph_\Sigma v(\cdot,-\log(-t_0)) \cap B_R(\bOh) = \emptyset$. This is a contradiction, when $R$ is large. Thus $T<\infty$, so the “blowup” condition

\[
\sup_\Sigma |v(\cdot,T)| = \psi_0.
\]

follows from (4) in Corollary <ref>.

The remaining statement (“Moreover...”) follows from Proposition <ref>.

§ EXISTENCE OF A ONE-SIDED ANCIENT FLOW

Fix $n\leq 6$ and a shrinker $(\Sigma, \Omega) \in  \cS''_n$ with nice stable ends. (Recall that this means that the ends of $\Sigma$ are smoothly asymptotic to cones or cylinders and the first eigenfunction of $L$ is bounded.) We construct here an ancient shrinker mean convex rescaled Brakke flow inside of $\Omega$ converging to $\Sigma$ (with multiplicity one) in the ancient past.

Recall that the unit normal to $\Sigma$ points into $\Omega$. By Proposition <ref>, the first eigenfunction $\varphi$ (with first eigenvalue $\mu$) of $L$ along $\Sigma$ has the property that  there is $\eps_0 >0$ so that for $\eps\in (0,\eps_0)$, the normal graph of $\eps \varphi$ (denoted here by $\Sigma_\eps$) is a smooth strictly shrinker mean convex hypersurface contained in $\Omega$.

We first construct barriers for the one-sided flow.[These barriers allows us to simplify certain arguments used in [22].]

There is $M =M(\Sigma)>0$ sufficiently large and $\tau_0 = \tau_0(\Sigma)$ sufficiently negative so that $Me^{-\mu\tau_0} \leq 1/2$ and so that defining functions

\[
v_\pm(\bx,\tau) : =  (e^{-\mu \tau}\pm M e^{-2\mu \tau}) \varphi(\bx),
\]

the hypersurfaces $\Sigma_\pm(\tau) : = \Graph_\Sigma v_\pm(\cdot,\tau)$ satisfy:

\[
(-\infty,\tau_0] \ni \tau \mapsto \Sigma_+(\tau)
\]

is a supersolution[Namely, a smooth solution to rescaled mean curvarure flow cannot make contact with $\Sigma_+$ from below (with respect to the unit normal induced by $\Sigma$). ] to rescaled mean curvature flow and

\[
(-\infty,\tau_0] \ni \tau \mapsto \Sigma_-(\tau)
\]

is a subsolution to rescaled mean curvature flow.

We consider $\Sigma_+$ for simplicity (the argument for $\Sigma_-$ is essentially identical). We first compute

\[
(\partial_\tau - L) v_+ =   - M  \mu e^{-2\mu \tau} \varphi = M  |\mu| e^{-2\mu \tau} \varphi
\]

By Corollary <ref>,

\[
(\nu_{\Sigma_+}\cdot\nu_\Sigma)^{-1} (\partial_{\tau}\bx_{\Gamma} - \bH -\tfrac 12 \bx_{\Gamma}) \cdot \nu_{\Gamma} = (\partial_{\tau} - L) u + E = M |\mu| e^{-2\mu\tau}\varphi + E,
\]

at $\bx_{\Gamma} = \bx + v_{+}(\bx,\tau)\nu_{\Gamma}(\bx)$ for  $E=v_{+}E_1 + E_2(\nabla v_{+},\nabla v_{+})$ with $E_1=O(e^{-\mu\tau}), E_2(\cdot, \cdot)=O(1)$ uniformly on $\Sigma$ as $\tau\to-\infty$. Note that

\begin{align*}
|E(\bx)| & \leq v_{+}(\bx) |E_{1}(\bx)| + |E_{2}(\nabla v_{+}(\bx),\nabla v_{+}(\bx))|\\
& \leq C e^{-\mu\tau} (e^{-\mu\tau} + M e^{-2\mu\tau}) \varphi(\bx) + C (e^{-\mu\tau} + M e^{-2\mu\tau})^{2} |\nabla \varphi(\bx)|^{2}\\
&  \leq C (1 + M^2 e^{-2\mu \tau})  e^{-2\mu\tau} \varphi(\bx).
\end{align*}

The final estimate uses $\varphi^{-1}|\nabla \varphi|^{2} \in L^{\infty}(\Sigma)$ as proven in part (3) of Lemma <ref>. Hence,

\begin{align*}
(\nu_{\Sigma_+}\cdot\nu_\Sigma)^{-1} (\partial_{\tau}\bx_{\Gamma} - \bH -\tfrac 12 \bx_{\Gamma}) \cdot \nu_{\Gamma}  & \geq (M|\mu| - C - C M^2 e^{-2\mu \tau})  e^{-2\mu \tau}\varphi
\end{align*}

Taking $M$ large and then $\tau_{0}$ sufficiently negative ensures that $M|\mu| - C - C M^2 e^{-2\mu \tau} >0$ for all $\tau \in (-\infty,\tau_0)$. This proves the assertion.

The following result is an extension of <cit.> to allow for asymptotically cylindrical ends.
We will denote $\eps_0'=\eps_0'(\Sigma)$ the constant from Lemma <ref> (denoted by $\eps_0$ there).

There is $\eps_0 = \eps_0(\Sigma) \in (0,\eps_0']$ so that for $\eps \in (0,\eps_0)$ there is a smooth mean curvature flow $\Sigma_\eps(t)$ for $t \in [-1,- (\eps/\eps_0)^{1/|\mu|}]$ with $\Sigma_\eps(-1) = \Sigma_\eps$ and $C=C(\Sigma)$ with the following properties:

* the flow $\Sigma_\eps(t)$ remains strictly shrinker mean convex to the inside with the bound

\[
2t H_{\Sigma_\eps(t)} + \mathbf{x}\cdot\nu_{\Sigma_\eps(t)} \geq C^{-1}\eps (1+|\mathbf{x}|^2 + 2n(t+1))^{\mu}
\]

for all $t \in [-1,- (\eps/\eps_0)^{1/|\mu|} ]$,

* the rescaled surface $\tfrac{1}{\sqrt{-t}}\Sigma_\eps(t)$ is the normal exponential graph over $\Sigma$ of a function $v_\eps(\cdot,t)$ with $v_\eps \in C^\infty(\Sigma \times [-1, - (\eps/\eps_0)^{1/|\mu|}])$ and $\|v_\eps(\cdot,t)\|_{C^0(\Sigma)} \leq 2\eps_0$
for $t \in  [-1,- (\eps/\eps_0)^{1/|\mu|} ]$,

* $\Vert v_\eps(\cdot,t)\Vert_{C^3(\Sigma)} \leq C $ for all $t \in [-1, - (\eps/\eps_0)^{1/|\mu|}]$, and

* the following estimate

\[
\frac{\eps}{2} e^{-\mu \tau} \varphi(\bx) \leq v_{\eps}(\bx,-e^{-\tau}) \leq 2 \eps e^{-\mu \tau} \varphi(\bx)
\]

holds for all $(\bx,\tau) \in \Sigma\times [0,\tfrac{1}{\mu} (\log(\eps) - \log(\eps_0))]$.

Fix $\psi^{*}=\psi^{*}(\Sigma)$ and $\psi_{0}=\psi_{0}(\Sigma)$ as in Corollary <ref>. Choose $\eps_{0} \in (0,\eps_{0}']$ (for $\eps_{0}'$ the constant from Lemma <ref> so that (i) $\eps_{0} \leq \Vert \varphi\Vert_{C^{2}(\Sigma)}^{-1}\psi^{*}$, (ii)  $\eps_{0} \Vert \varphi\Vert_{C^{0}(\Sigma)} (1+M \eps_{0}) < \psi_{0}$ for $M$ fixed in Lemma <ref> and (iii) $\eps_{0}\leq e^{-\mu\tau_{0}}$, for $\tau_{0}$ fixed in Lemma <ref>.

Condition (i) above and (the first part of) Proposition <ref> yield $T \in [1,\infty)$ and a graphical rescaled mean curvature flow $[0,T]\ni \tau \mapsto v_{\eps}(\cdot,\tau)$ with $v_{\eps}(\cdot,0) = \eps \varphi(\cdot)$ and so that

\begin{equation}\label{eq:pre-ancient-blowup-barrier-crt}
\sup_{\Sigma}|v_{\eps}(\cdot,T)|  = \psi_{0}.
\end{equation}

Choose $\bar \tau \leq \tau_0$ so that $e^{-\mu \bar \tau} = \eps$ (possible by (iii) above). Then we have that

\[
v_-(\bx,\bar \tau) =  \eps (1- M\eps) \varphi(\bx) \leq \eps \varphi(\bx) \leq  \eps (1+ M\eps) \varphi(\bx)  = v_+(\bx,\bar \tau)
\]

where $v_{\pm}$ are defined in Lemma <ref>. The Ecker–Huisken maximum principle <cit.> and Lemma <ref> yields

\[
v_-(\bx,\bar \tau + \tau)  \leq v_{\eps}(\bx,\tau) \leq  v_+(\bx,\bar \tau + \tau)
\]

for $\tau \leq T_{0} : = \min\{T,\tau_{0}-\bar \tau\}$.

