# Rotational symmetry of ancient solutions to fully nonlinear curvature flows

A. Cogo  Eberhard Karls Universität Tübingen, Fachbereich Mathematik, Auf der
Morgenstelle 10, 72076 Tübingen, Germany<EMAIL_ADDRESS>,
S. Lynch  Department of Mathematics, Imperial College London, London SW7 2AZ,
United Kingdom<EMAIL_ADDRESS>and  O. Vičánek Martínez
Eberhard Karls Universität Tübingen, Fachbereich Mathematik, Auf der
Morgenstelle 10, 72076 Tübingen, Germany<EMAIL_ADDRESS>tuebingen.de

###### Abstract.

We address the classification of ancient solutions to fully nonlinear
curvature flows for hypersurfaces. Under natural conditions on the speed of
motion, we show that every convex, noncollapsing, uniformly two-convex ancient
solution which is also noncompact is either a self-similarly shrinking
cylinder, or else is a rotationally symmetric translating soliton. For a large
class of flows this yields a complete classification of the blow-up limits
that can arise at a singularity of a solution which is compact, embedded and
two-convex.

###### Contents

  1. 1 Introduction
    1. 1.1 Main results
    2. 1.2 Overview of the proof
  2. 2 Preliminaries
    1. 2.1 Uniform parabolicity
    2. 2.2 Compactness of noncollapsing convex ancient solutions
    3. 2.3 Neck-cap decomposition
    4. 2.4 Differential Harnack inequality
    5. 2.5 Expansions for hypersurfaces close to a cylinder
    6. 2.6 Asymptotic cones
  3. 3 The parabolic blow-down limit
  4. 4 A barrier construction
    1. 4.1 Local existence
    2. 4.2 Global behaviour
  5. 5 Sharp asymptotics for the rescaled flow
    1. 5.1 Inner-outer estimates
    2. 5.2 Spectral analysis
  6. 6 The Neck Improvement Theorem
  7. 7 Rotational symmetry
  8. 8 Uniqueness of ancient solutions with rotational symmetry
  9. A Asymptotics for the bowl soliton

##  1\. Introduction

The study of singularity formation in geometric flows began with the
fundamental work of Hamilton [Ham82]. There it was shown that positively
curved three-manifolds become singular in finite time under the Ricci flow,
and are deformed to a constant curvature space in the process. This is an
example of a phenomenon which has since been observed in many other
situations—close to a singularity, solutions to geometric flows tend to become
highly symmetric. This allows for a very precise description of the geometric
structure of singularities; an essentially complete picture has now been
established eg. for the three-dimensional Ricci flow and for two-convex mean
curvature flow. In these settings it has been possible to extend the flow in
question beyond singularities using a surgery procedure (see [Per02a, Per02b]
and [HS09, BH16, HK17b]), enabling profound insights into questions in
geometry and topology.

A solution of a geometric flow is called ancient if it exists on a time
interval of the form $(-\infty,T]$. Ancient solutions arise as dilations of
singularities. Therefore, classification results for ancient solutions are of
central importance in understanding singularity formation. In this paper we
are concerned with the classification of ancient solutions to fully nonlinear
geometric flows. Consider a family of hypersurfaces
$M_{t}\subset\mathbb{R}^{n+1}$, $t\in(-\infty,T]$, which move with pointwise
normal speed equal to a symmetric function of their principal curvatures.
Under natural assumptions on the speed function, we show that if $M_{t}$ is
convex, noncollapsing, uniformly two-convex and noncompact then it is either a
self-similarly shrinking round $\mathbb{R}\times S^{n-1}$, or else is
rotationally symmetric and evolves by translation in a constant direction.
Brendle and Choi proved this when $M_{t}$ moves by mean curvature flow [BC19,
BC21] (see also [CHHW22]). Analogous results for ancient solutions of the
Ricci flow were obtained in [Bre20, BN23].

In certain situations our results yield a complete classification of the
possible ancient solutions that can arise by dilating a singularity. A key
example we have in mind is the flow which moves $M_{t}$ with speed

(1)
$\gamma(\lambda)=\bigg{(}\sum_{i<j}\frac{1}{\lambda_{i}+\lambda_{j}}\bigg{)}^{-1},$

where the $\lambda_{i}$ are the principal curvatures. This flow was first
considered by Brendle and Huisken [BH17]. They used it to prove a remarkable
classification result for compact Riemannian manifolds which have strictly
two-convex boundary and satisfy the curvature condition $R_{ikik}+R_{jkjk}\geq
0$. Namely, every such space is diffeomorphic to a standard ball or
1-handlebody.

###  1.1. Main results

Let $\Gamma\subset\mathbb{R}^{n}$ denote an open, symmetric (under
permutations), convex cone. Fix a function $\gamma:\Gamma\to\mathbb{R}$ which
is smooth, positive, symmetric, strictly increasing in each argument, and
homogeneous of degree one. We assume $\gamma$ is either convex or concave.

We also assume that $\Gamma$ contains the positive cone $\mathbb{R}^{n}_{+}$
and the cone $\\{0\\}\times\mathbb{R}_{+}^{n-1}$, and that both $\gamma$ and
$\lambda\mapsto\gamma(0,\lambda)$ are strictly inverse-concave functions.
(These conditions ensure that we have a splitting theorem [Lyn22b] and a
differential Harnack inequality [Lyn].) Note that inverse-concavity is
automatic if $\gamma$ is convex.

For a hypersurface $M$ with principal curvatures
$\lambda=(\lambda_{1},\dots,\lambda_{n})$ in $\Gamma$ we write
$G=\gamma(\lambda)$. The principal curvatures are the eigenvalues of the
second fundamental form $A$ with respect to the induced metric $g$. We label
these so that $\lambda_{1}\leq\dots\leq\lambda_{n}$. The outward unit normal
to $M$ is denoted $\nu$.

A $G$-flow is a smooth one-parameter family of embedded hypersurfaces $M_{t}$
in $\mathbb{R}^{n+1}$ which move inwards with pointwise velocity $-G\nu$. We
say that a $G$-flow is:

  1. (1)

convex if $M_{t}=\partial\Omega_{t}$, where $\Omega_{t}$ is convex;

  2. (2)

uniformly two-convex if there is a constant $\beta>0$ such that
$\lambda_{1}+\lambda_{2}\geq\beta H$, where $H$ is the mean curvature;

  3. (3)

noncollapsing if there is a constant $\alpha>0$ such that every point $x\in
M_{t}$ admits an inscribed ball of radius at least $\alpha G(x,t)$.

We can now state our main result.

######  Theorem 1.1.

Let $M_{t}$, $t\in(-\infty,T]$, be a convex ancient $G$-flow which is
noncollapsing and uniformly two-convex. Suppose also that $M_{t}$ is
noncompact. Then $M_{t}$ is either a self-similarly shrinking round
$\mathbb{R}\times S^{n-1}$, or else is a rotationally symmetric translating
soliton.

We say that $M_{t}$, $t\in(-\infty,T]$, is a translating soliton if there is a
constant vector $V\in\mathbb{R}^{n+1}$ such that $M_{t}=M_{T}+(t-T)V$. Up to
rigid motions of the ambient space and parabolic rescalings, there is a unique
rotationally symmetric translating $G$-flow (the ‘bowl’ soliton). This
solution was constructed rigorously in [Ren21].

As we mentioned above, Theorem 1.1 applies when $\gamma$ is the speed (1)
introduced in [BH17]. It also applies to the speeds introduced in [Lyn22a],
and when $\gamma$ is a ratio of elementary symmetric poynomials
$\sigma_{k}/\sigma_{k-1}$ such that $n\geq k+2$.

Under some further conditions on $\gamma$, Theorem 1.1 provides a complete
classification of the possible limits that can arise when we blow up a
singularity of a compact $G$-flow.

######  Corollary 1.2.

Suppose $\gamma$ is concave. Suppose also that $\Gamma$ is contained in the
two-positive cone $\\{\lambda:\min_{i<j}\lambda_{i}+\lambda_{j}>0\\}$, and
that $\gamma$ admits a continuous extension to $\bar{\Gamma}$ which vanishes
on $\partial\Gamma$. Consider a compact embedded hypersurface $M_{0}$ in
$\mathbb{R}^{n+1}$ whose principal curvatures lie in $\Gamma$ (this implies
strict two-convexity). Let $M_{t}$ denote the maximal evolution of $M_{0}$
which moves inwards with pointwise velocity $-G\nu$. Every blow-up of $M_{t}$
at the singular time is a self-similarly shrinking round $S^{n}$ or
$\mathbb{R}\times S^{n-1}$, or else is the bowl soliton.

Corollary 1.2 applies, for example, when $n\geq 3$, $\gamma$ is the speed (1),
and $\Gamma$ is the two-positive cone.

Let us indicate how Corollary 1.2 is proven. We write $C$ for positive
constant depending on $M_{0}$. Assuming $\gamma$ is concave, the parabolic
maximum principle implies that $\min_{M_{t}}G>C^{-1}$ and $\max_{M_{t}}H/G\leq
C$. Assuming that $\Gamma$ sits inside the two-positive cone and vanishes on
$\partial\Gamma$, the bound $H/G\leq C$ implies uniform parabolicity and
uniform two-convexity. The speed $G$ evolves according to

$\bigg{(}\frac{\partial}{\partial t}-\frac{\partial\gamma}{\partial
A_{ij}}\nabla_{i}\nabla_{j}\bigg{)}G=\frac{\partial\gamma}{\partial
A_{ij}}A_{i}^{k}A_{kj}G\geq\frac{1}{C}G^{3},$

so by the maximum principle a singularity must form in finite time. The flow
$M_{t}$ is noncollapsing by [ALM13], and satisfies a convexity estimate
(implying that blow-ups are convex) by [BH17] (see also [LL20]). The gradient
estimates in [BH17] (see also [Lyn20]) imply that if $G(p_{k},t_{k})\to\infty$
then the sequence

$G(p_{k},t_{k})(M_{t/G(p_{k},t_{k})^{2}+t_{k}}-p_{k})$

subconverges in $C^{\infty}_{\operatorname{loc}}$ to an ancient $G$-flow. This
ancient flow is either compact, in which case it is a sphere by [And07,
And94a], or else is noncompact and satisfies the hypotheses of Theorem 1.1.
The claim follows.

We remark that Corollary 1.2 still applies when $M_{t}$ is the boundary of a
region of a compact Riemannian $(n+1)$-manifold satisfying a curvature
condition. Namely we require that

$(R(e_{1},e_{n+1},e_{1},e_{n+1}),\dots,R(e_{n},e_{n+1},e_{n},e_{n+1}))\in\bar{\Gamma}$

for every orthonormal frame $\\{e_{i}\\}_{i=1}^{n+1}$. In particular,
Corollary 1.2 applies in the setting of [BH17]. Note this ambient curvature
condition is always satisfied if, for example, the sectional curvatures are
nonnegative. As in [BH17], we can also weaken the ambient curvature condition,
provided that we assume more pinching for the hypersurfaces—Corollary 1.2
applies if

$(R(e_{1},e_{n+1},e_{1},e_{n+1})+\kappa^{2},\dots,R(e_{n},e_{n+1},e_{n},e_{n+1})+\kappa^{2})\in\bar{\Gamma}$

for every orthonormal frame $\\{e_{i}\\}_{i=1}^{n+1}$, and the boundaries
$M_{t}$ are assumed to satisfy

$(\lambda_{1}-\kappa,\dots,\lambda_{n}-\kappa)\in\Gamma$

and move with velocity $-G_{\kappa}\nu$ where
$G_{\kappa}:=\gamma(\lambda_{1}-\kappa,\dots,\lambda_{n}-\kappa)$.

###  1.2. Overview of the proof

Let $M_{t}$ denote a noncompact, convex ancient $G$-flow which is
noncollapsing and uniformly two-convex. We write $\bar{M}_{\tau}$ for the
rescaled family of hypersurfaces $e^{\tau/2}M_{-e^{-\tau}}$, which move with
velocity $-(G-\frac{1}{2}\langle x,\nu\rangle)\nu$.

In Section 2 we collect preliminary results to be used throughout the paper.

In Section 3 we begin studying the behaviour of the hypersurfaces $M_{t}$ at
very early times. We show that as $\tau\to-\infty$, up to a choice of ambient
Euclidean coordinates, the family $\bar{M}_{\tau}$ converges in
$C^{\infty}_{\operatorname{loc}}$ to the cylinder $\Sigma=\mathbb{R}\times
S^{n-1}((2\gamma(0,1,\dots,1)^{1/2})$. The corresponding convergence statement
for mean curvature flow is usually proven using Huisken’s monotonicity
formula, but no such tool is available for fully nonlinear flows; the majority
of these certainly lack any kind of variational structure. Xu-Jia Wang found
another approach for convex ancient solutions to the mean curvature flow
[Wan11]. The idea is to extract a blow-down limit, and then translate it
infinitely far along its asymptotic cone. The limit of the translated
solutions is again an ancient solution, and splits off a line. Wang then
exploits the fact that lower-dimensional slices through a convex mean
curvature flow satisfy an avoidance principle to conclude that the translated
solution actually coincides with the blow-down. In particular, the blow-down
splits off a line. Since it is also uniformly two-convex, the blow-down is a
cylinder by [Hui84] (this also follows from [HS15, HH16]).

Unfortunately, for flows by other curvature functions, there is no avoidance
principle for lower-dimensional slices. The validity of such a principle for
the mean curvature flow traces back to the fact that the graphical mean
curvature operator is the Laplacian (and in particular is isotropic in the
Hessian) at points with horizontal tangent planes. In [Lyn22b] the second
author established a certain rigidity property for shrinking cylinders, and
used this to classify Type I convex ancient solutions to a large class of
fully nonlinear flows. Using this same rigidity property, we are able to adapt
Xu-Jia Wang’s arguments to show prove subsequential convergence of the family
$\bar{M}_{\tau}$ to a cylinder of radius $(2\gamma(0,1,\dots,1))^{1/2}$ whose
axis passes through the origin. This leaves open the possibility that
$\bar{M}_{\tau}$ might slowly rotate, so that along different subsequences we
might see cylinders with different axes arising in the limit. We rule out this
possibility by showing that the asymptotic cone of $\bar{M}_{\tau}$ is
independent of $\tau$. The subsequential convergence can then be upgraded to
full convergence to $\Sigma$.

The next step is to prove a sharp asymptotic estimate for the rate at which
$\bar{M}_{\tau}$ converges to $\Sigma$. For this we make use of certain
barrier solutions, which are constructed in Section 4. For each sufficiently
large $a>0$ we construct an embedded disc $\Sigma_{a}$ which is strictly
convex, rotationally symmetric, and generates a self-similarly shrinking
$G$-flow. Equivalently, each $\Sigma_{a}$ is a stationary solution of the
rescaled flow. We also establish rather precise asymptotics for these
solutions as $a\to\infty$, which allow us to use them as barriers for
$\bar{M}_{\tau}$. To describe these asymptotics let us express $\Sigma_{a}$ as
the set of points of the form

$(z,\Psi_{a}(z)\theta),\qquad z\in[L_{0},a],\;\theta\in S^{n-1},$

where $\Psi_{a}$ is a nonnegative concave function. We show that $L_{0}$ can
be fixed independently of $a$. The tip of $\Sigma_{a}$ is located at $(a,0)$,
and we have $\Psi_{a}(z)^{2}<2\gamma(0,1,\dots,1)$, meaning $\Sigma_{a}$ lies
inside the region bounded by $\Sigma$. In addition, there is a constant
$C=C(n,\gamma)$ such that, for every bounded $L$, when $a$ is sufficiently
large we have

$\Psi_{a}(z)^{2}\leq 2\gamma(0,1,\dots,1)\bigg{(}1-\bigg{(}1-C\frac{\log
a}{a^{2}}\bigg{)}\frac{z^{2}-C}{a^{2}}\bigg{)}.$

for $L_{0}\leq z\leq L$. We also have the lower bound

$\Psi_{a}(z)^{2}\geq
2\gamma(0,1,\dots,1)\bigg{(}1-\frac{z^{2}}{a^{2}}\bigg{)}$

for $L_{0}\leq z\leq a$. Analogous self-similarly shrinking solutions to the
mean curvature flow were constructed in [ADS19] and played a central role in
[BC19, BC21, ADS20]. Our construction is similar to that in [ADS19], but is
complicated by the fact that the profile function $\Psi_{a}$ now solves a
fully nonlinear ODE.

By the results of Section 3, we may express $\bar{M}_{\tau}$ as the graph of a
function $u(z,\theta,\tau)$ over the $z$-axis, and we have $u\to 0$ in
$C^{\infty}_{\operatorname{loc}}$ as $\tau\to-\infty$. In Section 5 we prove
the following sharp asymptotic estimate for $u$: for each $L<\infty$, we have

(2) $\sup_{|z|\leq L}|u(\cdot,\tau)|\leq O(e^{-\tau/2}).$

This is achieved by performing a spectral decomposition of $u$ with respect to
the linearised rescaled $G$-flow about the cylinder $\Sigma$, and determining
that this decomposition is dominated by positive modes as $\tau\to-\infty$.
The argument is one of the key steps in our analysis, so let us describe it in
more detail.

We show that $u$ satisfies

(3) $\frac{\partial u}{\partial\tau}=a\frac{\partial^{2}u}{\partial
z^{2}}+\frac{1}{2(n-1)}\Delta_{S^{n-1}}u-\frac{1}{2}z\frac{\partial
u}{\partial z}+u+E,$

where $a:=\dot{\gamma}^{1}(0,1,\dots,1)$ and $E=O(|u|^{2}+|\nabla
u|^{2}+|\nabla^{2}u|^{2})$. For each large $-\tau$ this holds in a large but
bounded subset of space. The linear elliptic operator on the right-hand side
of (3) admits a complete set of $L^{2}$-eigenfunctions with respect to the
measure $e^{-z^{2}/4a}\,dzd\theta$, and its eigenvalues are
$1-\frac{k}{2}-\frac{1}{2(n-1)}l(l+n-2)$, where $k,l\geq 0$ are integers.
After cutting off in space we decompose $u$ into pieces lying in the positive,
neutral and negative eigenspaces. These evolve as one would expect for a
solution to the linearised equation, up to error terms depending on $E$ and
the cutoff function. We show that all of these errors decay rapidly as
$\tau\to-\infty$ by employing the hypersurfaces $\Sigma_{a}$ as barriers. Here
the asymptotics for $\Sigma_{a}$ established in Section 4 play a decisive
role. A standard argument (due to Merle and Zaag) shows that either positive
or neutral modes dominate as $\tau\to-\infty$. The latter possibility is ruled
out using the fact that $\bar{M}_{\tau}$ is noncompact—this forces
$u(z,\theta,\tau)$ to be monotone in $z$, whereas no nonzero linear
combination of zero modes has this property. The estimate (2) follows.

The proof of (2) is broadly similar to the spectral analysis of convex ancient
solutions to the mean curvature flow carried out in [BC19, BC21] and [ADS19,
ADS20]. In these works sufficient decay of errors could be proven by using
Huisken’s monotonicity formula to derive a powerful reverse-Poincaré
inequality for the profile function $u$. Our approach using barriers is
instead informed by Brendle’s work on ancient $\kappa$-solutions to the
3-dimensional Ricci flow [Bre20]. We note however that error terms involving
the Hessian $|\nabla^{2}u|$ do not arise when considering the mean curvature
flow or Ricci flow, as these equations are quasilinear, whereas for fully
nonlinear equations such higher-order errors are inevitable.

In our setting we are able to control these higher-order errors, roughly
speaking, as follows. Using a gradient bound furnished by the barriers, we can
appeal to the interior $C^{2}$-estimate for radial graph solutions due to
Brendle–Huisken [BH17], followed by Krylov’s $C^{2,\alpha}$-estimate for
convex parabolic PDE [Kry82], and then use Schauder estimates to get uniform
bounds on higher derivatives. Next we use interpolation inequalities and
interior estimates for linear parabolic equations to obtain an improved bound
for the supremum of $|u|+|\nabla u|+|\nabla^{2}u|$. Crucially, this bound is
only slightly worse than linear in the $L^{2}$-norm of $u$ with respect to the
measure $e^{-z^{2}/4a}dzd\theta$ (see Lemma 5.8). This is sufficient to carry
out the Merle–Zaag argument. Note that all of the estimates just described
hold in a subset of space which grows at a definite rate as $\tau\to-\infty$.

With the sharp asymptotic (2) in hand, the remainder of the proof of Theorem
1.1 is quite similar to its counterpart for solutions of mean curvature flow
[BC19, BC21]. In Section 6 we prove the Neck Improvement Theorem for $G$-flows
(cf. [BC19, Theorem 4.4] and [BC21, Theorem 4.4]). Using this, we show that if
$M_{t}$ satisfies the hypotheses of Theorem 1.1 and is strictly convex then it
is rotationally symmetric (Section 7) and must, therefore, be a translating
soliton (Section 8). Combining this statement with the splitting theorem in
[Lyn22b], we obtain Theorem 1.1.

The arguments in Section 7 and Section 8 make use of a differential Harnack
inequality for noncompact convex solutions to fully nonlinear flows, which was
recently proven in [Lyn]. For solutions of mean curvature flow this inequality
is due to Hamilton [Ham95], and for compact convex solutions to fully
nonlinear flows it is due to Andrews [And94b].

##  2\. Preliminaries

In this section we collect auxhiliary results which will be needed in the
proof of Theorem 1.1.

We may view $\gamma$ as a symmetric function defined in $\Gamma$, or as an
$O(n)$-invariant function on the space of symmetric matrices with eigenvalues
in $\Gamma$. We alternate between these viewpoints. We write
$\dot{\gamma}^{i}(\lambda)$ and $\ddot{\gamma}^{ij}(\lambda)$ for derivatives
with respect to eigenvalues, so that

$\frac{d}{ds}\bigg{|}_{s=0}\gamma(\lambda+s\mu)=\dot{\gamma}^{i}(\lambda)\mu_{i},\qquad\frac{d}{ds}\bigg{|}_{s=0}\dot{\gamma}^{i}(\lambda+s\mu)=\ddot{\gamma}^{ij}(\lambda)\mu_{j},$

and write $\dot{\gamma}^{ij}(A)$ and $\ddot{\gamma}^{ij,kl}(A)$ for
derivatives with respect to matrix entries, so that

$\frac{d}{ds}\bigg{|}_{s=0}\gamma(A+sB)=\dot{\gamma}^{ij}(A)B_{ij},\qquad\frac{d}{ds}\bigg{|}_{s=0}\dot{\gamma}^{ij}(A+sB)=\ddot{\gamma}^{ij,kl}(A)B_{kl}.$

When $A$ is a diagonal matrix with entries $\lambda$, $\dot{\gamma}^{ij}(A)$
is also diagonal, with entries $\dot{\gamma}^{i}(\lambda)$.

