# On the Stability of Cylindrical Singularities of the Mean Curvature Flow

Jingxuan Zhang

###### Abstract

We study the rescaled mean curvature flow (MCF) of hypersurfaces that are
global graphs over a fixed cylinder of arbitrary dimensions. We construct an
explicit stable manifold for the rescaled MCF of finite codimensions in a
suitable configuration space. For any initial hypersurface from this stable
manifold, we construct a unique global solution to the rescaled MCF, and
derive precise asymptotics for these solutions that are valid for all time.
Using these asymptotics, we prove asymptotic stability of cylindrical
singularities of arbitrary dimensions under generic initial perturbations.

As a by-product, for any flow of hypersurfaces evolving according to the MCF
that enters this stable manifold at any time and first develops a singularity
at a subsequent time, we give a simple proof of the uniqueness of tangent
flow, first established by Colding and Minicozzi. Moreover, in this case we
show the unique singularity profile is determined by the hypersurface profile
when the flow enters the stable manifold.

For all results in this paper, there is no symmetry or solitonic assumption.

##  1 Introduction

Consider the mean curvature flow (MCF) for a family of hypersurfaces given by
immersions
$X(\cdot,t):\mathbb{R}^{n-k}\times\mathbb{R}^{k+1}\to\mathbb{R}^{n+1}$,
satisfying

$\partial_{t}X=-H(X)\nu(X).$ (1)

In this paper, we are interested in the dynamical behaviour of a solution $X$
to (1), which first develops a singularity at $0\in\mathbb{R}^{n+1},\,t=T>0$.

Equation (1) is invariant under rotation, translation, and parabolic
rescaling. Motivated by these symmetries, we consider the following time-
dependent rescaling for a solution to (1) as follows:

$X(x,\omega,t)=\lambda(t)g(t)Y(y(x,t),\omega,\tau)+(0,z(t)),$ (2)

where the immersion $Y$ is defined through this relation, and

$\displaystyle g(t)\in SO(n+1),\quad g(0,x^{\prime})=(0,x^{\prime})\text{ for
every $x^{\prime}\in\mathbb{R}^{k+1}$},$ (3) $\displaystyle
z(t)\in\mathbb{R}^{k+1},$ (4)
$\displaystyle\lambda(t):=\left(2\int_{t}^{T}a(t^{\prime})\,dt^{\prime}\right)^{1/2},\quad
a(t)>0,$ (5) $\displaystyle y(x,t):=\lambda(t)^{-1}x,$ (6)
$\displaystyle\tau(t):=\int_{0}^{t}\lambda(t^{\prime})^{-2}\,dt^{\prime}.$ (7)

Notice that we do not fix the parameter $\sigma:=(g,z,a)$, but rather regard
this as a path to be determined in the manifold

$\Sigma:=SO(n+1)\times\mathbb{R}^{k+1}\times\mathbb{R}_{>0}.$ (8)

The condition (5) shows that $\lambda(t)$ is uniquely determined by the path
$a(t)>0$. Indeed, this $\lambda(t)$ is the unique solution to the Cauchy
problem

$\dot{\lambda}(t)\lambda(t)=-a(t),\quad\lambda(T)=0.$

Here and in the remaining of this section, the dot denotes differentiation
w.r.t. the fast time $t$-variable. By definition, the terminal condition on
$\lambda$ ensures the rescaling (2) gives rise to a tangent flow
$Y=Y(y,\omega,\tau)$ in the microscopic variable $y$ and slow time variable
$\tau$.

Direct computation shows $X$ of the form (2) solves (1) if and only if the
pair $(\sigma,Y)$ solves

$\partial_{\tau}Y=-H(Y)\nu(Y)-a\left\langle y,\,\nabla_{y}\right\rangle
Y+aY-g^{-1}\partial_{\tau}gY-\lambda^{-1}g^{-1}\partial_{\tau}z.$ (9)

To get (9), one uses the relations $\lambda\dot{\lambda}=-a$,
$\lambda^{2}\dot{y}=ay$, and $\dot{\tau}=\lambda^{-2}$, which follow from
(5)-(7) respectively. We call (9) the rescaled mean curvature flow.

The rescaled MCF (9) has the following family of stationary solutions:

$\displaystyle Y_{a_{0}}\equiv\left(y,\sqrt{\frac{k}{a_{0}}}\omega\right),$
(10) $\displaystyle\sigma_{0}\equiv(g_{0},z_{0},a_{0})\in\Sigma.$ (11)

Geometrically, the three components in $\sigma_{0}$ consist of a rotation
$g_{0}$ of the cylindrical axis, a transversal translation $z_{0}$, and a
dilation by a factor of $\sqrt{k/a_{0}}$. The pair $(\sigma,Y)$ corresponds to
a cylinder with unit radius along the $y$ axis, transformed by the symmetry
$\sigma_{0}$ as in (2).

###  1.1 Main Results

In the remaining of this paper, we seek maximal solution $X$ to (1) on the
spatial domain $\mathbb{R}^{n-k}\times\mathbb{S}^{k}$ and the time interval
$0\leq t<T$, of the form

$X(x,\omega,t)=\lambda(t)g(t)\underbrace{\left(y(x,t),\left(\sqrt{\frac{k}{a(t)}}+\xi(y(x,t),\omega,\tau(t))\right)\omega\right)}_{\text{as
$Y$ in \eqref{1.1}}}+(0,z(t)).$ (12)

In terms of the blow-up variables $y,\,\tau$ from (6)-(7), this amounts to
finding a pair $(\sigma,Y)$ that solves the rescaled MCF (9) for all $\tau\geq
0$. Here $\sigma=\sigma(\tau)$ is a path of parameters. $Y=Y(\xi)$ is a flow
of graphs over a fixed cylinder, parametrized by a path of functions
$\xi(\cdot,\tau):\mathbb{R}^{n-k}\times\mathbb{S}^{k}\to\mathbb{R}_{\geq
0},\,\tau\geq 0$, as in (12).

With this convention as well as the rescaling (3)-(7) understood, the main
result of this paper is the following assertions about the rescaled MCF (9):

######  Theorem 1.

Let $X^{s}(a),\,s\geq 2,\,a>0$ be the weighted Sobolev space defined in (22).
There exists $0<\delta\ll 1$ s.th. the following holds:

  1. 1.

(Global existence) For every $a_{0}\geq 1/2+2\delta$, there exists a subspace
$\mathcal{S}\subset X^{s}(a_{0})$ with finite codimensions, together with a
map

$\Phi:\mathcal{B}_{\delta}\cap\mathcal{S}\to X^{s}\equiv
X^{s}(1/2),\quad\mathcal{B}_{\delta},\mathcal{S}\text{ as in
Definition~{}\ref{defn3.2}},$

satisfying

$\displaystyle\left\lVert\Phi(\eta_{0})\right\rVert_{s}$
$\displaystyle\lesssim\left\lVert\eta_{0}\right\rVert_{X^{s}}^{2},$ (13)
$\displaystyle\left\lVert\Phi(\eta_{0})-\Phi(\eta_{1})\right\rVert_{X^{s}}$
$\displaystyle\lesssim\delta\left\lVert\eta_{0}-\eta_{1}\right\rVert_{X^{s}},$
(14)

for every $\eta_{0},\,\eta_{1}\in\mathcal{B}_{\delta}\cap\mathcal{S}$, as well
as the following properties:

For every $\eta_{0}\in\mathcal{B}_{\delta}\cap\mathcal{S}$, there exists a
unique global solution $(\sigma,Y)$ to the rescaled MCF (9) with initial
configuration

$Y|_{\tau=0}=(y,(\sqrt{k/a_{0}}+\eta_{0}+\Phi(\eta_{0})\omega)\quad$

on $\mathbb{R}^{n-k}\times\mathbb{S}^{n-k}\times\mathbb{R}_{\geq 0}$.

Moreover, this solution $Y$ is uniquely determined by the decomposition

$Y=(y,(\sqrt{k/a}+\xi(y,\omega,\tau))\omega)\quad(y\in\mathbb{R}^{n-k},\,\omega\in\mathbb{S}^{k}),$
(15)

where $a=a(\tau)$ is a component of the path
$\sigma(\tau)=(g(\tau),z(\tau),a(\tau))\in\Sigma$, and $\xi=\xi(\cdot,\tau)$
is a path of functions on $\mathbb{R}^{n-k}\times\mathbb{S}^{n-k}$. The path
$(\sigma,\xi)$ lies in the space

$(\sigma,\xi)\in Lip([0,\infty),\Sigma)\times(C([0,\infty),X^{s})\cap
C^{1}([0,\infty),X^{s-2})).$

  2. 2.

(Effective dynamics) Moreover, the path $\sigma$ evolves according to the
following system of ODEs:

$\displaystyle\partial_{\tau}\sigma=\vec{F}(\sigma)\xi+\vec{M}(\sigma,\xi)\quad\text{
for a.e. }\tau,$ (16) $\displaystyle\sigma(0)=(\delta_{ij},0,a_{0})\in\Sigma,$
(17)

where the vector fields $\vec{F},\,\vec{M}$ are as in (49).

  3. 3.

(Dissipative estimates) Moreover, the path $\sigma(\tau)$ dissipates to
$\sigma(0)$, with the decay estimate

$\left\lvert\sigma(\tau)-\sigma(0)\right\rvert\lesssim\delta\left\langle\tau\right\rangle^{-1},\quad\tau\geq
0.$ (18)

Here and below, we write
$\left\langle\cdot\right\rangle:=(1+\left\lvert\cdot\right\rvert^{2})^{1/2}$.

Moreover, the function $\xi$ in the decomposition (15) is non-negative for all
$\tau$, and dissipates to zero, with the decay estimate

$\left\lVert\xi(\cdot,\tau)\right\rVert_{X^{s}}\leq\delta\left\langle\tau\right\rangle^{-2},\quad\tau\geq
0.$ (19)

The result above are obtained by studying a quasilinear PDE (27) in $\xi$,
coupled to a system of ODEs, (43)-(45) in the parameter $\sigma$. Up to a
rigid motion and a dilation, initial configuration (17) can be replaced with
any $\sigma_{0}\in\Sigma$.

In [MR2993752]*Thm. 4.31, Colding and Minicozzi have shown that the
cylindrical singularity of the MCF is $F$-unstable. In terms of the PDE (27),
this means that the static solution $\xi=0$ is linearly unstable. In view of
this, Theorem 1 above states that for a generic (i.e. finite codimensional)
class of initial perturbations, $\xi=0$ is asymptotically stable under the
evolution (27). This gives a justification of the generality of cylindrical
singularity based on the PDE ground.

Indeed, at the intuitive level, the fact that cylinders are $F$-unstable has
to do with the symmetry of the (rescaled) MCF. The broken symmetries of the
cylinder, e.g. transversal translation to the axis, generates some unstable
modes to the linearized operator at the cylinder,

$L(a):=-\Delta_{y}+a\left\langle
y,\,\nabla_{y}(\cdot)\right\rangle-\frac{a}{k}\Delta_{\omega}-2a.$

Here $a>0$ corresponds to the cylindrical radius. In view of this, the space
$\mathcal{S}$ in Theorem 1, defined in Definition 4 below, removes all the
zero-unstable modes of $L(a)$ from the configuration space.

Then, as customary in the analysis of solitons, one would like to write down a
number of ODEs, which governs the motion of a frame of symmetries that
eliminates the zero-unstable modes of $L(a)$ for all subsequent time. In our
situation, this gives rise to the ODEs (16)-(17) above. These ODEs are called
the modulation equations, and their use in the study of solitary wave dynamics
dates back to [MR1071238, MR1170476]

The twist here is that not all of the zero-unstable modes of $L(a)$ can be
eliminated by the modulation method. First, there is the translation along the
cylindrical axis, which incurs unstable modes protected by the symmetry of the
cylinder. Secondly, as pointed out in [MR3374960]*Sect. 3.2, there are certain
zero modes that are not due to any symmetries.

To eliminate these, following [MR2480603], we introduce the second order
correction $\Phi$, constructed explicitly in Definition 5 below. In
[MR2480603], Schlag has studied a $L^{2}$-supercritical nonlinear Schrödinger
equation, where some unstable mode arises which is not due to any symmetry.
Here we implement Schlag’s method, which is essentially a renormalization
technique, to study the rescaled MCF (27).