We claim that $T_{0}\geq \mu^{-1} \log (\eps/\eps_{0})$. Indeed, by (iii) (and $\mu<0$) we note that

\[
\mu^{-1} \log (\eps/\eps_{0}) \leq \tau_0 - \bar \tau.
\]

Thus, if $T_{0}< \mu^{-1} \log (\eps/\eps_{0})$ then $T_{0} = T$. Using (<ref>) and (ii) above, we thus find

\[
\psi_{0} = \sup_{\Sigma} |v_{\eps}(\cdot,T)| \leq \sup |v_{+}(\cdot,\bar \tau + \mu^{-1}\log(\eps/\eps_{0}))| \leq \eps_{0} \Vert \varphi\Vert_{C^{0}(\Sigma)} (1+M \eps_{0}) < \psi_{0} .
\]

This is a contradiction. The graph of $v_{\eps}(\cdot,\tau)$ yields the desired mean curvature flow (after rescaling), with the asserted existence time.

The shrinker mean convexity follows from the Ecker–Huisken maximum principle and Lemma <ref> exactly as in <cit.>. The remaining properties follow by construction. This finishes the proof.

Fix $(\Sigma,\Omega)\in \cS_n''$. Set $\tau_0 = |\mu|^{-1}\log \eps_0$. There is an ancient graphical shrinker mean convex rescaled mean curvature flow $  (-\infty,\tau_0] \ni \tau \mapsto \Graph_\Sigma v(\cdot,\tau)$  with  $\lambda(\Graph_\Sigma v(\cdot,\tau)) \leq F(\Sigma)$, $v(\bx,\tau) > 0$ for all $(\bx,\tau) \in \Sigma \times (-\infty,\tau_0]$, as well as the following properties for $\tau \in (-\infty,\tau_0]$:

* $\Vert v(\cdot,\tau)\Vert_{C^3(\Sigma)} \leq C$,

* $\frac{1}{2} e^{-\mu \tau} \varphi \leq v(\cdot, \tau) \leq 2 e^{-\mu \tau} \varphi$,

* $\Vert v(\cdot, \tau)\Vert_{C^k(\Sigma)} \leq C_k e^{-\mu \tau}$ for all $k\in \NN$, and

* $ |e^{\mu\tau} \langle v(\cdot, \tau), \varphi\rangle_W -a| \leq Ce^{-\mu \tau}$ for some fixed $a >0$.

For $\Sigma_\eps(t)$ defined in Proposition <ref>, we set

\[
S_\eps(\tau) : = e^{(\tau-\tau(\eps))/2} \Sigma_\eps(-e^{-\tau+\tau(\eps)})
\]

for $\tau \in [\tau(\eps),\tau_0]$, $\tau_0=|\mu|^{-1}\log\eps_0$, and $\tau(\eps) = |\mu|^{-1}\log\eps$. Observe that $S_\eps(\tau)$ is a rescaled mean curvature flow and $S_\eps(\tau)$ is the graph of $\hat v_\eps(\cdot,\tau) : = v_\eps(\cdot,-e^{-\tau+\tau(\eps)})$, defined in Proposition <ref>. Property (3) in Proposition <ref> yields $\Vert \hat v_\eps(\cdot,\tau)\Vert_{C^3(\Sigma)} \leq C$ for $\tau \in [\tau(\eps), \tau_0]$. Similarly, (4) in Proposition <ref> yields

\begin{equation}\label{eq:barrier_est}
\frac{1}{2} e^{-\mu \tau} \varphi(\bx) \leq \hat v_{\eps}(\bx, \tau) \leq 2 e^{-\mu \tau} \varphi(\bx)
\end{equation}

for $(\bx,\tau) \in\Sigma\times [\tau(\eps), \tau_0]$.

We can thus find $\eps_i\to 0$ so that $\hat v_\eps$ converges in $C^\infty_\textrm{loc}(\Sigma\times (-\infty,\tau_0])$ to $v(\cdot,\tau)$ with $\Vert v(\cdot,\tau) \Vert_{C^3(\Sigma)} \leq C$ for $\tau \in (-\infty,\tau_0]$ and so that $\Graph_\Sigma v(\cdot,\tau)$ is  rescaled mean curvature flow. This proves property (1). Property (2) follows by passing (<ref>) to the limit. Property (3) follows from this, combined with interior Schauder estimates applied to the (non-rescaled) mean curvature flow equation.

Finally, to prove (4), we recall that by <ref>, $v$ satisfies

$$ \partial_\tau u = Lu + E\, ,$$

where (3) implies that $|E| \leq C e^{-2\mu\tau}$. Since

$$ \partial_\tau \langle v,\varphi\rangle_W = -\mu \langle v,\varphi\rangle_W + \langle E,\varphi\rangle_W\, ,$$

the function $f(\tau):= e^{\mu \tau} \langle v,\varphi\rangle_W$ satisfies

$$ |\partial_\tau f| \leq C e^{-\mu \tau}\, .$$

Integrating this proves (4) (note that $a>0$ by (2)).

The shrinker mean convexity follows from the Ecker–Huisken maximum principle and Lemma <ref> exactly as in <cit.>. The remaining properties follow by construction. This finishes the proof.

§.§ The flow for all times $t<0$
In this section we extend the flow constructed previously to all (non-rescaled) times $t<0$. To do so, we must assume that $\Sigma \in \cS_n'''$ has nice, strictly stable ends. Recall that this means that the first eigenfunction $\varphi$ decays to $0$ at spatial infinity.

We note that the strictly stable end condition excludes cylinders $\hat\Sigma \times \RR$, $\hat\Sigma \in \cS_{n-1}$ compact. In the exact cylindrical case one can construct the ancient flow coming out of $\hat\Sigma$ (using the arguments discussed below) and then cross with $\RR$. As such, we will not discuss this case further in this section.

For later reference we also note the following basic stability result.

For every $\delta > 0$, $\Lambda>0$ there exists a $D \geq \delta^{-1}$ such that the following holds. Let $\cM$ be unit regular integral Brakke flow, such that $\lambda(\cM) \leq \Lambda$ and $\cM(-1)\lfloor B_{D}(\bOh)$ is a smooth graph over $\hat{\Sigma}_j \times \RR_{\bx_j}$ with the $C^3$-norm of the graph function bounded by $D^{-1}$. Then $\cM(t)$ remains a smooth graph over $(\hat{\Sigma}_j)_t \times \RR_{\bx_j}$ on $B_{\delta^{-1}}$ for all $t \in [-1, -\delta^2]$ with (rescaled) $C^3$-norm bounded by $\delta$.

By a direct contradiction argument, this follows from the uniqueness of the evolution of $\hat{\Sigma}_j \times \RR_{\bx_j}$.

We recall the definition of the parabolic dilation map

\[ \cF_\lambda : \RR^{n+1} \times \RR\to  \RR^{n+1} \times \RR, \qquad \cF_\lambda : (\bx,t) \mapsto (\lambda \bx,\lambda^2 t) \]

and also that $\ft(\bx,t) = t$ is the projection onto the time coordinate.

In the following we will consider $\eps_0=\eps_0(\Sigma)$ as in Proposition <ref>
and $t_0=-\eps_0^{-1/|\mu|}$.

For $(\Sigma,\Omega)\in \cS_n''$ and $\eps \in (0,\eps_0)$ there exists unit-regular integral Brakke flows (resp. weak set flows $\cK_\eps$) with initial condition $\cM_\eps(-1) = \cH^n\lfloor \Sigma_\eps$ (resp. $\cK_\eps(-1) = K_\eps$, the unique closed set with $K_\eps \subset \Omega$ and $\partial K_\eps = \Sigma_\eps)$
with the following properties:

* $\partial \cK_\eps \setminus \ft^{-1}(-1) \subset \supp \cM_\eps \subset \cK_\eps$

* for $t\in [-1, - (\eps/\eps_0)^{1/|\mu|}]$ we have $ \cM_\eps(t) = \cH^n\lfloor \Sigma_\eps(t)$, for $\Sigma_\eps(t)$ the flow constructed in Proposition <ref>.