Recall that we assume $\gamma$ is convex or concave. We also assume that the
functions $\gamma$ and $\lambda\mapsto\gamma(0,\lambda)$ are strictly inverse-
concave in $\mathbb{R}^{n}_{+}$ and $\\{0\\}\times\mathbb{R}_{+}^{n-1}$,
respectively. That is,

$\lambda\mapsto-\gamma(\lambda_{1}^{-1},\dots,\lambda_{n}^{-1})$

is strictly concave for $\lambda\in\mathbb{R}_{+}^{n}$, and

$(\lambda_{1},\dots,\lambda_{n-1})\mapsto-\gamma(0,\lambda_{1}^{-1},\dots,\lambda_{n-1}^{-1})$

is strictly concave for
$(\lambda_{1},\dots,\lambda_{n-1})\in\mathbb{R}_{+}^{n-1}$. When $\gamma$ is
convex these two inverse-concavity hypotheses are redundant.

We often use the notation

$F(\lambda,\mu)=\gamma(\lambda,\mu,\dots,\mu).$

###  2.1. Uniform parabolicity

Let $M_{t}$, $t\in I$, denote a $G$-flow. We say that $M_{t}$ is uniformly
parabolic if there is a constant $\delta>0$ such that

(4)
$\inf_{M_{t}}\operatorname{dist}(\tfrac{\lambda}{|\lambda|},\partial\Gamma)\geq\delta.$

for every $t\in I$. Uniform parabolicity implies that, with respect to any
orthonormal frame,

$C^{-1}g_{ij}\leq\dot{\gamma}^{ij}(A)\leq Cg_{ij},$

where $C$ is a positive constant depending only on $n$, $\gamma$ and $\delta$.

Recall that we assume $\gamma$ is positive in the cones $\mathbb{R}_{+}^{n}$
and $\\{0\\}\times\mathbb{R}_{+}^{n-1}$. Given these hypotheses, weak
convexity and uniform two-convexity imply uniform parabolicity. That is, if
$\lambda_{1}\geq 0$ and $\lambda_{1}+\lambda_{2}\geq\beta H$ at every point on
a $G$-flow, then there is a constant $\delta=\delta(n,\gamma,\beta)$ such that
(4) holds for $t\in I$.

######  Lemma 2.1.

Let $M_{t}$ be a convex, uniformly two-convex $G$-flow. Then $M_{t}$ is
uniformly parabolic.

###### Proof.

If not, let $(x_{k},t_{k})$ be a sequence such that
$\tfrac{\lambda(x_{k},t_{k})}{|\lambda(x_{k},t_{k})|}\to\partial\Gamma$. After
rescaling we may assume $|\lambda(x_{k},t_{k})|=1$. By hypothesis,
$\lambda_{1}(x_{k},t_{k})$ and $\lambda_{2}(x_{k},t_{k})$ both tend to zero.
But convexity implies

$\frac{1}{n}H(x_{k},t_{k})^{2}\leq|\lambda(x_{k},t_{k})|^{2}\leq
H(x_{k},t_{k})^{2},$

so this contradicts uniform two-convexity. ∎

###  2.2. Compactness of noncollapsing convex ancient solutions

The following interior curvature estimate was proven in [BH17, Proposition
5.1] for the speed
$\gamma(\lambda)=(\sum_{i<j}(\lambda_{i}+\lambda_{j})^{-1})^{-1}$. However the
same proof works for any $\gamma$ satisfying our hypotheses (see [Lyn22b] for
further discussion).

######  Lemma 2.2.

Let $M_{t}=\partial\Omega_{t},t\in[-T,0],$ be a convex $G$-flow which is
$\delta$-uniformly parabolic. If $B(0,r)$ is contained in $\Omega_{0}$, then

$\sup_{B(0,R)\times[-T/2,0]}|A|\leq
Cr^{-3}R^{3}\Big{(}r^{-1}+T^{-1/2}\Big{)},$

where $C$ depends only on $n$, $\gamma$, $\delta$ and $R/r$.

As a consequence of Lemma 2.2 and the avoidance principle, noncollapsing
$G$-flows satisfy the following curvature estimate (cf. [HK17a]). For the
proof we refer to [Lyn20, Theorem 4.17] (note however that the argument there
simplifies considerably when $M_{t}$ is convex).

Here and throughout the paper we only consider ancient $G$-flows which exist
for $t\in(-\infty,0]$. This is purely for convenience, and can always be
arranged by shifting the time variable.

######  Proposition 2.3.

Let $M_{t}=\partial\Omega_{t}$, $t\in(-\infty,0]$, be a convex ancient
$G$-flow which is $\delta$-uniformly parabolic and $\alpha$-noncollapsing.
Suppose $x_{0}\in M_{t_{0}}$. We then have

$C^{-1}G(x_{0},t_{0})\leq G(x,t)\leq CG(x_{0},t_{0}),$

for $(x,t)\in
B_{LG(x_{0},t_{0})^{-1}}(x_{0})\times[t_{0}-L^{2}G(x_{0},t_{0})^{-2},t_{0}]$,
where the constant $C$ depends only on $n$, $\gamma$, $\delta$, $\alpha$ and
$L$.

As a consequence of the curvature estimate we have the following compactness
theorem for sequences of convex ancient $G$-flows which are uniformly
parabolic and noncollapsing.

######  Proposition 2.4.

Let $M_{t}^{k}$, $t\in(-\infty,0]$, be a sequence of convex ancient $G$-flows.
Suppose each $M_{t}^{k}$ is $\delta$-uniformly parabolic and
$\alpha$-noncollapsing, where $\delta$ and $\alpha$ are independent of $k$.
Suppose in addition that there exists an $R<\infty$ such that $M_{0}^{k}\cap
B(0,R)$ is nonempty for every $k$ and

$\liminf_{k\to\infty}\sup_{M_{0}^{k}\cap B(0,R)}(G_{k}+G_{k}^{-1})>0.$

Then there is a convex ancient $G$-flow $M_{t}$, $t\in(-\infty,0]$, such that
after passing to a subsequence

$M_{t}^{k}\to
M_{t}\;\;\text{in}\;\;C^{\infty}_{\operatorname{loc}}(\mathbb{R}^{n+1}\times(-\infty,0]).$

###### Proof.

The assumptions imply that, for each $k$, there is a point $p_{k}$ satisfying
$|p_{k}|\leq R$ and a ball

$B(p_{k},r)\subset\Omega_{0}^{k},$

where $r>0$ is independent of $k$. We may therefore apply the interior
curvature bound of Lemma 2.2 to conclude that, in every compact subset of
spacetime, $|A_{k}|$ is bounded independently of $k$. Moreover, by Proposition
2.3, in every compact subset of spacetime $G_{k}$ is bounded from below by a
positive constant which is independent of $k$.

With the uniform upper bound for $|A_{k}|$, we can deduce uniform
$C^{2,\alpha}$-estimates in compact subsets of spacetime by expressing
$M_{t}^{k}$ locally as a graph and (since $\gamma$ is convex or concave)
appealing to the Hölder estimate for for concave/convex parabolic PDE proven
in [Kry82]. Using the upper bound for $|A_{k}|$ and the uniform lower bound
for $G_{k}$, we can bootstrap using the Schauder estimates to get uniform
derivative bounds up to any order in compact subsets of spacetime. Since the
hypersurfaces $M_{t}^{k}$ are radial graphs over $\partial B(p_{k},r)$, a
standard application of the Arzela–Ascoli shows that the flows subconverge in
$C^{\infty}_{\operatorname{loc}}$. ∎

###  2.3. Neck-cap decomposition

We have the following consequence of the splitting theorem in [Lyn22b].

######  Lemma 2.5.

Let $M_{t}$, $t\in(-\infty,0]$, be a convex, noncollapsing, uniformly two-
convex ancient $G$-flow. If there is a time $t\leq 0$ and a point in $M_{t}$
at which $\lambda_{1}=0$, then $M_{t}$ is a self-similarly shrinking
$\mathbb{R}\times S^{n-1}$.

###### Proof.

Suppose $\lambda_{1}$ vanishes at some point in spacetime. By the splitting
theorem [Lyn22b, Proposition A.2],

$M_{t}=\mathbb{R}\times M_{t}^{\perp}$

for $t\leq 0$, where $M_{t}^{\perp}$ is a family of hypersurfaces in
$\mathbb{R}^{n}$. If $n-1\geq 2$ then $M_{t}^{\perp}$ is uniformly convex, and
hence compact by [Ham94], so Theorem 1.7 in [LL20] implies that
$M_{t}^{\perp}$ is a self-similarly shrinking sphere. If $n-1=1$, since we are
assuming noncollapsing, the classification result in [BLT20] implies that
$M_{t}^{\perp}$ is a self-similarly shrinking circle. This completes the
proof. ∎

Let $M_{t}$ be a $G$-flow. For $\bar{x}\in M_{\bar{t}}$ we define

$\displaystyle\hat{\mathcal{P}}(\bar{x},\bar{t},L,\theta):=\Big{\\{}(x,t):\,$
$\displaystyle x\in
B_{g(t)}(\bar{x}(t),\gamma(0,1,\dots,1)LG(\bar{x},\bar{t})^{-1}),$
$\displaystyle\;t\in[\bar{t}-\gamma(0,1,\dots,1)^{2}\theta
G(\bar{x},\bar{t})^{-2},\bar{t}]\Big{\\}},$

where $\bar{x}(t)$ is the unique curve satisfying

$\frac{d}{dt}\bar{x}(t)=-G(\bar{x}(t),t)\nu(\bar{x}(t),t),\qquad\bar{x}(\bar{t})=\bar{x}.$

######  Definition 2.6.

A point $(\bar{x},\bar{t})$ is said to lie at the center of an
$\varepsilon$-neck if, after shifting $(\bar{x},\bar{t})$ to $(0,0)$,
parabolically rescaling to arrange $G(0,0)=(\gamma(0,1,\dots,1)/2)^{1/2}$, and
applying a rigid motion, the region $\hat{\mathcal{P}}(0,0,100,100)$ is
$\varepsilon$-close to the shrinking cylinder

$\mathbb{R}\times S^{n-1}\big{(}(2\gamma(0,1,\dots,1)(1-t))^{1/2}\big{)}$

in the $C^{10}$-norm. That is, for each $t\in[-2\cdot 10^{4},0]$, the
hypersurface $B_{g(t)}(0,\sqrt{2}\cdot 100)$ is a normal graph over
$\mathbb{R}\times S^{n-1}\big{(}(2\gamma(0,1,\dots,1)(1-t))^{1/2}\big{)}$, and
the $C^{10}$-norm of the height function is at most $\varepsilon$.

Let $M_{t}$ be a noncompact, strictly convex ancient $G$-flow. Then $M_{t}$
has a single end at each time. The following result asserts that if $M_{t}$ is
also noncollapsing and uniformly two-convex, then this end consists of necks.
For solutions of the mean curvature flow this was proven in [HK17b,
Proposition 3.4].

######  Lemma 2.7.

Let $M_{t}$, $t\in(-\infty,0]$, be a convex ancient $G$-flow which is
$\alpha$-noncollapsing and $\beta$-uniformly two-convex. For every
$\varepsilon_{1}$, $\varepsilon_{2}>0$ there is a positive constant
$R=R(n,\gamma,\alpha,\beta,\varepsilon_{1},\varepsilon_{2})$ with the
following property. If $x_{1}$ and $x_{2}$ are points in $M_{0}$, neither of
which lies at the center of an $\varepsilon_{1}$-neck, then one of the
following holds:

  1. (1)

$\max\\{G(x_{1},0),G(x_{2},0)\\}|x_{1}-x_{2}|\leq R$.

  2. (2)

$M_{t}$ is compact and bounds a region of diameter at most
$(1+\varepsilon_{2})|x_{1}-x_{2}|$.

###### Proof.

Suppose the claim is false for some $\varepsilon_{1}$ and $\varepsilon_{2}$.
Then there is a sequence of convex ancient $G$-flows
$M_{t}^{k}=\partial\Omega_{t}^{k}$, $t\in(-\infty,0]$, which are
$\alpha$-noncollapsing and $\beta$-uniformly two-convex, and have the
following property. There are points $x_{1}^{k}$ and $x_{2}^{k}$ in
$M_{0}^{k}$, neither of which lies at the center of an $\varepsilon_{1}$-neck,
such that

$\max\\{G(x_{1},0),G(x_{2},0)\\}|x_{1}-x_{2}|\to\infty$

and

$\operatorname{diam}(\Omega_{0}^{k})\geq(1+\varepsilon_{2})|x_{1}-x_{2}|.$

After performing a parabolic rescaling, we may assume that
$|x_{1}^{k}-x_{2}^{k}|=1$. We then have
$\max\\{G(x_{1}^{k},0),G(x_{2}^{k},0)\\}\to\infty$. Consequently, after
passing to a subsequence, the sets $\bar{\Omega}_{0}^{k}$ Hausdorff-converge
to a closed convex set of dimension at most $n$ (this follows from the
interior curvature estimate, Lemma 2.2). Let us denote this set $K$.

We claim that the dimension of $K$ is 1 (that is, $K$ is an interval). If $K$
has dimension at least 2 then there is a ‘cross’ in $K$ consisting of
intervals $\overline{y_{1}y_{2}}$ and $\overline{y_{3}y_{4}}$ which intersect
transversally at a point $y_{0}$. Approximate the points $y_{i}$ by sequences
$y_{i}^{k}$ in $M_{0}^{k}$, and let $T_{k}$ denote the convex hull of the set
$\\{y_{i}^{k}\\}$. We have $T_{k}\subset\bar{\Omega}_{0}^{k}$. After
performing a parabolic rescaling to arrange $G(y_{0}^{k},0)=1$, we may apply
Proposition 2.4 to extract a limiting $G$-flow which is uniformly two-convex
for $t\in(-\infty,0]$. But after rescaling the sets $\partial T_{k}$ converge
to a 2-plane which makes interior contact with the limiting flow at time
$t=0$. This contradicts uniform two-convexity. So we conclude that $K$ has
dimension at most 1. Moreover, since $|x_{1}^{k}-x_{2}^{k}|=1$, $K$ is not a
point. Therefore, the dimension of $K$ is 1.

If $x_{1}^{k}$ and $x_{2}^{k}$ converge to the endpoints of $K$ then for
sufficiently large $k$ we have
$\operatorname{diam}(\Omega_{0}^{k})<1+\varepsilon_{2}$, which is impossible
by hypothesis. Therefore, we may assume without loss of generality that
$x_{2}^{k}$ converges to an interior point of $K$. Let us now parabolically
rescale so that $G(x_{2}^{k},0)=1$ and extract a limiting $G$-flow using
Proposition 2.4. Arguing as above, we see that the limit makes interior
contact with a line at time $t=0$. Therefore, by Lemma 2.5, $x_{2}^{k}$ lies
at the center of an $\varepsilon_{1}$-neck for all large $k$. This is a
contradiction. ∎

The neck-cap decomposition implies that we have bounded curvature on bounded
time intervals.

######  Lemma 2.8.

Let $M_{t}=\partial\Omega_{t}$, $t\in(-\infty,0]$, be a noncompact, convex
ancient $G$-flow which is noncollapsing and uniformly two-convex. Then for
each $T<\infty$, $\sup_{M_{t}}G$ is bounded from above independently
$t\in[-T,0]$.

###### Proof.

Let $B$ be an open ball in $\Omega_{0}$. Then $B$ is contained in $\Omega_{t}$
for all $t\leq 0$. Suppose there is a sequence of times $t_{k}\in[-T,0]$ and
points $x_{k}\in M_{t_{k}}$ such that $G(x_{k},t_{k})\to\infty$. Then
$|x_{k}|\to\infty$, so for any $\varepsilon>0$ we may apply Lemma 2.7 to
conclude that $(x_{k},t_{k})$ lies at the center of an $\varepsilon$-neck for
all large $k$. Since $G(x_{k},t_{k})\to\infty$, the radius of the neck must
tend to zero as $k\to\infty$. On the other hand, by convexity and
noncompactness, $\Omega_{t_{k}}$ contains a ray emanating from the center of
$B$. As we slide $B$ along this ray it must remain inside $\Omega_{t_{k}}$ and
eventually pass through the neck containing $x_{k}$. This is impossible if the
neck is too thin, so we have reached a contradiction. ∎

###  2.4. Differential Harnack inequality

Every strictly convex ancient $G$-flow which is noncollapsing and uniformly
two-convex satisfies the following differential Harnack inequality. In
addition, if equality is ever attained then the $G$-flow is a translating
soliton.

######  Proposition 2.9.

Let $M_{t}$ be a strictly convex ancient $G$-flow which is uniformly two-
convex and noncollapsing. We then have

(5) $\partial_{t}G+2\langle\nabla G,V\rangle+A(V,V)\geq 0$

at every point $x\in M_{t}$ and for every $V\in T_{x}M_{t}$. Equivalently,

(6) $\partial_{t}G-A^{-1}(\nabla G,\nabla G)\geq 0$

for every $x\in M_{t}$. The last inequality is strict at every point in
spacetime unless $M_{t}$ is a translating soliton.

###### Proof.

We first observe that (5) is equivalent to (6); indeed, we have

$2\langle\nabla G,V\rangle+A(V,V)\geq-A^{-1}(\nabla G,\nabla G)$

with equality exactly when $V=-A^{-1}(\nabla G)$. When $M_{t}$ is compact, (6)
was proven in [And94b]. When $M_{t}$ is noncompact we have bounded curvature
on bounded time intervals by Lemma 2.8. Therefore, Theorem 1.2 in [Lyn] gives
the inequality (5). If equality is attained in (6) then $M_{t}$ is a
translating soliton by [Lyn, Corollary 1.3]. ∎

As a consequence of the Harnack inequality we obtain a curvature bound which
is valid for all times.

######  Lemma 2.10.

Let $M_{t}$ be a strictly convex ancient $G$-flow which is uniformly two-
convex and noncollapsing. Then $\sup_{M_{t}}G$ is bounded from above
independently of $t$.

###### Proof.

If $\lambda_{1}$ vanishes somewhere then $M_{t}$ is a shrinking cylinder by
Lemma 2.5, in which case the claim is immediate, so assume $A>0$.

We have $\sup_{M_{0}}G<\infty$. This is immediate if $M_{0}$ is compact, and
follows from Lemma 2.8 if $M_{t}$ is noncompact. On the other hand, the
Harnack inequality says that $G$ is nondecreasing at each point in spacetime.
With this the claim is proven. ∎

###  2.5. Expansions for hypersurfaces close to a cylinder

Let

$\Sigma=\mathbb{R}\times S^{n-1}(r)$

and consider a smooth function $u:\Sigma\to\mathbb{R}$. We consider a
hypersurface of the form

$M=\operatorname{graph}u=\\{x+u(x)\nu_{\Sigma}(x):x\in\Sigma\\},$

where $\nu_{\Sigma}$ is the outward unit normal to $\Sigma$. We assume
throughout that the quantity

$\sup_{\Sigma}\,(r^{-1}|u|+|\nabla^{\Sigma}u|+r|\nabla^{2}_{\Sigma}u|)$

is small. Our goal is to compute expansions for certain geometric quantities
on $M$ in terms of their counterparts on $\Sigma$. We write $a=b+O(c)$ when
$|a-b|\leq K|c|$ for a positive constant $K$ depending only on $n$ and
$\gamma$.

Let $g$ and $A$ denote the induced metric and second fundamental form of $M$.
Let $g_{\Sigma}$ and $h$ denote the induced metric and second fundamental form
of $\Sigma$. We will not distinguish between tensors on $M$ and their
pullbacks to $\Sigma$ via the map $x\mapsto x+u(x)\nu_{\Sigma}(x)$.

######  Lemma 2.11.

We have the expansions

$\displaystyle g$
$\displaystyle=g_{\Sigma}+2uh+O(r^{-2}|u|^{2}+|\nabla^{\Sigma}u|^{2})$
$\displaystyle\nu$
$\displaystyle=\nu_{\Sigma}-\nabla^{\Sigma}u+O(r^{-2}|u|^{2}+|\nabla^{\Sigma}u|^{2})$
$\displaystyle A$
$\displaystyle=h-\nabla^{2}_{\Sigma}u+uh^{2}+O(r^{-3}|u|^{2}+r^{-1}|\nabla^{\Sigma}u|^{2}).$

The proof of Lemma 2.11 is straightforward, so we omit it.

Let $\\{e_{i}\\}$ be an orthonormal frame of eigenvectors for $h$. Such a
frame will typically fail to be orthonormal with respect to $g$. In the
following lemma we smoothly transform $\\{e_{i}\\}$ to a frame which does have
this property.

######  Lemma 2.12.

There is a smooth field of matrices $\alpha_{i}^{k}$ on $\Sigma$ such that the
vectors $\alpha_{i}^{k}e_{k}$ satisfy

$g(\alpha_{i}^{k}e_{k},\alpha_{j}^{l}e_{l})=\delta_{ij}$

and

$\alpha_{i}^{k}=\delta_{i}^{k}-uh_{i}^{k}+O(r^{-2}|u|^{2}+|\nabla^{\Sigma}u|^{2}).$

###### Proof.

For $s\in[0,1]$ we define

$g_{ij}(s):=\delta_{ij}+2uh_{ij}+u^{2}h^{2}_{ij}+s\nabla^{\Sigma}_{i}u\nabla^{\Sigma}_{j}u.$

We then define $\alpha_{i}^{k}(s)$ to be the solution of the initial value
problem

$g_{jk}(s)\frac{d}{ds}\alpha_{i}^{k}(s)=-\frac{1}{2}\alpha_{i}^{m}(s)\nabla^{\Sigma}_{m}u\nabla^{\Sigma}_{j}u,\qquad(\delta_{k}^{i}+uh_{k}^{i})\alpha_{j}^{k}(0)=\delta^{i}_{j}$

This ensures that

$\frac{d}{ds}(\alpha_{i}^{k}\alpha_{j}^{l}g_{kl})=0\qquad\text{and}\qquad\alpha_{i}^{k}(0)\alpha_{j}^{l}(0)g_{kl}(0)=\delta_{ij},$

and hence $\alpha_{i}^{k}(1)$ a smooth field of matrices on $\Sigma$
satisfying $\alpha_{i}^{k}(1)\alpha_{j}^{l}(1)g_{kl}=\delta_{ij}$.
Straightforward computations show that

$\alpha_{i}^{k}(1)=\alpha_{i}^{k}(0)+O(|\nabla^{\Sigma}u|^{2})=\delta_{i}^{k}-uh_{i}^{k}+O(r^{-2}|u|^{2}+|\nabla^{\Sigma}u|^{2}).$

∎

Let $\lambda$ and $\lambda_{\Sigma}$ denote the principal curvatures of $M$
and $\Sigma$, respectively. We label these in increasing order. Recall that we
assume $\Gamma$ contains $\\{0\\}\times\mathbb{R}_{+}^{n}$. It follows that
$\lambda_{\Sigma}\in\Gamma$. We define
$G_{\Sigma}(x)=\gamma(\lambda_{\Sigma}(x))$. Suppose $\lambda$ lies in
$\Gamma$ and define $G(x)=\gamma(\lambda(x))$.