Let $\Phi$ be the map constructed in Theorem 5. In terms of invariant manifold
theory, the set

$M=M_{\delta,a_{0}}=\left\\{X:(x,\omega)\mapsto(x,(\sqrt{k/a_{0}}+\eta+\Phi(\eta))\omega)\mid\eta\in\mathcal{B}_{\delta}\cap\mathcal{S}\right\\}$
(20)

can be viewed as a stable manifold for (1), or, more appropriately, for the
rescaled MCF (9). We discuss the geometric property of $M$ in Section 6. In
this paper, our focus is on the “shadow” of a flow generated by an initial
configuration in $M$ under the rescaled MCF, namely (16)-(17), and dissipative
estimates of the form (18)-(19). Indeed, $M$ is a stable manifold precisely in
the sense that the infinite dimensional flow of hypersurfaces is well-
approximated by the finite system of ODEs (16)-(17) in the space $\Sigma$.

In physical terms, Theorem 1 is an adiabatic theory for MCF near a cylindrical
singularity. In the physics literature, especially in classical field theory,
the method of adiabatic approximation involves the decomposition of a flow
into a slowly evolving main part, together with a small fluctuation. In our
situation, for a rescaled flow starting in $M$, the adiabatic approximation
are the ODEs (16), which govern the motion of a main part (a transformed
cylinder by symmetry) moving at the speed of $O(\delta)$. The fluctuation is
the remainder $\xi$.

In view of the original MCF, the adiabatic dynamics (16)-(17) of $\sigma$
implies the following:

######  Corollary 1 (uniqueness of tangent flow, c.f. [MR3374960]*Thm. 0.2).

Fix $0<\delta\ll 1$ and $a_{0}\geq 1/2+2\delta$. Suppose $X(\cdot,t)$ is a
maximal solution to (1) that first develops a singularity at
$(0,T)\in\mathbb{R}^{n+1}\times\mathbb{R}_{>0}$. Suppose up to a dilation by a
positive factor, at some $t_{0}<T$ the hypersurface $X(\cdot,t_{0})$ lies in
the stable manifold $M_{\delta,a_{0}}$ given in (20). Then the tangent flow of
$X$ is unique.

Moreover, the limit cylinder is parametrized by $(y,\sqrt{k/a_{0}}\omega)$
with $y\in\mathbb{R}^{n-k}$ and $\omega\in\mathbb{S}^{k}$.

The uniqueness of tangent flow is established in [MR3349836, MR3374960,
choi2021ancient] under various assumptions. In these papers, the authors show
that assuming a flow develops a cylindrical singularity of a certain profile,
then the latter is independent of the tangent flow.

The point of Corollary 1 is that for a flow starting from $M$, we have
explicit information of the limit of the tangent flow given the initial
profile. In fact, the uniqueness of tangent flow is a by-product of the
equations (16)-(17). By assumption, Corollary 1 applies only to surfaces that
at some $t_{0}<T$ that are $X^{s}(a_{0})$-close to a cylinder. This is as
expected for an adiabatic theory, whose validity relies on the fact that the
evolving configurations are near equilibira.

Theorem 1 and the corollary are proved in Section 3.3.

###  1.2 Historical Remarks

In a series of papers [MR2465296, MR3397388, MR3803553, MR4303943,
gang2021dynamics, zhou2021nondegenerate], Zhou Gang and several co-authors
have developed an adiabatic theory for cylindrical singularity of certain
dimensions. Here, as remarked earlier, by adiabatic theory we mean the method
of slow motion approximation. Indeed, the key technique in those papers is the
decomposition of a solution evolving according to (1) into a slowly evolving
main part of the form

$V_{a,B}=\sqrt{\frac{k+\tfrac{1}{2}\left\langle By,\,y\right\rangle}{a}},$
(21)

together with a remainder. Here $a(t)>0$, and $B(t)$ is a path of symmetric
real matrix of size $n-k$ to be determined. (These play the role of $\sigma$
in (12).) Then one studies (1) in a suitable configuration space, and tries to
derive dissipative estimate for the remainder. This line of thought is an
essential motivation to the present paper.

Ultimately, the adiabatic theory of [MR2465296, MR3397388, MR3803553,
MR4303943, gang2021dynamics, zhou2021nondegenerate] has to do with the zero-
unstable modes of the linearized operator at a cylinder of radius $a>0,$ as
mentioned earlier. Indeed, consider the expression (21). This $V$ solves the
static equation

$-a\left\langle y,\,\nabla_{y}\right\rangle v+av-kv^{-1}=0,$

which is obtained by ignoring all time derivatives and all second order
spatial derivatives in the graphical form of the rescaled MCF, (27). (In
multi-scale analysis and classical field theory, solution of such truncated
equation is often called the adiabatic solution.) For $\left\lvert
B\right\rvert\ll 1$, expanding $V$ in $y$ gives
$V=\sqrt{k/a}(1+\tfrac{1}{4k}\left\langle By,\,y\right\rangle+O(\left\lvert
B\right\rvert^{2}\left\lvert y\right\rvert^{4}))$, where the first term
corresponds to the unstable modes with eigenvalue $-2a$ of the operator
$L(a)$, and the second term, a homogeneous quadratic polynomial in $y$,
corresponds to the non-symmetry zero modes mentioned earlier. This expansion
of $V$ is valid for $\left\lvert\left\langle
By,\,y\right\rangle\right\rvert\ll 1$, which holds on a bounded region for
sufficiently small $\left\lvert B\right\rvert$. On such a region, it is
possible to write down a system of modulation equations for $B$ so as to
eliminate the effect of the non-symmetry zero modes. The analysis of the
dynamics of the matrix $B$ constitutes the most difficult part in [MR2465296,
MR3397388, MR3803553, MR4303943, gang2021dynamics, zhou2021nondegenerate].
Then there is another modulation equation for $a$ to eliminate the scaling
instability, together with some center manifold analysis to control the rest
of zero-unstable modes. The authors of the above mentioned papers have
developed an extension-improvement method (what they call the “bootstrap
machines”) to make this rigorous. This method culminates in a proof of the
mean convexity conjecture with $\mathbb{R}^{3}\times\mathbb{S}^{1}$
singularity in [MR4303943].

In the past decade, in a series of papers [MR2993752, MR3374960, MR3349836,
MR3381496, MR3489702, MR3521319, MR3602529, MR3662439, MR4176547,
chodosh2020mean, chodosh2021mean, choi2021ancient, choi2021ancient2], a group
of prominent geometers have developed some remarkable regularity results for
cylindrical singularities, using various methods of geometric analysis and
geometric measure theory. Here we single out [MR3374960], in which Colding and
Minicozzi developed the Łojasiewicz inequalities for the rescaled MCF. Some
estimates from that paper are used for the developments in Sects. 2-6, though
the Łojasiewicz inequalities themselves are not used.

The parabolic rescaling (5)-(7) sends a fixed spacetime neighbourhood on which
the MCF (1) is posed to a growing region in the blow-up variables
$y\in\mathbb{R}^{n-k}$ and $\tau\geq 0$, on which (9) is posed. This results
in some unfavourable issues due to non-compactness in the analysis of the
latter equation. To overcome these issues, many results in [MR2993752,
MR3374960, MR3349836, MR3381496, MR3489702, MR3521319, MR3602529, MR3662439,
MR4176547, chodosh2020mean, chodosh2021mean, choi2021ancient,
choi2021ancient2] are set in the Gaussian weighted Sobolev space (e.g.
[MR2993752, MR3374960]). This, together with the extensive use of geometric
measure theory, makes it difficult to apply the regularity theory from these
papers to derive precise asymptotics of the form (16)-(19) for the rescaled
MCF, which are explicit and valid uniformly in time.

In [MR2465296, MR3397388, MR3803553, MR4303943, gang2021dynamics,
zhou2021nondegenerate], in order to derive precise and uniform asymptotics,
Zhou Gang and various co-authors study the rescaled MCF in the blow-up
variables in a weighted $C^{k}$-space, with a norm of the form
$\left\lVert\phi\right\rVert=\left\lVert\left\langle
y\right\rangle^{-l}\partial_{y}^{\alpha}\partial_{\omega}^{\beta}\phi\right\rVert_{L^{\infty}}$
for a function $\phi$ defined on a fixed cylinder, multi-indices
$\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert\leq k$,
$\left\langle y\right\rangle:=(1+\left\lvert y\right\rvert^{2})^{1/2}$, and
some power $l\geq 1$. Using this norm, these authors are able to obtain
$C^{2}$-control over a frowing region of size roughly $\tau^{1/2}$ at time
$\tau$ in the microscopic scale. This amounts to $C^{2}$-control over a fixed
spatial neighbourhood in the original scale. To get such control, the authors
of the aforementioned papers have developed several novel propagator estimates
in the polynomially weighted norm given above. This kind of propagator
estimates are used in the present paper.

In contrast, using the Łojasiewicz inequalities from [MR3374960], one has
$C^{2}$-control only over a region of size $\sqrt{\log\tau}$ on the rescaled
hypersurface, which amounts to a shrinking region in the original, macroscopic
scale. This limitation is due to the fact that control over Gaussian weighted
Sobolev norm is, in general, not sufficient to conclude any uniform control,
since one loses almost all asymptotic data. In particular, we lack embedding
theorems of Gaussian weighted Sobolev spaces into Hölder spaces.

On the other hand, since the adiabatic solution $V$ in (21) is not in fact a
stationary solution to the rescaled MCF, the regularity theory of Colding-
Minicozzi, based on the analysis of Huisken’s $F$-functional (the Gaussian
weighted volome), is not very useful for the result in [MR2465296, MR3397388,
MR3803553, MR4303943, gang2021dynamics, zhou2021nondegenerate]. Indeed, unlike
the cylinder, this $V$ from (21) has nothing to do with the $F$-functional.
This fact results in considerable technical complications in those papers.

In the present paper, we are only interested in the asymptotic singularity
profile of (1), and therefore all the analysis in the subsequent sections are
set in the Gaussian weighted Sobolev space. By the first Łojasiewicz
inequality from [MR3374960]*Thm. 2.54, Thm. 5.3, it is not hard to see that
the remainder estimate (19) implies $C^{2}$-control over a region of size
$\sqrt{\log\tau}$ on the rescaled hypersurfaces, but this does not correspond
to a fixed region on the original scale. Hence, we do not conclude any uniform
estimate in this paper, but leave this to some future work.

Recently, invariant manifold theory for MCF is studied in [sun2021initial,
sun2021initial2]. In these papers, the authors have constructed
stable/central/unstable manifolds for MCF near a closed singularity, and
obtained some results about unstable initial perturbation for conical
singularity. These works are along the line of [MR1000974, MR2439610,
MR1445489, MR1675237, MR1160925], in which invariant manifold theory for
semilinear parabolic equations are studid extensively. In this paper we use
different methods and have rather different focus, as mentioned earlier.

###  1.3 Organization of the Paper

The purpose of the present paper is to lay out a framework in which one has
precise asymptotics and effective dynamics near a cylindrical singularity of
arbitrary dimensions.

Uniform control (in time) of this kind is achieved in the main result, Theorem
1, for a generic class of initial configurations. The latter is the manifold
$M$ given in (20), constructed explicitly in Definition 5. This is generic in
the sense that it has finite codimensions in the configuration space.

In this paper we mostly consider the rescaled equation (9). To keep track of
the precise dynamics of the hypersurfaces, we restrict ourselves to graphs
over cylinder. To use some of the regularity theory of Colding-Minicozzi, we
study the rescaled MCF in the Gaussian weighted Sobolev space $X^{s}(a)$. The
graphical equation thus obtained and the configuration spaces are defined in
Section 2.

In Section 2.2, we collect some known results regarding the linearized
operator $L(a)$ at a cylinder. In Section 2.3, we develop the first order
correction that eliminates the destabilizing effect of the zero-unstable modes
arising from broken symmetry. This is done by imposing a certain set of
modulation equations, namely (43)-(45).

In Section 3, we define the key map $\Phi$ in Theorem 1. This is defined in
terms of the unique fixed points of a family of maps
$\Psi=\Psi(\cdot,\eta_{0})$, also introduced in that section. In Section 3.3,
we prove the main theorem and its corollaries, assuming the key ingredients
from Sects. 4-6.