Now, assume that $\Sigma \in \cS_n'''$ has strictly stable ends and define the dilated flows

\[
\hat\cM_\eps = \cF_{\eps^{-1/(2|\mu|)}}(\cM_\eps) \text{ resp. } \hat\cK_\eps = \cF_{\eps^{-1/(2|\mu|)}}(\cK_\eps).
\]

There is a monotonically non-decreasing function $R(t)$ on $(-\infty,0)$ with $R(t) \equiv 0$ for $t \in (-\infty,t_0]$ independent of $\eps \in (0,\eps_0)$ so that

\begin{equation}\label{eq:rescaled-pre-ancient-weak-reg-infty}
\hat\cM_\eps(t) \lfloor (\RR^{n+1}\setminus B_{R(t)}) \text{ and } \partial\hat\cK_\eps(t) \cap (\ft^{-1}(t)\setminus B_{R(t)})
\end{equation}

are the same, smooth, strictly shrinker mean convex mean curvature flow, which we will denote by $\hat\Sigma_\eps(t)$; moreover $\tfrac{1}{\sqrt{-t}} \hat\Sigma_\eps(t)$  can be written as a small $C^2$-graph over a subset of $\Sigma$.

Approximating $\Sigma_\eps$ by smooth compact hypersurfaces whose level set flow does not fatten, we may use [55] to find a sequence of compact unit regular integral Brakke flows $\cM_{\eps, \rho}$ and weak set flows $\cK_{\eps,\rho}$ satisfying

\begin{equation}\label{eq:approx_flows}
\supp \cM_{\eps,\rho} \cap \ft^{-1}((-1,\infty)) = \partial\cK_{\eps,\rho}\cap \ft^{-1}((-1,\infty))
\end{equation}

with $\cM_{\eps,\rho}$ converging to a Brakke flow $\cM_\eps$ as Brakke flows and $\cK_{\eps,\rho}$ converging to a weak set flow $\cK_\eps$ in the local Hausdorff sense (the flows $\cM_\eps,\cK_\eps$ are easily seen to have the asserted initial conditions). See  <cit.> for the (standard) details of this construction.

We recall here the argument from <cit.> proving (1). For $X \in \partial\cK_\eps\setminus \ft^{-1}(-1)$, there is $X_i \in \partial\cK_{\eps,\rho_i}\setminus \ft^{-1}(-1) = \supp\cM_{\eps,\rho_i}\setminus \ft^{-1}(-1)$ converging to $X$. The monotonicity formula guarantees that $X \in \supp\cM_\eps$. This proves the first inclusion in (1). For the second inclusion, take $X \in \supp\cM_\eps$. The same argument yields $X_i \in \cK_{\eps,\rho_i}$ with $X_i \to X$. Thus $X \in \cK_\eps$ since $\cK_\eps$ is closed.

Using pseudolocality, unit-regularity and uniqueness of smooth solutions to mean curvature flow with bounded curvature <cit.>, it follows that $\cM_\eps(t) = \cH^n\lfloor \Sigma_\eps(t)$ as long as $\Sigma_\eps(t)$ has bounded curvature (which holds at least until $t = -(\eps/\eps_0)^{1/|\mu|}$). Thus the flows satisfy (2) as well.

It thus remains to prove (<ref>) under the assumption that $\Sigma \in \cS_n'''$ has strictly stable ends. We note that

\[
\cM_\eps(t_0) = \cH^n\lfloor \hat\Sigma_\eps(t_0)
\]

where $(-t_0)^{-1/2} \hat\Sigma_\eps(t_0) = (\eps/\eps_0)^{-1/2|\mu|} \Sigma_\eps((\eps/\eps_0)^{1/|\mu|}) = \Graph_\Sigma v_\eps(\cdot, (\eps/\eps_0)^{1/|\mu|})$ where

\[
v_\eps(\bx,(\eps/\eps_0)^{1/2|\mu|}) \leq 2 \eps (\eps/\eps_0) \varphi(\bx) = 2 \eps_0 \varphi(\bx)
\]

(this follows from (4) in Proposition <ref>). In particular, the strongly stable end hypothesis (and (3) in Lemma <ref>) implies that for any $\bx_i \in \hat\Sigma_\eps(t_0)$, $(-t_0)^{-1/2} \hat\Sigma_\eps(t_0) - \bx_i$ converges in $C^3_\textrm{loc}$ (up to passing to a subsequence and applying a fixed rotation) to either $\RR^n$ (on the conical part) or $\hat\Sigma_j\times \RR$ (one of the finitely many cylindrical ends of $\Sigma$). In particular, by applying pseudolocality (in the conical case) or stability of cylinders Lemma <ref> (in the cylindrical case) we find $R(t)$ monotonically non-decreasing so that

\[
\hat\cM_\eps(t) \lfloor (\RR^{n+1}\setminus B_{R(t)})
\]

is a smooth mean curvature flow, which we will denote by $\hat\Sigma_\eps(t)$; moreover $\tfrac{1}{\sqrt{-t}} \hat\Sigma_\eps(t)$  can be written as a small $C^2$-graph over a subset of $\Sigma$. Unit regularity at the level of the approximating flows $\cM_{\eps,\rho}$ guarantees that the same thing holds for $\cK_\eps$.

That $[-\eps^{-1/|\mu|},t_0]\mapsto \hat\Sigma_\eps(t)$ is strictly shrinker mean convex follows from Proposition <ref>. Finally, we can use Ilmanen's localized avoidance principle (cf. Theorem <ref>) and the Ecker–Huisken maximum principle as in <cit.> to see that $\hat\Sigma_\eps(t)$ is weakly shrinker mean convex for all $t \in [-\eps^{-1/|\mu|},0)$.  Strict mean convexity then follows from the strong maximum principle and the barriers constructed in property (4) of Proposition <ref>. This completes the proof.

In <cit.> (i.e., for $\Sigma$ with only asymptotically conical ends), the estimate $|\bx| |A| + |\bx|^2 |\nabla A| + |\bx|^3 |\nabla^2 A|\leq C$ for the flow $\hat\Sigma_\eps$ is proven and used in several places. Such an estimate will not hold for asymptotically cylindrical ends, so in any reference below to  <cit.>, such an estimate should be replaced by $|A| + |\nabla A| + |\nabla^2A|\leq C$. Fortunately, this does not affect the logic of any of the arguments referred to below.

Fix $\Sigma \in \cS'''_n$.
There exists an ancient unit regular integral Brakke flow $\cM$ and a corresponding weak set flow $\cK$ with the following properties:

* $\partial \cK \subset \supp \cM \subset \cK$,

* $\lambda(\cM) \leq F(\Sigma)$,

* there is a continuous, monotonically increasing function $R(t)$ on $(-\infty,0)$,  with $R(t)=0$ for $t \in (-\infty , t_0]$, $t_0<0$, such that

$$ \cM(t) \lfloor (\RR^{n+1}\setminus B_{R(t)}(\bOh)) \text{ and } \partial\cK \cap (\ft^{-1}(t)\setminus B_{R(t)}(\bOh)) $$

are the same, smooth, strictly shrinker mean convex mean curvature flow, which we will denote by $\Sigma(t)$; moreover $(-t)^{-1/2} \Sigma(t)$  can be written as a small $C^3$-graph over a subset of $\Sigma$,

* for $t \in (-\infty ,t_0]$,  $ (-t)^{-1/2}\Sigma(t) =\Graph_\Sigma v(\cdot, \tau)$, $t = -e^{-\tau}$, for $v$ as in Proposition <ref>.

The proof of Proposition <ref> yields $\eps_i\to0$ so that $[-\eps^{1/|\mu|},t_0]\ni t\mapsto \hat\Sigma_\eps(t)$ converges in $C^\infty_\textrm{loc}$ to $(-\infty,t_0] \ni t\mapsto (-t)^{-1/2} \Graph_\Sigma v(\cdot, -\log(-t)) : = \Sigma(t)$  for $v$ as in Proposition <ref>. Proposition <ref> allows us to find $\Sigma(t)$ a flow in $\RR^{n+1}\setminus B_{R(t)}(\bOh)$ so that (3) is satisfied.[A priori, $\Sigma(t)$ must only be weakly shrinker mean convex, but by  Proposition <ref> it is strictly shrinker mean convex for $t \in (-\infty,t_0]$ so the strong maximum principle guarantees strict shrinker mean convexity of $\Sigma(t)$. ] Letting the weak flows $\hat\cM_{\eps_i},\hat\cK_{\eps_i}$ pass to a subsequential limit we find an ancient unit regular integral Brakke flow $\cM$ and weak set flow $\cK$ which (thanks to Proposition <ref>) satisfy the various assertions.