With respect to the frame $\\{e_{i}\\}$, the principal curvatures
$\lambda_{\Sigma}$ are the eigenvalues of the matrix with entries
$h(e_{i},e_{j})$. Therefore, we may write $G_{\Sigma}=\gamma(h)$.

######  Lemma 2.13.

We have

$G={G_{\Sigma}}-\dot{\gamma}^{ij}(h)\nabla^{\Sigma}_{i}\nabla^{\Sigma}_{j}u-\dot{\gamma}^{ij}(h)h^{2}_{ij}u+O(r^{-3}|u|^{2}+r^{-1}|\nabla^{\Sigma}u|^{2}+r|\nabla^{2}_{\Sigma}u|^{2}).$

This may also be written as

$\displaystyle G$
$\displaystyle=G_{\Sigma}-\dot{\gamma}^{1}(0,1,\dots,1)\frac{\partial^{2}u}{\partial
z^{2}}-\frac{1}{n-1}\frac{\gamma(0,1,\dots,1)}{r^{2}}r^{2}\Delta_{S^{n-1}_{r}}u-\frac{\gamma(0,1,\dots,1)}{r^{2}}u$
$\displaystyle+O(r^{-3}|u|^{2}+r^{-1}|\nabla^{\Sigma}u|^{2}+r|\nabla^{2}_{\Sigma}u|^{2}),$

where $z$ denotes a Euclidean coordinate on the $x_{1}$-axis.

###### Proof.

Let $\alpha_{i}^{j}$ denote the smooth field of matrices constructed in Lemma
2.12, so that the vectors $\alpha_{i}^{j}e_{j}$ are orthonormal with respect
to $g$. With respect to the frame $\alpha_{i}^{j}e_{j}$, $A$ is represented by
the matrix $B_{ij}:=\alpha_{i}^{k}\alpha_{j}^{l}A(e_{k},e_{l})$. The
eigenvalues of $B$ are equal to $\lambda$, so we may write $G=\gamma(B)$.

Let us define

$\varepsilon_{ij}=B_{ij}-h_{ij}+\nabla^{\Sigma}_{i}\nabla^{\Sigma}_{j}u+h^{2}_{ij}u,$

and observe that $\varepsilon_{ij}=\varepsilon_{ji}$. It follows that the
function

$f(t):=\gamma(h_{ij}+t(-D_{i}D_{j}u-h^{2}_{ij}u+\varepsilon_{ij}))$

is well-defined and smooth in $t$. Moreover,

$f^{\prime}(1)-f^{\prime}(0)=G-G_{\Sigma}-\dot{\gamma}^{ij}(h)(-D_{i}D_{j}u-h_{ij}^{2}u+\varepsilon_{ij}),$

so we have

$|G-G_{\Sigma}-\dot{\gamma}^{ij}(h)(-\nabla^{\Sigma}_{i}\nabla^{\Sigma}_{j}u-h^{2}_{ij}u+\varepsilon_{ij})|\leq\sup_{t\in[0,1]}|f^{\prime\prime}(t)|.$

Using the fact that $\gamma$ is 1-homogeneous we obtain

$\displaystyle|G-G_{\Sigma}-\dot{\gamma}^{ij}(h)(-\nabla^{\Sigma}_{i}$
$\displaystyle\nabla^{\Sigma}_{j}u-h^{2}_{ij}u+\varepsilon_{ij})|$
$\displaystyle=O(r^{-3}|u|^{2}+r^{-1}|\nabla^{\Sigma}u|^{2}+r|\nabla^{2}_{\Sigma}u|^{2}+r|\varepsilon|^{2}).$

Now we use Lemma 2.11 and Lemma 2.12 to deduce that

$\displaystyle|\varepsilon|$
$\displaystyle=O(r^{-3}|u|^{2}+r^{-1}|\nabla^{\Sigma}u|^{2}+r|\nabla^{2}_{\Sigma}u|^{2}).$

This proves the first part of the claim.

Next we use the fact that $h$ is diagonal with respect to $\\{e_{i}\\}$ in
order to compute

$\displaystyle\dot{\gamma}^{ij}(h)\nabla^{\Sigma}_{i}\nabla^{\Sigma}_{j}u$
$\displaystyle=\dot{\gamma}^{1}(0,r^{-1},\dots,r^{-1})\frac{\partial^{2}u}{\partial
z^{2}}$
$\displaystyle+\frac{1}{n-1}\sum_{i=2}^{n}\dot{\gamma}^{i}(0,r^{-1},\dots,r^{-1})\Delta_{S^{n-1}_{r}}u+O(|\nabla^{\Sigma}u|^{2}).$

Here we have used symmetry to express

$\dot{\gamma}^{j}(0,r^{-1},\dots,r^{-1})=\frac{1}{n-1}\sum_{i=2}^{n}\dot{\gamma}^{i}(0,r^{-1},\dots,r^{-1})$

for $j\geq 2$. Since $\gamma$ is homogeneous of degree one we have

$\dot{\gamma}^{i}(0,r^{-1},\dots,r^{-1})=\dot{\gamma}^{i}(0,1,\dots,1),$

and hence

$\sum_{i=2}^{n}\dot{\gamma}^{i}(0,r^{-1},\dots,r^{-1})=\gamma(0,1,\dots,1).$

The same reasoning yields

$\dot{\gamma}^{ij}(h)h^{2}_{ij}u=r^{-2}\sum_{i=2}^{n}\dot{\gamma}^{i}(0,r^{-1},\dots,r^{-1})u=r^{-2}\gamma(0,1,\dots,1)u.$

The claim follows. ∎

Let $S$ be a symmetric $(0,2)$-tensor on $M$. After fixing a $g$-orthonormal
frame, so that $A$ and $B$ can be identified with symmetric matrices, we
define

$\operatorname{tr}_{\gamma}(S):=\frac{d}{dt}\bigg{|}_{t=0}\gamma(A+tS).$

The right-hand side is independent of the frame we chose, so
$\operatorname{tr}_{\gamma}(S)$ is a well defined smooth function on $M$.

######  Lemma 2.14.

Given a symmetric $(0,2)$-tensor $S$ on $M$, we have

$\operatorname{tr}_{\gamma}(S)=\dot{\gamma}^{ij}(h)S_{ij}+O(r^{-1}|u|+|\nabla^{\Sigma}u|+r|\nabla^{2}_{\Sigma}u|)|S|.$

In particular, when $S$ is the Hessian of a function $f$ we have

$\displaystyle\operatorname{tr}_{\gamma}(\nabla^{2}f)$
$\displaystyle=\dot{\gamma}^{1}(0,1,\dots,1)\frac{\partial^{2}u}{\partial
z^{2}}+\frac{1}{n-1}\frac{\gamma(0,1,\dots,1)}{r^{2}}r^{2}\Delta_{S^{n-1}_{r}}u$
$\displaystyle+O(r^{-1}|u|+|\nabla^{\Sigma}u|+r|\nabla^{2}_{\Sigma}u|)|\nabla^{2}f|+O(|\nabla^{\Sigma}u|)|\nabla
f|$

where $z$ denotes a Euclidean coordinate on the $x_{1}$-axis.

###### Proof.

As in the proof of Lemma 2.13, let us write $B$ for the symmetric matrix with
entries $B_{ij}=\alpha_{i}^{k}\alpha_{j}^{l}A(e_{k},e_{l})$. We have

$\operatorname{tr}_{\gamma}(S)=\frac{d}{dt}\bigg{|}_{t=0}\gamma(\alpha_{i}^{k}\alpha_{j}^{l}(A_{kl}+tS_{kl}))=\dot{\gamma}^{ij}(B)\alpha_{i}^{k}\alpha_{j}^{l}S_{kl}.$

Once again, we define an auxiliary function

$f(t):=\dot{\gamma}^{ij}(h+t(-\nabla^{2}_{\Sigma}u-h^{2}u+\varepsilon))\tilde{S}_{ij},$

where

$\varepsilon_{ij}:=B_{ij}-h_{ij}+\nabla^{\Sigma}_{i}\nabla^{\Sigma}_{j}u+h^{2}_{ij}u,$

and $\tilde{S}_{ij}:=\alpha_{i}^{k}\alpha_{j}^{l}S_{kl}$. Then since

$|\operatorname{tr}_{\gamma}(S)-\dot{\gamma}^{ij}(h)\tilde{S}_{ij}|\leq\sup_{t\in[0,1]}|f^{\prime}(t)|$

and

$f^{\prime}(t)=\gamma^{ij,kl}(h+t(-\nabla^{2}_{\Sigma}u-(h^{2}u+\varepsilon))\tilde{S}_{ij}(-\nabla^{\Sigma}_{k}\nabla^{\Sigma}_{l}u-h^{2}_{kl}u+\varepsilon_{kl}),$

we deduce that

$|\operatorname{tr}_{\gamma}(S)-\dot{\gamma}^{ij}(h)\tilde{S}_{ij}|\leq
O(r^{-1}|u|+|Du|+r|D^{2}u|)|\tilde{S}|.$

The first part of the claim now follows from Lemma 2.12. The second part
follows as in the proof of Lemma 2.13. ∎

###  2.6. Asymptotic cones

Let $\Omega$ be an open convex subset of $\mathbb{R}^{n+1}$. We define the
asymptotic cone $C(\Omega)$ of $\Omega$ to be the Hausdorff-limit of the
sequence $k^{-1}\bar{\Omega}$. When $\Omega$ is bounded, $C(\Omega)=\\{0\\}$.
When $\Omega$ is unbounded, $C(\Omega)$ is a closed cone. If we translate
$\Omega$ so that it contains the origin, then $C(\Omega)$ is precisely the
union of all the rays emanating from the origin which are contained in
$\bar{\Omega}$.

######  Lemma 2.15.

Let $M_{t}=\partial\Omega_{t}$, $t\in(-\infty,0]$, be a noncompact, convex
ancient $G$-flow. The asymptotic cone $C(\Omega_{t})$ is independent of $t$.

###### Proof.

Since the hypersurfaces $M_{t}$ evolve inwards, we have
$\Omega_{t}\subset\Omega_{s}$ whenever $s<t$. It follows that
$C(\Omega_{t})\subset C(\Omega_{s})$. In particular, $C(\Omega_{0})\subset
C(\Omega_{t})$ for $t\leq 0$.

Now consider an arbitrary time $\bar{t}<0$, and an arbitrary ray $R$ contained
in $C(\Omega_{\bar{t}})$. We may assume without loss of generality that
$0\in\Omega_{0}$. We then have
$C(\Omega_{\bar{t}})\subset\bar{\Omega}_{\bar{t}}$, and hence
$R\subset\Omega_{\bar{t}}$. Let $\tilde{t}$ denote the supremum of all the
times $t$ for which $R\subset\Omega_{t}$. Let $B$ be an open ball with center
at $0$ such that $B\subset\Omega_{0}$. Sliding $B$ along $R$ and appealing to
the avoidance principle, we see that unless $\tilde{t}=0$, there is a short
time period following $\tilde{t}$ in which $R\subset\Omega_{t}$. So
$\tilde{t}=0$, and hence $R\subset\bar{\Omega}_{0}$. It follows that $R\subset
C(\Omega_{0})$. Since $R$ was arbitrary, we conclude that
$C(\Omega_{\bar{t}})\subset\Omega_{0}$.

We have shown that

$C(\Omega_{t})\subset C(\Omega_{0})\subset C(\Omega_{t})$

for every $t\leq 0$. This proves the claim. ∎

##  3\. The parabolic blow-down limit

Let $M_{t}=\partial\Omega_{t}$, $t\in(-\infty,0]$, be a convex ancient
$G$-flow which is noncollapsing and uniformly two-convex. In this section we
study the possible parabolic blow-down limits of $M_{t}$. We show that the
only possibilities are self-similarly shrinking sphere and cylinder solutions.

We first prove that the region $\Omega_{t}$ contains shrinking ball with
radius on the order of $(-t)^{1/2}$ (cf. [Wan11, Theorem 2.2] and [BLL21,
Proposition 4.3]). Without loss of generality we may assume $0\in M_{0}$.

######  Lemma 3.1.

There is a constant $\eta>0$ such that $B(0,\eta(-t)^{1/2})\subset\Omega_{t}$
whenever $-t$ is sufficiently large.

###### Proof.

We proceed by contradiction. Supposing the claim is false, we can find a
sequence of times $t_{k}\to-\infty$ such that the rescaled flows
$\tilde{M}_{t}^{k}:=(-t_{k})^{-1/2}M_{(-t_{k})t}$ satisfy

$\operatorname{dist}(0,\tilde{M}_{-1}^{k})\to 0.$

The function $f_{k}(t):=\operatorname{dist}(0,\tilde{M}_{t}^{k})$ is locally
Lipschitz, and for almost every $t$ we have

$f_{k}^{\prime}(t)\leq-\sup\\{\tilde{G}_{k}(x,t):|x|=f_{k}(t)\\}.$

Therefore, since

$-\int^{0}_{-1}f_{k}^{\prime}(s)\,ds=\operatorname{dist}(0,\tilde{M}^{k}_{-1})\to
0,$

there are sequences $s_{k}\in[-1,0]$ and $x_{k}\in\tilde{M}_{s_{k}}^{k}$ such
that $|x_{k}|=f_{k}(s_{k})$ and $\tilde{G}_{k}(x_{k},s_{k})\to 0$. By the
noncollapsing property, $\tilde{M}^{k}_{s_{k}}$ admits an inscribed ball at
$x_{k}$ whose radius becomes arbitrarily large as $k\to\infty$. Using the
avoidance principle we conclude that for some sequence of points satisfying
$|p_{k}|\leq 10$, the ball $B(p_{k},1)$ is contained in the region bounded by
$\tilde{M}_{t}^{k}$ for all $t\leq 0$. This contradicts Lemma 2.2, since

$\tilde{G}_{k}(0,0)=(-t_{k})^{1/2}G(0,0)\to\infty.$

∎

Let us fix an arbitrary sequence $a_{k}\to 0$. We show that the rescaled flows
$M^{k}_{t}:=a_{k}M_{a_{k}^{-2}t}$ converge smoothly along a subequence.

######  Lemma 3.2.

Possibly after passing to a subsequence, the rescaled flows $M^{k}_{t}$
converge in
$C^{\infty}_{\operatorname{loc}}(\mathbb{R}^{n+1}\times(-\infty,0))$ to a
limiting flow $\hat{M}_{t}=\partial\hat{\Omega}_{t}$.

###### Proof.

As a consequence of Lemma 3.1, we have
$B(0,\eta(-t)^{1/2})\subset\Omega_{t}^{k}$ when $t<s$ and $k$ is sufficiently
large depending on $s$.

Using this fact we will show that $\inf_{B(0,R)}G_{k}(\cdot,s)>0$ for every
$s<0$ and $R<\infty$. If this is not the case then, by Proposition 2.3,
$G_{k}\to 0$ uniformly in every compact subset of
$\mathbb{R}^{n+1}\times(-\infty,s]$. Using the noncollapsing assumption we see
that $\Omega_{s}^{k}$ converges to a halfspace in the Hausdorff topology. But
since $G_{k}\to 0$ locally uniformly in $\mathbb{R}^{n+1}\times(-\infty,s]$,
$\Omega^{k}_{t}$ converges to the same halfspace for every $t\leq s$. This
contradicts $B(0,\eta(-t)^{1/2})\subset\Omega^{k}_{t}$.

To complete the argument we use the lower bound for $G^{k}$ and Proposition
2.4 to establish subconvergence in $\mathbb{R}^{n+1}\times(-\infty,s]$ for
each $s>0$. The claim follows by sending $s\to 0$ and passing to a diagonal
subsequence. ∎

We have the following characterisation of $\hat{M}_{t}$ in case it is compact.
(We do not make use of this result, it is only included for the sake of
completeness.)

######  Lemma 3.3.

Suppose $\hat{M}_{t}$ is compact. Then $M_{t}$ is a self-similarly shrinking
sphere. In particular, $\hat{M}_{t}$ is a self-similarly shrinking sphere.

###### Proof.

Since $\hat{M}_{t}$ is compact, the splitting theorem [Lyn22b, Proposition
A.2] implies $\hat{\lambda}_{1}>0$ on $\hat{M}_{t}$ for every
$t\in(-\infty,0)$. In particular, there is a positive constant $\varepsilon$
such that $\hat{\lambda}_{1}\geq\varepsilon\hat{H}$. Therefore, for
sufficiently large $k$, $\lambda_{1}\geq\tfrac{\varepsilon}{2}H$ on the
hypersurface $M_{-a_{k}^{-2}}$. If $\gamma$ is concave and inverse-concave,
this inequality is preserved by the flow [And07], and hence
$\lambda_{1}\geq\tfrac{\varepsilon}{2}H$ on $M_{t}$ for all $t\leq 0$. Using
[LL20, Theorem 1.7] we conclude that $M_{t}$ is a shrinking sphere solution.

We proceed similarly if $\gamma$ is convex rather than concave, but work with
$\lambda_{1}/G$ in place of $\lambda_{1}/H$. A maximum principle argument
shows that positive lower bounds for this ratio are preserved by the flow. ∎

Suppose now that $\hat{M}_{t}$ is noncompact. Proposition 4.5 in [Lyn22b]
implies the following lemma.

######  Lemma 3.4.

If $\hat{M}_{t}$ is such that

$\hat{M}_{t}\subset\\{x\in\mathbb{R}^{n+1}:x_{1}^{2}+\dots+x_{n}^{2}\leq-2\gamma(0,1,\dots,1)t\\}$

for every $t<0$, then

$\hat{M}_{t}=\\{x\in\mathbb{R}^{n+1}:x_{1}^{2}+\dots+x_{n}^{2}=-2\gamma(0,1,\dots,1)t\\}$

for $t<0$.

Using this lemma, we show that $\hat{M}_{t}$ is a self-similarly shrinking
cylinder.

######  Lemma 3.5.

Suppose $\hat{M}_{t}$ is noncompact. Up to a choice of Euclidean coordinates,
we then have

$\hat{M}_{t}=\\{x\in\mathbb{R}^{n+1}:x_{1}^{2}+\dots+x_{n}^{2}=-2\gamma(0,1,\dots,1)t\\}$

for $t<0$.

###### Proof.

Since $\hat{M}_{t}$ is noncompact, the asymptotic cone $C(\hat{\Omega}_{t})$
is noncompact. Moreover $C(\hat{\Omega}_{t})$ is equal to
$C:=C(\hat{\Omega}_{-1})$ for every $t<0$ by Lemma 2.15. Lemma 2.7 implies
that the dimension of $C$ is one. That is, $C$ is either a ray or a line. If
$C$ is a line then $\hat{M}_{t}$ splits off a line, and the claim follows from
Lemma 2.5. So suppose $C$ is a ray.

Consider the set $\hat{M}_{0}:=\cap_{t<0}\hat{\Omega}_{t}$. This set is closed
and convex. A straightward argument using Lemma 2.2 and the avoidance
principle shows that

$\sup_{\hat{M}_{t}\cap B(0,1)}\hat{G}\to\infty$

as $t\to 0$. Therefore, using Lemma 2.7, we conclude that the dimension of
$\hat{M}_{0}$ is at most one. On the other hand $C\subset\hat{\Omega}_{t}$ for
$t<0$. It follows that $C\subset\hat{M}_{0}$. From this we conclude that the
dimension of $\hat{M}_{0}$ is equal to one, and that every point in $C$ is
reached by $\hat{M}_{t}$ as $t\to 0$

Let $x_{j}\in C$ be a sequence of points going to $\infty$, and consider the
shifted flows $\hat{M}_{t}-x_{j}$. We know that $\hat{M}_{t}-x_{j}$ reaches
the origin as $t\to 0$. Therefore, by the avoidance principle, for each $t<0$
there is a point in $\hat{M}_{t}-x_{j}$ whose distance to the origin is at
most $\sqrt{-2\gamma(1,\dots,1)t}$. Moreover, the ball $B(0,\eta(-t)^{1/2})$
is contained in $\hat{\Omega}_{t}$, and hence in $\hat{\Omega}_{t}-x_{j}$, for
every $t<0$. This is due to Lemma 3.1.

Using these facts, we can appeal to Proposition 2.4 to extract a smooth limit
of the shifted flows in
$C^{\infty}_{\operatorname{loc}}(\mathbb{R}^{n+1}\times(-\infty,0))$. Denote
the limiting flow by $\tilde{M}_{t}$. By construction, $\tilde{M}_{t}$
encloses the line through the origin parallel to $C$. Therefore,
$\tilde{M}_{t}$ splits off a line and, up to a choice of Euclidean
coordinates, Lemma 2.5 implies

$\tilde{M}_{t}=\\{x\in\mathbb{R}^{n+1}:x_{1}^{2}+\dots+x_{n}^{2}=-2\gamma(0,1,\dots,1)t\\}.$

But $\tilde{M}_{t}$ also encloses $\hat{M}_{t}$, so using Lemma 3.4 we
conclude that

$\hat{M}_{t}=\\{x\in\mathbb{R}^{n+1}:x_{1}^{2}+\dots+x_{n}^{2}=-2\gamma(0,1,\dots,1)t\\}.$

This completes the proof. ∎

With these preparations out of the way, we can now prove the main result of
this section.

We assume that the hypersurfaces $M_{t}$ are noncompact. The asympotic cones
$C(\Omega_{t})$ are constant in time and equal to $C:=C(\Omega_{0})$ by Lemma
2.15. Lemma 2.7 implies that $C$ is either a ray or a line. Up to a rotation
of coordinates, we may assume that $C$ lies in the $x_{n+1}$-axis.

Let $\bar{M}_{\tau}$ denote the rescaled family of hypersurfaces
$e^{\tau/2}M_{-e^{-\tau}}$. These move with velocity $-(G-\frac{1}{2}\langle
x,\nu\rangle)\nu$.

######  Proposition 3.6.

As $\tau\to-\infty$ the rescaled hypersurfaces $\bar{M}_{\tau}$ converge in
$C^{\infty}_{\operatorname{loc}}$ to the cylinder

$\\{x\in\mathbb{R}^{n+1}:x_{1}^{2}+\dots+x_{n}^{2}=2\gamma(0,1,\dots,1)\\}.$

###### Proof.

It suffices to show that, for every sequence $\tau_{k}\to\infty$,
$\bar{M}_{\tau_{k}}$ admits a subsequence which converges to the cylinder

$\\{x\in\mathbb{R}^{n+1}:x_{1}^{2}+\dots+x_{n}^{2}=2\gamma(0,1,\dots,1)\\}.$

Lemma 3.5 guarantees subconvergence to a cylinder of radius
$\sqrt{2\gamma(0,1,\dots,1)}$ with axis through the origin. But since
$C\subset\Omega_{t}$ for $t\leq 0$, the axis of every such limiting cylinder
is the $x_{n+1}$-axis. The claim follows. ∎

##  4\. A barrier construction

In this section we prove the following statement.