Indeed, Theorem 1 relies on a fixed point scheme for the map $\Psi$. In
Section 4, we show that this map $\Psi$ goes from a suitable configuration
space $\mathcal{A}_{\delta}$ into itself. This section contains the heart of
the matter, namely an appropriate second order correction that eliminates the
effect of the remaining zero-unstable modes that persist the modulation. These
remaining modes correspond to translation symmetry $Y(y,\omega)\mapsto
Y(y+y^{\prime},\omega)$ along the cylindrical axis, which is not broken by the
cylinder, and the non-symmetry zero modes mentioned earlier.

In Section 5, we show that $\Psi$ is a contraction in $\mathcal{A}_{\delta}$.
In Section 6, we prove the estimates (13)-(14), which show that the set $M$ in
(20) forms a Lipschitz graph over $\mathcal{B}_{\delta}\cap\mathcal{S}$ of
size $\delta$, and therefore can be treated as a manifold.

In Appendix A, we prove some elementary results that are nonetheless crucial
for the developments in Sects. 4-6. In Appendix B, we prove that the map
$\Psi$ from Section 3.1 is well-defined. In Appendix C, we collect certain
technical nonlinear estimates.

### Notation

Throughout the paper, the notation $A\lesssim B$ means that there is a
constant $C>0$ independent of time and any small parameter in the problem,
such that $A\leq CB$. For two vectors $A,\,B$ in Banach spaces $X,\,Y$
respectively, the notation $A=O_{Y}(B)$ means $\left\lVert
A\right\rVert_{X}\lesssim\left\lVert B\right\rVert_{Y}$.

##  2 The modulation equations: first-order correction

In this section we develop the first order correction for the rescaled mean
curvature flow.

By the relations (2) and (12), as far as Theorem 1 is concerned, it suffices
to consider (9) in the unknown pair $(\sigma,\xi)$, entering the equation
through (12).

There are various advantage of studying (9) instead of (1). This is typical
when one is interested in blow-up solutions to evolution equations, such as in
the study of critical solitary wave dynamics where in general solution blows
up with various profiles in finite time. See for instance [MR2656972]. For
one, (9) has global solutions in time. (Actually, Theorem 3 below proves
global well-posedness for (9) for a large class of initial configurations.)
Thus we will mostly work with (9) and only return to (1) in Section 3.3, when
we derive geometric consequences of the effective dynamics for the original
flow. Moreover, since Theorem 1 only concerns with solutions that are normal
graphs over $\mathbb{R}^{n-k}\times\mathbb{S}^{k}$ of the form (12), when we
study (9) below, we reparametrize $Y$ as $Y=(x,v(y,\omega,\tau)\omega)$ and
analyze the behaviour of the path of functions
$v(\cdot,\tau):\mathbb{R}^{n-k}\times\mathbb{S}^{k}\to\mathbb{R},\,\tau\geq
0$.

To this end, following [MR2993752, MR3374960], we introduce a family of
Gaussian weighted Sobolev spaces.

######  Definition 1.

For $s\geq 0,a>0$, define the space

$X^{s}(a):=H^{s}(\mathbb{R}^{n-k}_{y}\times\mathbb{S}^{k}_{\omega}\,,\mathbb{R};\,\rho_{a}),\quad\rho_{a}:=e^{-a\left\lvert
y\right\rvert^{2}/2}\,d\mu.$ (22)

Here $d\mu$ is the canonical measure on
$\mathbb{R}^{n-k}\times\mathbb{S}^{k}$.

The space $X^{s}(a)$ is equipped with the weighted inner product

$\left\langle\phi,\,\psi\right\rangle_{a}=\int\phi\psi\rho_{a}\quad(\phi,\psi\in
X^{s}(a)).$ (23)

The induced norm by this inner product on $X^{s}(a)$ is denoted by
$\left\lVert\cdot\right\rVert_{s,a}$. For simplicity, we write $X^{s}\equiv
X^{s}(1/2)$ with norm $\left\lVert\cdot\right\rVert_{s}$. The space $X^{s}$
has standard Gaussian measure $e^{-\left\lvert y\right\rvert^{2}/4}d\mu$
centered at $y=0$. (The choice of a pivot $a=1/2$ can be replaced by any other
number.) For a linear map $L:X^{s}(a)\to X^{r}(b)$, we denote the operator
norm of $L$ as $\left\lVert L\right\rVert_{s,a\to r,b}$.

For $s\leq r$ and $0<b\leq a$, there holds the trivial continuous embedding

$X^{r}(b)\subset X^{s}(a).$ (24)

In Appendix A, we consider the issue of estimating the weaker $X^{s}(b)$-norm
in terms of the $X^{s}(a)$-norm, with $a\geq b$.

Following [MR1030675], we introduced the weighted volume, or the
$F$-functional, for a normal graph $v=v(y,\omega)$ over cylinder as

$F_{a}(v):=\int_{S}\rho_{a}\,d\mu_{S},\quad
S=S(v):=\left\\{(y,v(y,\omega)\omega):y\in\mathbb{R}^{n-k},\,\omega\in\mathbb{S}^{k}\right\\}.$
(25)

Using Sobolev inequalities, one can show that this functional is $C^{2}$ as a
Fréchet-differentiable map on the space $X^{s}(a)$ with $a>0,\,s\geq 2$.
Indeed, denotes $F_{a}^{\prime}(v)$ the $X^{0}(a)$-gradient of $F_{a}$ at $v$.
Explicitly, we have

$F_{a}^{\prime}(v)=-\Delta_{y}v+a\left\langle y,\,\nabla_{y}\right\rangle
v-v^{-2}\Delta_{\omega}v-av+kv^{-1}+N_{1}(v),$ (26)

where $N_{1}(v):X^{s}(a)\to X^{s-2}(a)$ is a quasilinear elliptic operator,
given explicitly in (155). Expression (26) can be derived from the first
variation formula in [MR2993752], and the graphical MCF equation obtained in
[MR1100206]. The nonlinear map $F_{a}^{\prime}$ is $C^{1}$ from $X^{s}(a)\to
X^{s-2}(a)$.

###  2.1 The graphical equations

In this subsection we consider (9) in the space $X^{s}(a)$ with $a>0,\,s\geq
2$.

  * •

In the remaining of this paper, unless otherwise stated, the dot denotes
$\tfrac{\partial}{\partial\tau}$.

######  Lemma 1.

$X$ of the form (12) satisfies the MCF (1) if and only if the pair
$(\sigma,\xi)$ satisfies

$\dot{\xi}=-F_{a}^{\prime}(\sqrt{k/a}+\xi)-\partial_{\sigma}W(\sigma)\dot{\sigma},$
(27)

where $F_{a}^{\prime}(v)$ is as in (26), and $W:\Sigma\to X^{s}(a)$ is given
by

$W(\sigma):=\sqrt{k/a}+g_{n-k+l,j}\omega^{l}y^{j}+\left\langle
z,\,\lambda^{-1}\omega\right\rangle,$ (28)

###### Proof.

Write $v=\sqrt{k/a}+\xi$. Recall from the Introduction that $X$ of the form
(12) satisfies the MCF (1) if and only if the rescaled hyersurface
$Y=(y,v(y,\omega)\omega,\tau)$ from (2) solves (9). Following [MR2465296,
MR3397388, MR3803553, MR4303943, gang2021dynamics, zhou2021nondegenerate], we
derive the evolution of $v$ from (9).

Taking inner product on both sides of (9) with the vector
$(0,\omega)\in\mathbb{R}^{n+1}$, we find

$\dot{v}=-\left\langle H(v)\nu(v),\,(0,\omega)\right\rangle-a\left\langle
y,\,\nabla_{y}\right\rangle
v+av-\left\langle\dot{g}(y,v\omega),\,(0,\omega)\right\rangle-\lambda^{-1}\left\langle\dot{z},\,\omega\right\rangle.$
(29)

Here $H(v)$ and $\nu(v)$ denotes the mean curvature and the unit normal vector
at the point $(y,v(y,\omega)\omega)$ respectively. To get (29), we use the
fact that $\left\langle
g^{-1}(\cdot),\,\cdot\right\rangle=\left\langle\cdot,\,g(\cdot)\right\rangle$
as $g\in SO(n+1)$, and the requirement $g(0,x^{\prime})=(0,x^{\prime})$ for
every $x^{\prime}\in\mathbb{R}^{k+1}$.

The first term in the r.h.s. is calculated e.g. in [MR3803553]*Appendix A,
whose derivation is routine. This, together with the following two terms,
constitute the r.h.s. of (26)

Differentiating the condition (3), we find $\dot{g}(0,x^{\prime})=0$. Thus the
penultimate term in the r.h.s. of (29) further simplifies as

$\left\langle\dot{g}(y,v\omega),\,(0,\omega)\right\rangle=\dot{g}_{n-k+l,j}\omega^{l}y^{j}\quad(1\leq
j\leq n-k,\,1\leq l\leq k+1).$

Lastly, moving the $\tau$-derivative of $\sqrt{k/a}$ in the l.h.s. of (29)
(recall $v(\tau)=\sqrt{k/a(\tau)}+\xi(\tau))$, gives (27). ∎

######  Remark 1.

For future use, here we note that for every $s\geq 0$ and $b>0$, the function
$W:\Sigma\to X^{s}(b)$ is smooth and bounded on the region

$\left\\{(g,z,a)\in\Sigma:\lambda z\leq C,\,a\geq c\right\\},$

for any fixed constants $c,\,C>0$.

###  2.2 Linearized operator at $Y_{a}$

The only nonlinear term in (27) is contained in the map $F^{\prime}_{a}$,
which is the (normal) $X^{0}(a)$-gradient of the $F$-functional, (25). In this
subsection we linearize this map around the stationary solution
$v\equiv\sqrt{k/a}$, which corresponds to a cylinder with radius $\sqrt{k/a}$
and along the $y$ axis. Then we study the zero-unstable modes of the
linearized operator.

All of the results in this section are known by far. See for instance
[MR2993752]*Sec.5, [MR3374960]*Sec. 3.2.

######  Lemma 2.

The linearized operator of $F^{\prime}_{a}(v)$ at the critical point
$v\equiv\sqrt{k/a}$ is given by

$L(a):=-\mathcal{L}_{a}-\frac{a}{k}\Delta_{\omega}-2a,$ (30)

where $\mathcal{L}_{a}:=\Delta_{y}-a\left\langle
y,\,\nabla_{y}(\cdot)\right\rangle$ is the drift Laplacian on the weighted
space $X^{s}(a)$.

Moreover, the operator $L(a)$ is self-adjoint in $X^{s}(a)$ w.r.t. the inner
product from (23), and is bounded from $X^{s}(a)\to X^{s-2}(a)$. The spectrum
of $L(a)$ is purely discrete, and the only non-positive eigenvalues, together
with the associated eigenfuncitons, are given by

$\displaystyle-2a,\quad\text{with eigenfunction
}\Sigma^{(0,0)(0,0,0)}(a):=-\frac{\sqrt{k}}{2}a^{-3/2},$ (31)
$\displaystyle-a,\quad\text{with eigenfunctions
}\Sigma^{(0,1)(0,0,l)}(a):=\lambda^{-1}\omega,$ (32)
$\displaystyle-a,\quad\text{with eigenfunctions
}\Sigma^{(1,0)(i,0,0)}(a):=\frac{1}{\left\lVert
y^{i}\right\rVert_{0,a}^{2}}y^{i},$ (33) $\displaystyle 0,\quad\text{with
eigenfunctions }\Sigma^{(1,1)(i,0,l)}(a):=y^{i}\omega^{l},\quad$ (34)
$\displaystyle 0,\quad\text{with eigenfunctions
}\Sigma^{(2,0)(i,j,0)}(a):=\frac{1}{\left\lVert
ay^{i}y^{j}-\delta_{ij}\right\rVert_{0,a}^{2}}(ay^{i}y^{j}-\delta_{ij}).$ (35)

Here and in the remaining of this paper, the indices $l=1,\ldots,k+1$ and
$i,j=1,\ldots,n-k$.

Moreover, the functions in (31)-(34) are mutually orthogonal w.r.t. the inner
product (23).

######  Remark 2.

Notice that we do not normalize (31),(32) and (34). The point is that these
correspond to various components of the Fréchet derivative
$\partial_{\sigma}W(\sigma)$, where $W$ is as in (28).