We can now combine the previous results with Ilmanen's localised avoidance principle to extend the strict shrinker mean convexity to the whole flow for $t<0$ (cf. <cit.>). Recall that $\cK^\circ$ denotes the interior of $\cK$.

The ancient unit regular integral Brakke flow and the corresponding weak set flow $(\cM, \cK)$ from the previous proposition have the following properties: let $\check\cK = \cK \cap \ft^{-1}((-\infty, 0))$, then

* we have $d((\bOh,0), \check\cK) > 0$,

* for $\lambda > 1$ we have $\cF_\lambda(\check\cK)\subset \check\cK^{\circ}$ and $\supp\cM \cap \cF_\lambda(\check\cK) = \emptyset$.

Since the flow $t\mapsto \Sigma(t)$ is strictly shrinker mean convex we have that $\Sigma(t) \cap \Sigma = \emptyset$ for all $t<0$. Thus, using Ilmanen's localised avoidance principle, Theorem <ref>,  we see that $\cM(t) \cap \Sigma = \emptyset$ for all $t<0$. Applying Ilmanen's localised avoidance principle another time, we thus see that $d((\bOh,0), \supp \cM) > 0$, which by Proposition <ref> implies $(1)$.

To prove (2), we note that strict shrinker mean convexity of the exterior flow $\Sigma(t)$ guarantees that for $\lambda>1$, $\supp\cM$ and $\cF_{\lambda}(\check \cK)$ are disjoint outside of a set $D$ in space-time which has $D \cap \mathfrak{t}^{-1}([a,b])$ compact for any $a<b<0$. Thus, we may apply Ilmanen's localized avoidance principle, Theorem <ref>, to show that $\supp \cM$ and $\cF_{\lambda}(\check\cK)$ are indeed disjoint. Using again Proposition <ref> this completes the proof of (2).

We can replace <cit.> with Proposition <ref> and Proposition <ref> for all the results in <cit.> to carry over. This yields the following result, replacing <cit.>.[Property (6) follows from <cit.>.]

For $n\leq 6$, fix $\Sigma \in \cS'''_n$.
The flows $(\cM,\cK)$ constructed in Proposition <ref> have the following properties for $t<0$:

* $\cM(t) = \cH^{n}\lfloor \partial\cK(t)$,

* $\partial\cK(t) \subset \sqrt{-t}\Omega$ for all $t<0$,

* $\sing\cM \cap\{t<0\}$ has parabolic Hausdorff dimension $\leq n-1$ and for $t<0$, $\sing\cM(t)$ has spatial Hausdorff dimension $\leq n-1$,

* any limit flow at $X=(\bx,t)$ with $t<0$ is weakly convex on the regular part and all tangent flows are multiplicity one generalized cylinders,

* any singular point has a (strict) mean-convex neighborhood, and

* if $n=3,4$, $\ft(\sing\cM)\cap\{t<0\}$ has measure zero.

§.§ Conditional uniqueness results
Proposition <ref> implies the following conditional uniqueness result.

For $n\leq 6$, fix $\Sigma \in \cS'''_n$.
Let $(\cM,\cK)$ be as constructed in Proposition <ref>. Let $(\check \cM(t))_{-\infty < t < \infty}$ be an ancient unit-regular integral Brakke flow such that there is $T<0$ so that

\begin{equation} \label{eq:one.sided.assumption}
\check \cM(t) = \cM(t) \text{ for all } t \leq  T.
\end{equation}

Then $\check \cM(t) = \cM(t)$ for $t<0$.

We can assume that
$T<0$ is sufficiently negative so that for $t \leq T$, $\cM(t) = \cH^{n}\lfloor \Sigma(t)$, where $\Sigma(t)$ is the smooth flow from Proposition <ref> (3).

Below, we consider only times $t<0$. As in <cit.>, Ilmanen's localized avoidance principle (Theorem <ref>) combined with Ecker–Huisken's maximum principle at infinity <cit.>, we see that $(\supp \check\cM)_{t}$ is disjoint from $\cF_{\lambda}(\supp \cM)$ for $\lambda \not = 1$. This implies that, $(\supp \check\cM)_{t} \subset \supp\cM$.

Finally, since $\reg\cM$ is connected for by (5) in Proposition <ref> and <cit.>, we see that $\check\cM(t) = \cM(t)$ in $(\sing \cM)^{c}$ (using the unit-regularity of $\cM$ and $\check\cM$). This completes the proof.

Under a global graphicality assumption, we can also ensure uniqueness. Note that this result holds with just the (non-strict) stable end hypothesis.
For $\Sigma \in \cS_n''$, consider two ancient rescaled mean curvature flows $\Sigma_i(\tau)$ for $\tau \in (-\infty, \tau_0), i=1,2$, such that $\Sigma_i(\tau) = \Graph_\Sigma u_i(\cdot, \tau)$. There exists $\eta = \eta(\Sigma)>0$ such that if

\begin{equation}
\label{eq:estimate-decay}
\|u_i(\cdot, \tau)\|_{C^3(\Sigma)} \leq C e^{-\mu\tau}
\end{equation}

for $\tau \in (-\infty, \tau_0)$ and $i =1,2$ as well as

$$ \lim_{\tau \to -\infty} e^{\mu \tau} \| w \|_W =0\, ,$$

where $w =u_1-u_2$, then $w\equiv 0$.

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

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

From  Lemma <ref> we see that we can write

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

We take the $L^2_W$ dot product of (<ref>) with $w$ to find

\begin{align*}
\bangle{E^w,w}_W & =   \bangle{wF,w}_W + \bangle{\nabla_\Sigma w\cdot \bF,w}_W + \bangle{\nabla^2_\Sigma w\cdot \cF,w}_W\\
& \leq C e^{-\mu\tau} \Vert w\Vert_{W,1}^2 + \bangle{\nabla^2_\Sigma w\cdot \cF,w}_W,
\end{align*}

where we recall that $\Vert f\Vert_{W,1}^2 = \Vert f\Vert_W^2 + \Vert \nabla_\Sigma f\Vert_W^2$ is the weighted Sobolev norm. To handle the remaining term, we integrate by parts. We obtain

\[
\bangle{\nabla^2_\Sigma w\cdot \cF,w}_W = O(\Vert w\Vert_{W,1}^2 \Vert \cF\Vert_{C^1}) +  \int_\Sigma w\, (\bx^T \otimes \nabla_\Sigma w) \cdot \mathcal{F} \, e^{-\tfrac{|\bx|^2}{4}}\, d\mathcal{H}^n.
\]

Thanks to the estimate $\Vert \cF\Vert_{C^1}\leq Ce^{-\mu\tau}$, the first term is as desired. Again by Lemma <ref> we see that

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

which combining with the previous estimates implies

\begin{equation*} \label{eq:one.sided.decay.uniqueness.w.dot.error}
|\langle w(\cdot, \tau), E^w(\cdot, \tau) \rangle_W| \leq C_2 e^{-\mu \tau} \Vert w(\cdot, \tau) \Vert^2_{W,1}, \; \tau \leq \tau_0,
\end{equation*}

This replaces <cit.>. The rest of the argument is identical to the remainder of the proof of <cit.>.

§ BARRIERS ALONG THE CYLINDRICAL ENDS

In this section we consider $(\Sigma,\Omega)\in \cS_n'$. (Recall that this means that $\Sigma = \partial\Omega$ and the ends of $\Sigma$ are  either asymptotic to smooth cones or cylinders over smooth compact shrinkers.) Angenent–Daskalopoulos–Šešum have recently [1], constructed a self-shrinker foliation inside and outside of the end of a round cylinder $\Sigma = \SS^{n-1}(\sqrt{2(n-1)})\times \RR$ using ODE techniques (see also [61]). This foliation plays an important role in the study of ancient solutions with cylindrical tangent flow at $t=-\infty$ in [1, 2, 5, 6, 25, 26, 27]. The goal of this section is to (partially) generalize this to the case of shrinkers with nice ends.