######  Proposition 4.1.

There is a positive constant $L_{0}=L_{0}(n,\gamma)$ such that, for each
sufficiently large $a>0$, there exists a function
$\Psi_{a}:[L_{0},a]\to\mathbb{R}$ with the following properties:

  * (i)

$\Psi_{a}(z)\geq 0$ with equality at $z=a$ and
$\Psi_{a}(z)^{2}<2\gamma(0,1,\dots,1)$ for $z\in[L_{0},a]$.

  * (ii)

$\Psi_{a}$ is strictly decreasing and strictly concave.

  * (iii)

The set of points

$\\{(z,\Psi_{a}(z)\theta):z\in(L_{0},a],\;\theta\in S^{n-1}\\}$

is a smooth hypersurface which solves the $\gamma$-shrinker equation,

(7) $G=\tfrac{1}{2}\langle x,\nu\rangle.$

  * (iv)

For each $z\in[L_{0},a]$ we have

$\Psi_{a}(z)^{2}\geq
2\gamma(0,1,\dots,1)\bigg{(}1-\frac{z^{2}}{a^{2}}\bigg{)}.$

  * (v)

There is a positive constant $C=C(n,\gamma)$ with the following property. For
any given $L>L_{0}$, the inequality

$\Psi_{a}(z)^{2}\leq 2\gamma(0,1,\dots,1)\bigg{(}1-\bigg{(}1-C\frac{\log
a}{a^{2}}\bigg{)}\frac{z^{2}-C}{a^{2}}\bigg{)}.$

holds for $z\in[L_{0},L]$, provided that $a$ is sufficiently large depending
on $L$.

Consider a strictly concave smooth function $h:(0,R)\to\mathbb{R}$. The
hypersurface

$\\{(h(r),r\theta):r\in(0,R),\;\theta\in S^{n-1}\\}$

solves (7) if and only if $h$ solves the ODE

$\gamma\bigg{(}-\frac{h_{rr}}{1+h_{r}^{2}},-\frac{h_{r}}{r},\dots,-\frac{h_{r}}{r}\bigg{)}=\frac{1}{2}(h-rh_{r}).$

For a given $a>0$, consider the function $\psi:(0,Ra)\to\mathbb{R}$ defined by

$h(r)=a-\frac{1}{a}\psi(ar),$

and set $\rho=ar$. Then $h$ solves (4) if and only if $\psi$ solves

(8)
$\gamma\bigg{(}\frac{\psi_{\rho\rho}}{1+\psi_{\rho}^{2}},\frac{\psi_{\rho}}{\rho},\dots,\frac{\psi_{\rho}}{\rho}\bigg{)}=\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi).$

As a first step towards proving Proposition 4.1, we will construct solutions
of (8) which satisfy the initial condition $(\psi,\psi_{\rho})\to 0$ as
$\rho\to 0$. Some of our arguments take inspiration from [Ren21], where (8)
was studied in the case $a=\infty$ (which corresponds to translating, rather
than shrinking, $G$-flows).

###  4.1. Local existence

It will be convenient to rewrite (8). To this end we consider the function
$F:\mathbb{R}^{2}_{+}\to\mathbb{R}_{+}$ given by

$\displaystyle F(x,y):=\gamma(x,y,\dots,y).$

Since $\gamma$ is increasing in each argument we have $\frac{\partial
F}{\partial x}>0$, and consequently the set

$\\{F(x,y)-z=0\,:\,(x,y,z)\in\mathbb{R}_{+}^{3}\\}$

is a smooth hypersurface. Moreover, by the implicit function theorem there is
an open set $U\subset\mathbb{R}^{2}_{+}$ and a smooth function
$f:U\to\mathbb{R}$ such that $F(x,y)-z=0$ if and only if $(y,z)\in U$ and
$x=f(y,z)$. In particular, $F(f(y,z),y)=z$ for all $(y,z)\in U$. We have

$\frac{\partial f}{\partial y}(y,z)=-\frac{\frac{\partial F}{\partial
y}(f(y,z),y)}{\frac{\partial F}{\partial x}(f(y,z),y)},\qquad\frac{\partial
f}{\partial z}(y,z)=\frac{1}{\frac{\partial F}{\partial x}(f(y,z),y)},$

so $f$ is strictly decreasing in its first argument and strictly increasing in
its second. If $\psi$ is such that

$\bigg{(}\frac{\psi_{\rho}}{\rho},\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi)\bigg{)}\in
U,$

then $\psi$ solves (8) if and only if

(9)
$\psi_{\rho\rho}=(1+\psi_{\rho}^{2})f\bigg{(}\frac{\psi_{\rho}}{\rho},\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi)\bigg{)}.$

Let $Q\in(0,\infty]$ denote $\lim_{x\to\infty}F(x,1)$. For example, when
$\gamma(\lambda)=\lambda_{1}+\dots+\lambda_{n}$ we have $Q=\infty$, whereas
for the speed
$\gamma(\lambda)=(\sum_{i<j}(\lambda_{i}+\lambda_{j})^{-1})^{-1}$ we have
$Q=\frac{1}{4(n-1)(n-2)}$. Observe that $f(y,z)<\infty$ if and only if
$z/y=F(f(1,z/y),1)<Q$. Moreover, $f(y,z)>0$ if and only if
$z/y=F(f(1,z/y),1)>F(0,1)$. Therefore, $U$ is the open cone

$U=\\{(y,z)\in\mathbb{R}^{2}_{+}:F(0,1)<z/y<Q\\}.$

If $\psi$ is a $C^{2}$ solution of (8) such that $\psi_{\rho\rho}>0$ then we
can always rewrite (8) in the form (9).

The possibility that $Q<\infty$ is one reason why the construction we carry
out in this section is more subtle than its counterpart for the mean curvature
flow. Geometrically, if $\gamma$ is such that $Q<\infty$, a convex
hypersurface on which $G$ is bounded can have principal curvatures which are
arbitrarily large. This cannot happen if $\gamma$ is uniformly elliptic on the
hypersurface. The following lemma establishes, in particular, uniform
ellipticity of $\gamma$ on strictly convex solutions to (8).

Given a solution $\psi$ of (8), let us define

$\Lambda:=\frac{\rho\psi_{\rho\rho}}{\psi_{\rho}(1+\psi_{\rho}^{2})}.$

Thinking of $\psi$ is the profile of a surface of rotation, $\Lambda$ is equal
to the radial principal curvature divided by the principal curvature in the
direction of rotation.

######  Lemma 4.2.

Let $\psi$ be a solution of (8) on the interval $[\rho_{0},\rho_{1}]$ such
that $\psi_{\rho}>0$ and $\psi_{\rho\rho}>0$. Suppose
$\rho_{1}<\sqrt{2F(0,1)}a$. There is a constant $C$ depending only on $n$ and
$\gamma$ such that

$\Lambda(\rho)\leq\max\\{C,\Lambda(\rho_{0})\\}$

for each $\rho\in[\rho_{0},\rho_{1}]$.

###### Proof.

Let
$B(\rho):=\rho\psi_{\rho}^{-1}(\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi))$.
Since $\psi_{\rho\rho}>0$ we can write (8) in the form (9), which simply says
that

$\Lambda=f(1,B).$

We compute

$\displaystyle\rho
B_{\rho}=\bigg{(}\frac{\rho^{2}}{2a^{2}}-B\bigg{)}(1+\psi_{\rho}^{2})f(1,B)+B.$

Let us define $2\varepsilon_{0}:=1-\frac{F(0,1)}{Q}$. If
$\rho<\sqrt{2F(0,1)}a$ is such that

$B(\rho)>\max\\{\tfrac{F(0,1)}{1-\varepsilon_{0}},F(1/\varepsilon_{0},1)\\}$

then $f(1,B(\rho))>1/\varepsilon_{0}$, and hence

$\rho
B_{\rho}<(F(0,1)-(1-\varepsilon_{0})B-\varepsilon_{0}B)(1+\psi_{\rho}^{2})f(1,B)+B<0.$

It follows that

$\max_{\rho_{0}\leq\rho\leq\rho_{1}}B(\rho)\leq\max\\{B(\rho_{0}),\tfrac{F(0,1)}{1-\varepsilon_{0}},F(1/\varepsilon_{0},1)\\}.$

Since $\tfrac{F(0,1)}{1-\varepsilon_{0}}$ and $F(1/\varepsilon_{0},1)$ are
both less than $Q$, we may define a finite constant

$C:=\max\Big{\\{}f\left(1,\tfrac{F(0,1)}{1-\varepsilon_{0}}\right),f(1,F(1/\varepsilon_{0},1))\Big{\\}}.$

Since $f$ is increasing in its second argument,

$\max_{\rho_{0}\leq\rho\leq\rho_{1}}f(1,B(\rho))\leq\max\\{C,f(1,B(\rho_{0}))\\}.$

The claim follows. ∎

In addition to the upper bound in Lemma 4.2, we have the following lower bound
for $\Lambda$.

######  Lemma 4.3.

Let $\psi$ be a solution of (9) on the interval $[\rho_{0},\rho_{1}]$ such
that $\psi_{\rho}>0$ and $\psi_{\rho\rho}>0$. There is a constant $c$
depending only on $n$ and $\gamma$ such that

$\Lambda\geq\min\\{c(1+\psi_{\rho}(\rho_{1})^{2})^{-1},\Lambda(\rho_{0})\\},$

for each $\rho\in[\rho_{0},\rho_{1}]$.

###### Proof.

We have

$\rho
B_{\rho}=\bigg{(}\frac{\rho^{2}}{2a^{2}}-B\bigg{)}(1+\psi_{\rho}^{2})f(1,B)+B>(1-Mf(1,B))B,$

where $M:=(1+\psi_{\rho}(\rho_{1})^{2})$. This implies that $B_{\rho}>0$
whenever $B<F(M^{-1},1)$. Therefore,

$\min_{\rho_{0}\leq\rho\leq\rho_{1}}B\geq\min\\{B(\rho_{0}),F(M^{-1},1)\\}$

and hence

$\min_{\rho_{0}\leq\rho\leq\rho_{1}}f(1,B)\geq\min\\{f(1,B(\rho_{0})),f(1,F(M^{-1},1))\\}.$

There is a constant $c$ depending only on $n$ and $\gamma$ such that

$f(1,F(M^{-1},1))\geq cM^{-1},$

so the claim follows by insertion of $\Lambda=f(1,B)$. ∎

To construct solutions of (9) which satisfy $(\psi,\psi_{\rho})\to 0$ as
$\rho\to 0$, we will solve a sequence of initial value problems and take a
limit. To do so we will need appropriate barriers.

The following two lemmas establish that $w=\frac{\theta}{4F(1,1)}\rho^{2}$ is
a lower barrier for solutions of (9) when $\theta\in(\frac{F(1,1)}{Q},1)$.

######  Lemma 4.4.

Fix a constant $\frac{F(1,1)}{Q}<\theta<1$ and let
$w=\frac{\theta}{4F(1,1)}\rho^{2}$. We then have

$w_{\rho\rho}<(1+w_{\rho}^{2})f\bigg{(}\frac{w_{\rho}}{\rho},\frac{1}{2}\bigg{)}$

for all $\rho\geq 0$.

###### Proof.

Since

$F(0,1)\frac{w_{\rho}}{\rho}=\frac{F(0,1)}{2F(1,1)}<\frac{1}{2}<\frac{Q}{2F(1,1)}=Q\frac{w_{\rho}}{\rho},$

we have $(\frac{w_{\rho}}{\rho},\frac{1}{2})\in U$ for all $\rho\geq 0$.
Observe that

$F\bigg{(}f\Big{(}\frac{w_{\rho}}{\rho},\frac{1}{2}\Big{)},\frac{w_{\rho}}{\rho}\bigg{)}=\frac{1}{2}$

and

$F\bigg{(}\frac{w_{\rho\rho}}{1+w_{\rho}^{2}},\frac{w_{\rho}}{\rho}\bigg{)}=\frac{\theta}{2F(1,1)}F\bigg{(}\frac{1}{1+w_{\rho}^{2}},1\bigg{)}.$

Since $F$ is increasing in each of its arguments, we see that

$w_{\rho\rho}<(1+w_{\rho}^{2})f\bigg{(}\frac{w_{\rho}}{\rho},\frac{1}{2}\bigg{)}$

if and only if

$\frac{\theta}{2F(1,1)}F\bigg{(}\frac{1}{1+w_{\rho}^{2}},1\bigg{)}<\frac{1}{2},$

or equivalently

$F\bigg{(}\frac{1}{1+w_{\rho}^{2}},1\bigg{)}<\theta^{-1}F(1,1).$

The last inequality is always true when $\theta<1$. ∎

######  Lemma 4.5.

Let $\psi\in C^{2}([\rho_{0},\rho_{1}))$ be a solution of (9) satisfying
$\psi_{\rho\rho}>0$. Let $w=\tfrac{\theta}{4F(1,1)}\rho^{2}$ for some constant
$\theta\in(\frac{F(1,1)}{Q},1)$. If at $\rho=\rho_{0}$ we have

$\psi\geq w,\qquad\psi_{\rho}\geq w_{\rho},\qquad\rho\psi_{\rho}-\psi\geq 0,$

then $\psi_{\rho}\geq w_{\rho}$ and $\rho\psi_{\rho}-\psi\geq 0$ for
$\rho\in[\rho_{0},\rho_{1})$.

###### Proof.

Let
$s:=\sup\\{\rho\in(\rho_{0},\rho_{1}):\psi_{\rho}(\rho)>w_{\rho}(\rho)\\}$. At
$\rho=\rho_{0}$ we have

$w_{\rho\rho}<(1+w_{\rho}^{2})f\bigg{(}\frac{w_{\rho}}{\rho},\frac{1}{2}\bigg{)}\leq(1+\psi_{\rho}^{2})f\bigg{(}\frac{\psi_{\rho}}{\rho},\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi)\bigg{)}=\psi_{\rho\rho},$

so since $\psi_{\rho}(\rho_{0})\geq w_{\rho}(\rho_{0})$, we deduce that
$s>\rho_{0}$. With the aim of deriving a contradiction, suppose $s<\rho_{1}$.
We then have

$\psi_{\rho}(\rho)>w_{\rho}(\rho)$

for $\rho<s$, and

$\psi_{\rho}(s)=w_{\rho}(s),\qquad\psi_{\rho\rho}(s)\leq w_{\rho\rho}(s).$

In addition, since

$(\rho\psi_{\rho}-\psi)_{\rho}=\rho\psi_{\rho\rho}>0$

for $\rho\in[\rho_{0},s)$, and $\rho\psi_{\rho}-\psi\geq 0$ at
$\rho=\rho_{0}$, we have

$\rho\psi_{\rho}-\psi_{\rho}\geq 0$

for all $\rho\in[\rho_{0},\rho_{1})$. We conclude that, at $\rho=s$,

$\psi_{\rho\rho}=(1+\psi_{\rho}^{2})f\bigg{(}\frac{\psi_{\rho}}{\rho},\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi)\bigg{)}\geq(1+w_{\rho}^{2})f\bigg{(}\frac{w_{\rho}}{\rho},\frac{1}{2}\bigg{)}>w_{\rho\rho}.$

This is a contradiction, so $s=\rho_{1}$. ∎

Next we provide upper barriers for solutions of (9).

######  Lemma 4.6.

Let $\psi\in C^{2}([\rho_{0},\rho_{1}))$ be a solution of (9) satisfying
$\psi_{\rho\rho}>0$. Let $W=\tfrac{\Theta}{4F(1,1)}\rho^{2}$ for some constant
$\Theta>\frac{F(1,1)}{F(0,1)}$. If at $\rho=\rho_{0}$ we have

$\psi\leq W,\qquad\psi_{\rho}<W_{\rho},$

and

$\rho_{1}^{2}<2a^{2}(F(0,1)-F(1,1)\Theta^{-1}),$

then $\psi_{\rho}\leq W_{\rho}$ for $\rho\in[\rho_{0},\rho_{1})$.

###### Proof.

First observe that

$s:=\sup\\{\rho\in[\rho_{0},\rho_{1}):W_{\rho}(\rho)>\psi_{\rho}(\rho)\\}$

satisfies $s>\rho_{0}$. With the aim of deriving a contradiction, suppose that
$s<\rho_{1}$. We then have

$W_{\rho}(\rho)-\psi_{\rho}(\rho)>0$

for $\rho\in(\rho_{0},s)$, and

$W_{\rho}(s)=\psi_{\rho}(s).$

Since $\psi_{\rho\rho}>0$ we have
$\big{(}\frac{\psi_{\rho}}{\rho},\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi)\big{)}\in
U$ and, in particular,

$\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi)>F(0,1)\frac{\psi_{\rho}}{\rho}.$

At $\rho=s$ this yields

$\frac{1}{2}+\frac{1}{2a^{2}}(sW_{\rho}(s)-\psi(s))>F(0,1)\frac{W_{\rho}(s)}{s},$

which we rearrange to obtain

$s^{2}>2a^{2}(F(0,1)-F(1,1)\Theta^{-1}).$

This contradicts our assumption that
$\rho_{1}^{2}<2a^{2}(F(0,1)-F(1,1)\Theta^{-1})$. ∎

We now have all the tools needed to construct solutions of (9).

######  Proposition 4.7.

For each $a>0$ there is a strictly convex function

$\psi\in C^{\infty}((0,\sqrt{2F(0,1)}a))$

which solves (9) with the initial conditions $(\psi,\psi_{\rho})\to 0$ as
$\rho\to 0$. Moreover, there is a constant $C>0$ depending only on $n$ and
$\gamma$ such that $\psi_{\rho}\geq C\rho$ for every
$\rho\in(0,\sqrt{2F(0,1)}a)$, and $\psi_{\rho}\leq C\varepsilon^{-1}\rho$ for
every $\rho\in(0,\sqrt{(2-\varepsilon)F(0,1)}a)$.

###### Proof.

Fix a constant $\theta\in(\frac{F(1,1)}{Q},1)$ and set
$w=\frac{\theta}{4F(1,1)}\rho^{2}$. Let $\rho_{k}$ be a decreasing sequence of
positive numbers such that $\rho_{k}\to 0$ as $k\to\infty$. By the
Picard–Lindelöf theorem, for each $k$ there exists some $R_{k}>\rho_{k}$ such
that (9) admits a maximal strictly convex solution $\psi^{k}\in
C^{2}([\rho_{k},R_{k}))$ satisfying

$\psi^{k}(\rho_{k})=w(\rho_{k}),\qquad\psi^{k}_{\rho}(\rho_{k})=w_{\rho}(\rho_{k}).$

By Lemma 4.5 we have

$\psi^{k}\geq\frac{\theta}{4F(1,1)}\rho^{2},\qquad\psi^{k}_{\rho}\geq\frac{\theta}{2F(1,1)}\rho,\qquad\rho\psi^{k}_{\rho}-\psi^{k}\geq
0$

for every $\rho\in[\rho_{k},R_{k})$. Moreover, by Lemma 4.6, if
$\Theta>\frac{F(1,1)}{F(0,1)}$ then we have

$\psi^{k}\leq\frac{\Theta}{4F(1,1)}\rho^{2},\qquad\psi^{k}_{\rho}\leq\frac{\Theta}{2F(1,1)}\rho,$

for every $\rho\in[\rho_{k},R_{k})$ such that
$\rho^{2}<2a^{2}(F(0,1)-F(1,1)\Theta^{-1})$. Finally, since at $\rho=\rho_{k}$
we have

$\frac{\rho}{\psi^{k}_{\rho}}\bigg{(}\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi^{k}_{\rho}-\psi^{k})\bigg{)}=\frac{F(1,1)}{\theta}+\frac{\rho_{k}^{2}}{4a^{2}},$

for all sufficiently large $k$ we may appeal to Lemma 4.2 to obtain

$\psi^{k}_{\rho\rho}\leq
C(1+|\psi^{k}_{\rho}|^{2})\frac{\psi^{k}_{\rho}}{\rho}$

for every $\rho\in[\rho_{k},R_{k})$ such that $\rho<\sqrt{2F(0,1)}a$, where
$C$ depends only on $n$, $\gamma$ and $\theta$. Consequently, we have

$\psi^{k}_{\rho\rho}\leq C^{\prime}\Theta^{3}(1+\rho^{2})$

for every $\rho\in[\rho_{k},R_{k})$ such that
$\rho^{2}<2a^{2}(F(0,1)-F(1,1)\Theta^{-1})$, where $C^{\prime}$ depends only
on $n$, $\gamma$ and $\theta$.

Suppose $R_{k}<\sqrt{2F(0,1)}a$. By the estimates above, when $k$ is
sufficiently large we have

$\sup_{\rho_{k}\leq\rho<R_{k}}|\psi^{k}|+|\psi^{k}_{\rho}|+|\psi^{k}_{\rho\rho}|<\infty,$

and Lemma 4.3 gives

$\inf_{\rho_{k}\leq\rho<R_{k}}\psi_{\rho\rho}^{k}>0.$

Using (9), we conclude that

$\sup_{\rho_{k}\leq\rho<R_{k}}|\psi^{k}_{\rho\rho\rho}|<\infty.$

Therefore, the Picard-Lindelöf theorem can be used to extend $\psi^{k}$ beyond
$R_{k}$ so that it is still a strictly convex solution of (9). This
contradicts maximality, so we must have $R_{k}\geq\sqrt{2F(0,1)}a$.

We now take a limit of the solutions $\psi^{k}$ as $k\to\infty$. We have shown
that $\|\psi^{k}\|_{C^{2}(I)}$ is bounded independently of $k$ for every
compact subinterval $I\subset(0,\sqrt{2F(0,1)}a)$, and using (9) we can bound
$\|\psi^{k}\|_{C^{3}(I)}$ in terms of $\|\psi^{k}\|_{C^{2}(I)}$. Therefore, by
a standard diagonal argument, $\psi^{k}$ subconverges to a limit $\psi\in
C^{2}_{\operatorname{loc}}((0,\sqrt{2F(0,1)}a))$. The function $\psi$ is
strictly convex as a consequence of Lemma 4.3, and solves (9), so it is
smooth. Moreover, we have

$\psi\geq\frac{\theta}{4F(1,1)}\rho^{2},\qquad\psi_{\rho}\geq\frac{\theta}{2F(1,1)}\rho$

for $\rho\in(0,\sqrt{2F(0,1)}a)$, and

$\psi\leq\frac{\Theta}{4F(1,1)}\rho^{2},\qquad\psi_{\rho}\leq\frac{\Theta}{2F(1,1)}\rho$

for $\rho\in(0,\sqrt{2(F(0,1)-F(1,1)\Theta^{-1})}a)$. In particular,
$(\psi,\psi_{\rho})\to 0$ as $\rho\to 0$. ∎

Next we extend the solutions constructed in Proposition 4.7 to the origin.