###### Proof.

Here we only remark, after [MR3374960, MR2465296, MR3397388, MR3803553,
MR4303943], that the eigenvalues and eigenfunctions in (31)-(35) can be
deduced from the spectral properties of the two components in (30). Indeed,
the drift Laplacian $-\mathcal{L}_{a}\geq 0$ has

$\displaystyle\text{eigenvalue }0,\quad\text{with eigenfunction
}\phi^{0,0,0}\equiv 1,$ (36) $\displaystyle\text{eigenvalue }a,\quad\text{with
eigenfunctions }\phi^{1,i,0}:=y^{i},$ (37) $\displaystyle\text{eigenvalue
}2a,\quad\text{with eigenfunctions }\phi^{2,i,j}:ay^{i}y^{j}-\delta_{ij}.$
(38)

The spherical Laplacian $\Delta_{\mathbb{S}^{k}}\geq 0$ has

$\displaystyle\text{eigenvalue }0,\quad\text{with eigenfunction }Y^{0,0}\equiv
1,$ (39) $\displaystyle\text{eigenvalue }k,\quad\text{with eigenfunctions
}Y^{1,l}:=\omega^{l}.$ (40)

Since the operator $L(a)$ in (30) is decoupled, (36)-(40) give the following
eigenfunctions of $L(a)$:

$\Sigma^{(m,n)(i,j,k)}=\phi^{m,i,j}Y^{n,l}.$

These correspond to the only non-positive eigenvalues of $L(a)$. ∎

Thus, with $L_{a}$ given in (30), we can rewrite (27) as

$\dot{\xi}=-L(a)\xi-N(a,\xi)-\partial_{\sigma}W(\sigma)\dot{\sigma},$ (41)

where the nonlinearity $N(a,\xi)$ is defined by this expansion. This map is
calculated explicitly in (154). In Section C, we study some key properties of
the nonlinearity.

###  2.3 The modulation equation

In this subsection we study (41), which is equivalent to (27) and the rescaled
MCF, (9). Equation (41) is the central subject in the remaining sections.

It is important to note that some, but not all of the zero-unstable modes of
the linearized operator $L(a)$ are due to broken symmetries. Indeed, for
$\Sigma^{(m,n)}(a)$ with $(m,n)=(0,0),(0,1),(1,1)$, there exists a path
$\sigma(s)\in\Sigma$ (the manifold defined in (8)) s.th.

$\Sigma^{(m,n)(i,j,l)}(a)=\partial_{s}|_{s=0}T_{\sigma(s)}Y_{a}.$

Here $T_{\sigma}=T_{g,z,a}$ denotes the action of symmetry on the graph
function, derived from (2), and $Y_{a}$ is the cylinder from (10). For
example, take $\sigma(s)=(g_{0},z(s),a_{0})$ with a path
$z(s)\in\mathbb{R}^{k+1},\,\partial_{s}|_{s=0}z(s)=\lambda^{-1}e^{1}$. Then

$\partial_{s}|_{s=0}T_{\sigma(s)}Y_{a_{0}}=\partial_{s}|_{s=0}(\sqrt{k/a_{0}}+z_{l}(s)\omega^{l})=\Sigma^{(0,1),(0,0,1)}(a_{0}).$

Consequently, with a proper choice of $\sigma$ as a function of $\xi$, we can
eliminate these modes. This is the content of the next lemma.

######  Lemma 3.

Suppose the pair $(\sigma,\xi)$ is a global solution to (27), s.th.

$\left\langle\xi(0),\,\Sigma^{(m,n)}(a(0))\right\rangle_{a(0)}=0$

for $(m,n)=(0,0),(0,1),(1,1)$. Then $\xi$ satisfies the orthogonality
condition

$\left\langle\xi(\tau),\,\Sigma^{(m,n)}(a(\tau))\right\rangle_{a(\tau)},\quad\tau\geq
0,\;(m,n)=(0,0),(0,1),(1,1),$ (42)

if and only if $\sigma=(g,z,a)$ satisfies the modulation equations:

$\displaystyle\begin{split}\left\lVert\Sigma^{(1,1)(j,0,l)}(a)\right\rVert^{2}_{0,a}\dot{g}_{n-k+l,j}=&-\left\langle
N(a,\xi),\,\Sigma^{(1,1)(j,0,l)}(a)\right\rangle_{a},\end{split}$ (43)
$\displaystyle\begin{split}\left\lVert\Sigma^{(0,1)(0,0,l)}(a)\right\rVert^{2}_{0,a}\dot{z}_{l}=&a\left\langle\xi,\,\Sigma^{(0,1)(0,0,l)}(a)\right\rangle_{a}\\\
&-\left\langle N(a,\xi),\,\Sigma^{(0,1)(0,0,l)}(a)\right\rangle_{a}\\\
&+\left\langle\xi,\,\partial_{\tau}\Sigma^{(0,1)(0,0,l)}(a)\right\rangle_{a},\end{split}$
(44)
$\displaystyle\begin{split}\left\lVert\Sigma^{(0,0)}(a)\right\rVert^{2}_{0,a}\dot{a}=&2a\left\langle\xi,\,\Sigma^{(0,0)}(a)\right\rangle_{a}\\\
&-\left\langle
N(a,\xi),\,\Sigma^{(0,0)}(a)\right\rangle_{a}+\left\langle\xi,\,\partial_{\tau}\Sigma^{(0,0)}(a)\right\rangle_{a}.\end{split}$
(45)

######  Remark 3.

The terminology modulation equations is common in the study of solitary wave
dynamics, e.g. [MR2094474, MR2351368, MR1071238, MR1170476], and refers to
equations of the form (43)-(45), derived from a condition as (42). We adopt
the same terminology here.

###### Proof.

Differentiating both sides of (42) w.r.t. $\tau$, we find that (42) holds for
all $\tau\geq 0$ if and only if

  1. 1.

(42) holds for $\tau=0$;

  2. 2.

For $\tau>0$, there holds

$\left\langle\dot{\xi}(\tau),\,\Sigma^{(m,n)}(a(\tau))\right\rangle_{a(\tau)}=-\left\langle\xi(\tau),\,\partial_{\tau}\Sigma^{(m,n)}(a(\tau))\right\rangle_{a(\tau)}.$
(46)

The first point is in the assumption. For the second point, at time $\tau$,
taking the $a(\tau)$-weighted inner product (see (23)) of both sides of (41),
using the spectral property of $L(a)$ shown in Lemma 2, together with the
orthogonality among $\Sigma^{(m,n)}$ as $L(a)$ is self-adjoint in $X^{s}(a)$,
we conclude that (46) is equivalent to (45). ∎

From (43)-(45) we see that if we have a priori control of $\xi$ in
$X^{s}$-norm, then we can ensure that the path $\sigma(\tau)$ stays
approximately constant. This is the usual low velocity condition for an
adiabatic theory.

##  3 The quadratic correction $\Phi$ and its properties

In this section we define the key map $\Phi$ in Theorem 1, which originates
from a fixed point scheme. Then we prove Theorem 1, assuming the results in
Sects. 4-6. These results are inspired by [MR2480603] and we use some of the
notations there.

From now on we will often work with $X^{s}(a)$ spaces with different $a$. In
order to facilitate this effort, we introduce the following notion of
admissible paths:

######  Definition 2.

Fix $0<\delta\ll 1$. A path $\sigma(\tau)=(g(\tau),z(\tau),a(\tau))\in\Sigma$
is admissible if the following holds:

  1. 1.

The map $\tau\mapsto\sigma(\tau)$ is continuous and locally Lipschitz.

  2. 2.

The entire path $a(\tau)$ lies in the $\delta$-neighbourhood of $a(0)$, i.e.

$\sup_{\tau\geq 0}\left\lvert a(\tau)-a(0)\right\rvert\leq\delta.$

  3. 3.

Initially, $a(0)\geq 2\delta+1/2$;

  4. 4.

The path $z(\tau)<\sqrt{k/a}e^{\int^{\tau}a}$, or equivalently
$z(\tau)/\lambda(\tau)<\sqrt{k/a}$ for all $\tau$.

Item 1 above is a standard regularity requirement to make sense of the
velocity of the path $\sigma$. Items 2-3, which are the most important ones,
are imposed for several reasons. First, we want to study (27) in the space
$X^{s}\equiv X^{s}(1/2)$, while from time to time we consider the stronger
norm $\left\lVert\cdot\right\rVert_{s,a(\tau)}$. These conditions ensure
$a(\tau)\geq 1/2$ and therefore $X^{s}\subset X^{s}(a(\tau))$ for all $\tau$
by (24). Secondly, for some estimates, e.g. to bound a projection into an
eigenspace of $L(a)$, we need $a$ to stay within a fixed range, so that these
estimates can be made independent of $a$. Thirdly, for some other estimates,
e.g. the nonlinear estimate (156), we need $a(\tau)$ to stay a fixed distance
away from the pivotal number $1/2$. Item 4 above is imposed so that the
function $W$ in (28) remains bounded.

###  3.1 The definition of $\Psi$

In this subsection, we introduce another key map $\Psi$. This map goes into
the definition of the map $\Phi$ in Theorem 1.

Consider the following linear equation obtained by freezing coefficients in
(41) and the modulation equations (43)-(45) at a fixed path

$(\sigma^{(0)},\xi^{(0)})\in
Lip([0,\infty),\Sigma)\times(C([0,\infty),X^{s})\cap
C^{1}([0,\infty),X^{s-2})),$ (47)

given as follows:

$\displaystyle\dot{\xi}=-L(a^{(0)})\xi-N(a^{(0)},\xi^{(0)})-\partial_{\sigma}W(\sigma^{(0)})\dot{\sigma},$
(48)
$\displaystyle\dot{\sigma}=\vec{F}(\sigma^{(0)})\xi+\vec{M}(\sigma^{(0)},\xi^{(0)}),$
(49)

where $\vec{F},\,\vec{M}$ are defined by isolating the linear and nonlinear
terms in $\xi$ respectively in (43)-(45). For instance, the last entry of
$\vec{F}(\sigma^{(0)})\xi$ is given by

$-\frac{1}{\left\lVert\Sigma^{(0,0)}(a^{(0)})\right\rVert_{a^{(0)}}^{2}}\left(2a^{(0)}\left\langle\xi,\,\Sigma^{(0,0)}(a^{(0)})\right\rangle_{a^{(0)}}+\left\langle\xi,\,\partial_{\tau}\Sigma^{(0,0)}(a^{(0)})\right\rangle_{a^{(0)}}\right),$

and the last entry of $\vec{M}(\sigma^{(0)},\xi^{(0)})$ is given by

$-\frac{1}{\left\lVert\Sigma^{(0,0)}(a^{(0)})\right\rVert_{a^{(0)}}^{2}}\left\langle
N(a^{(0)},\xi^{(0)}),\,\Sigma^{(0,0)}(a^{(0)})\right\rangle_{a^{(0)}}.$

To (48)-(49) we associate the initial configuration

$\displaystyle\sigma(0)=(\delta_{ij},0,a_{0})\quad\text{for some fixed
}a_{0}>1/2,$ (50)
$\displaystyle\xi(0)=\eta_{0}+\beta_{i}\Sigma^{(1,0)(i,0,0)}(a_{0})+\gamma_{ij}\Sigma^{(2,0),(i,j,0)}(a_{0}),$
(51)

where $\eta_{0}\in X^{s}(a_{0})$ is fixed, and and
$\beta_{i},\gamma_{ij}\in\mathbb{R}$ are to be chosen later as functions of
$\eta_{0}$.

In the remaining of this paper, the central object is the Cauchy problem
(48)-(51). In Appendix B, we show the linear system (48)-(49) is equivalent to
a single linear equation

$\dot{\xi}+L(a^{(0)})\xi=-\mathcal{W}(\sigma^{(0)})\xi-\tilde{N}(a^{(0)},\xi^{(0)}),$
(52)

where the maps $\mathcal{W},\,\tilde{N}$ are defined in (146). We also show
the Cauchy problem (48)-(51) has a unique global solution in the space (47).