§.§ Setup and statement of barrier construction

After a rotation, we will assume that $\Sigma$ has a cylindrical end in the direction $\be = (0,\dots,0,1)$ modeled on $\hat\Sigma \times \RR$. More precisely, we consider $\hat\Sigma^{n-1} \subset \RR^n$ a smoothly embedded closed self-shrinker, so that the surfaces $\Sigma - s\be$ converge smoothly on compact sets to $\hat\Sigma\times \RR$ as $s\to\infty$. We will use $(\bomega,z) \in \RR^n\times \RR$ as coordinates on $\RR^{n+1}$. We assume that $\rho>0$ is chosen such that $\hat\Sigma \subset B_{\rho}(\bOh) \subset \RR^n$.

Fix $\hat \Omega \subset \RR^n$ so that $\partial\hat\Omega = \hat\Sigma$ and $\Omega - s\be$ converges to $\hat\Omega\times \RR$ as $s\to\infty$. Set $\Omega_Z : = \Omega \cap \{z\geq Z\}$, and $\Sigma_Z = \Sigma \cap \{z\geq Z\}$. For $\delta>0$, we define

\begin{equation}\label{eq:move-hat-sigma-inside-by-delta-barriers}
\hat\Sigma_\delta : = \{\bomega + \delta \hat\varphi(\bomega) \nu_{\hat\Sigma}(\bomega) : \bomega \in \hat\Sigma \}\subset \RR^n.
\end{equation}

Let $\hat\Omega_\delta \subset  \hat\Omega$ denote the region (possibly unbounded) to the “inside” of $\hat\Sigma_\delta$ (as determined by the inwards pointing unit normal). Note that $\hat\Sigma_\delta$ is smooth and strictly shrinker mean convex (to the inside) and $F(\hat\Sigma_\delta) < F(\hat\Sigma)$ for $\delta \in (0,\delta_0)$, where $\delta_0 = \delta_0(\hat\Sigma)$.

By Lemma <ref>, there is $\bar Z = \bar Z(\Sigma)$ large and a smooth function $\tilde v : \hat\Sigma \times \{z\geq \bar Z\} \to \RR$ so that

\begin{equation}\label{eq:cyl.end.graph.cyl}
\Sigma \cap (B_{2\rho}(\bOh) \times [\bar Z,\infty)) = \{ (\bomega,z) + \tilde v(\bomega,z) \nu_{\hat\Sigma}(\bomega): (\bomega,z) \in\hat\Sigma \times[ \bar Z,\infty)\}\, .
\end{equation}

We have $|\nabla^k_{\hat\Sigma\times \RR} \tilde v| = o(1)$ as $z\to\infty$ for all $k\geq 0$. Fix $\hat \mu < 0$ the first eigenvalue of $L_{\hat\Sigma\times \RR}$.

For $Z_{0}\geq \bar Z$ we assume there is $[-\infty,T_{0}] \ni \tau \mapsto \tilde v_{\tau} \in C_\textrm{loc}^{\infty}(\hat\Sigma \times [Z_{0},\infty))$ so that

\begin{equation}\label{eq:barrier-w-tau-decay-to-end-at-infty}
\Vert \tilde v_\tau\Vert_{C^3;\hat\Sigma\times [Z_0,\infty)} = o(1) \textrm{ as $\tau\to-\infty$},
\end{equation}

and so that

\[
\Graph_{\hat\Sigma \times [Z_{0},\infty)} \tilde v_{\tau}(\cdot)\textrm{ is a rescaled mean curvature flow}.
\]

(An important special case is $\tilde v_{\tau}\equiv \tilde v$, so $\Graph_{\hat\Sigma \times [Z_{0},\infty)} \tilde v_{\tau}(\cdot)$ is the chosen end of $\Sigma$, but we will also consider the case that $\tilde v_{\tau}$ is the end of the ancient flow constructed in Proposition <ref>.)

Taking $Z_{0}$ larger if necessary we can assume that there is $\vartheta_{0} > 0$ so that

\begin{equation}\label{eq:sep.cyl.ends.from.other.ends}
\Sigma \cap \{z>Z_{0}\} \cap \{t\bx : \bx \in \partial B_{1}(\bOh), d(\bx,\be) < \vartheta_{0}, t>0\} = \Sigma \cap (B_{2\rho}(\bOh)\times (Z_{0},\infty)).
\end{equation}

We will use this later to separate the chosen cylindrical ends from all other ends.

Consider the setup described above. There is $Z_1 \geq  Z_{0}$, $T_{1}\leq T_{0}$, $\alpha > 2|\hat \mu|$, $\eta_0 > 0 $, and $h_{0}>0$ so that for $\eta \in (0,\eta_0)$, there is $\vartheta = \vartheta(\eta)\in (0,\vartheta_{0})$ with the following properties. For $T_{2}\leq T_{1}$, suppose that that $(-\infty,T_{2}]\ni \tau\mapsto \cM(\tau)$ is a rescaled Brakke flow in $\Omega \cap \{z > Z_{1}\} \subset \RR^{n+1}$ so that

* for 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

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

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 $\tau \to -\infty$.

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

\begin{multline*}
\supp \cM(\tau) \cap  \{Z_{1}<z<\eta^{-\kappa}\} \subset\\
\{ (\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{multline*}

for $\kappa = (1+\alpha)^{-1} \in (0,1)$.

The construction of these barriers will follow by welding a barrier (constructed using the linearized shrinker equation) that covers a long portion of the end of $\Sigma$ to a “translator” that approximates a one-sided rescaled mean curvature flow coming out of $\hat\Sigma$. We will prove Theorem <ref> in Section <ref> after constructing these pieces.

§.§ Translator barriers

As we recall in Appendix <ref>, Hershkovits–White have recently defined a notion of Brakke flow with an ambient vector field  (cf. <cit.>). In particular, a rescaled mean curvature flow is a $\frac{\bx}{2}$-flow in their framework. Below, we will consider a related object, a $\frac{\bomega}{2}$-flow. These arise in the elliptic regularization construction of rescaled mean curvature flow.

Consider a compact Riemannian manifold $(M^{n},g)$, a gradient vector field $\bX = \nabla f$ on $M$, and a hypersurface $\Gamma^{n} \subset M\times [0,\infty)$ so that $\bH + \bX^{\perp} + \lambda \be^{\perp} = 0$, where $\be=\partial_{z}$ is the unit vector field on $M\times \RR$ in the $\RR$ direction. Assume that $\Gamma \cap M\times \{0\} = \partial\Gamma$. For any $0\leq a<b$ so that $\Gamma$ intersects $\{z\in\{a,b\}\}$ transversely, we have

\[
\int_{\Gamma\cap\{a\leq z\leq b\}} e^{-f} \leq (b-a+\lambda^{-1}) \int_{\partial\Gamma} e^{-f}.
\]

\[
\int_{\Gamma(a,b)}|\be^{\perp}|_{g}^{2} e^{-f} \leq \lambda^{-1}  \int_{\partial\Gamma} e^{-f}.
\]

Set $\Gamma(a,b) : = \Gamma \cap \{a\leq z\leq b\}$ and $\Gamma(s) = \Gamma \cap\{z=s\}$. By Sard's theorem, $\Gamma(a,b)$ is a smooth hypersurface with boundary for a.e. $0<a<b$. We now follow the proof of local mass bounds for the translators obtained in elliptic regularization <cit.> (see also <cit.>).

We will abuse notation slightly and write $g$ for the product metric on $(M,g)\times \RR$. Then, for $a<b$ with $\Gamma$ transversal to $\{z=a\}\cup\{z=b\}$, we find

\begin{align*}
\int_{\Gamma(a,b)} g(-\bX^{T},\be) e^{-f}
& = \int_{\Gamma(a,b)} \Div_{\Gamma,g}(e^{-f}\be)\\
& = \int_{\Gamma(a,b)} g(-\bH,\be) e^{-f} + \int_{\Gamma(b)} |\be^{T}|_{g} e^{-f} - \int_{\Gamma(a)} |\be^{T}|_{g} e^{-f}\\
& = \int_{\Gamma(a,b)} g(\bX^{\perp} + \lambda \be^{\perp},\be) e^{-f} + \int_{\Gamma(b)} |\be^{T}|_{g} e^{-f} - \int_{\Gamma(a)} |\be^{T}|_{g} e^{-f}.
\end{align*}

Observe that

\[
g(\bX^\perp + \bX^T,e) = g(\bX,e) = 0,
\]

so, we find

\begin{equation}\label{eq:translator-barriers-loc-mass-bounds}
\int_{\Gamma(a)} |\be^T|_g e^{-f} = \int_{\Gamma(a,b)} \lambda |\be^\perp|_g^2 e^{-f}  + \int_{\Gamma(b)} |\be^T|_g e^{-f} .
\end{equation}