######  Lemma 4.8.

For each $a>0$ the solution $\psi$ constructed in Proposition 4.7 can be
extended to a function in $C^{\infty}([0,\sqrt{2F(0,1)}a))$. Moreover,

$\psi_{\rho\rho}(0)=\frac{1}{2F(1,1)}.$

###### Proof.

Let $\beta(\rho)=\frac{\psi_{\rho}}{\rho}$. We have

$\displaystyle\rho\beta_{\rho\rho}=\rho\bigg{(}\frac{\psi_{\rho\rho\rho}}{\rho}-2\frac{\psi_{\rho\rho}}{\rho^{2}}+2\frac{\psi_{\rho}}{\rho^{3}}\bigg{)}>\psi_{\rho\rho\rho}$

and hence

$\displaystyle\rho\beta_{\rho\rho}$
$\displaystyle>2\psi_{\rho}\,\psi_{\rho\rho}\,\,f\left(\frac{\psi_{\rho}}{\rho},\,\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi)\right)$
$\displaystyle+(1+\psi_{\rho}^{2})\frac{\partial f}{\partial
y}\bigg{(}\frac{\psi_{\rho}}{\rho},\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi)\bigg{)}\beta_{\rho}$
$\displaystyle+\frac{\rho}{2a^{2}}(1+\psi_{\rho}^{2})\frac{\partial
f}{\partial
z}\bigg{(}\frac{\psi_{\rho}}{\rho},\frac{1}{2}+\frac{1}{2a^{2}}(\rho\psi_{\rho}-\psi)\bigg{)}\psi_{\rho\rho}.$

The right-hand side is positive at any point where $\beta_{\rho}<0$.
Therefore, $\beta_{\rho}$ is either negative for all sufficiently small $\rho$
or positive for all sufficiently small $\rho$. In particular, $\beta$
converges as $\rho\to 0$, and hence $\psi$ extends to a $C^{2}$ function on
the interval $[0,\sqrt{2F(0,1)}a)$. Using (8) we conclude that
$\psi_{\rho\rho}(0)=\frac{1}{2F(1,1)}$.

We thus have that the function

$u(x):=a-a^{-1}\psi(a|x|)$

is of class $C^{2}$ in $B^{n-1}_{\sqrt{2F(0,1)}}(0)$. Moreover, $u$ solves the
shrinker equation,

$\frac{1}{2}\,(x,u(x))\cdot\frac{(-Du,1)}{\sqrt{1+|Du|^{2}}}=\gamma\bigg{(}-\bigg{(}\delta^{ik}-\frac{D^{i}uD^{k}u}{1+|Du|^{2}}\bigg{)}\frac{D_{k}uD_{j}u}{\sqrt{1+|Du|^{2}}}\bigg{)}\bigg{)}.$

Standard theory for fully nonlinear elliptic PDE now implies that $u$ is of
class $C^{\infty}$ (see for example Lemma 17.16 in [GT01]). In particular,
$\psi$ extends to a function in $C^{\infty}([0,\sqrt{2F(0,1)}a))$.

∎

###  4.2. Global behaviour

For each $a>0$, let $\psi_{a}$ be a solution to (8) as in Proposition 4.7. We
will typically suppress the dependence on $a$ and simply write
$\psi=\psi_{a}$. We define

$h(r):=a-a^{-1}\psi(ar)$

and write $v(z)$ for the inverse of $h$, so that $r=v(h(r))$. Note that since
$\psi$ is defined for $\rho\in[0,\sqrt{2F(0,1)}a)$ we have $0\leq
v<\sqrt{2F(0,1)}$. For each $a>0$ the function $v$ is positive, strictly
decreasing, strictly concave, and satisfies

$\gamma\bigg{(}-\frac{v_{zz}}{1+v_{z}^{2}},\frac{1}{v},\dots,\frac{1}{v}\bigg{)}=\frac{v}{2}-\frac{z}{2}v_{z},\qquad
v(a)=0.$

We are interested in the asymptotic behaviour of $v$ as $a\to\infty$.
Following [ADS19], we prove upper and lower bounds for the quantity

$\displaystyle
w(z):=z\frac{d}{dz}\log(2F(0,1)-v^{2})=\frac{-2zvv_{z}}{2F(0,1)-v^{2}},$

and then integrate to obtain estimates for $v$.

######  Lemma 4.9.

We have $w(z)>2$.

###### Proof.

We first show that $w(z)\to 2\tfrac{F(1,1)}{F(0,1)}$ as $z\to a$. Observe that
when $z=h(\rho/a)$ we have

$\displaystyle
w(z)=-\frac{zv(z)v_{z}(z)}{F(0,1)-\frac{v(z)^{2}}{2}}=\frac{\left(1-\frac{\psi(\rho)}{a^{2}}\right)}{F(0,1)-\frac{\rho^{2}}{2a^{2}}}\frac{\rho}{\psi_{\rho}(\rho)}.$

Using Lemma 4.8, we conclude that

$\displaystyle\lim_{z\to a}w=\frac{1}{F(0,1)}\lim_{\rho\to
0}\frac{\rho}{\psi_{\rho}}=\frac{1}{F(0,1)}\frac{1}{\psi_{\rho\rho}(0)}=2\frac{F(1,1)}{F(0,1)}.$

Since $2\tfrac{F(1,1)}{F(0,1)}>2$, if the claim is false then there is a value
$z_{1}$ such that $w(z_{1})=2$ and $w_{z}(z_{1})\geq 0$. But we have

$zw_{z}=w-w^{2}\bigg{(}\frac{1}{2}+\frac{F(0,1)}{v^{2}}\bigg{)}-(1+v_{z}^{2})\frac{2z}{v_{z}(z)}f\bigg{(}\frac{1}{v},\frac{1}{2}(v-zv_{z})\bigg{)},$

and $w(z_{1})=2$ implies that $\frac{1}{2}(v-zv_{z})=\tfrac{F(0,1)}{v}$, which
in turn gives

$f\bigg{(}\frac{1}{v},\frac{1}{2}(v-zv_{z})\bigg{)}=\frac{1}{v}f(1,F(0,1))=0$

at $z=z_{1}$. It follows that

$w_{z}=-\frac{4F(0,1)}{zv^{2}}<0$

at $z=z_{1}$, which is a contradiction. ∎

Lemma 4.9 immediately implies the following estimate for $v$.

######  Lemma 4.10.

We have

$v(z)^{2}\geq 2F(0,1)\left(1-\frac{z^{2}}{a^{2}}\right).$

###### Proof.

The estimate $w>2$ says exactly that

$\frac{d}{dz}\log(2F(0,1)-v^{2})>\frac{2}{z}.$

Integrating this inequality from $z$ to $a$ gives the claim. ∎

Next we derive an upper bound for $w$, using the following inequality.

######  Lemma 4.11.

There is a constant $c>0$ depending only on $n$ and $\gamma$ such that

(10) $\displaystyle zw_{z}\geq
w-w^{2}\bigg{(}\frac{1}{2}+\frac{F(0,1)}{v^{2}}\bigg{)}+cz^{2}(1+v_{z}^{2})(w-2).$

###### Proof.

We first note that

$-\frac{wz}{v_{z}}f\left(\frac{1}{v},\frac{1}{2}(v-zv_{z})\right)=\frac{2z^{2}}{2F(0,1)-v^{2}}f\bigg{(}1,\frac{2F(0,1)-v^{2}}{2}\bigg{(}\frac{w}{2}-1\bigg{)}+F(0,1)\bigg{)}.$

To estimate the right-hand side we write

$f(1,b+F(0,1))=f(1,b+F(0,1))-f(1,F(0,1))=\int_{0}^{b}\frac{\partial
f}{\partial z}(1,t+F(0,1))\,dt$

and observe that since $\gamma$ is one-homogeneous,

$\displaystyle\inf_{t\in[0,\infty)}\frac{\partial f}{\partial z}(1,t+F(0,1))$
$\displaystyle=\inf_{t\in[0,\infty)}\dot{\gamma}^{1}(t,1,\dots,1)^{-1}$
$\displaystyle=\inf_{t\in[0,\infty)}\dot{\gamma}^{1}\bigg{(}\frac{t}{n-1+t},\frac{1}{n-1+t},\dots,\frac{1}{n-1+t}\bigg{)}^{-1}.$

The last infimum is over a compact subset of $\Gamma$ (the segment connecting
$(0,\frac{1}{n-1},\dots,\frac{1}{n-1})$ with $(1,0,\dots,0)$), so

$c:=\inf_{t\in[0,\infty)}\frac{\partial f}{\partial z}(1,t+F(0,1))$

is a positive constant depending only on $n$ and $\gamma$, and we have

$f\bigg{(}1,\frac{2F(0,1)-v^{2}}{2}\bigg{(}\frac{w}{2}-1\bigg{)}+F(0,1)\bigg{)}\geq
c\frac{2F(0,1)-v^{2}}{2}\bigg{(}\frac{w}{2}-1\bigg{)}.$

Therefore,

$-\frac{wz}{v_{z}}f\left(\frac{1}{v},\frac{1}{2}(v-zv_{z})\right)\geq
cz^{2}(w-2),$

and the claim follows when we combine this inequality with

$zw_{z}=w-w^{2}\bigg{(}\frac{1}{2}+\frac{F(0,1)}{v^{2}}\bigg{)}-(1+v_{z}^{2})\frac{2z}{v_{z}}f\bigg{(}\frac{1}{v},\frac{1}{2}(v-zv_{z})\bigg{)}$

and $w>2$. ∎

Let $\zeta:\mathbb{R}_{+}\to\mathbb{R}$ denote the unique positive solution to

$\zeta_{\rho\rho}=(1+\zeta_{\rho}^{2})f\bigg{(}\frac{\zeta_{\rho}}{\rho},\frac{1}{2}\bigg{)}$

which satisfies $(\zeta,\zeta_{\rho})\to 0$ as $\rho\to 0$. We will make use
of the following asymptotic expansions for $\zeta$ and $\zeta_{\rho}$, which
are derived in Appendix A:

$\zeta(M)=\frac{1}{4F(0,1)}M^{2}-2\dot{F}^{1}(0,1)\log M+o(\log(M))$

and

$\zeta_{\rho}(M)=\frac{1}{2F(0,1)}M-2\dot{F}^{1}(0,1)M^{-1}+o(M^{-1})$

as $M\to\infty$, where $\dot{F}^{1}(0,1)=\dot{\gamma}^{1}(0,1,\dots,1)$.

######  Lemma 4.12.

As $a\to\infty$ we have $\psi\to\zeta$ in
$C^{\infty}_{\operatorname{loc}}(\mathbb{R}_{+})$. In particular, if $M$ is
sufficiently large,

$\psi(M)\to\frac{1}{4F(0,1)}M^{2}-\dot{F}^{1}(0,1)\log M+o(\log(M))$

and

$\psi_{\rho}(M)\to\frac{1}{2F(0,1)}M-2\dot{F}^{1}(0,1)M^{-1}+o(M^{-1})$

as $a\to\infty$.

###### Proof.

The claim follows if we can show that every sequence $a_{k}\to\infty$ admits a
subsequence $a_{k_{i}}$ along which $\psi_{a_{k_{i}}}\to\zeta$ in
$C^{\infty}_{\operatorname{loc}}(\mathbb{R}_{+})$. So consider an arbitrary
sequence $a_{k}\to\infty$. Recall that there is a positive constant $C$
depending only on $n$ and $\gamma$ such that
$C^{-1}\leq(\psi_{a_{k}})_{\rho}/\rho\leq C$ holds in
$(0,\sqrt{F(0,1)}a_{k})$. Using Lemma 4.2, Lemma 4.3 and the equation (9), we
conclude that

$\sup_{k}\|\psi_{a_{k}}\|_{C^{3}(I)}<\infty$

for every compact interval $I\subset\mathbb{R}_{+}$. Therefore, after passing
to a subsequence, we have that $\psi_{a_{k}}$ converges to some limit
$\bar{\psi}$ in $C^{2,\alpha}_{\operatorname{loc}}(\mathbb{R}_{+})$. The
function $\bar{\psi}$ is strictly convex and satisfies

$\bar{\psi}_{\rho\rho}=(1+\bar{\psi}_{\rho}^{2})f\bigg{(}\frac{\bar{\psi}_{\rho}}{\rho},\frac{1}{2}\bigg{)}.$

In particular, $\bar{\psi}$ is smooth. Moreover, $\bar{\psi}$ satisfies the
initial conditions $(\bar{\psi},\bar{\psi}_{\rho})\to 0$ as $\rho\to 0$. But
$\zeta$ is the unique function with this properties, so $\bar{\psi}=\zeta$, as
required. ∎

Let us fix a large positive constant $M$ and set $z_{M,a}=a-a^{-1}\psi(M)$. By
Lemma 4.12, if $M$ is sufficiently large we have $a^{-1}z_{M,a}\to 1$ as
$a\to\infty$. The following lemma concerns the asymptotic behaviour of
$w(z_{M,a})$ as $a\to\infty$. (Note that since $z_{M,a}\sim a$, this is really
a statement about the behaviour of our self-shrinking solutions near their
tips.)

######  Lemma 4.13.

If $M$ is sufficiently large then we have

$w(z_{M,a})\to\frac{1}{F(0,1)}\frac{M}{\zeta_{\rho}(M)}=2+8F(0,1)^{2}M^{-2}+O(M^{-4})$

as $a\to\infty$.

###### Proof.

We first note that $z_{M,a}=h(M/a)$ and hence $v(z_{M,a})=M/a$. It follows
that

$z_{M,a}\,v(z_{M,a})\to M,\qquad v(z_{M,a})^{2}\to 0$

as $a\to\infty$. Also,

$v_{z}(z_{M,a})=-\psi_{\rho}(M)^{-1}\to-\zeta_{\rho}(M)^{-1}$

as $a\to\infty$. Combining these facts we see that

$w(z_{M,a})=\frac{-2z_{M,a}v(z_{M,a})v_{z}(z_{M,a})}{2F(0,1)-v(z_{M,a})^{2}}\to\frac{1}{F(0,1)}\frac{M}{\zeta_{\rho}(M)}$

as $a\to\infty$. We conclude using the expansion

$\frac{1}{\zeta_{\rho}(M)}=\frac{1}{\frac{1}{2F(0,1)}M-2\dot{F}^{1}(0,1)M^{-1}+O(M^{-2})}=\frac{2F(0,1)}{M}+8F(0,1)^{3}M^{-3}+O(M^{-4}).$

∎

We define

$w_{1}(z)=\frac{1}{z^{2}}+\frac{1}{a^{2}-z^{2}}$

and set $\bar{w}=2+Kw_{1}$, where $K:=\max\\{1,6F(0,1),17c^{-1}\\}$ and $c$ is
the constant appearing in Lemma 4.11.

######  Lemma 4.14.

If $M$ is sufficiently large, and $a$ is sufficiently large depending on $M$,
then we have

$w(z_{M,a})<\bar{w}(z_{M,a}).$

###### Proof.

Recalling that $z_{M,a}=a-a^{-1}\psi(M)$, we observe that

$a^{2}-z_{M,a}^{2}\to 2\zeta(M)$

as $a\to\infty$, and hence

$\bar{w}(z_{M,a})\to
2+\frac{K}{2\zeta(M)}=2+2F(0,1)KM^{-2}+O(M^{-4})\,\log(M)$

as $a\to\infty$. On the other hand,

$w(z_{M,a})\to 2+8F(0,1)^{2}M^{-2}+O(M^{-4}),$

Since $K>4F(0,1)$, the claim follows. ∎

######  Proposition 4.15.

If $M$ is sufficiently large, and $a$ is sufficiently large depending on $M$,
then we have

$w(z)\leq\bar{w}(z)$

for $\sqrt{K}<z<z_{M,a}$.

###### Proof.

We claim that

$z\overline{w}_{z}<\bar{w}-\overline{w}^{2}\left(\frac{1}{2}+\frac{F(0,1)}{v^{2}}\right)+cv^{2}(1+v_{z}^{2})(\overline{w}-2)$

for $\sqrt{K}<z<z_{M,a}$.

If $a$ is sufficiently large then we have

$z\bar{w}_{z}=-\frac{2K}{z^{2}}+\frac{2Kz^{2}}{(a^{2}-z^{2})^{2}}\leq\frac{2K}{a^{2}-z_{M,a}^{2}}z^{2}w_{1}\leq\frac{2K}{\zeta(M)}z^{2}w_{1}.$

If $M$ is large enough then we have $2K<\zeta(M)$, and so conclude that

$z\bar{w}_{z}<z^{2}w_{1},$

for $z<z_{M,a}$ and all sufficiently large $a$.

Next we use $\bar{w}=2+Kw_{1}$ to obtain

$cz^{2}(1+v_{z}^{2})(\overline{w}-2)\geq cKz^{2}w_{1},$

and use $v^{2}\geq 2F(0,1)(1-z^{2}/a^{2})$ to obtain

$\frac{1}{2}+\frac{F(0,1)}{v^{2}}\leq\frac{1}{2}+\frac{a^{2}}{2(a^{2}-z^{2})}\leq
z^{2}w_{1}.$

Together these inequalities give

$\overline{w}-\bigg{(}\frac{1}{2}+\frac{F(0,1)}{v^{2}}\bigg{)}\overline{w}^{2}+cz^{2}(1+v_{z}^{2})(\overline{w}-2)\geq-z^{2}\bar{w}^{2}w_{1}+cKz^{2}w_{1},$

and so since, for large $M$, we have

$\bar{w}=2+\frac{K}{z}+\frac{K}{a^{2}-z^{2}}\leq
2+\frac{K}{z^{2}}+\frac{K}{\zeta(M)}\leq 4$

for $\sqrt{K}\leq z\leq z_{M,a}$, we may conclude that

$\overline{w}-\bigg{(}\frac{1}{2}+\frac{F(0,1)}{v^{2}}\bigg{)}\overline{w}^{2}+cz^{2}(1+v_{z}^{2})(\overline{w}-2)\geq-16z^{2}w_{1}+cKz^{2}w_{1}\geq
z^{2}w_{1}$

for $\sqrt{K}\leq z\leq z_{M,a}$.

Putting all of this together, we see that

$z\overline{w}_{z}<\bar{w}-\overline{w}^{2}\left(\frac{1}{2}+\frac{F(0,1)}{v^{2}}\right)+cv^{2}(1+v_{z}^{2})(\overline{w}-2)$

for $\sqrt{K}\leq z\leq z_{M,a}$. In the previous lemma we showed that, if $a$
is sufficiently large, we have

$w(z_{M,a})<\bar{w}(z_{M,a}).$

The claim now follows from a simple comparison principle argument. ∎

######  Proposition 4.16.

If $M$ is sufficiently large, and $a$ is sufficiently large depending on $M$,
then we have

$v^{2}\leq 2F(0,1)\bigg{(}1-\bigg{(}1-C\frac{\log
a}{a^{2}}\bigg{)}\frac{z^{2}-K/2}{a^{2}}\bigg{)}$

for $\sqrt{K}<z<z_{M,a}$, where $C$ depends only on $K$ and $M$.

###### Proof.

We use $w=z\tfrac{d}{dz}\log(2F(0,1)-v^{2})$ and the estimate in Lemma 4.15 to
obtain

$\displaystyle\frac{\log(2F(0,1)-M^{2}/a^{2})}{\log(2F(0,1)-v(z)^{2})}$
$\displaystyle=\int_{z}^{z_{M,a}}\frac{w(\eta)}{\eta}\,d\eta$
$\displaystyle\leq\log\bigg{(}\frac{z_{M,a}^{2}}{z^{2}}\bigg{)}+\frac{K}{2z^{2}}+\frac{K}{2a^{2}}\log\bigg{(}\frac{z_{M,a}^{2}(a^{2}-z^{2})}{z^{2}(a^{2}-z_{M,a}^{2})}\bigg{)}$
$\displaystyle\leq\log\bigg{(}\frac{z_{M,a}^{2}}{z^{2}}\bigg{)}+\frac{K}{2z^{2}}+\frac{C}{a^{2}}\log
a-\frac{K}{2a^{2}}\log(z^{2}(a^{2}-z_{M,a}^{2})),$

where $C$ depends only on $K$ and $M$. If $M$ is so large that $\zeta(M)>1/2$
then we have

$a^{2}-z_{M,a}^{2}=2\psi(M)-a^{-2}\psi(M)^{2}>1$

for all sufficiently large $a$, and hence the last term on the right is
negative for $z>\sqrt{K}>1$. We thus have

$\frac{\log(2F(0,1)-M^{2}/a^{2})}{\log(2F(0,1)-v(z)^{2})}\leq\log\bigg{(}\frac{z_{M,a}^{2}}{z^{2}}\bigg{)}+\frac{K}{2z^{2}}+\frac{C}{a^{2}}\log
a$

for $\sqrt{K}<z\leq z_{M,a}$ whenever $a$ is sufficiently large. We rearrange
and use $e^{-x}\geq 1-x$ to obtain

$2F(0,1)-v^{2}\geq\bigg{(}2F(0,1)-C\frac{\log
a}{a^{2}}\bigg{)}\frac{z^{2}-K/2}{a^{2}}$

for $\sqrt{K}<z<z_{M,a}$, where $C$ depends only on $M$ and $K$. This implies
the claim. ∎

We can finally prove the main result of this section, Proposition 4.1.

###### Proof of Proposition 4.1.

Let $M$ be large enough so fixed so that Proposition 4.16 applies. It then
suffices to take $\Psi_{a}(z)=v(z)$ and $L_{0}=\sqrt{K}+1$. ∎

##  5\. Sharp asymptotics for the rescaled flow

Let $M_{t}=\partial\Omega_{t}$, $t\in(-\infty,0]$, be a strictly convex
ancient $G$-flow which is noncollapsing and uniformly two-convex. In addition,
we assume that $M_{t}$ noncompact. We define a family of rescaled
hypersurfaces $\bar{M}_{\tau}=e^{\tau/2}M_{-e^{-\tau}}$. These move with
pointwise normal speed $-(G-\tfrac{1}{2}\langle x,\nu\rangle)$. Proposition
3.6 implies that, as $\tau\to-\infty$, $\bar{M}_{\tau}$ converges in
$C^{\infty}_{\operatorname{loc}}$ to a cylinder $\Sigma$ of radius
$\sigma:=\sqrt{2\gamma(0,1,\dots,1)}$ whose axis passes through the origin.
Without loss of generality we may assume the axis of $\Sigma$ is the
$x_{1}$-axis, which we equip with the coordinate $z$. That is,

$\Sigma:=\\{(z,\sigma\theta):z\in\mathbb{R},\;\theta\in S^{n-1}\\}.$

Recall that the asymptotic cone of $M_{t}$ (and hence of $\bar{M}_{\tau}$) is
constant, by Lemma 2.15, and of dimension 1, by Lemma 2.7. Given that
$\bar{M}_{\tau}\to\Sigma$, the asymptotic cone is a subset of the
$x_{1}$-axis. Since $M_{t}$ is strictly convex, we may assume without loss of
generality that the asymptotic cone is $\\{(z,0):z\leq 0\\}$. For each $\tau$
we define

$\bar{z}(\tau):=\sup\\{z\in\mathbb{R}:(z,0)\in
e^{\tau/2}\Omega_{-e^{-\tau}}\\}.$

Observe that $\bar{z}(\tau)\to\infty$ as $\tau\to-\infty$. We may define a
function $u(z,\theta,\tau)$ such that $\bar{M}_{\tau}$ coincides with the set
of points $(z,\sigma\theta)+u(z,\theta,\tau)\theta$ in the region of space
where $z<\bar{z}(\tau)$. Since $\bar{M}_{\tau}$ is strictly convex and
noncompact we have

$u>0,\qquad\frac{\partial u}{\partial z}<0,\qquad\frac{\partial^{2}u}{\partial
z^{2}}<0.$

In this section we establish the following sharp asymptotic estimate for $u$.