Now, we set up a contraction scheme in the following space:

######  Definition 3.

Fix $0<\delta\ll 1$. The space
$\mathcal{A}_{\delta}=\mathcal{A}_{\delta}^{\sigma}\times\mathcal{A}_{\delta}^{\xi}$
consists of

$(\sigma,\xi)\in Lip([0,\infty),\Sigma)\times(C([0,\infty),X^{s})\cap
C^{1}([0,\infty),X^{s-2})),$

s.th. the following holds:

  1. 1.

$\sigma(0)$ is as in (50), with $a_{0}\geq\tfrac{1}{2}+2\delta$;

  2. 2.

For some fixed $c_{0}>0$, there hold the decay estimates

$\displaystyle\left\lvert\dot{\sigma}(\tau)\right\rvert\leq
c_{0}\delta\left\langle\tau\right\rangle^{-2},\quad\tau\geq 0,$ (53)
$\displaystyle\left\lVert\xi(\tau)\right\rVert_{s}\leq\delta\left\langle\tau\right\rangle^{-2},\quad\tau\geq
0;$ (54)

  3. 3.

Let $b:=\tfrac{1}{2}-4\delta.$ There holds the pivot condition from Lemma 4,
i.e.

$\left\lVert\xi(\tau)\right\rVert^{2}_{s,b}\leq c,\quad\tau\geq 0,$ (55)

for some fixed $c>0$.

Clearly, $\mathcal{A}_{\delta}$ is not empty, as the pair
$(\sigma,\xi)\equiv(\sigma(0),0)$ lies in this space.

Item 1 above amounts to fixing an initial cylindrical coordinate. Up to a
rigid motion in $\mathbb{R}^{n+1}$ and an initial dilation, (50) can be
replaced by any other parameters. The decay conditions in Item 2, (53)-(54),
are the most important ones here, for the following reasons. First, these
correspond to the claimed decay properties from Theorem 1, and finding a fixed
point in $\mathcal{A}_{\delta}$ amounts to constructing a global solution to
(52). Secondly, if $\sigma\in\mathcal{A}_{\delta}^{\sigma}$, then $\sigma$ is
admissible as in Definition 2, and we have the convenient properties mentioned
earlier. Item 4 has to do with the interpolation from Lemma 4, whose
importance is explained in Section A. Of course, if (55) holds, then
$\xi(\tau)\in X^{s}(b)$. But we never use this fact other than a pivot
condition for deriving estimates in $X^{s}$.

Consider the solution map

$\Psi:(\sigma^{(0)},\xi^{(0)})\mapsto\text{ the unique solution $(\sigma,\xi)$
to \eqref{3.4}-\eqref{3.7}}.$ (56)

In Lemma 8, we show this map is well-defined. Hereafter we want to show that

  1. 1.

$\Psi(\mathcal{A}_{\delta})\subset\mathcal{A}_{\delta}$;

  2. 2.

$\Psi:\mathcal{A}_{\delta}\to\mathcal{A}_{\delta}$ is a contraction w.r.t. a
suitable norm on $\mathcal{A}_{\delta}$.

In Section 4, we show the first point holds for appropriately chosen initial
configurations (51). In Section 5 we show the second point holds if $\delta\ll
1$.

######  Remark 4.

Consider the initial configuration (51). The constants $\beta_{i}$ and
$\gamma_{ij}$ are to be determined later as a function of $\eta_{0}\in
X^{s}(a_{0})$. This way the map $\Psi$ depends only on the function
$\eta_{0}$, i.e. $\Psi=\Psi(\cdot,\eta_{0})$.

Indeed, if $\eta_{0}=0$, then it is easy to see that the fixed point of
$\Psi(\cdot,0)$ is just the vector $(\sigma,\xi)\equiv(\sigma(0),0)$ in
$\mathcal{A}_{\delta}$. This corresponds to the trivial static solution of
(41), namely the cylinder of radius $\sqrt{k/a_{0}}$. On the other hand, if
$\eta_{0}\neq 0$, then in general $\Psi(0,\eta_{0})\neq 0$ by (56).

For simplicity of notation, most of the time we do not write
$\Psi=\Psi(\cdot,\eta_{0})$ but let this be understood.

###  3.2 The definition of $\Phi$

For a function $\eta$, the map $\Phi$ is defined in terms of the fixed point
of $\Psi=\Psi(\cdot,\eta)$ (see Remark 4). But first, we need to introduce a
suitable parameter space for $\Phi$.

######  Definition 4.

Let $a_{0}>1/2$ be as in (50). Let $0<\delta\ll 1,\,0<b<1/2$ be as in
Definition 3. Let $C>0$ be a large constant depending on the absolute,
implicit constant in (92).

The space

$\mathcal{B}_{\delta}\subset X^{s}(b)\subset X^{s}\subset X^{s}(a_{0})$

consists of all functions $\eta=\eta(y,\omega)$ s.th.

$\displaystyle\left\lVert\eta\right\rVert_{s}<\delta,$ (57)
$\displaystyle\left\lVert\eta\right\rVert_{s,b}^{2}\leq c/2,\quad\text{where
the number $c>0$ is as in \eqref{3.3'}},$ (58) $\eta(y,\omega)\geq\delta$ and
$\left\lvert\eta(y,\omega)\right\rvert\geq C\delta^{2}\left\lvert
y\right\rvert^{2}$ for $\left\lvert y\right\rvert^{2}\geq C^{-1}\delta^{-1}$.
(59)

The space $\mathcal{S}=X^{s}(a_{0})$ is the orthogonal complement to all the
zero-unstable modes in (31)-(35) of the linearized operator $L(a_{0})$ w.r.t.
the $X^{0}(a_{0})$-inner product defined in (23).

We will explain the the meaning of this parameter space below. For now,
observe that the codimension of $\mathcal{S}$ is the sum of the multipliciteis
of all the non-positive eigenvalues of $L(a_{0})$, which equals to

$\text{codim}\,\mathcal{S}=n+2+\frac{(n-k)(n-k+3)}{2}.$

Notice that, as pointed out in [MR3374960]*Eqn. (3.44), not all of the zero-
unstable modes in (31)-(35) are linearly independent.

Now we define the key map $\Phi$ in Theorem 1.

######  Definition 5 (the quadratic correction $\Phi$).

For $\eta_{0}\in\mathcal{B}_{\delta}\cap\mathcal{S}$ as in Definition 4,
define

$\Phi(\eta_{0}):=\beta_{i}(\eta_{0})\Sigma^{(1,0)(i,0,0)}(a_{0})+\gamma_{ij}(\eta_{0})\Sigma^{(2,0),(i,j,0)}(a_{0}),$
(60)

where $\beta_{i},\,\gamma_{ij}$ are defined in Theorem 3.

######  Remark 5.

Consider the requirements from Definition 4. In view of the quadratic
estimates (13), for sufficiently small $\delta$, conditions (57)-(58) ensure
the compatibility to impose the initial condition
$\xi(0)=\eta_{0}+\Phi(\eta_{0})$ for a path
$\xi\in\mathcal{A}_{\delta}^{\xi}$. (I.e. this ensures
$\left\lVert\xi(0)\right\rVert_{s}\leq\delta$ for sufficiently small
$\delta$). Condition (59) has to to with the geometric interpretation of
Theorem 1. Indeed, for sufficiently small $\delta$, by the definition (60)
above and the formulae (33), (35), condition (59) ensures the function
$\eta_{0}+\Phi(\eta_{0})\geq 0$ on the entire cylinder. In this case, we can
interpret this function as a normal graph over the the cylinder, parametrizing
the hypersurface

$Y_{0}=(y,(\sqrt{k/a_{0}}+\eta_{0}+\Phi(\eta_{0}))\omega),\quad
y\in\mathbb{R}^{n-k},\,\omega\in\mathbb{S}^{k}.$

By the avoidance principle for MCF, it follows that a flow satisfying (9) with
initial configuration $Y_{0}$ above remains to be a normal graph over the
cylinder of radius $\sqrt{k/a_{0}}$ for all time. This justifies the geometric
meaning for the analysis of PDEs in terms of the graph function $\xi$ in the
remaining sections.

Using the modulation equations from Section 2.3, we can ensure that a flow
generated by a function $\eta_{0}\in\mathcal{B}_{\delta}\cap\mathcal{S}$ under
(41) remains orthogonal at all $\tau\geq 0$ to all of the zero-unstable modes
that are due to broken symmetry (see a discussion at the beginning of Section
2.3). Yet, this does not suffice to give any dissipative estimate because of
the remaining zero-unstable modes that cannot be eliminated by the modulation
equations, namely $\Sigma^{(1,0)}$ (horizontal translation) and
$\Sigma^{(2,0)}$ (shape instability). The former is protected by the symmetry
of cylinder lying along the $y$-axis. The latter is not due to any symmetry.

###  3.3 Proof of the Main Results

In this subsection we assume Theorem 2-Theorem 4 hold. Then we prove the main
results from Section 1.

###### Proof of Theorem 1, assuming Theorem 2-Theorem 4.

By construction, the fixed point $(\sigma,\xi)$ in $\mathcal{A}_{\delta}$ of
the map $\Psi$ solves the graphical rescaled MCF (41), coupled to the
modulation equations (43)-(45), with initial configuration given by

$\sigma(0)=(\delta_{ij},0,a_{0}),\quad\xi(0)=\eta_{0}+\Phi(\eta_{0}).$

This amounts to a global solution to the rescaled MCF (9) satisfying

$\dot{\sigma}=\vec{F}(\sigma)\xi+\vec{M}(\sigma,\xi),\quad
Y(t)=(y,(\sqrt{k/a(t)}+\xi(t))\omega),$ (61)

together with the initial configuration

$\sigma(0)=(\delta_{ij},0,a_{0}),\quad
Y(0)=(y,(\sqrt{k/a_{0}}+\eta_{0}+\Phi(\eta_{0}))\omega).$ (62)

Estimates (13)-(14) are the content of Theorem 4. Thus Parts 1-2 of the
theorem is proved. The positivity of $\xi$ follows from the fact that
$\xi(0)\geq 0$, which holds by (59), and the avoidance principle for MCF, c.f.
Remark 5. The remainder estimates (18)-(19) come from the decay condition in
the space $\mathcal{A}_{\delta}$ in which $\Psi$ is a contraction, namely
(53)-(54).

∎

###### Proof of Corollary 1.

Without loss of generality, suppose $t_{0}=0$. Let $X$ be a maximal solution
to (1), and suppose for some $\lambda_{0}>0$, the hypersurface
$\lambda_{0}^{-1}X(\cdot,0)\in M$ with $M$ given in (20). Then, by Theorem 1,
$X$ is of the form (12) for all $t<T$, and $X$ is uniquely determined by the
pair $\sigma(\tau(t)),\xi(y(x,t),\omega,\tau(t))$ constructed in Theorem 3
below.

In terms of the slow time variable $\tau$, the Cauchy problem for $\sigma$ in
(16)-(17) uniquely determines the axis and radius of the limit cylinder given
the initial profile. This system of ODEs does not depend on taking any
particular sequence of $\lambda\to\infty$ in the rescaling procedure, and the
solution to the Cauchy problem is unique by Theorem 3. Thus Corollary 1
follows. ∎

##  4 The mapping property of $\Psi$

In this section we prove that for appropriately chosen constants
$\beta_{i},\,\gamma_{ij}$ in (51) the $\Psi$ maps from $\mathcal{A}_{\delta}$
into itself.

Recall that $\Psi$ maps a fixed path
$(\sigma^{(0)},\xi^{(0)})\in\mathcal{A}_{\delta}$ to the global solution to
(52) with initial configuration (50)-(51), constructed explicitly in Lemma 8.

######  Theorem 2.

For every $\eta_{0}\in\mathcal{B}_{\delta}\cap\mathcal{S}$ as in Definition 4
and every fixed path $(\sigma^{(0)},\xi^{(0)})\in\mathcal{A}_{\delta}$, there
exist unique coefficients $\beta_{i},\,\gamma_{ij}$, depending on the choice
of $\sigma^{(0)},\,\xi^{(0)}$ only, s.th. the solution to (48)-(51) lies in
$\mathcal{A}_{\delta}$.