In particular $s\mapsto \int_{\Gamma(s)} |\be^T|_g e^{-f}$ is non-increasing so

\[
\int_{\Gamma(s)} |\be^T| e^{-f} \leq \int_{\partial \Gamma} e^{-f}
\]

for $s>0$ so that $\Gamma$ intersects $\{z=s\}$ transversely. Combined with (<ref>) we find

\begin{align*}
\int_{\Gamma(a,b)}|\be^{\perp}|_{g}^{2} e^{-f} & = \lambda^{-1} \int_{\Gamma(a,b)}\lambda |\be^{\perp}|^{2}_{g} e^{-f}\\
& \leq  \lambda^{-1}  \int_{\Gamma(a)} |\be^{T}|_{g} e^{-f}\\
& \leq  \lambda^{-1}  \int_{\partial\Gamma} e^{-f}
\end{align*}

proving the second equation. Similarly,

\begin{align*}
\int_{\Gamma(a,b)} e^{-f} & = \int_{\Gamma(a,b)} |\be^\perp|_g^2 e^{-f} + \int_{\Gamma(a,b)} |\be^T|_g^2 e^{-f} \\
& \leq \lambda^{-1} \int_{\partial \Gamma }  e^{-f} + \int_{\Gamma(a,b)} |\be^T|_g^2 e^{-f} \\
& = \lambda^{-1} \int_{\partial \Gamma }   e^{-f} + \int_a^b \int_{\Gamma(s)} |\be^T|_g e^{-f} \\
& \leq (b-a+ \lambda^{-1}) \int_{\partial \Gamma } e^{-f},
\end{align*}

proving the first equation. This completes the proof.

For $\lambda>0$ and $\delta\in (0,\delta_0)$, there exists a smooth hypersurface $\Gamma_\delta^\lambda \subset \RR^{n+1}$ with $\partial\Gamma^{\lambda}_{\delta} = \hat\Sigma_\delta \times\{0\}\subset \RR^{n+1}$ so that $\mathring \Gamma_\delta^\lambda \subset \hat \Omega_\delta \times [0,\infty)$ and the mean curvature of $\Gamma_\delta^\lambda$ satisfies

\[
\bH + \tfrac{\bomega^\perp}{2} + \lambda \be^\perp = 0.
\]

The hypersurfaces  $\mathring\Gamma^\lambda_\delta$ are graphical over $\hat\Omega_\delta\times \{0\}$. Furthermore, the hypersurfaces

\[
\Gamma^\lambda_\delta(t) : = \Gamma_\delta^\lambda - \lambda t \be.
\]

have normal velocity $\bH + \tfrac{\bomega^{\perp}}{2}$ and are thus $\tfrac{\bomega}{2}$-flows. As $\lambda\to\infty$, the flows converge in the sense of Brakke $\tfrac{\bomega}{2}$-flows to $\cH^n \lfloor (\partial^*\Omega_\delta(t)\times \RR)$ where $\Omega_\delta(t)$ is the interior of the rescaled mean curvature flow of $\hat\Sigma_\delta$.

We consider the case that $\hat\Omega$ is unbounded (the bounded case is similar but easier). For $R_{j}\to\infty$, consider $(M,g_{j})$ where $M=\SS^{n}$ and $g_{j}$ contains a region isometric to $B_{2R_{j}}(\bOh) \subset \RR^{n}$. We assume that $\hat\Sigma \subset B_{R_{j}}(\bOh)$. Choose a gradient vector field $\bX_{j}$ on $(M,g)$ so that $\bX_{j} = \tfrac 12 \bomega$ on the region isometric to $B_{R_{j}}(\bOh)$ (recalling that $\bomega$ is the coordinate on $\RR^{n}$). Note that $(M,g_{j},\bX_{j},\bOh)$ converges in a pointed sense to $(\RR^{n},g_{\RR^{n}},\tfrac 12 \bomega,\bOh)$.

Consider the region $\hat\Omega_{\delta,j} \subset M$ that agrees with $\hat\Omega_{\delta}$ in the $B_{R_{j}}(\bOh)$ region and $\partial \hat\Omega_{\delta,j} = \hat\Sigma_{\delta}$  and currents $\Gamma$ with support in the closure of $\hat\Omega_{\delta,j} \times \RR$ and $\partial \Gamma = \hat\Sigma_{\delta}\times \{0\}$. Note that by construction, $\hat\Omega_{\delta,j}$ is  strictly $\bX_{j}$-mean convex to the inside and thus $\partial \hat\Omega_{\delta,j}\times \RR$ acts as barrier for constrained minimization. Hence, by modifying the arguments used in elliptic regularization to $\bX$-flows (cf. [43, 24, 54, 55, 46, 51, 52]) for each $\lambda >0$,  there exists a smooth hypersurface $\Gamma_{j} \subset M\times [0,\infty)$ with (i) $\partial\Gamma_{j} = \hat\Sigma_{\delta}\times \{0\}$, (ii) $\Gamma_{j}$ is graphical over $\hat\Omega_{\delta,j}\times\{0\}$ and (iii) $\Gamma_{j}$ satisfies the $\bX_{j}$-translator equation $\bH + \bX^{\perp}_{j} + \lambda \be^{\perp} = 0$.

Choose $f_{j} \in C^{\infty}(M)$ so that $\nabla f_{j} =\bX_{j}$ and $f_{j} = \tfrac 1 4 |\bomega|^{2}$ in $B_{R_{j}}(\bOh)$. Lemma <ref> implies that for almost every $a<b$, it holds that

\[
\int_{\Gamma_{j} \cap \{a\leq z \leq b\}} e^{-f_{j}} d\mu_{g_{j}}\leq (b-a+\lambda^{-1}) \int_{\partial\Gamma} e^{-\frac{|\bomega|^{2}}{4}} d\mu_{\RR^{n}}.
\]

In particular, for all $R>0$, it holds that

\[
\limsup_{j\to\infty} |\Gamma_{j} \cap (B_{R}(\bOh)\times \{0\leq z \leq R\} )|_{g_{\RR^{n+1}}} < \infty.
\]

We can now pass to a subsequential limit as $j\to\infty$ by following the arguments in <cit.> (one should replace convexity in  <cit.> by the fact that the region $\hat \Omega_{j}$ is mean convex with respect to the metric $e^{-2f_{j}/n}g^{(j)}$ and that $\Gamma_j$ is a constrained minimizer in $\hat\Omega_{\delta,j} \times \RR$) but otherwise the argument is unchanged; the Frankel property for self shrinkers (see for example <cit.>) prevents cylinders from forming as in <cit.>). This yields the smooth hypersurface $\Gamma^{\lambda}_{\delta}$ graphical over $\hat\Omega_{\delta}\times \{0\}$ with $\bH + \frac{\bomega^{\perp}}{2} + \lambda\be^{\perp} = 0$.

Using Lemma <ref>, the remaining assertions follow exactly as in the construction of (mean convex) mean curvature flow via elliptic regularization (cf. <cit.> or <cit.>). Indeed, $\Gamma^{\lambda}_{\delta}-\lambda t\be$ defines an integral unit regular Brakke $\frac{\bomega}{2}$-flow and using the local area estimates that follow from Lemma <ref>, we can take a subsequential limit as $\lambda\to\infty$. Since $\bomega$ is $z$-translation invariant, it is easy to show that the limit is of the form $\mu(t)\times \RR$ where $\mu(t)$ is an integral unit regular Brakke $\frac{\bomega}{2}$-flow on $\RR^{n}$.

Finally, writing  $\pi : \SS^{n}\times  \RR \to\SS^n$ for the projection to $\SS^{n}$, we compute

\begin{align*}
& \int_{\pi(\Gamma^{\lambda}_{\delta,j}\cap \{0<z<b\})}e^{-f_{j}}d\mu_{g^{(j)}}   \leq \int_{\Gamma^{\lambda}_{\delta,j}\cap \{0<z<b\}} |\be^{\perp}|_{g^{(j)}} e^{-f_{j}}d\mu_{g^{(j)}} \\
& \leq \left( \int_{\Gamma^{\lambda}_{\delta,j}\cap \{0<z<b\}} |\be^{\perp}|_{g^{(j)}}^{2} e^{-f_{j}}d\mu_{g^{(j)}}  \right)^{1/2} (b+\lambda^{-1})^{1/2} \left( \int_{\partial\hat\Omega_{\delta,j} } e^{-f_j} d\mu_{g^{(j)}} \right)^{1/2}\\
& \leq \lambda^{-1/2} (b+\lambda^{-1})^{1/2} \left( \int_{\partial\hat\Omega_{\delta,j} } e^{-f_j} d\mu_{g^{(j)}} \right).
\end{align*}

Sending $j\to\infty$ and then $\lambda\to\infty$, we can follow <cit.> or <cit.> to find that $\mu(0) = \cH^{n-1}\lfloor \partial^{*}\hat\Omega_{\delta}$. Since $\partial^{*}\Omega_{\delta}$ is $\tfrac{\bomega}{2}$-mean convex to the inside, any integral unit regular Brakke $\tfrac{\bomega}{2}$-flow starting at $\partial^{*}\hat\Omega_{\delta}$ agrees with the rescaled mean curvature flow of $\hat\Sigma_{\delta}$.