######  Proposition 5.1.

For every $L<\infty$ we have

$\sup_{|z|\leq L}|u(\cdot,\tau)|\leq O(e^{\tau/2})$

as $\tau\to-\infty$.

In other words, there is a positive constant $K$ such that

$M_{t}\cap\\{|z|\leq L(-t)^{1/2}\\}\subset\\{(z,y):||y|-(-2\gamma(0,1,\dots
1)t)^{1/2}|\leq K\\}$

whenever $-t$ is sufficiently large.

We first prove that the maximum of $|u|$ for $|z|\leq L$ controls $u$ and its
derivatives in a subset which grows as $\tau\to-\infty$. To obtain these
‘inner-outer’ estimates, we employ the hypersurfaces $\Sigma_{a}$ as barriers.
We then use the inner-outer estimates to conduct a spectral analysis of the
linearised PDE for $u$. To do so we must cut off in space. Both linearising
and cutting off introduce error terms, but we are be able to show that these
decay rapidly as $\tau\to-\infty$. Proposition 5.1 is then proven by showing
that the spectral decomposition is dominated by positive modes.

We take inspiration from [ADS19, BC19, BC21], which concern ancient solutions
to the mean curvature flow, and Brendle’s work on ancient solutions to the
Ricci flow [Bre20] (see also [BN23]). Our analysis is the first of its kind
for a fully nonlinear geometric flow. The main new difficulty which we need to
overcome is the presence of error terms depending on the Hessian of $u$.

###  5.1. Inner-outer estimates

For each sufficiently large $a>0$ we write

$\Psi_{a}:[L_{0},a]\to\mathbb{R}$

for the function referred to in Proposition 4.1. Since $u(\cdot,\tau)\to 0$ on
compact subsets as $\tau\to-\infty$, for every $a>0$ the inequality

$\Psi_{a}(z)<\sqrt{2\gamma(0,1,\dots,1)}+u(z,\theta,\tau)$

holds for $L_{0}\leq z\leq a$ and sufficiently large $-\tau$. Since $\Psi_{a}$
is the profile of a self-similarly shrinking $G$-flow, which is a stationary
solution to the rescaled flow, the parabolic maximum principle implies the
following.

######  Lemma 5.2.

Suppose $z_{0}\geq L_{0}$ and $\tau_{0}<0$ are such that

$\Psi_{a}(z_{0})<\sqrt{2\gamma(0,1,\dots,1)}+u(z_{0},\theta,\tau)$

for every $\tau\leq\tau_{0}$. We then have

$\Psi_{a}(z)\leq\sqrt{2\gamma(0,1,\dots,1)}+u(z,\theta,\tau)$

for $z\in[z_{0},a]$ and $\tau\leq\tau_{0}$.

Let $L>L_{0}$ be an arbitrary large constant. For each integer $k\geq 0$ we
define

$\delta_{k}:=\sup_{\tau\leq-k}\sup_{|z|\leq L}|u(\cdot,\tau)|.$

Since $\bar{M}_{\tau}$ is strictly convex we have $\delta_{k}>0$. Since
$\delta_{k}\to 0$ as $k\to\infty$, after passing to a subsequence we may
assume $\delta_{k}\leq 10^{-6}$. We fix a small constant $r>0$. Choosing
$r=10^{-4}$ will prove to be sufficient.

In this section we write $\nabla$ for the Levi-Civita connection on
$\mathbb{R}\times S^{n-1}$. We use $C$ to denote a positive constant which is
independent of $k$. The value of $C$ may increase as we proceed.

######  Lemma 5.3.

When $k$ is sufficiently large we have the estimates

$|u|+\bigg{|}\frac{\partial u}{\partial z}\bigg{|}\leq
C\delta_{k}^{1-2r},\qquad|\nabla^{S^{n-1}}u|\leq C\delta_{k}^{1/2-2r}$

for $|z|\leq 8\delta_{k}^{-r}$ and $\tau\leq-k$. Moreover, we have

$|u|+\bigg{|}\frac{\partial u}{\partial z}\bigg{|}\leq C\delta_{k}$

for $|z|\leq L+2$ and $\tau\leq-k$.

###### Proof.

Recall from Proposition 4.1 that when $a$ is large we have

$\Psi_{a}(z)\leq\sqrt{2\gamma(0,1,\dots,1)\bigg{(}1-\bigg{(}1-C\frac{\log
a}{a^{2}}\bigg{)}\frac{z^{2}-C}{a^{2}}\bigg{)}}$

for $z\in[L_{0},L]$, where $C=C(n,\gamma)$. Let $z_{0}=L_{0}+\sqrt{C}$ and
$a=\Lambda\delta_{k}^{-1/2}$, where

$\Lambda^{2}:=\sqrt{\frac{\gamma(0,1,\dots,1)}{8}}(L_{0}^{2}+2L_{0}\sqrt{C}).$

We then have

$\sqrt{2\gamma(0,1,\dots,1)}-\delta_{k}>\Psi_{a}(z_{0})$

whenever $k$ is sufficiently large. We may assume without loss of generality
that $L>z_{0}$. Since $|u|\leq\delta_{k}$ for $|z|\leq L$ and $\tau\leq-k$,

$\sqrt{2\gamma(0,1,\dots,1)}+u(z_{0},\theta,\tau)\geq\sqrt{2\gamma(0,1,\dots,1)}-\delta_{k}>\Psi_{a}(z_{0})$

for $\tau\leq-k$. Using Lemma 5.2 and Proposition 4.1 we obtain

$\sqrt{2\gamma(0,1,\dots,1)}+u(z,\theta,\tau)\geq\Psi_{a}(z)\geq\sqrt{2\gamma(0,1,\dots,1)\bigg{(}1-\frac{z^{2}}{a^{2}}\bigg{)}}$

for $z\in[z_{0},a]$ and $\tau\leq-k$. In particular, for all sufficiently
large $k$, we have

$u\geq-C\delta_{k}^{1-2r}$

for $z\in[z_{0},9\delta_{k}^{-r}]$ and $\tau\leq-k$. Moreover,

$u\geq-C\delta_{k}$

for $z\in[L+2,L+3]$ and $\tau\leq-k$.

Since $\frac{\partial u}{\partial z}<0$ we conclude that

$-C\delta_{k}^{1-2r}\leq u(z,\theta,\tau)\leq u(0,\theta,\tau)\leq\delta_{k}$

for $z\in[0,9\delta_{k}^{-r}]$ and $\tau\leq-k$. The mean value theorem then
gives

$\inf_{8\delta_{k}^{-r}\leq z\leq 9\delta_{k}^{-r}}\frac{\partial u}{\partial
z}(z,\theta,\tau)\geq-C\delta_{k}^{1-r}$

for $\tau\leq-k$, and so since $\frac{\partial u}{\partial z}<0$ and
$\frac{\partial^{2}u}{\partial z^{2}}<0$,

$\bigg{|}\frac{\partial u}{\partial z}\bigg{|}\leq C\delta_{k}^{1-r}$

for $z\leq 8\delta_{k}^{-r}$ and $\tau\leq-k$. Using this inequality and the
fundamental theorem of calculus we obtain

$|u|\leq C\delta_{k}^{1-2r}$

for $|z|\leq 8\delta_{k}^{-r}$ and $\tau\leq-k$.

Similarly, since $u\geq-C\delta_{k}$ for $z\in[L+2,L+3]$, we have

$\bigg{|}\frac{\partial u}{\partial z}\bigg{|}\leq C\delta_{k}$

for $z\leq L+2$ and $\tau\leq-k$. It follows that

$|u|+\Big{|}\frac{\partial u}{\partial z}\Big{|}\leq C\delta_{k}$

for $|z|\leq L+2$ and $\tau\leq-k$.

Since $\bar{M}_{\tau}$ is convex, a straightforward geometric argument using
the bound $|u|\leq C\delta_{k}^{1-2r}$ gives

$|\nabla^{S^{n-1}}u|\leq C\delta_{k}^{1/2-r}$

for $|z|\leq 8\delta_{k}^{-r}$ and $\tau\leq-k$. This completes the proof. ∎

Next we obtain a stronger bound for $\nabla^{S^{n-1}}u$, and establish an
estimate for the Hessian of $u$, in the region where $|z|\leq
5\delta_{k}^{-r}$ and $\tau\leq-k$.

######  Lemma 5.4.

When $k$ is sufficiently large we have $|\nabla u|\leq C\delta_{k}^{1-6r}$ and
$|\nabla^{2}u|\leq C\delta_{k}^{1-6r}$ for $|z|\leq 5\delta_{k}^{-r}$ and
$\tau\leq-k$.

###### Proof.

Consider a time $\tau_{0}\leq-k$, and set $t_{0}=-e^{-\tau_{0}}$. Lemma 5.3
implies that

$M_{t_{0}}\cap\\{|z|\leq
8\delta_{k}^{-r}(-t_{0})^{1/2}\\}\subset\\{(z,y):||y|-(-2\gamma(0,1,\dots,1)t_{0})^{1/2}|\leq\delta_{k}^{1-2r}(-t_{0})^{1/2}\\}.$

In particular, if $k$ is large and $|z|\leq 7\delta_{k}^{-r}(-t_{0})^{1/2}$,
then

$B((z,0),\tfrac{99}{100}(-2\gamma(0,1,\dots,1)t_{0})^{1/2})\subset\Omega_{t_{0}}.$

Using the interior curvature estimate of Proposition 2.2, we conclude that

$(-t_{0})^{1/2}|A_{M_{t_{0}}}|\leq C$

at every point in $M_{t_{0}}\cap\\{|z|\leq 7\delta_{k}^{-r}\sqrt{-t_{0}}\\}$.
Moreover, the noncollapsing property implies

$(-t_{0})^{1/2}G_{M_{t_{0}}}\geq\frac{1}{C}$

at every point in $M_{t_{0}}\cap\\{|z|\leq 7\delta_{k}^{-r}(-t_{0})^{1/2}\\}$
It follows that $|A_{\bar{M}_{\tau_{0}}}|\leq C$ and
$G_{\bar{M}_{\tau_{0}}}\geq 1/C$ on $\bar{M}_{\tau_{0}}\cap\\{|z|\leq
7\delta_{k}^{-r}\\}$. Since $\tau_{0}\leq-k$ was chosen arbitrarily we
conclude that $|A_{\bar{M}_{\tau}}|\leq C$ and $G_{\bar{M}_{\tau}}\geq 1/C$
for $|z|\leq 7\delta_{k}^{-r}$ and $\tau\leq-k$ whenever $k$ is sufficiently
large.

Using $|A|\leq C$, a straightforward computation shows that when $k$ is
sufficiently large we have

$|\nabla^{2}u|\leq C$

for $|z|\leq 7\delta_{k}^{-r}$ and $\tau\leq-k$. Recall that the hypersurfaces
$\bar{M}_{\tau}$ move with normal speed $-(G(x,\tau)-\tfrac{1}{2}\langle
x,\nu(x,\tau))$. This implies that $u$ solves a fully nonlinear parabolic PDE
which is convex or concave in $\nabla^{2}u$ (depending on whether $\gamma$ is
convex or concave) and uniformly parabolic by Lemma 2.1. We also have

$|u_{\tau}|+|\nabla^{2}u|+|\nabla u|+|u|\leq C$

for $|z|\leq 7\delta_{k}^{-r}$ and $\tau\leq-k$. Therefore, the parabolic
Hölder estimate for convex/concave fully nonlinear parabolic equations due to
[Kry82] and the parabolic Schauder estimates imply that

$|\nabla^{m}u|\leq C(m)$

for each integer $m\geq 2$ in the region where $|z|\leq 7\delta_{k}^{-r}-m$
and $\tau\leq-k$. In this step we also need the bound $\bar{G}\geq 1/C$, since
this implies bounds on the higher derivatives of $\gamma$.

Given a smooth function $\varphi(z,\theta)$ which is compactly supported in
the region $|z-z_{0}|<2$, we have the interpolation inequality

$\sup_{|z-z_{0}|\leq 2}|\nabla\varphi|\leq C(m)\bigg{(}\sup_{|z-z_{0}|\leq
2}|\varphi|\bigg{)}^{1-1/m}\bigg{(}\sup_{|z-z_{0}|\leq
2}|\nabla^{m}\varphi|\bigg{)}^{1/m}.$

We apply this with $\varphi(z,\theta)=\eta(z)u(z,\theta,\tau)$, where $\eta$
is an appropriate cutoff function, and use $|\nabla^{j}u|\leq C(m)$ for $j\leq
m$ to obtain

$\sup_{|z-z_{0}|\leq 1}|\nabla u|\leq C(m)\sup_{|z-z_{0}|\leq
3}|u|+C(m)\bigg{(}\sup_{|z-z_{0}|\leq
3}|u|\bigg{)}^{1-1/m}\bigg{(}\sup_{|z-z_{0}|\leq
3}|\nabla^{m}u|\bigg{)}^{1/m}.$

Fix an integer $m_{0}\geq 1/r$. Consider a point $|z_{0}|\leq
7\delta_{k}^{-r}-m_{0}-3$ and a time $\tau\leq-k$. Applying the last
inequality with $m=m_{0}$, and using Lemma 5.3 and $|\nabla^{j}u|\leq
C(m_{0})$ for $j\leq m_{0}$, we obtain

$\sup_{|z-z_{0}|\leq 1}|\nabla u|(z,\theta,\tau)\leq
C\delta_{k}^{1-2r}+C\delta_{k}^{(1-r)(1-2r)}\leq C\delta_{k}^{1-4r}.$

It follows that for $k$ sufficiently large we have $|\nabla u|\leq
C\delta_{k}^{1-4r}$ for $|z|\leq 6\delta_{k}^{-r}$.

Now consider a point $|z_{0}|\leq 6\delta_{k}^{-r}-m_{0}-3$ and a time
$\tau\leq-k$. The same kind argument as above yields

$\displaystyle\sup_{|z-z_{0}|\leq 1}$
$\displaystyle|\nabla^{2}u|(z,\theta,\tau)$
$\displaystyle\leq\sup_{|z-z_{0}|\leq 3}|\nabla u|+\bigg{(}\sup_{|z-z_{0}|\leq
3}|\nabla u|\bigg{)}^{1-1/m_{0}}\bigg{(}\sup_{|z-z_{0}|\leq
4}|\nabla^{m_{0}+1}u|\bigg{)}^{1/m_{0}}.$

Inserting $|\nabla u|\leq C\delta_{k}^{1-4r}$ on the right-hand side, we
conclude that when $k$ is sufficiently large, $|\nabla^{2}u|\leq
C\delta_{k}^{1-6r}$ for $|z|\leq 5\delta_{k}^{-r}$ and $\tau\leq-k$. ∎

We have shown that $u$ and its first two spatial derivatives are small in the
region where $|z|\leq 5\delta_{k}^{-r}$ and $\tau\leq-k$. It follows that $u$
solves a linear parabolic equation in this region, up to quadratic error
terms.

######  Lemma 5.5.

When $k$ is sufficiently large we have

$|u_{\tau}-\mathcal{L}u|\leq C(|u|^{2}+|\nabla u|^{2}+|\nabla^{2}u|^{2})$

for $|z|\leq 5\delta_{k}^{-r}$ and $\tau\leq-k$, where

$\mathcal{L}u:=a\frac{\partial^{2}u}{\partial
z^{2}}+\frac{1}{2(n-1)}\Delta_{S^{n-1}}u-\frac{1}{2}z\frac{\partial
u}{\partial z}+u$

and $a:=\dot{\gamma}^{1}(0,1,\dots,1)$.

###### Proof.

Recall the notation $\sigma=\sqrt{2\gamma(0,1,\dots,1)}$. Given
$(z,\theta)\in\mathbb{R}\times S^{n-1}$, let us write $x=(z,\sigma\theta)$ for
the corresponding point in $\Sigma$ and $y=x+u(z,\theta,\tau)\theta$ for the
corresponding point in $\bar{M}_{\tau}$. We have

$\displaystyle u_{\tau}(z,\theta,\tau)$
$\displaystyle=-\frac{1}{\nu_{\Sigma}(x)\cdot\nu(y)}\,G(y,\tau)+\frac{1}{2}\frac{1}{\nu_{\Sigma}(x)\cdot\nu(y)}y\cdot\nu(y).$

For $\tau\leq-k$ and $|z|\leq 5\delta_{k}^{-r}$ we may apply Lemma 2.11 to the
function

$(z,\sigma\theta)\mapsto u(z,\theta,\tau),$

and so obtain

$\nu(y)=\nu_{\Sigma}(x)-\nabla u(z,\theta,\tau)+O(|u|^{2}+|\nabla u|^{2}).$

We thus have

$\displaystyle u_{\tau}(z,\theta,\tau)$
$\displaystyle=-G(y,\tau)-\frac{1}{2}x\cdot\nabla
u(z,\theta,\tau)+\frac{1}{2}u(z,\theta,\tau)$
$\displaystyle+\frac{1}{2}x\cdot\nu_{\Sigma}(x)+O(|u|^{2}+|\nabla u|^{2}).$

Lemma 2.13 now gives

$\displaystyle u_{\tau}$ $\displaystyle=a\frac{\partial^{2}u}{\partial
z^{2}}+\frac{1}{2(n-1)}\Delta_{S^{n-1}}u-\frac{1}{2}x\cdot\nabla u+u$
$\displaystyle+\frac{1}{2}x\cdot\nu_{\Sigma}-G_{\Sigma}(x)+O(|u|^{2}+|\nabla
u|^{2}+|\nabla^{2}u|^{2}),$

for $\tau\leq-k$ and $|z|\leq 5\delta_{k}^{-r}$. The claim follows upon
insertion of the identities

$x\cdot\nabla u=z\frac{\partial u}{\partial z}\qquad\text{
and}\qquad\frac{1}{2}x\cdot\nu_{\Sigma}-G_{\Sigma}=0.$

∎

Using Lemma 5.5, we obtain the following improved estimate for $\nabla u$ in
the region $|z|\leq L+1$.

######  Lemma 5.6.

When $k$ is sufficiently large we have $|\nabla u|\leq C\delta_{k}$ for
$|z|\leq L+1$ and $\tau\leq-k$.

###### Proof.

Let us define

$\tilde{\mathcal{L}}u:=a\frac{\partial^{2}u}{\partial
z^{2}}+\frac{1}{2(n-1)}\Delta_{S^{n-1}}u.$

We then have

$|u_{\tau}-\tilde{\mathcal{L}}u|\leq\bigg{|}\frac{1}{2}z\frac{\partial
u}{\partial z}\bigg{|}+|u|+C(|u|^{2}+|\nabla u|^{2}+|\nabla^{2}u|^{2})$

for $|z|\leq 5\delta_{k}^{-r}$ and $\tau\leq-k$. Using Lemma 5.3 and Lemma 5.4
we conclude that

$|u_{\tau}-\tilde{\mathcal{L}}u|\leq C\delta_{k}$

for $|z|\leq L+2$ and $\tau\leq-k$. Given any $(z_{0},\theta_{0},\tau_{0})$
such that $|z_{0}|\leq L+1$ and $\tau\leq-k$, we claim that if $d=d(n)$ is
sufficiently small, then

(11) $\displaystyle|\nabla^{S^{n-1}}u|(z_{0},\theta_{0},\tau_{0})\leq
Cd^{-1}\sup_{\tau_{0}-d^{2}\leq\tau\leq\tau_{0}}\sup_{|z-z_{0}|\leq
d}|u|+Cd\sup_{\tau_{0}-d^{2}\leq\tau\leq\tau_{0}}\sup_{|z-z_{0}|\leq
d}|u_{\tau}-\tilde{\mathcal{L}}u|.$

The estimate also holds with $|\nabla u|$ on the left-hand side, but we don’t
need this since we showed in Lemma 5.3 that $|\frac{\partial u}{\partial
z}|\leq C\delta_{k}$ for $|z|\leq L+2$ and $\tau\leq-k$.

The estimate (11) is a standard result in the theory of linear parabolic PDE
(cf. Section 3.4 in [GT01]), but we provide a proof for the sake of
completeness. We only describe the case $n\geq 3$ (the case $n=2$ follows the
same lines but is simpler).

We choose geodesic normal coordinates $x^{i}$ for $S^{n-1}$ such that
$\theta_{0}$ coincides with the origin. We consider the cube
$Q:=\\{|z-z_{0}|\leq d\\}\cap\\{|x^{i}|\leq d\\}$ and the half-cube
$Q^{\prime}:=Q\cap\\{x^{1}\geq 0\\}$. We fix $d$ small enough (depending only
on $n$) so that the functions

$f(x)=\sum_{i=2}^{n-1}(x^{i})^{2}+2x^{1}((n-1)d-(n-2)x^{1}),\qquad
h(x)=x^{1}(d-x_{1}),$

satisfy

$\Delta_{S^{n-1}}f\leq-(n-2),\qquad\Delta_{S^{n-1}}h\leq-1$

in $Q^{\prime}$. In the parabolic region
$Q^{\prime}\times[\tau_{0}-d^{2},\tau_{0}]$ we define

$\psi(z,x,\tau):=\frac{\alpha}{d^{2}}\bigg{(}\frac{1}{2}(z-z_{0})^{2}+\frac{2(n-1)}{n-2}(1+a)f(x)+\tau_{0}-\tau\bigg{)}+2(n-1)\beta
h(x),$

where $\alpha$ and $\beta$ are some positive constants to be chosen in a
moment. We then have

$\psi_{\tau}-\tilde{\mathcal{L}}\psi\geq\beta$

in $Q^{\prime}\times[\tau_{0}-d^{2},\tau_{0}]$.