Moreover, there hold the quadratic estimates

$\displaystyle\left\lvert\beta_{i}(\sigma^{(0)},\xi^{(0)})\right\rvert\lesssim\delta^{2},$
(63)
$\displaystyle\left\lvert\gamma_{ij}(\sigma^{(0)},\xi^{(0)})\right\rvert\lesssim\delta^{2}.$
(64)

Moreover, for two given paths $\sigma^{(m)},\,\xi^{(m)},m=1,2$, there hold the
Lipschitz estimates

$\displaystyle\left\lvert\beta_{i}(\sigma^{(0)},\xi^{(0)})-\beta_{i}(\sigma^{(1)},\xi^{(1)})\right\rvert\lesssim\delta\left\lVert(\sigma^{(0)},\xi^{(0)})-(\sigma^{(1)},\xi^{(1)})\right\rVert,$
(65)
$\displaystyle\left\lvert\gamma_{ij}(\sigma^{(0)},\xi^{(0)})-\gamma_{ij}(\sigma^{(1)},\xi^{(1)})\right\rvert\lesssim\delta\left\lVert(\sigma^{(0)},\xi^{(0)})-(\sigma^{(1)},\xi^{(1)})\right\rVert,$
(66)

where the norm in the r.h.s. is defined in (90).

###### Proof.

1\. Let $(\sigma,\xi):=\Psi(\sigma^{(0)},\xi^{(0)})$. We want to check this
pair $(\sigma,\xi)$ satisfies (53)-(55). In view of the regularity result
Lemma 9, it remains to check the decay conditions (54) only.

Indeed, condition (54) does not hold in general, because in principle the
solution construct in Lemma 8 may not decay due to the zero-unstable modes
that are not eliminated by the modulation equations. Once the decay condition
(54) for $\xi$ is proved, we also get the velocity bound (53) for $\sigma$,
from the equation (49).

2\. Let $P^{(m,n)}(\tau)$ be the projection onto the span of the zero-unstable
modes
$\operatorname{span}\left\\{\Sigma^{(m,n)(i,j,l)}(a^{(0)}(\tau))\right\\}$ of
$L(a^{(0)}(\tau))$, and write $\xi^{(m,n)}:=P^{(m,n)}\xi$. Let $Q:=1-\sum
P^{(m,n)}$ and write $\xi_{S}=Q\xi=\xi-\sum\xi^{(m,n)}$. Here the superscript
$S$ refers to stable projection.

By the orthogonality condition (145), we know $\xi^{(m,n)}=0$ for
$(m,n)=(0,0),(0,1),(1,1)$. Now we expand

$\xi=\xi_{S}+\sum_{m=1,2}\xi^{(m,0)},$

and plug this expansion into (52). Using the orthogonality among various
eigenfunctions of the self-adjoint operator $L(a^{(0)}(\tau))$ in the space
$X^{s}(a^{(0)}(\tau))$ (notice that there is no essential spectrum because of
the compact $\mathbb{S}^{k}$ factor in the domain), we find that equation (52)
is equivalent to the following system:

$\displaystyle\dot{\xi}_{S}-QL(a^{(0)})\xi_{S}=-QN(a^{(0)},\xi^{(0)}),$ (68)
$\displaystyle\dot{\beta}_{i}-a^{(0)}\beta_{i}=f_{i},$ (69)
$\displaystyle\dot{\gamma}_{ij}=h_{ij},$ (70)

where

$\displaystyle f_{i}=-\left\langle
N(a^{(0)},\xi^{(0)}),\,\Sigma^{(1,0)(i,0,0)}(a^{(0)})\right\rangle_{a^{(0)}},$
(71) $\displaystyle h_{ij}=-\left\langle
N(a^{(0)},\xi^{(0)}),\,\Sigma^{(2,0)(i,j,0)}(a^{(0)})\right\rangle_{a^{(0)}}.$
(72)

(Notice that $\Sigma^{(1,0)}$ and $\Sigma^{(2,0)}$ are already normalized.) To
get (71)-(72), which are the projections of the nonlinearity into the
eigenspaces of $P^{(1,0)}$ and $P^{(2,0)}$ respectively, we use the fact that
since $\partial_{\sigma}W(\sigma^{(0)})$ maps into
$\operatorname{ran}P^{(0,0)}\oplus\operatorname{ran}P^{(0,1)}\oplus\operatorname{ran}P^{(1,1)}$,
the terms in (52) involving $\mathcal{W}=\partial_{\sigma}W\vec{F}$ and
$\partial_{\sigma}W\vec{M}$ all drop out after taking projection into the
other eigenspaces.

The initial configurations associated to (68)-(70) are the three terms in
(51), respectively. Since (68)-(70) are already decoupled, in what follows we
consider the Cauchy problem for $\xi_{S},\,\beta_{i},\,\gamma_{ij}$
separately.

First, for (68), consider the following representation formula of the solution
$\xi_{S}(\tau)$ by Duhamel’s principle:

$\xi_{S}(\tau)=e^{-\tau\bar{L}}\xi_{S}(0)-\int_{0}^{\tau}e^{-(\tau-\tau^{\prime})\bar{L}}QN(a^{(0)}(\tau^{\prime}),\xi^{(0)}(\tau^{\prime}))\,d\tau^{\prime}.$
(73)

Here $\bar{L}:=QL(a^{(0)})Q$. Since at $\tau=0$, the function
$\xi_{S}(0)=\eta_{0}$ is perpendicular to all the zero-unstable modes of
$L(a^{(0)}(0))$ in the space $X^{s}(a_{0})$, and for all $\tau>0$ the operator
$\bar{L}$ is uniformly coercive on $X^{s}(a^{(0)}(\tau))$ (which follows from
Item 2 in Definition 2), it follows from standard parabolic theory (see
Section B for details) that there exists $c>0$ depending on $n,k$ only s.th.

$\left\lVert\xi_{S}(\tau)\right\rVert_{s,a(\tau)}\leq
e^{-c\tau}\left\lVert\xi_{S}(0)\right\rVert_{s-2,a(\tau)},\quad\tau\geq 0.$
(74)

By the interpolation inequality in Lemma 4, it follows

$\left\lVert\xi_{S}(\tau)\right\rVert_{s}\leq\delta
e^{-c^{\prime}\tau},\quad\tau\geq 0,$ (75)

where $c^{\prime}$ depends on the constant $c$ in (74) only, assuming
$\beta_{i},\,\gamma_{ij}=O(\delta^{2})$ in (51).

Next, for (69), by Lemma 5, we find that $\beta_{i}(\tau)$ remains bounded for
all $\tau$ if and only if the initial configuration is given by

$\beta_{i}(0)=-\int_{0}^{\infty}f_{i}(\tau^{\prime})\lambda(\tau^{\prime})^{-1}\,d\tau^{\prime}\quad\left(\lambda(\tau)=e^{\int_{0}^{\tau}a(\tau^{\prime})}\,d\tau^{\prime}\right).$
(76)

This integral indeed converges, since $\lambda(\tau)\geq 1$, and the nonlinear
estimate (156) shows $\left\lvert f_{i}(\tau)\right\rvert\leq\left\lVert
P^{(1,0)}N\right\rVert_{0,a^{(0)}}\lesssim\left\lVert\xi^{(0)}(\tau)\right\rVert_{s}^{2}\lesssim\left\langle\tau\right\rangle^{-4}.$
For this estimate, we use the fact that $a^{(0)}$ is admissible, c.f.
Definition 2, together with the decay condition (54).

(76) determines the choice of $\beta_{i}$ in (51). Since $f_{i}$ is a
functional of $\sigma^{(0)},\xi^{(0)}$, this choice of $\beta_{i}$ is a
function of the fixed path $\sigma^{(0)},\xi^{(0)}$.

Suppose (76) is satisfied. Then $\beta_{i}(\tau)$ is given by

$\beta_{i}(\tau)=-\int_{\tau}^{\infty}f(\tau^{\prime})\frac{\lambda(\tau)}{\lambda(\tau^{\prime})}\,d\tau^{\prime}.$
(77)

See Lemma 5 for details. For all $\tau\geq 0$, this function satisfies

$\displaystyle\left\lvert\beta_{i}(\tau)\right\rvert$
$\displaystyle\leq\int_{\tau}^{\infty}\left\lvert
f_{i}(\tau^{\prime})\right\rvert
e^{-\int_{\tau}^{\tau^{\prime}}a}\,d\tau^{\prime}$ (78)
$\displaystyle\leq\int_{\tau}^{\infty}\left\lvert
f_{i}(\tau^{\prime})\right\rvert d\tau^{\prime}$ $\displaystyle\leq
C\delta^{2}\left\langle\tau\right\rangle^{-3}.$

Here the last inequality follows from (54) and the estimate $\left\lvert
f_{i}\right\rvert\leq\left\lVert
P^{(1,0)}N\right\rVert_{0,a^{(0)}}\lesssim\left\lVert\xi^{(0)}\right\rVert_{s}^{2}\leq\delta^{2}\left\langle\tau\right\rangle^{-4},$
due to the nonlinear estimate (156). For the latter, we need the lower bound
from Item 3 in the admissibility conditions listed in Definition 2, which
holds for $\sigma^{(0)}\in\mathcal{A}_{\delta}^{\sigma}$. In (78), the
constant $C$ depends only on the constant in the estimate (156), which in turn
depends only on the dimension $n$.

We conclude from (78) that

$\left\lvert\beta_{i}(0)\right\rvert\lesssim\delta^{2}.$ (79)

This gives (63).

By (78), the elliptic estimate (150), and the interpolation inequality (133),
we conclude

$\left\lVert
P^{(1,0)}\xi(\tau)\right\rVert_{s}\leq\delta\left\langle\tau\right\rangle^{-2},$
(80)

provided $\delta$ is sufficiently small, in the sense that
$\delta^{1/3}Cc^{1/12}c_{1}\leq 1$ for the constants $C$ from (78), $c$ the
pivot condition (132), and $c_{1}$ from some elliptic estimate, respectively.
Notice that the path $\xi$ satisfies the pivot condition (132), as we show in
Lemma 9. This justifies the use of the interpolation (150).

Lastly, for (70), we note that $\gamma_{ij}(\tau)\to 0$ as $\tau\to 0$ if and
only if

$\gamma_{ij}(0)=-\int_{0}^{\infty}h_{ij}.$ (81)

This defines the choice of $\gamma_{ij}$ in (51). Moreover, if (81) holds,
then

$\gamma_{ij}(\tau)=-\int_{\tau}^{\infty}h_{ij},$ (82)

and there holds

$\left\lvert\gamma_{ij}(\tau)\right\rvert\leq
C^{\prime}\delta^{2}\left\langle\tau\right\rangle^{-3}.$ (83)

Here the constant $C^{\prime}$ is a constant depending on the estimate (156),
similar to the one from (78) (but possibly larger). From here we conclude (64)
by setting $\tau=0$. Shrinking $\delta$ if necessary, we can also deduce from
(83) the following estimate, whose derivation is identical to (80):

$\left\lVert
P^{(2,0)}\xi(\tau)\right\rVert_{s}\leq\delta\left\langle\tau\right\rangle^{-2}.$
(84)

Combining (75), (80), (84), we conclude that with the unique choices (80),
(82) of $\beta_{i},\,\gamma_{ij}$, as functions of $\xi^{(0)},\sigma^{(0)}$,
the solution $\xi$ to (52) with initial condition (51) satisfies the decay
condition (54). For reason described at the beginning of the proof, this
implies (53) holds for $\sigma$. This proves the the image
$(\sigma,\xi)=\Psi(\sigma^{(0)},\xi^{(0)})$ lies in $\mathcal{A}_{\delta}$.

3\. Now we prove the estimates (65)-(66). For simplicity we only prove (65)
because the other is completely analogous.

Write $U^{m}:=(\sigma^{(m)},\xi^{(m)}),\,m=0,1$ for two paths in
$\mathcal{A}_{\delta}$, and $\beta^{(m)}_{i}=\beta_{i}(U^{m})$ for the
constants in the initial configurations associated to these paths, as
constructed above.

By (76), we find

$\left\lvert\beta_{i}^{(1)}-\beta_{i}^{(0)}\right\rvert\leq\int_{0}^{\infty}\left\lvert
f_{i}^{(1)}(\tau^{\prime})-f_{i}^{(0)}(\tau^{\prime})\right\rvert\,d\tau^{\prime},$
(85)

where $f_{i}^{(m)}$ is given by (71).