Fix $\delta \in (0,\delta_{0})$ and $\lambda > \tfrac 12 Z_{0}$ sufficiently large so that $\hat\Omega_{\delta}\times[2\lambda,\infty) \subset \Omega$.
Then, there is $\vartheta = \vartheta(\lambda,\delta)>0$, $R = R(\lambda,\delta)>0$ so that

\[
(\Gamma_{\delta}^{\lambda} + 2 \lambda \be) \cap \{ t\bx : \bx \in \partial B_{1}(\bOh), d(\bx,\be) < \vartheta, t>R\} = \emptyset
\]

We begin by assuming that $\hat\Omega$ is unbounded. (We will consider the easier case when $\hat\Omega$ is bounded below.)

Note that the shrinker mean curvature of $\Gamma_{\delta}^{\lambda} + 2 \lambda \be$ satisfies

\begin{equation}\label{eq:mean.curv.trans.shift}
\bH + \tfrac 12 \bx^\perp  = (\tfrac 12 z - \lambda ) \be^\perp
\end{equation}

at $(\bomega,z) \in \Gamma_{\delta}^{\lambda} + 2 \lambda \be$ and such a point must have $z\geq 2\lambda$. Consider the open set $Q \subset \{z>2\lambda\}$ with

\[
\partial Q =  (\RR^{n} \setminus \hat\Omega_{\delta}) \times \{2\lambda\}  \cup (\Gamma_{\delta}^{\lambda} + 2 \lambda \be).
\]

Note that $Q$ is weakly shrinker mean convex to the inside in the sense of Meeks–Yau <cit.>. This is obvious along the planar boundary, while along the translator portion of the boundary we can use (<ref>) and the fact that $\Gamma^\lambda_\delta$ is graphical over $\hat\Omega_{\delta}\times\{0\}$ as proven in Lemma <ref>.

Following <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 $Q \cap B_{2k}(\bOh)$ is mean convex with respect to $g_{k}$ for $k$ large and $\Sigma \cap Q \cap B_{2k}(\bOh)$ is a minimal surface with respect to $g_{k}$. Write $A_{k} : = (\RR^{n+1}\setminus\overline{ \Omega}) \cap Q \cap B_{2k}(\bOh)$ for the “inside” of $\Sigma \cap Q \cap B_{2k}(\bOh)$. Choose a Caccioppoli set $B_{k}$ in $Q\cap B_{2k}(\bOh)$ with (i) $A_{k}\subset B_{k}$, (ii) $\overline B_{k} \cap \partial(Q \cap B_{2k}(\bOh)) = \overline A_{k} \cap \partial(Q \cap B_{2k}(\bOh))$ and (iii) $B_{k}$ minimizes $g_{k}$-perimeter among Caccioppoli sets satisfying (i) and (ii). Set $\Upsilon_{k} : = \partial^{*} B_{k}$.

Taking a subsequence, we can thus find $\Upsilon = \partial^{*}B$ for $B \subset Q$ locally finite perimeter with $\overline B \cap \{z=\lambda\} = (\RR^{n+1}\setminus \Omega) \cap\{z=\lambda\}$, so that $\Upsilon$ has at worst a co-dimension $7$ singular set. It is thus straightforward to prove that $t\mapsto \sqrt{-t} \Upsilon$ is a (integral) Brakke flow. Recall that $\sqrt{0}\Upsilon$ is a conical closed set (not necessarily smooth) and satisfies

\[
\sqrt{0}\Upsilon \cap \be [1,\infty) = \emptyset
\]

by Ilmanen's avoidance principle (Theorem <ref>) applied to the disjoint flows $\sqrt{-t}\Upsilon$ and $\sqrt{-t}\Sigma$. In particular, there is $\vartheta_{1}$ so that

\[
\sqrt{0} \Upsilon \cap \{t \bx : \bx \in \partial B_{1}(\bOh), d(\bx,\be) < \vartheta_{1}, t>0 \} = \emptyset.
\]

This is easily seen to imply the assertion (when $\hat\Omega$ is unbounded). In the bounded case, a similar proof works with the conclusion that $\Upsilon$ (and thus $\Gamma$) is bounded. This completes the proof.

§.§ Long barriers

We continue with the setup of Theorem <ref>.

For constants $h_0>0$ to be fixed small below and $\alpha > \max\{ 2|\hat \mu|,Z_{0}\}$ to be fixed large below (depending on $\Sigma$ and the function $\tau \mapsto \tilde v_{\tau}(\cdot)$), for any $\eta \in (0,\alpha^{-\alpha} h_0)$ fixed we consider the function

\[
U(\bomega,z) =  \eta z^\alpha \hat\varphi(\bomega).
\]

Define $R=R(h_0,\eta,\alpha)>\alpha$ so that $\eta R^\alpha = h_0$. Then define

\[
(-\infty,T_{1}]\ni \tau \mapsto \Xi_{\tau} : = \{(\bomega,z) + (\tilde v_{\tau}(\bomega,z) + U(\bomega,z)) \nu_{\hat \Sigma}(\bomega) : (\bomega,z) \in \hat \Sigma \times [\alpha,R]\}.
\]

As long as $T_{1}\leq T_{0}$ is sufficiently negative, $h_0$ is small, and $\alpha$ is large (depending only on $\Sigma$ and $\tilde v_{\tau}$), we see that $\Xi_{\tau}$ is a smooth hypersurface (with boundary) contained in $\Omega$. We will write $\nu_{\Xi_{\tau}}$ for the “upwards pointing” unit normal, i.e., $\nu_{\Xi_{\tau}}\cdot\nu_{\hat\Sigma} > 0$. Define

\[
v_{\Xi_{\tau}} = (\nu_{\Xi_{\tau}} \cdot \nu_{\hat\Sigma})^{-1}
\]

along $\hat\Sigma\times [\alpha,R]$, where $\nu_{\Xi_{\tau}}$ is evaluated at $(\bomega,z) + (\tilde v_{\tau}(\bomega,z) + U(\bomega,z)) \nu_{\hat \Sigma}(\bomega)$, for $(\bomega,z) \in \hat \Sigma \times [\alpha,R]$, as usual. Note that the normal speed of $\Xi_{\tau}$ is given by

\[
v_{\Xi_{\tau}}^{-1}(\partial_{\tau} \tilde v_{\tau})
\]

and the scalar (rescaled) mean curvature satisfies

\[
\bH_{\Xi_{\tau}} + \tfrac 12 \bx^\perp = (H_{\Xi_{\tau}} + \tfrac 12 \bx\cdot\nu_{\Xi_{\tau}}) \nu_{\Xi_{\tau}}.
\]

As such, the next proposition says that $\tau \mapsto \Xi_{\tau}$ is a barrier for rescaled mean curvature flows pinched between $\Xi_{\tau}$ and $\Sigma$.

There is $T_{1}'\leq T_{0}$, $h_0$ small, and $\alpha_0 >\max\{2|\hat \mu|, Z_{0}\}$ large so that for $\alpha \geq \alpha_0$ and $\eta \in (0,\alpha^{-\alpha}h_0)$, it holds that

\[
v_{\Xi_{\tau}}^{-1}(\partial_{\tau} \tilde v_{\tau}) -  (H_{\Xi_{\tau}} + \tfrac 12 \bx\cdot\nu_{\Xi_{\tau}}) > 0.
\]

We begin by observing that,

\begin{equation}\label{eq:bendy-barrer-function-higher-derivatives}
|\nabla_{\hat\Sigma\times \RR}^k U|(\bomega,z) \leq C_k \eta z^\alpha \hat\varphi(\bomega)
\end{equation}

for $z \geq \alpha$ and $k\geq 0$, (recall that $|\hat \mu|\geq 1$). This will be used in several places below to control various error terms.