In the set $Q^{\prime}\times[\tau_{0}-d^{2},\tau_{0}]$ we define

$\varphi(z,x,\tau)=\frac{1}{2}(u(z,x,\tau)-u(z,Rx,\tau)),$

where $R:S^{n-1}\to S^{n-1}$ is the reflection map across the great sphere
$\\{x^{1}=0\\}$. We then have

$\sup_{Q^{\prime}\times[\tau_{0}-d^{2},\tau_{0}]}|\varphi_{\tau}-\tilde{\mathcal{L}}\varphi|=\sup_{Q\times[\tau_{0}-d^{2},\tau_{0}]}|u_{\tau}-\tilde{\mathcal{L}}u|.$

It is easy to see that for

$\alpha=\sup_{Q\times[\tau_{0}-d^{2},\tau_{0}]}|u|,\qquad\beta=\sup_{Q\times[\tau_{0}-d^{2},\tau_{0}]}|u_{\tau}-\tilde{\mathcal{L}}u|,$

the parabolic maximum principle implies $\psi\pm\varphi\geq 0$ in
$Q^{\prime}\times[\tau_{0}-d^{2},\tau_{0}]$. At the point
$(z_{0},\theta_{0},\tau_{0})$, this inequality yields

$|\nabla_{1}u|(z_{0},\theta_{0},\tau_{0})=\lim_{x^{1}\to
0}\bigg{|}\frac{\varphi(z_{0},x^{1},0,\dots,0,\tau_{0})}{x^{1}}\bigg{|}\leq
Cd^{-1}\alpha+Cd\beta.$

We obtain the same estimate for $\nabla_{i}u$ simply by relabelling the
coordinates $x^{i}$. This proves (11).

Since $d$ depends only on $n$, as a consequence of (11) we have

$|\nabla^{S^{n-1}}u|(z_{0},\theta_{0},\tau_{0})\leq C\delta_{k}.$

We also have $|\frac{\partial u}{\partial z}|\leq C\delta_{k}$ at
$(z_{0},\theta_{0},\tau_{0})$ by Lemma 5.4. Since $z_{0}$ was chosen
arbitrarily in the region $|z|\leq L+1$, we conclude that $|\nabla u|\leq
C\delta_{k}$ for $|z|\leq L+1$ and $\tau\leq-k$. ∎

###  5.2. Spectral analysis

Let us denote by $\mathcal{H}$ the space of functions $v:\mathbb{R}\times
S^{n-1}\to\mathbb{R}$ such that

$\|v\|_{\mathcal{H}}^{2}:=\int_{\mathbb{R}\times
S^{n-1}}e^{-z^{2}/4a}v^{2}\,dzd\theta<\infty.$

The operator $\mathcal{L}$ is symmetric with respect to the inner product
$\langle\cdot,\cdot\rangle_{\mathcal{H}}$. Therefore, there exists an
orthogonal basis for $\mathcal{H}$ consisting of eigenfunctions of
$\mathcal{L}$. Let us write $H_{k}$ for the $k$-th Hermite polynomial, and fix
a basis $Y_{l}^{m}$ of eigenfunctions for $\Delta_{S^{n-1}}$ such that

$\Delta_{S^{n-1}}Y_{l}^{m}=-l(l+n-2)Y_{l}^{m}.$

A basis of eigenfunctions for $\mathcal{L}$ is then given by
$H_{k}(z/2\sqrt{a})Y_{l}^{m}(\theta)$, where $k$ and $l$ range over the
nonnegative integers. We have

$\bigg{(}a\frac{\partial^{2}}{\partial
z^{2}}-\frac{1}{2}z\frac{\partial}{\partial
z}\bigg{)}H_{k}\bigg{(}\frac{z}{2\sqrt{a}}\bigg{)}=-\frac{k}{2}.$

Therefore, the eigenvalues of $\mathcal{L}$ are given by
$1-\frac{k}{2}-\frac{l(l+n-2)}{2(n-1)}$, where $k$ and $l$ are nonnegative
integers. Up to scaling, the eigenfunctions with positive eigenvalues are of
the form $1$, $z/\sqrt{a}$ and $\theta^{i}$, $1\leq i\leq n$. The
eigenfunctions with eigenvalue equal to zero are of the form $z^{2}/a-1$ and
$z\theta^{i}/\sqrt{a}$, $1\leq i\leq n$. Let $P_{+}$, $P_{0}$ and $P_{-}$
denote the orthogonal projection operators onto the positive, zero and
negative eigenspaces. These projections satisfy

(12)
$\langle\mathcal{L}P_{+}v,P_{+}v\rangle_{\mathcal{H}}\geq\frac{1}{2}\|P_{+}v\|_{\mathcal{H}}^{2},\;\;\;\langle\mathcal{L}P_{0}v,P_{0}v\rangle_{\mathcal{H}}=0,\;\;\;\langle\mathcal{L}P_{-}v,P_{-}v\rangle_{\mathcal{H}}\leq-\frac{1}{2}\|P_{-}v\|_{\mathcal{H}}^{2}.$

We are going to show that $P_{+}u$ dominates $P_{0}u$ and $P_{-}u$ in the
region where $|z|\leq\delta_{k}^{-r}$ and $\tau\leq-k$. To this end, let
$\xi:\mathbb{R}\to\mathbb{R}$ be a smooth cutoff function such that $\xi(s)=1$
for $s\in[-1/2,1/2]$, $\xi(s)=0$ for $s\not\in[-1,1]$, and
$s\xi^{\prime}(s)\leq 0$ for $s\in\mathbb{R}$. We define

$\displaystyle\gamma_{j}$
$\displaystyle:=\sup_{\tau\in[-j-1,-j]}\int_{\mathbb{R}\times
S^{n-1}}e^{-z^{2}/4a}|u(z,\theta,\tau)\chi(\delta_{j}^{r}z)|^{2}\,dzd\theta$
$\displaystyle\gamma_{j}^{+}$
$\displaystyle:=\sup_{\tau\in[-j-1,-j]}\int_{\mathbb{R}\times
S^{n-1}}e^{-z^{2}/4a}|P_{+}(u(z,\theta,\tau)\chi(\delta_{j}^{r}z))|^{2}\,dzd\theta$
$\displaystyle\gamma_{j}^{0}$
$\displaystyle:=\sup_{\tau\in[-j-1,-j]}\int_{\mathbb{R}\times
S^{n-1}}e^{-z^{2}/4a}|P_{0}(u(z,\theta,\tau)\chi(\delta_{j}^{r}z))|^{2}\,dzd\theta$
$\displaystyle\gamma_{j}^{-}$
$\displaystyle:=\sup_{\tau\in[-j-1,-j]}\int_{\mathbb{R}\times
S^{n-1}}e^{-z^{2}/4a}|P_{-}(u(z,\theta,\tau)\chi(\delta_{j}^{r}z))|^{2}\,dzd\theta.$

We have $C^{-1}\gamma_{j}\leq\gamma_{j}^{+}+\gamma_{j}^{0}+\gamma_{j}^{-}\leq
C\gamma_{j}$ and Lemma 5.3 gives $\gamma_{j}\leq C\delta_{j}^{2-4r}$. In
particular, $\gamma_{j}\to 0$ as $j\to\infty$.

The following result is similar to [Bre20, Lemma 3.12].

######  Lemma 5.7.

We have

$\displaystyle\gamma_{j+1}^{+}$ $\displaystyle\leq
e^{-1}\gamma_{j}^{+}+C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}+C\exp(-\delta_{j}^{-2r}/64a),$
$\displaystyle|\gamma_{j+1}^{0}-\gamma_{j}^{0}|$ $\displaystyle\leq
C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}+C\exp(-\delta_{j}^{-2r}/64a),$
$\displaystyle\gamma_{j+1}^{-}$ $\displaystyle\geq
e\gamma_{j}^{-}-C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}-C\exp(-\delta_{j}^{-2r}/64a),$

where

$E_{j}:=\sup_{\tau\in[-j-2,-j]}\int_{|z|\leq\delta_{j}^{-r}}e^{-z^{2}/4a}|u_{\tau}-\mathcal{L}u|^{2}.$

###### Proof.

Let us write $\hat{u}(z,\theta,\tau)=u(z,\theta,\tau)\,\chi(\delta_{j}^{r}z)$.
We first note that

$\sup_{\tau\in[-j-2,-j]}\int_{\mathbb{R}\times
S^{n-1}}e^{-z^{2}/4a}|\hat{u}|^{2}\leq\gamma_{j}+\gamma_{j+1}.$

Combining this with (12) and using Hölder’s inequality we obtain

$\displaystyle\frac{d}{d\tau}\int e^{-z^{2}/4a}|P_{+}\hat{u}|^{2}$
$\displaystyle\geq\int
e^{-z^{2}/4a}|P_{+}\hat{u}|^{2}-2(\gamma_{j}+\gamma_{j+1})^{1/2}\bigg{(}\int
e^{-z^{2}/4a}|\hat{u}_{\tau}-\mathcal{L}\hat{u}|^{2}\bigg{)}^{\frac{1}{2}},$
$\displaystyle\bigg{|}\frac{d}{d\tau}\int
e^{-z^{2}/4a}|P_{0}\hat{u}|^{2}\bigg{|}$ $\displaystyle\leq
2(\gamma_{j}+\gamma_{j+1})^{1/2}\bigg{(}\int
e^{-z^{2}/4a}|\hat{u}_{\tau}-\mathcal{L}\hat{u}|^{2}\bigg{)}^{\frac{1}{2}},$
$\displaystyle\frac{d}{d\tau}\int e^{-z^{2}/4a}|P_{-}\hat{u}|^{2}$
$\displaystyle\leq-\int
e^{-z^{2}/4a}|P_{-}\hat{u}|^{2}+2(\gamma_{j}+\gamma_{j+1})^{1/2}\bigg{(}\int
e^{-z^{2}/4a}|\hat{u}_{\tau}-\mathcal{L}\hat{u}|^{2}\bigg{)}^{\frac{1}{2}}$

for $\tau\in[-j-2,-j]$.

Since $\chi(s)$ is constant for $|s|\leq 1/2$ and $|s|\geq 1$, and
$|\chi(s)|\leq 1$ for $s\in\mathbb{R}$, we have

$\displaystyle\int e^{-z^{2}/4a}|\hat{u}_{\tau}-\mathcal{L}\hat{u}|^{2}$
$\displaystyle\leq
C\int_{|z|\leq\delta_{j}^{-r}}e^{-z^{2}/4a}|u_{\tau}-\mathcal{L}u|^{2}+C\int_{\delta_{j}^{-r}/2\leq|z|\leq\delta_{j}^{-r}}e^{-z^{2}/4a}(|u|^{2}+|\nabla
u|^{2})$ $\displaystyle\leq CE_{j}+C\exp(-\delta_{j}^{-2r}/32a)$

for $\tau\in[-j-2,-j]$. It follows that

$\displaystyle\frac{d}{d\tau}\int e^{-z^{2}/4a}|P_{+}\hat{u}|^{2}$
$\displaystyle\geq\int
e^{-z^{2}/4a}|P_{+}\hat{u}|^{2}-C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}-C\exp(-\delta_{j}^{-2r}/64a),$
$\displaystyle\bigg{|}\frac{d}{d\tau}\int
e^{-z^{2}/4a}|P_{0}\hat{u}|^{2}\bigg{|}$ $\displaystyle\leq
2(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}+C\exp(-\delta_{j}^{-2r}/64a),$
$\displaystyle\frac{d}{d\tau}\int e^{-z^{2}/4a}|P_{-}\hat{u}|^{2}$
$\displaystyle\leq-\int
e^{-z^{2}/4a}|P_{-}\hat{u}|^{2}+C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}+C\exp(-\delta_{j}^{-2r}/64a)$

for $\tau\in[-j-2,-j]$. Integrating over $[\tau-1,\tau]$ we obtain

$\displaystyle\int e^{-z^{2}/4a}|P_{+}\hat{u}(\cdot,\tau-1)|^{2}$
$\displaystyle\leq e^{-1}\int
e^{-z^{2}/4a}|P_{+}\hat{u}(\cdot,\tau)|^{2}+C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}+C\exp(-\delta_{j}^{-2r}/64a),$
$\displaystyle\bigg{|}\int e^{-z^{2}/4a}|P_{0}\hat{u}(\cdot,\tau-1)|^{2}-\int
e^{-z^{2}/4a}|P_{0}\hat{u}(\cdot,\tau)|^{2}\bigg{|}$ $\displaystyle\leq
C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}+C\exp(-\delta_{j}^{-2r}/64a),$
$\displaystyle\int e^{-z^{2}/4a}|P_{-}\hat{u}(\cdot,\tau-1)|^{2}$
$\displaystyle\geq e\int
e^{-z^{2}/4a}|P_{-}\hat{u}(\cdot,\tau)|^{2}-C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}-C\exp(-\delta_{j}^{-2r}/64a).$

for $\tau\in[-j-1,-j]$.

Define $\tilde{u}(z,\tau)=u(z,\tau)\chi(\delta_{j+1}^{r}z)$. Observing that

$\displaystyle\int e^{-z^{2}/4a}|\tilde{u}(z,\tau-1)$
$\displaystyle-\hat{u}(z,\tau-1)|^{2}\leq C\exp(-\delta_{j}^{-2r}/64a)$

for all $\tau\in[-j-1,j]$, we conclude that

$\displaystyle\int e^{-z^{2}/4a}|P_{+}\tilde{u}(\cdot,\tau-1)|^{2}-e^{-1}\int
e^{-z^{2}/4a}|P_{+}\hat{u}(\cdot,\tau)|^{2}$ $\displaystyle\hskip
56.9055pt\leq
C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}+C\exp(-\delta_{j}^{-2r}/64a),$
$\displaystyle\bigg{|}\int
e^{-z^{2}/4a}|P_{0}\tilde{u}(\cdot,\tau-1)|^{2}-\int
e^{-z^{2}/4a}|P_{0}\hat{u}(\cdot,\tau)|^{2}\bigg{|}$ $\displaystyle\hskip
56.9055pt\leq
C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}+C\exp(-\delta_{j}^{-2r}/64a),$
$\displaystyle\int e^{-z^{2}/4a}|P_{-}\tilde{u}(\cdot,\tau-1)|^{2}-e\int
e^{-z^{2}/4a}|P_{-}\hat{u}(\cdot,\tau)|^{2}$ $\displaystyle\hskip
56.9055pt\geq-C(\gamma_{j}+\gamma_{j+1})^{1/2}E_{j}^{1/2}-C\exp(-\delta_{j}^{-2r}/64a).$

for $\tau\in[-j-1,-1]$. Taking the supremum over $\tau\in[-j-1,-j]$, we arrive
at the claim. ∎

We now define

$\Gamma_{k}:=\sup_{j\geq k}\gamma_{j},\qquad\Gamma_{k}^{+}:=\sup_{j\geq
k}\gamma_{j}^{+},\qquad\Gamma_{k}^{0}:=\sup_{j\geq
k}\gamma_{j}^{0},\qquad\Gamma_{k}^{-}:=\sup_{j\geq k}\gamma_{j}^{-}.$

We have $C^{-1}\Gamma_{k}\leq\Gamma_{k}^{+}+\Gamma_{k}^{0}+\Gamma_{k}^{-}\leq
C\Gamma_{k}$ and $\Gamma_{k}\to 0$ as $k\to\infty$. Since

$\displaystyle E_{j}$
$\displaystyle=\sup_{\tau\in[-j-2,-j]}\int_{|z|\leq\delta_{j}^{-r}}e^{-z^{2}/4a}|u_{\tau}-\mathcal{L}u|^{2}$
$\displaystyle\leq
C\sup_{\tau\in[-j-2,-j]}\Big{(}\sup_{|z|\leq\delta_{j}^{-r}}|u|^{4}+|\nabla
u|^{4}+|\nabla^{2}u|^{4}\Big{)},$

as a consequence of Lemma 5.3 and Lemma 5.4 we have

$E_{j}\leq C\delta_{j}^{4-24r}$

whenever $j$ is sufficiently large. In particular,

$F_{k}:=\sup_{j\geq k}E_{j}$

satisfies $F_{k}\leq C\varepsilon_{k}^{4-24r}$, where
$\varepsilon_{k}:=\sup_{j\geq k}\delta_{j}$. Note that $\varepsilon_{k}\to 0$
as $k\to\infty$. Lemma 5.7 immediately implies

$\displaystyle\Gamma_{k+1}^{+}$ $\displaystyle\leq
e^{-1}\Gamma_{k}^{+}+C\Gamma_{k}^{1/2}F_{k}^{1/2}+C\exp(-\delta_{k}^{-2r}/64a),$
$\displaystyle|\Gamma_{k+1}^{0}-\Gamma_{k}^{0}|$ $\displaystyle\leq
C\Gamma_{k}^{1/2}F_{k}^{1/2}+C\exp(-\delta_{k}^{-2r}/64a),$
$\displaystyle\Gamma_{k+1}^{-}$ $\displaystyle\geq
e\Gamma_{k}^{-}-C\Gamma_{k}^{1/2}F_{k}^{1/2}-C\exp(-\delta_{k}^{-2r}/64a).$

In the next lemma we improve our estimate for the error term $F_{k}$.

######  Lemma 5.8.

For sufficiently large $k$ we have $\delta_{k}\leq C\Gamma_{k}^{1/2}$ and

$F_{k}\leq C\varepsilon_{k}^{2-24r}\Gamma_{k}.$

###### Proof.

Since

$\sup_{|z|\leq L}|u(z,\theta,\tau)|\to 0$

as $\tau\to-\infty$, for each $k$ there is a point
$(z_{0},\theta_{0},\tau_{0})$ such that $|z_{0}|\leq L$, $\tau_{0}\leq-k$, and

$|u(z_{0},\theta_{0},\tau_{0})|=\sup_{\tau\leq-k}\sup_{|z|\leq
L}|u(z,\theta,\tau)|=\delta_{k}.$

Lemma 5.6 provides a constant $C_{0}$ such that $|\nabla u|\leq
C_{0}\delta_{k}$ for $|z|\leq L+1$ and $\tau\leq-k$. Using this estimate we
obtain

$\inf\Big{\\{}|u(z,\theta,\tau_{0})|:\operatorname{dist}_{\mathbb{R}\times
S^{n-1}}((z,\theta),(z_{0},\theta_{0}))\leq\tfrac{1}{100}\min\\{1,C_{0}^{-1}\\}\Big{\\}}\geq\frac{1}{C}\delta_{k}.$

It follows that

$\delta_{k}^{2}\leq C\int_{|z|\leq L+1}|u|^{2}(\cdot,\tau_{0})\,dzd\theta.$

There exists an integer $j\geq k$ such that $\tau_{0}\in[-j-1,-j]$. For this
$j$, as a result of the last inequality we have

$\delta_{k}^{2}\leq C\sup_{\tau\in[-j-1,-j]}\int_{\mathbb{R}\times
S^{n-1}}e^{-z^{2}/4a}|u(z,\theta,\tau)\chi(\delta_{j}^{r}z)|^{2}\,dzd\theta=C\gamma_{j},$

at least if $k$ is sufficiently large. Therefore,

$\delta_{k}^{2}\leq C\sup_{j\geq k}\gamma_{j}\leq C\Gamma_{k}$

for every sufficiently large $k$. Combining this inequality with $F_{k}\leq
C\varepsilon_{k}^{4-24r}$, and assuming $k$ is sufficiently large, we obtain

$F_{k}\leq C\varepsilon_{k}^{4-24r}=C\varepsilon_{k}^{2-24r}\sup_{j\geq
k}\delta_{j}^{2}\leq C\varepsilon_{k}^{2-24r}\sup_{j\geq k}\Gamma_{j}\leq
C\varepsilon_{k}^{2-24r}\Gamma_{k}.$

Here we have used the fact that $\Gamma_{k}$ is nonincreasing in $k$. This
completes the proof. ∎

Lemma 5.8 tells us that $\delta_{k}\leq C\Gamma_{k}^{1/2}$ and $F_{k}\leq
C\varepsilon_{k}\Gamma_{k}$. Moreover, for sufficiently large $k$ we have

$\exp(-\delta_{k}^{-2r}/64a)\leq\delta_{k}^{5/2}\leq
C\delta_{k}^{1/2}\Gamma_{k}\leq C\varepsilon_{k}^{1/2}\Gamma_{k}.$

Consequently, for large $k$ we have

$\displaystyle\Gamma_{k+1}^{+}$ $\displaystyle\leq
e^{-1}\Gamma_{k}^{+}+C\varepsilon_{k}^{1/2}\Gamma_{k}$
$\displaystyle|\Gamma_{k+1}^{0}-\Gamma_{k}^{0}|$ $\displaystyle\leq
C\varepsilon_{k}^{1/2}\Gamma_{k}$ $\displaystyle\Gamma_{k+1}^{-}$
$\displaystyle\geq e\Gamma_{k}^{-}-C\varepsilon_{k}^{1/2}\Gamma_{k},$

and in particular

$\displaystyle\Gamma_{k+1}^{+}$ $\displaystyle\leq
e^{-1}\Gamma_{k}^{+}o(1)\Gamma_{k}$
$\displaystyle|\Gamma_{k+1}^{0}-\Gamma_{k}^{0}|$ $\displaystyle\leq
o(1)\Gamma_{k}$ $\displaystyle\Gamma_{k+1}^{-}$ $\displaystyle\geq
e\Gamma_{k}^{-}-o(1)\Gamma_{k}.$

This system of inequalities implies that the decomposition
$\Gamma_{k}=\Gamma_{k}^{+}+\Gamma_{k}^{0}+\Gamma_{k}^{-}$ is dominated either
by $\Gamma_{k}^{+}$ or $\Gamma_{k}^{0}$ as $k\to\infty$. This kind of result
was used by Merle and Zaag in [MZ98].

######  Lemma 5.9.

As $k\to\infty$ we either have $\Gamma_{k}^{0}+\Gamma_{k}^{-}\leq
o(1)\Gamma_{k}^{+}$ or $\Gamma_{k}^{+}+\Gamma_{k}^{-}\leq o(1)\Gamma_{k}^{0}$.

###### Proof.

The claim was proven by Brendle in [Bre20, Lemma 3.1]. We recall Brendle’s
proof for the convenience of the reader.

Since $\Gamma_{k}^{-}$ is nonincreasing we have
$\Gamma_{k}^{-}\geq\Gamma_{k+1}^{-}\geq e\Gamma_{k}^{-}-o(1)\Gamma_{k}$. It
follows that $\Gamma_{k}^{-}\leq o(1)(\Gamma_{k}^{+}+\Gamma_{k}^{0})$.