$f_{i}^{(m)}=l_{i}^{(m)}N(U^{m}),$ (86)

where $N$ is as in (48), and $l_{i}^{(m)}:X^{s-2}\to\mathbb{R}$ is the linear
functional given by
$\phi\mapsto\left\langle\phi,\,\Sigma^{(1,0)(i,0,0)}(a^{(m)})\right\rangle$.

Consider the difference of the r.h.s. of (86), which we write as

$l_{i}^{(1)}N(U^{1})-l_{i}^{(0)}N(U^{0})=(l_{i}^{(1)}-l_{i}^{(0)})N(U^{1})+l_{i}^{(0)}(N(U^{1})-N(U^{0})).$
(87)

Since the paths $\sigma^{(m)}$ are admissible (see Definition 2), we have the
uniform estiamte $\left\lVert
l_{i}^{(m)}\right\rVert_{X^{s-2}\to\mathbb{R}}\lesssim\left\lvert
a^{(m)}\right\rvert$ independent of $m,\,i$ and $\tau$. Using this, we can
bound the first term in the r.h.s. of (87) by
$\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert\left\lVert
N(U^{1})\right\rVert_{s-2}$. By (156) and (54), this implies

$\left\lvert(l_{i}^{(1)}-l_{i}^{(0)})N(U^{1})\right\rvert\lesssim\delta^{2}\left\langle\tau\right\rangle^{-4}\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert.$
(88)

By the Lipschitz estimate (99) (see Section C for the proof) and (53), the
second term in the r.h.s. of (87) can be bounded as

$\displaystyle\left\lvert l_{i}^{(0)}(N(U^{1})-N(U^{0}))\right\rvert$
$\displaystyle\lesssim\left\lVert
l_{i}^{(m)}\right\rVert_{X^{s-2}\to\mathbb{R}}\left\lVert
N(U^{1})-N(U^{0})\right\rVert_{s-2}$ (89)
$\displaystyle\lesssim\delta\left\langle\tau\right\rangle^{-2}\left(\left\lVert\xi^{(1)}-\xi^{(0)}\right\rVert_{s}+\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert\right).$

Combining (88)-(89) in (87), and plugging the result back to (85), we find

$\left\lvert\beta_{i}^{(1)}-\beta_{i}^{(0)}\right\rvert\lesssim\delta\left(\int\left\langle\tau^{\prime}\right\rangle^{-2}\,d\tau^{\prime}\right)\left\lVert
U^{1}-U^{0}\right\rVert.$

This gives (65). The same argument gives (66).

∎

##  5 The contraction property of $\Psi$

In this section we prove the map
$\Psi:\mathcal{A}_{\delta}\to\mathcal{A}_{\delta}$ defines a contraction
w.r.t. the following norm:

$\left\lVert(\sigma,\xi)\right\rVert=\sup_{\tau\geq
0}(c_{0}^{-1}\left\langle\tau\right\rangle\left\lvert\sigma(\tau)-\tau(0)\right\rvert+\left\langle\tau\right\rangle^{2}\left\lVert\xi(\tau)\right\rVert_{s}).$
(90)

Here $c_{0}>0$ is the constant in (53). This defines a norm equivalent to the
uniform one in the space $C([0,\infty),\Sigma\times X^{s})$, in which the the
space $\mathcal{A}_{\delta}$ from Definition 3 is the closed ball of size
$O(\delta)$ around zero w.r.t. the norm (90). This shows
$(\mathcal{A}_{\delta},\left\lVert\cdot\right\rVert)$ is complete.

Now we show $\Psi$ is a contraction in
$(\mathcal{A}_{\delta},\left\lVert\cdot\right\rVert)$. This essentially
implies all of the statements in Theorem 1.

######  Theorem 3.

The map $\Psi:\mathcal{A}_{\delta}\to\mathcal{A}_{\delta}$ defined in (56) is
a contraction with respect to the norm (90), satisfying

$\left\lVert\Psi(U^{1})-\Psi(U^{2})\right\rVert\leq\delta^{1/2}\left\lVert
U^{1}-U^{0}\right\rVert\quad(U^{1},\,U^{2}\in\mathcal{A}_{\delta}).$ (91)

Consequently, there exist unique constants

$\beta_{i}(\eta_{0}),\,\gamma_{ij}(\eta_{0})=O(\delta^{2}),$ (92)

s.th. there exists a unique solution to the nonlinear system (41), (43)-(45)
with initial condition (50) and this choice of $\beta_{i}$, $\gamma_{ij}$ in
(51).

######  Remark 6.

As a by-product, this proves a global well-posedness result for the rescaled
MCF (9). The admissible set of initial configurations forms a finite-
codimensional manifold in $X^{s}$. The global solutions constructed here does
not necessarily arise from rescaling a given solution to (1).

###### Proof.

1\. For simplicity, write $U^{m}:=(\sigma^{(m)},\xi^{(m)}),\,m=0,1$ for two
elements in $\mathcal{A}_{\delta}$, and
$U^{m+2}:=\Psi(\sigma^{(m)},\xi^{(m)})$.

Using (48)-(51), we find that the difference $U^{3}-U^{2}$ is the unique
solution to the following linear Cauchy problem:

$\displaystyle\begin{split}\partial_{\tau}(\xi^{(3)}-\xi^{(2)})+L(a^{(0)})(\xi^{(3)}-\xi^{(2)})=&V(\sigma^{(0)},\sigma^{(1)})\xi^{(3)}\\\
&-(\dot{\sigma}^{(3)}-\dot{\sigma}^{(2)})\partial_{\sigma}W(\sigma^{(0)})\\\
&-\dot{\sigma}^{(3)}(\partial_{\sigma}W(\sigma^{(1)})-\partial_{\sigma}W(\sigma^{(0)}))\\\
&-N(a^{(1)},\xi^{(1)})+N(a^{(0)},\xi^{(0)}),\end{split}$ (93)
$\displaystyle\begin{split}(\xi^{(3)}-\xi^{(2)})|_{\tau=0}=&(\beta^{(1)}_{i}-\beta^{(0)}_{i})\Sigma^{(1,0)(i,0,0)}(a_{0})\\\
&+(\gamma_{ij}^{(1)}-\gamma_{ij}^{(0)})\Sigma^{(2,0)(i,j,0)}(a_{0}).\end{split}$
(94)

In (93), the operator $V$ is defined as the difference between the two
linearized operators as $V:=L(a^{(1)})-L(a^{(0)})$. In (94), the numbers
$\beta_{i}^{(m)},\,\gamma_{ij}^{(m)},\,m=0,1$ are the unique functions of
$U^{m}$ constructed in Theorem 1. To get (93)-(94), as before, we use the
equivalence between the system (48)-(51) in $U^{m}$ and the single equation in
$\xi^{(m)}$, i.e. (52) with (51).

2\. The claim now is that we have the following point-wise estimate for the
solution to (93)-(94):

$\left\lVert\xi^{(3)}(\tau)-\xi^{(2)}(\tau)\right\rVert_{s}\lesssim\delta\left\langle\tau\right\rangle^{-2}\left\lVert
U^{1}-U^{2}\right\rVert,\quad\tau\geq 0.$ (95)

To derive (95), we consider the projections of $\xi^{(3)}-\xi^{(2)}$ under
$P^{(m,n)}_{\mu}(\tau):=P^{(m,n)}(a^{(\mu)}(\tau)),\,\mu=1,2$ and
$Q_{\mu}=1-\sum P^{(m,n)}_{i}$. These operators are defined in the proof of
Theorem 2.

For $(m,n)=(0,0),(0,1),(1,1)$, it suffices to use the orthogonality condition
(145). Indeed, since $P_{i}^{(m,n)}\xi^{\mu+2}=0$ for $\mu=1,2$, we find we
have

$\displaystyle\left\lVert P_{0}^{(m,n)}(\xi^{(3)}-\xi^{(2)})\right\rVert_{s}$
$\displaystyle=\left\lVert P_{0}^{(m,n)}\xi^{(3)}\right\rVert_{s}$ (96)
$\displaystyle=\left\lVert(P^{(m,n)}_{1}-P^{(m,n)}_{0})\xi^{(3)}\right\rVert_{s}$
$\displaystyle\lesssim\delta\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert\left\langle\tau\right\rangle^{-2}.$

In the last step we use the fact that $\left\lVert
P_{1}-P_{2}\right\rVert_{s\to
s}\lesssim\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert$, together with the
condition (54) for $\xi^{(3)}$.

To bound the other projections of $\xi^{(3)}-\xi^{(2)}$, we need to use the
equation (93). First, we bound each of the terms in the r.h.s. of (93). On a
bounded domain in $X^{s},\,s\geq 2$, by the $C^{2}$-regularity of the
$F$-functional, the perturbation operator $V(\sigma^{(0)},\sigma^{(1)})$
satisfies the Lipschitz estimate $\left\lVert V\right\rVert_{s\to
s-2}\lesssim\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert$. (Alternative,
one can get this using (144).) This, together with the estimate (54) for
$\xi^{(3)}$, implies that the first term in the r.h.s. of (93) satisfies

$\left\lVert
V(\sigma^{(0)},\sigma^{(1)})\xi^{(3)}\right\rVert_{s-2}\lesssim\delta\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert\left\langle\tau\right\rangle^{-2}.$
(97)

The second term in the r.h.s. of (93) vanishes in the equation for
$P_{0}^{(m,n)}(\xi^{(3)}-\xi^{(2)})$ with $(m,n)\neq(0,0),(0,1),(1,1)$ and
$Q_{0}(\xi^{(3)}-\xi^{(2)})$, because the operator
$\partial_{\sigma}W(\sigma^{(0)})$ maps into the space
$\operatorname{ran}P_{0}^{(0,0)}\oplus\operatorname{ran}P^{(0,1)}_{0}\oplus\operatorname{ran}P^{(1,1)}_{0}$.
Thus we do not need to consider this term.

As discussed in Remark 1 earlier, the operator $\partial_{\sigma}W$ is locally
Lipschitz from the tangent bundle $T\Sigma\to X^{s}$. This, together with the
estimate (53) for $\left\lvert\dot{\sigma}^{(3)}\right\rvert$ , implies that
the third term in the r.h.s. of (93) satisfies

$\left\lVert\dot{\sigma}^{(3)}(\partial_{\sigma}W(\sigma^{(1)})-\partial_{\sigma}W(\sigma^{(0)}))\right\rVert_{s}\lesssim\delta\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert\left\langle\tau\right\rangle^{-2}.$
(98)

Lastly, in Appendix C, we show the estimate

$\left\lVert
N(a^{(1)},\xi^{(1)})-N(a^{(0)},\xi^{(0)})\right\rVert_{s-2}\lesssim\delta\left\langle\tau\right\rangle^{-2}\left(\left\lVert\xi^{(1)}-\xi^{(0)}\right\rVert_{s}+\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert\right),$
(99)

c.f. (159). So far, this has the weakest decay property.

3\. We conclude from (98)-(99) that

$\left\lVert\text{r.h.s. of
\eqref{5.3}}\right\rVert_{s-2}\lesssim\delta\left\langle\tau\right\rangle^{-2}\left(\left\lVert\xi^{(1)}-\xi^{(0)}\right\rVert_{s}+\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert\right).$
(100)

A similar estimate holds if we project both sides of (93) with $Q_{0}$ and
$P_{0}^{m,0},\,m=1,2$.