Combining (<ref>) and (<ref>) we see that Lemma <ref> applies,[Here we just use the part of Lemma <ref> estimating $F,\bF,\mathcal{F}$, not the second half of the lemma.] yielding

\begin{equation}\label{eq:application-relative-shrinker-mean-curvature-to-long-barrier}
v_{\Xi_{\tau}} (H_{\Xi_{\tau}} + \tfrac 12 \bx\cdot\nu_{\Xi_{\tau}}) - v_{\Graph \tilde v_{\tau}} (H_{\Graph \tilde v_{\tau}} + \tfrac 12 \bx\cdot\nu_{\Graph \tilde v_{\tau}}) = L_{\hat\Sigma\times \RR} U + E
\end{equation}

\begin{equation}\label{eq:application-relative-shrinker-mean-curvature-to-long-barrier-ERROR}
|E(\bx)| \leq C (|U(\bx)| + |\nabla_{\Sigma\times\RR} U(\bx)| + |\nabla^{2}_{\Sigma\times\RR} U(\bx)|)
\end{equation}

and $v_{\Graph \tilde v_{\tau}} = (\nu_{\Graph \tilde v_{\tau}}\cdot\nu_{\hat\Sigma})^{-1}$. Because $\tau \mapsto \Graph \tilde v_{\tau}$ is a rescaled mean curvature flow, we find

\begin{equation}\label{eq:long-barrier-w-tau-is-rescaled-MCF}
\partial_{\tau} \tilde v_{\tau} = v_{\Graph \tilde v_{\tau}} (H_{\Graph \tilde v_{\tau}} + \tfrac 12 \bx\cdot\nu_{\Graph \tilde v_{\tau}}),
\end{equation}

so combining (<ref>) and (<ref>) we find

\[
\partial_{\tau} \tilde v_{\tau} - v_{\Xi_{\tau}}  (H_{\Xi_{\tau}} + \tfrac 12 \bx\cdot\nu_{\Xi_{\tau}}) = - L_{\hat\Sigma\times \RR} U - E.
\]

We now compute

\[
- L_{\hat\Sigma\times \RR}U =  \left( \tfrac \alpha 2 -  \alpha(\alpha - 1) z^{-2} - |\hat\mu| \right)\eta z^\alpha \hat\varphi(\bomega),
\]

so when combined with (<ref>) and (<ref>) we find

\[
- L_{\hat\Sigma\times \RR} U - E \geq \left( \tfrac \alpha 2 -  \alpha(\alpha - 1) z^{-2} - |\hat\mu| - C\right)\eta z^\alpha \hat\varphi(\bomega)
\]

for $C$ independent of $z,\alpha,\tau$. Recalling that $z \geq \alpha$ on the region defining $\Xi_{\tau}$, we can thus take $\alpha$ sufficiently large to complete the proof.

§.§ The proof of Theorem <ref>

We continue with the setup of Theorem <ref>.

We fix $\alpha_0,h_0>0$, $T_{1}'\leq T_{0}$ using Proposition <ref>. It follows that for any $\alpha \geq\alpha_0$ and $\eta \in (0,\alpha^{-\alpha}h_0)$, defining $R$ by $\eta R^\alpha = h_0$, the family of hypersurfaces

\[
(-\infty,T_{1}']\ni\tau \mapsto \Xi_{\tau} : = \{(\bomega,z) + (\tilde v_{\tau}(\bomega,z) + U(\bomega,z)) \nu_{\hat \Sigma}(\bomega) : (\bomega,z) \in \hat \Sigma \times [\alpha,R]\} \subset \Omega.
\]

is a supersolution to mean curvature flow.

Define $\delta>0$ so that if $\hat \Sigma_\delta(t)$ is the rescaled mean curvature flow in $\hat\Omega\subset\RR^n$ with $\hat \Sigma_\delta(0) = \hat\Sigma_\delta$ (defined in (<ref>)) then $[0,3] \ni t \mapsto \Sigma_\delta(t)$ is smooth and

\begin{equation}\label{eq:barr.defn.delta.small.welding}
\hat\Sigma_\delta(t) \cap \hat \Omega_{h_0} = \emptyset
\end{equation}

The existence of such a $\delta>0$ follows from the fact that as $\delta\to 0$, the flows $t\mapsto \hat \Sigma_\delta(t)$ converge smoothly (on compact subsets of space-time) to the static flow given by $\hat\Sigma$ (since $\hat \Sigma$ is a shrinker). We now fix $\alpha\geq \alpha_0$ so that

\begin{equation}\label{eq:barr.defn.alpha.big.welding}
h_0 2^{-\alpha} < \delta/2
\end{equation}

and $\hat \Omega_\delta \times \{\alpha\} \subset \Omega$.

Now recall the surfaces $\Gamma^{\lambda}_\delta$ as defined in Lemma <ref>. We set

\begin{equation}
\Gamma : = \Gamma^{R/4}_\delta + \tfrac R 2 \be.
\end{equation}

We choose the unit normal to $\Gamma$ so that $\nu_\Gamma \cdot \be > 0$. Note that the shrinker mean curvature of $\Gamma$ satisfies

\begin{align*}
\bH_\Gamma + \tfrac 12 \bx^\perp
& = (\tfrac 12 z - \tfrac R 4 ) \be^\perp
\end{align*}

\[
H_\Gamma + \tfrac 12 \bx \cdot \nu_\Gamma = (\tfrac 12 z - \tfrac R 4 ) \be \cdot \nu_\Gamma \geq 0.
\]

Below, we will choose a family of regions $Q_{\tau}$ so that parts of $\Xi_{\tau}$ and $\Gamma$ are contained in $\partial Q_{\tau}$. The unit normal that $\Xi_{\tau}$ inherits from its graphical nature will be outwards pointing relative to $Q_{\tau}$.

There is a unique choice of open set $Q_\Gamma$ so that

\begin{equation*}
\partial Q_\Gamma = ( (\Omega \cap \{z=\tfrac R 2\}) \setminus (\hat\Omega_\delta \times \{\tfrac R 2\}))
\cup (\Sigma \cap (B_{2\rho}(\bOh)\times [\tfrac R2 ,\infty))) \cup \Gamma
\end{equation*}

and so that $Q_\Gamma$ is shrinker mean convex (to the inside) in the generalized sense allowing for the corners of $\partial Q_\Gamma$ (cf.  [65]). Similarly, there is a unique choice of families of open sets $\tau \mapsto Q_{\Xi_{\tau}}$ so that

\begin{multline*}
\partial Q_{\Xi_{\tau}} = \{(\bomega,z) + (\tilde v_{\tau}(\bomega,z)+sU(\bomega,z)) \nu_{\hat \Sigma}(\bomega) : \bomega \in \hat \Sigma, z \in \{\alpha,R\}, s \in [0,1]\} \\
\cup  (\Sigma \cap (B_{2\rho}(\bOh)\times [\alpha ,R]))  \cup \Xi_{\tau}.
\end{multline*}

We now consider the boundaries of $\Xi_{\tau},\Gamma$ at $z=R,z=R/2$ respectively and show that they do not affect the barrier argument if $\Xi,\Gamma$ are welded together appropriately. Define

\[
\tau \mapsto Q_{\tau} = (Q_{\Xi_{\tau}} \cap (Q_\Gamma \cup \{z \leq \tfrac R2\})) \cup (Q_\Gamma \cap (Q_{\Xi_{\tau}} \cup \{z \geq R\})).
\]

This will be the fundamental barrier region. The following proposition will be used to verify that as long as the parameter $\eta$ is chosen small in the definition of $\Xi_{\tau}$ then $\partial Q_{\tau}$ will serve as a barrier for a rescaled mean curvature flow in $Q_{\tau}$. In the next lemma it is important to recall that $\nu_{\Xi_{\tau}}$ was chosen to be the graphical unit normal, so it points out of $Q_{\tau}$.

There is $\eta_{0} \in (0,\alpha^{-\alpha}h_0)$ sufficiently small, $T_1 \leq T_1'$ sufficiently negative, and $\gamma_{0} > 0$ so that if $\eta \in (0,\eta_0)$, $\tau \in (-\infty,T_{1}]$ and $\bx \in B_{\gamma_{0}}(\Xi_{\tau}) \cap Q_{\tau}$, writing $\by \in \Xi_{\tau}$ a closest point to $\bx$, then

(1a) $\by \not \in \partial\Xi_{\tau} \cap \{z>\tfrac R 2 \}$, and

(1b) $(\by - \bx) \cdot  \nu_{\Xi_{\tau}}(\by) >  0$.