We define subsets of the positive real numbers $I$ and $J$ as follows:

$\displaystyle\alpha\in I\qquad$
$\displaystyle\iff\qquad\\{k:\Gamma_{k}^{0}<\alpha\Gamma_{k}^{+}\\}\text{ is
finite},$ $\displaystyle\alpha\in J\qquad$
$\displaystyle\iff\qquad\\{k:\Gamma_{k}^{0}\geq\alpha\Gamma_{k}^{+}\\}\text{
is infinite}.$

Clearly $I\subset J$. We claim that $e^{1/2}\alpha\in I$ whenever $\alpha\in
J$. To see this, suppose that $\alpha\in J$. There is a large integer $k_{0}$
depending on $\alpha$ such that

$\displaystyle\Gamma_{k+1}^{+}$ $\displaystyle\leq
e^{-1}\Gamma_{k}^{+}+\frac{1}{2(1+\alpha)}(e^{-1/2}-e^{-1})(\Gamma_{k}^{+}+\Gamma_{k}^{0}),$
$\displaystyle|\Gamma_{k+1}^{0}-\Gamma_{k}^{0}|$
$\displaystyle\leq\frac{\alpha}{(1+\alpha)}(1-e^{-1/2})(\Gamma_{k}^{+}+\Gamma_{k}^{0})$

for $k\geq k_{0}$. Since $\alpha\in J$, there is a $k_{1}\geq k_{0}$ depending
on $\alpha$ such that $\Gamma_{k}^{0}\geq\alpha\Gamma_{k}^{+}$ for $k\geq
k_{1}$. We therefore have

$\displaystyle\Gamma_{k+1}^{0}-e^{1/2}\alpha\Gamma_{k+1}^{+}$
$\displaystyle\geq\Gamma_{k}^{0}-e^{-1/2}\alpha\Gamma_{k}^{+}-\frac{\alpha}{1+\alpha}(1-e^{-1/2})(\Gamma_{k}^{+}+\Gamma_{k}^{0})$
$\displaystyle=\bigg{(}1-\frac{\alpha}{1+\alpha}(1-e^{-1/2})\Bigg{)}(\Gamma_{k}^{0}-\alpha\Gamma_{k}^{+})$

for $k\geq k_{1}$. Since the right-hand side is nonnegative we conclude that
$\Gamma_{k+1}^{0}\geq e^{1/2}\alpha\Gamma_{k+1}^{+}$. Proceeding inductively,
we find that $\Gamma_{k}^{0}\geq e^{1/2}\alpha\Gamma_{k}^{+}$ for every
$k>k_{1}$, and hence $e^{1/2}\alpha\in I$.

We therefore conclude that either $J=\emptyset$ or $I=(0,\infty)$. If
$I=(0,\infty)$ then $\Gamma_{k}^{+}\leq o(1)\Gamma_{k}^{0}$ and hence
$\Gamma_{k}^{+}+\Gamma_{k}^{-}\leq o(1)\Gamma_{k}^{0}$. On the other hand, if
$J=\emptyset$ then $\Gamma_{k}^{0}\leq o(1)\Gamma_{k}^{+}$ and hence
$\Gamma_{k}^{0}+\Gamma_{k}^{-}\leq o(1)\Gamma_{k}^{+}$. ∎

We now establish that $\Gamma_{k}^{0}+\Gamma_{k}^{-}\leq o(1)\Gamma_{k}^{+}$.
The argument makes use of the fact that $\bar{M}_{\tau}$ is noncompact—recall
that this allowed us to assume $\tfrac{\partial u}{\partial z}<0$.

######  Proposition 5.10.

As $k\to\infty$ we have $\Gamma_{k}^{0}+\Gamma_{k}^{-}\leq
o(1)\Gamma_{k}^{+}$.

###### Proof.

By Lemma 5.9, if the claim is false then $\Gamma_{k}^{+}+\Gamma_{k}^{-}\leq
o(1)\Gamma_{k}^{0}$. Suppose this is the case. Fix a sequence $\tau_{k}\leq-k$
such that

$\hat{u}_{k}(z,\theta):=\chi(\delta_{k}^{r}z)u(z,\theta,\tau_{k})$

satisfies

$\|\hat{u}_{k}\|_{\mathcal{H}}=\Gamma_{k}^{1/2}.$

Since $\Gamma_{k}^{+}+\Gamma_{k}^{-}\leq o(1)\Gamma_{k}^{0}$, the sequence
$\hat{u}_{k}/\|\hat{u}_{k}\|_{\mathcal{H}}$ converges to a limit
$\hat{u}(z,\theta)$ with respect to the norm $\|\cdot\|_{\mathcal{H}}$.
Passing to a further subsequence, we may assume that
$\hat{u}_{k}/\|\hat{u}_{k}\|_{\mathcal{H}}\to\hat{u}$ pointwise almost
everywhere in $\mathbb{R}\times S^{n-1}$. Since $\frac{\partial u}{\partial
z}<0$ we conclude that for each fixed $\theta$ the function
$z\mapsto\hat{u}(z,\theta)$ is nonincreasing. On the other hand, the limit
$\hat{u}$ lies in the zero-eigenspace of $\mathcal{L}$, which is to say that

$\hat{u}(z,\theta)=\alpha\bigg{(}\frac{z^{2}}{a}-2\bigg{)}+\beta_{i}\frac{z}{\sqrt{a}}\theta^{i},$

for some constants $\alpha$ and $\beta_{i}$. The right-hand side is monotone
in $z$ if and only if $\alpha=0$ and $\beta_{i}=0$ for each $1\leq i\leq n$.
With this we have reached a contradiction, since
$\|\hat{u}\|_{\mathcal{H}}=1$. ∎

We are now able to prove Proposition 5.1.

###### Proof of Proposition 5.1.

Using Proposition 5.10 we conclude that

$\Gamma_{k+1}^{+}\leq
e^{-1}\Gamma_{k}^{+}+C\varepsilon_{k}^{1/2}\Gamma_{k}^{+}\leq
e^{-1/2}\Gamma_{k}^{+}$

whenever $k$ is sufficiently large. Iterating this estimate we find that
$\Gamma_{k}^{+}\leq O(e^{-k/2})$, and hence $\Gamma_{k}\leq O(e^{-k/2})$.
Using the estimate

$\varepsilon_{k}=\sup_{j\geq k}\delta_{j}\leq C\sup_{j\geq
k}\Gamma_{j}^{1/2}\leq C\Gamma_{k}^{1/2}$

we obtain $\varepsilon_{k}\leq O(e^{-k/4})$ and hence

$\Gamma_{k+1}^{+}\leq e^{-1}\Gamma_{k}^{+}+Ce^{-k/8}\Gamma_{k}^{+}.$

Iterating this estimate we obtain $\Gamma_{k}^{+}\leq O(e^{-k})$ and hence
$\Gamma_{k}\leq O(e^{-k})$.

We have

$\sup_{\tau\leq-k}\sup_{|z|\leq L}|u|=\delta_{k}.$

Using $\delta_{k}\leq C\Gamma_{k}^{1/2}$ we conclude that

$\sup_{\tau\leq-k}\sup_{|z|\leq L}|u|\leq O(e^{-k/2}).$

The claim follows. ∎

We conclude this section with an important consequence of Proposition
5.1—namely, that $\max_{M_{t}}G$ is bounded from below by a positive constant
which is independent of time.

A straightforward computation using Proposition 4.1 shows that if $L>L_{0}$ is
sufficiently large depending on $n$ and $\gamma$, and $a$ is sufficiently
large depending on $L$, then we have

(13) $\Psi_{a}(L)-\Psi_{a}(L-1)<-a^{-2}.$

Moreover, Proposition 5.1 implies there is a positive constant $K=K(L)$ such
that

$M_{t}\cap\Big{\\{}|z|\leq
L(-t)^{1/2}\Big{\\}}\subset\Big{\\{}(z,y):\Big{|}|y|-(-2\gamma(0,1,\dots
1)t)^{1/2}\Big{|}\leq K\Big{\\}}$

whenever $-t$ is sufficiently large.

######  Proposition 5.11.

For $K$ as above we have $\max_{M_{t}}G\geq-1/8K$ for every $t\leq 0$.

###### Proof.

The proof is a direct adaptation of [BC19, Proposition 3.3]. We make use of
the hypersurfaces

$\Sigma_{a}=\\{(y,z):y=\Psi_{a}(z)\theta,\,\theta\in S^{n-1},\,L_{0}\leq z\leq
a\\}$

constructed in Section 4. The hypersurfaces

$\displaystyle\Sigma_{a,t}$ $\displaystyle:=(-t)^{1/2}\Sigma_{a}-(0,Ka^{2})$
$\displaystyle=\\{(y,z):y=(-t)^{1/2}\Psi_{a}((-t)^{-1/2}(z+Ka^{2}))\theta,\,\theta\in
S^{n-1},$ $\displaystyle\hskip 170.71652pt(-t)^{1/2}L_{0}-Ka^{2}\leq
z\leq(-t)^{1/2}a-Ka^{2}\\}$

form a $G$-flow. As $t\to-\infty$ the rescaled hypersurfaces
$(-t)^{-1/2}M_{t}$ converge in $C^{\infty}_{\operatorname{loc}}$ to the
cylinder $\\{|y|^{2}=2\gamma(0,1,\dots,1)\\}$, and the hypersurfaces

$(-t)^{-1/2}(\Sigma_{a,t}\cap\\{z\geq(-t)^{1/2}L_{0}\\})$

converge to a compact subset of $\\{|y|^{2}<2\gamma(0,1,\dots,1)\\}$. So when
$-t$ is sufficiently large depending on $a$ we have

$\Sigma_{a,t}\cap\\{z\geq(-t)^{1/2}L_{0}\\}\subset\Omega_{t},$

where $\Omega_{t}$ is the open region bounded by $M_{t}$.

Since $\Psi_{a}(z)$ is concave, as a consequence of (13) we have

$\displaystyle\Psi_{a}(L+Ka^{2}(-t)^{-1/2})$
$\displaystyle\leq\Psi_{a}(L)+(\Psi_{a}(L)-\Psi_{a}(L-1))Ka^{2}(-t)^{-1/2}$
$\displaystyle<(2\gamma(0,1,\dots,1))^{1/2}-K(-t)^{-1/2}$

for $-t\geq 4K^{2}a^{2}$ and $a$ sufficiently large. On the other hand,

$\\{(y,z):|y|\leq(-2\gamma(0,1,\dots,1)t)^{1/2}-K,\,z=(-t)^{1/2}L\\}\subset\Omega_{t}\cap\\{z=(-t)^{1/2}L\\}$

when $-t$ is sufficiently large. It follows that for $-t\geq 4K^{2}a^{2}$ and
$a$ sufficiently large we have

$\Sigma_{a,t}\cap\\{z=(-t)^{1/2}L\\}\subset\Omega_{t}\cap\\{z=(-t)^{1/2}L\\}.$

Therefore, the parabolic maximum principle implies that

$\Sigma_{a,t}\cap\\{z\geq(-t)^{1/2}L\\}\subset\Omega_{t}$

for $-t\geq 4K^{2}a^{2}$ and $a$ sufficiently large. When $t=-4K^{2}a^{2}$ the
tip of $\Sigma_{a,t}$ is located at $(0,Ka^{2})=(0,-t/4K)$. From this we
conclude that $M_{t}\cap\\{z\geq-t/4K\\}$ is nonempty for all sufficiently
large $-t$.

It follows that $\sup_{M_{t}}G\geq-1/8K$ at some sequence of times tending to
$-\infty$. On the other hand, the differential Harnack inequality implies that
$G$ is strictly increasing in time at every point. The claim follows. ∎

##  6\. The Neck Improvement Theorem

In this section we generalise the Neck Improvement Theorem for solutions of
mean curvature flow [BC21] to hypersurfaces moving with speed $G$.

To begin with we recall the following definition from [BC21].

######  Definition 6.1.

A set of vector fields
$\mathcal{K}=\\{K_{\alpha}:1\leq\alpha\leq\tfrac{1}{2}n(n-1)\\}$ on
$\mathbb{R}^{n+1}$ is a normalised set of rotation vector fields if there
exists an orthonormal basis
$\\{J_{\alpha}:1\leq\alpha\leq\tfrac{1}{2}n(n-1)\\}$ of
$\operatorname{so}(n)\subset\operatorname{so}(n+1)$, a matrix $S\in O(n+1)$
and a point $q\in\mathbb{R}^{n+1}$ such that

$\displaystyle K_{\alpha}(x)=SJ_{\alpha}S^{-1}(x-q).$

Next we define a notion of quantitative almost-rotational symmetry.

######  Definition 6.2.

Consider a $G$-flow $M_{t}$. A point $\bar{x}\in M_{\bar{t}}$ is called
$\varepsilon$-symmetric if there exists a normalised set of rotation vector
fields $\mathcal{K}=\\{K_{\alpha}:1\leq\alpha\leq\tfrac{1}{2}n(n-1)\\}$ such
that

$\max_{\alpha}|K_{\alpha}|\,G\leq 10\gamma(0,1,\dots,1)$

at the point $(\bar{x},\bar{t})$ and

$\max_{\alpha}|\langle K_{\alpha},\nu\rangle|\,G\leq\varepsilon$

in the neighbourhood $\hat{\mathcal{P}}(\bar{x},\bar{t},10,100)$.

We now establish the Neck Improvement Theorem for $G$-flows. In fact, the
proof reduces to the proof for mean curvature flow, since up to a change of
variables the linearised $G$-flow over a shrinking neck is identical to the
linearised mean curvature flow over a shrinking neck. We compute the
linearisation and then refer to [BC21] for the rest of the argument.

######  Theorem 6.3.

Let $M_{t}$ be a uniformly parabolic $G$-flow. There exists a large constant
$L$ (depending on $n$, $\gamma$ and
$\operatorname{dist}(\lambda/|\lambda|,\partial\Gamma)$), and a small constant
$\varepsilon_{1}$ depending on $L$, with the following property. Let
$(\bar{x},\bar{t})$ be a point in spacetime such that every point in the
parabolic neighbourhood $\hat{\mathcal{P}}(\bar{x},\bar{t},L,L^{2})$ lies at
the center of an $\varepsilon_{1}$-neck and is $\varepsilon$-symmetric for
some $\varepsilon\leq\varepsilon_{1}$. Then $(\bar{x},\bar{t})$ is
$\tfrac{\varepsilon}{2}$-symmetric.

###### Proof.

Without loss of generality, we assume $\bar{t}=-1$. After parabolically
rescaling we can arrange that $G(\bar{x},-1)=(\gamma(0,1,\dots,1)/2)^{1/2}$.
Since every point in the parabolic neighbourhood
$\hat{\mathcal{P}}(\bar{x},\bar{t},L,L^{2})$ lies at the center of an
$\varepsilon_{1}$-neck, we can then approximate $M_{t}$ by the cylinder

$S^{n-1}((-2\gamma(0,1,\dots,1)t)^{1/2})\times\mathbb{R}$

in $\hat{\mathcal{P}}(\bar{x},\bar{t},L,L^{2})$, up to errors which are
bounded by $C(L)\varepsilon_{1}$ in the $C^{10}$-norm.

For every $(x_{0},t_{0})\in\hat{\mathcal{P}}(\bar{x},-1,L,L^{2})$ there exists
a normalised set of rotation vector fields
$\\{K_{\alpha}^{(x_{0},t_{0})}:1\leq\alpha\leq\frac{1}{2}n(n-1)\\}$ such that
$\max_{\alpha}|K_{\alpha}|G\leq 10\gamma(0,1,\dots,1)$ at $(x_{0},t_{0})$ and

$\max_{\alpha}|\langle K_{\alpha}^{(x_{0},t_{0})},\nu\rangle|G\leq\varepsilon$

in $\hat{\mathcal{P}}(x_{0},t_{0},10,100)$. As in [BC21, Theorem 4.4], we may
assume without loss of generality that

(14)
$\displaystyle\sup_{B(0,10\gamma(0,1\dots,1)L)}\max_{\alpha}|K^{(\bar{x},-1)}_{\alpha}-K^{(x_{0},t_{0})}_{\alpha}|\leq
C(L)\varepsilon$

for each $(x_{0},t_{0})\in\hat{\mathcal{P}}(\bar{x},-1,L,L^{2})$. This implies
that

$|\langle K^{(\bar{x},-1)},\nu\rangle|\leq C(L)\varepsilon$

at each point in $\hat{\mathcal{P}}(\bar{x},-1,L,L^{2})$. We may also assume
that the common axis of rotation of the vector fields
$K_{\alpha}^{(\bar{x},-1)}$ is the $x_{n+1}$-axis (that is,
$K_{\alpha}=J_{\alpha}x$ for some orthonormal basis
$\\{J_{\alpha}:1\leq\alpha\leq\tfrac{1}{2}n(n-1)\\}$ of
$\operatorname{so}(n)\subset\operatorname{so}(n+1)$) and that $\bar{x}$ lies
in the hyperplane $\\{x_{n+1}=0\\}$. Let us define
$\bar{K}_{\alpha}:=K^{(\bar{x},-1)}_{\alpha}$.

Consider $M_{t}$ as a graph over the $x_{n+1}$-axis, which we equip with the
coordinate $z$, so that

$\displaystyle\Big{\\{}\big{(}r(\theta,z,t)\theta,z\big{)}:\theta\in
S^{n-1},\,z\in\left[-\tfrac{3}{4}L,\tfrac{3}{4}L\right]\Big{\\}}\subset
M_{t}.$

Exactly as in [BC21], we have

(15)
$|\langle\bar{K}_{\alpha},\nu\rangle+\operatorname{div}_{S^{n-1}}(r(\theta,z,t)J_{\alpha}\theta)|\leq
C(L)\varepsilon_{1}\varepsilon$

in $\hat{\mathcal{P}}(\bar{x},-1,L,L^{2})$.

Fix $\alpha$ and set
$\bar{u}_{\alpha}:=\langle\overline{K}_{\alpha},\nu\rangle$. With respect to
any orthonormal frame on $M_{t}$, $\bar{u}_{\alpha}$ satisfies

$\frac{\partial\bar{u}_{\alpha}}{\partial
t}=\dot{\gamma}^{ij}(\nabla_{i}\nabla_{j}\bar{u}_{\alpha}+A_{i}^{k}A_{kj}\bar{u}_{\alpha}).$

Define $a:=\dot{\gamma}^{1}(0,1,\dots,1)$. By Lemma 2.14 we have

$\displaystyle\bigg{|}\dot{\gamma}^{ij}\nabla_{i}\nabla_{j}\bar{u}_{\alpha}-a\frac{\partial^{2}\bar{u}_{\alpha}}{\partial
z^{2}}-\frac{1}{(-2(n-1)t)}\Delta_{S^{n-1}}\bar{u}_{\alpha}\bigg{|}\leq
C(L)\varepsilon_{1}(|\nabla\bar{u}_{\alpha}|+|\nabla^{2}\bar{u}_{\alpha}|).$

Standard interior estimates for parabolic equations imply
$|\nabla\bar{u}_{\alpha}|+|\nabla^{2}\bar{u}_{\alpha}|\leq C(L)\varepsilon$,
and hence

$\bigg{|}\dot{\gamma}^{ij}\nabla_{i}\nabla_{j}\bar{u}_{\alpha}-a\frac{\partial^{2}\bar{u}_{\alpha}}{\partial
z^{2}}-\frac{1}{(-2(n-1)t)}\Delta_{S^{n-1}}\bar{u}_{\alpha}\bigg{|}\leq
C(L)\varepsilon_{1}\varepsilon$

for $|z|\leq\frac{L}{4}$ and $t\in[-\frac{L^{2}}{16},-1]$. Lemma 2.11 and
Lemma 2.14 imply

$\bigg{|}\dot{\gamma}^{ij}A_{i}^{k}A_{kj}\bar{u}_{\alpha}-\frac{1}{(-2t)}\bar{u}_{\alpha}\bigg{|}\leq
C(L)\varepsilon_{1}\varepsilon.$

Putting this all together, we obtain

$\bigg{|}\frac{\partial\bar{u}_{\alpha}}{\partial
t}-a\frac{\partial^{2}\bar{u}_{\alpha}}{\partial
z^{2}}-\frac{1}{(-2(n-1)t)}\Delta_{S^{n-1}}\bar{u}_{\alpha}-\frac{1}{(-2t)}\bar{u}_{\alpha}\bigg{|}\leq
C(L)\varepsilon_{1}\varepsilon$

for $|z|\leq\frac{L}{4}$ and $t\in[-\frac{L^{2}}{16},-1]$.

Let us define $u_{\alpha}(\theta,z,t):=\bar{u}_{\alpha}(\theta,a^{-1/2}z,t)$.
We then have

(16) $\bigg{|}\frac{\partial u_{\alpha}}{\partial
t}-\frac{\partial^{2}u_{\alpha}}{\partial
z^{2}}-\frac{1}{(-2(n-1)t)}\Delta_{S^{n-1}}u_{\alpha}-\frac{1}{(-2t)}u_{\alpha}\bigg{|}\leq
C(L)\varepsilon_{1}\varepsilon$

for $|z|\leq\tfrac{a^{1/2}}{4}L$ and $t\in[-\frac{L^{2}}{16},-1]$. In
addition, (15) gives

(17)
$|u_{\alpha}(\theta,z,t)+\operatorname{div}_{S^{n-1}}(r(\theta,z,t)J_{\alpha}\theta)|\leq
C(L)\varepsilon_{1}\varepsilon$

for $|z|\leq a^{1/2}L$ and $t\in[-L^{2},-1]$.

Notice that the linear operator on the left-hand side of (16) is completely
independent of the speed $\gamma$. We may therefore appeal to Step 5 of [BC21,
Theorem 4.4] to conclude that, for each $1\leq\alpha\leq\tfrac{1}{2}n(n-1)$,
there are vectors $A_{\alpha}$ and $\tilde{B}_{\alpha}$ such that
$|A_{\alpha}|\leq C(L)\varepsilon$, $|\tilde{B}_{\alpha}|\leq C(L)\varepsilon$
and

$|u_{\alpha}(\theta,z,t)-\langle
A_{\alpha},\theta\rangle-\langle\tilde{B}_{\alpha},\theta\rangle z|\leq
CL^{-1/(n-1)}\varepsilon+C(L)\varepsilon_{1}\varepsilon$

for $|z|\leq 20\gamma(0,1,\dots,1)a^{1/2}$ and
$t\in[-400\gamma(0,1,\dots,1)^{2},-1]$. Undoing the rescaling $z\to a^{-1/2}z$
and setting $B_{\alpha}:=a^{1/2}\tilde{B}_{\alpha}$, we obtain

$|\langle\bar{K}_{\alpha},\nu\rangle-\langle A_{\alpha},\theta\rangle-\langle
B_{\alpha},\theta\rangle z|\leq
CL^{-1/(n-1)}\varepsilon+C(L)\varepsilon_{1}\varepsilon$

for $|z|\leq 20\gamma(0,1,\dots,1)$ and
$t\in[-400\gamma(0,1,\dots,1)^{2},-1]$. Step 7 and Step 8 of [BC21] show that