Below we will use (100) as follows: Suppose $P$ is one of the projections
$Q_{0},\,P_{0}^{m,0},\,m=1,2$. Consider the integral over the r.h.s. of (100),

$\int_{0}^{\tau}\left\langle\tau^{\prime}\right\rangle^{-2}\left(\left\lVert
P(\xi^{(1)}(\tau^{\prime})-\xi^{(0)}(\tau^{\prime}))\right\rVert_{s}+\left\lvert\sigma^{(1)}(\tau^{\prime})-\sigma^{(0)}(\tau^{\prime})\right\rvert\right)\,d\tau^{\prime}.$
(101)

The integrand is bounded by

$\left\langle\tau\right\rangle^{-3}\sup_{0\leq\tau^{\prime}\leq\tau}\left(\left\langle\tau^{\prime}\right\rangle\left\lVert
P(\xi^{(1)}(\tau^{\prime})-\xi^{(0)}(\tau^{\prime}))\right\rVert_{s}+\left\langle\tau^{\prime}\right\rangle\left\lvert\sigma^{(1)}(\tau^{\prime})-\sigma^{(0)}(\tau^{\prime})\right\rvert\right).$

Plugging this to (101), and factoring the constant out, we find the estimate

$\int_{0}^{\tau}\left\langle\tau^{\prime}\right\rangle^{-2}\left(\left\lVert
P(\xi^{(1)}(\tau^{\prime})-\xi^{(0)}(\tau^{\prime}))\right\rVert_{s}+\left\lvert\sigma^{(1)}(\tau^{\prime})-\sigma^{(0)}(\tau^{\prime})\right\rvert\right)\,d\tau^{\prime}\\\
\lesssim\left\langle\tau\right\rangle^{-2}\left\lVert(P(\xi^{(1)}-\xi^{(0)}),\sigma^{(1)}-\sigma^{(0)})\right\rVert,$
(102)

where the norm on the r.h.s. is defined in (90). The same argument gives the
estimate

$\int_{\tau}^{\infty}\left\langle\tau^{\prime}\right\rangle^{-2}\left(\left\lVert
P(\xi^{(1)}(\tau^{\prime})-\xi^{(0)}(\tau^{\prime}))\right\rVert_{s}+\left\lvert\sigma^{(1)}(\tau^{\prime})-\sigma^{(0)}(\tau^{\prime})\right\rvert\right)\,d\tau^{\prime}\\\
\lesssim\left\langle\tau\right\rangle^{-2}\left\lVert(P(\xi^{(1)}-\xi^{(0)}),\sigma^{(1)}-\sigma^{(0)})\right\rVert.$
(103)

4\. Now consider the equation for $Q_{0}(\xi^{(3)}-\xi^{(2)})$, which can be
derived from (93) as (68) from (52). Applying Duhamel’s principle to this
equation, we find

$\displaystyle\left\lVert Q_{0}(\xi^{(3)}-\xi^{(2)})\right\rVert_{s}\leq$
$\displaystyle\left\lVert
e^{-\tau\bar{L}}(Q_{0}(\xi^{(3)}-\xi^{(2)})(0))\right\rVert_{s}$ (104)
$\displaystyle+\delta\int_{0}^{\tau}\left\lVert
e^{-(\tau-\tau^{\prime})\bar{L}}\right\rVert_{s-2,s}\left\langle\tau^{\prime}\right\rangle^{-2}\Bigl{(}\left\lVert
Q_{0}(\xi^{(1)}(\tau^{\prime})-\xi^{(0)}(\tau^{\prime}))\right\rVert_{s-2}$
$\displaystyle+\left\lvert\sigma^{(1)}(\tau^{\prime})-\sigma^{(0)}(\tau^{\prime})\right\rvert\Bigr{)}\,d\tau^{\prime},$

where $\bar{L}:=Q_{0}L(a^{(0)})Q_{0}:X^{s}\to X^{s-2}$, and
$e^{-\tau\bar{L}}:X^{s-2}\to X^{s}$ is the propagator of $\bar{L}$.

We consider the two terms in the r.h.s. of (104). Since the stable projection
of the initial configuration (94) vanishes, the first term in the r.h.s. of
(104) drops out.

Since $Q_{0}$ projects onto the stable modes of $L$, the restriction $\bar{L}$
is uniformly coercive, with
$\left\langle\bar{L}\phi,\,\phi\right\rangle_{a^{(0)}}\geq
c\left\lVert\phi\right\rVert_{0,a^{(0)}}^{2}$ for some $c>0$ depending on the
decay property (53) only and any $\phi\in X^{s}(a^{(0)}),\,s\geq 2$. This and
the interpolation from Lemma 4 gives the propagator estimate $\left\lVert
e^{-\tau\bar{L}}\right\rVert_{s-2\to s}\lesssim 1$. Hence, the second term in
(104) reduces to an intergral of the form (101). By (102), we conclude

$\left\lVert
Q_{0}(\xi^{(3)}-\xi^{(2)})(\tau)\right\rVert_{s}\lesssim\left\langle\tau\right\rangle^{-2}\left\lVert(Q_{0}(\xi^{(1)}-\xi^{(0)}),\sigma^{(1)}-\sigma^{(0)})\right\rVert,\quad\tau\geq
0.$ (105)

Next, consider the term $P_{0}^{(1,0)}(\xi^{(3)}-\xi^{(2)})$. As in the proof
of Theorem 2, the evolution of this part amounts to the following system of
ODEs:

$\dot{\tilde{\beta}}_{i}-a^{(0)}\tilde{\beta_{i}}=\tilde{f_{i}},\quad
i=1,\ldots,n-k$ (106)

where

$\displaystyle\tilde{f_{i}}=$ $\displaystyle-\left\langle
V(\sigma^{(0)},\sigma^{(1)})\xi^{(3)},\,\Sigma^{(1,0)(i,0,0)}(a^{(0)})\right\rangle_{a^{(0)}}$
(107)
$\displaystyle-\left\langle\dot{\sigma}^{(3)}(\partial_{\sigma}W(\sigma^{(1)})-\partial_{\sigma}W(\sigma^{(0)})),\,\Sigma^{(1,0)(i,0,0)}(a^{(0)})\right\rangle_{a^{(0)}}$
$\displaystyle-\left\langle
N(a^{(1)},\xi^{(1)})-N(a^{(0)},\xi^{(0)}),\,\Sigma^{(1,0)(i,0,0)}(a^{(0)})\right\rangle_{a^{(0)}},$

and the initial configuration is $\beta_{i}^{(1)}-\beta_{i}^{(0)}$, c.f. (94).

Since $\xi^{(3)}-\xi^{(2)}$ remains bounded for all time, so do the
coefficients $\tilde{\beta_{i}}$ in (106). Thus Lemma 5 implies

$\displaystyle\left\lVert P_{0}^{(1,0)}(\xi^{(3)}-\xi^{(2)})\right\rVert_{s}$
(108) $\displaystyle\lesssim$
$\displaystyle\sum_{i=1}^{n-k}\left\lvert\tilde{\beta}_{i}\right\rvert\leq\sum_{i=1}^{n-k}\int_{\tau}^{\infty}\left\lvert\tilde{f}_{i}(\tau^{\prime})\right\rvert\frac{\lambda(\tau)}{\lambda(\tau^{\prime})}\,d\tau^{\prime}$
$\displaystyle\lesssim$
$\displaystyle\delta\sum_{i=1}^{n-k}\int_{\tau}^{\infty}\left\langle\tau^{\prime}\right\rangle^{-2}\Bigl{(}\left\lVert
P^{(1,0)(i,0,0)}(\xi^{(1)}(\tau^{\prime})-\xi^{(0)}(\tau^{\prime}))\right\rVert_{s}$
$\displaystyle+\left\lvert\sigma^{(1)}(\tau^{\prime})-\sigma^{(0)}(\tau^{\prime})\right\rvert\Bigr{)}e^{-(\tau^{\prime}-\tau)/2}\,d\tau^{\prime}.$

In the first step we use an elliptic estimate of the form (150). In the last
step we use (100) and the formula (107). Since (108) can be bounded by an
integral as in the l.h.s. of (103), we conclude

$\left\lVert P_{0}^{(1,0)}(\xi^{(3)}-\xi^{(2)})\right\rVert_{s}\\\
\lesssim\delta\left\langle\tau\right\rangle^{-2}\left\lVert(P_{0}^{(1,0)}(\xi^{(1)}-\xi^{(0)}),\sigma^{(1)}-\sigma^{(0)})\right\rVert,$
(109)

where the norm on the r.h.s. is (90).

The argument for estimating $P^{(2,0)}_{0}(\xi^{(3)}-\xi^{(2)})$ is completely
analogous. In this case the implicit constant may be larger because we do not
have a factor as $e^{-\int_{\tau}^{\tau^{\prime}}a}$ inside the integral that
defines $\gamma_{ij}$ (see (82) and (138)). We record this estimate:

$\left\lVert
P_{0}^{(2,0)}(\xi^{(3)}-\xi^{(2)})\right\rVert_{s}\lesssim\delta\left\langle\tau\right\rangle^{-2}\left\lVert(P_{0}^{(2,0)}(\xi^{(1)}-\xi^{(0)}),\sigma^{(1)}-\sigma^{(0)})\right\rVert.$
(110)

Collecting (96), (105), (109)-(110) gives (95).

5\. It remains now to prove a similar estimate for $\sigma^{(3)}-\sigma^{(2)}$
as (95). To this end, we use the fact that this difference satisfies the
Cauchy problem

$\displaystyle\begin{split}\partial_{\tau}(\sigma^{(3)}-\sigma^{(2)})=&\vec{F}(\sigma^{(1)})(\xi^{(3)}-\xi^{(2)})\\\
&+(\vec{F}(\sigma^{(1)})-\vec{F}(\sigma^{(0)}))\xi^{(2)}+\vec{M}(\sigma^{(1)},\xi^{(1)})-\vec{M}(\sigma^{(0)},\xi^{(0)}),\end{split}$
(111)
$\displaystyle\begin{split}\sigma^{(3)}-\sigma^{(2)}|_{\tau=0}=&(0,0,0)\in\Sigma.\end{split}$
(112)

By the definition of $\vec{F}$ in Section 3.1, using Cauchy-Schwartz we find
that the first term in the r.h.s. of (111) satisfies

$\displaystyle\left\lvert\vec{F}(\sigma^{(1)})(\xi^{(3)}-\xi^{(2)})\right\rvert$
(113) $\displaystyle\lesssim$
$\displaystyle\left(\left\lvert\sigma^{(1)}\right\rvert+\left\lvert\dot{\sigma}^{(1)}\right\rvert\right)\left\lVert\xi^{(3)}-\xi^{(2)}\right\rVert_{0}$
$\displaystyle\lesssim$
$\displaystyle\delta^{2}\left\langle\tau\right\rangle^{-3}\left\lVert
U^{1}-U^{2}\right\rVert,$

For the last inequality we use (95).

By the definition of $\vec{F}$ and $\vec{N}$ (see (49)), the two terms in the
second line of (111) satisfy similar bound as (98)-(99), respectively. Indeed,
we find

$\displaystyle\left\lvert\vec{F}(\sigma^{(1)})-\vec{F}(\sigma^{(0)}))\xi^{(2)}\right\rvert$
$\displaystyle\lesssim\delta\left\langle\tau\right\rangle^{-2}\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert\left\langle\tau\right\rangle^{-2},$
(114)
$\displaystyle\left\lvert\vec{M}(\sigma^{(1)},\xi^{(1)})-\vec{M}(\sigma^{(0)},\xi^{(0)})\right\rvert$
$\displaystyle\lesssim\delta\left\langle\tau\right\rangle^{-2}\left(\left\lVert\xi^{(1)}-\xi^{(0)}\right\rVert_{s}+\left\lvert\sigma^{(1)}-\sigma^{(0)}\right\rvert\right).$
(115)

We conclude from (113)-(115) that

$\left\lvert\partial_{\tau}(\sigma^{(3)}-\sigma^{(2)})\right\rvert\lesssim\delta\left\langle\tau\right\rangle^{-2}\left(\left\lVert\xi^{(1)}(\tau)-\xi^{(0)}(\tau)\right\rVert_{s}+\left\lvert\sigma^{(1)}(\tau)-\sigma^{(0)}(\tau)\right\rvert\right).$
(116)

Integrating (116) in time and using the zero initial condition (112), we find
the uniform bound

$\left\lvert\sigma^{(3)}(\tau)-\sigma^{(2)}(\tau)\right\rvert\lesssim\left\langle\tau\right\rangle^{-1}\left\lVert
U^{1}-U^{0}\right\rVert.$ (117)

This, together with (95), gives the desired contraction property after taking
supremum over $\tau$, namely

$\left\lVert U^{3}-U^{2}\right\rVert\leq\delta^{1/2}\left\lVert
U^{1}-U^{0}\right\rVert,$ (118)

provided $\delta^{1/2}$ is sufficiently small, so as to cancel the implicit
constants in (95) and (117).

6\. By Theorem 2, (118) and Lemma 6, for every $\eta_{0}$ satisfying the
assumption of Lemma 8, we get a unique fixed point $U\in\mathcal{A}_{\delta}$
by iterating $\Psi$ on a point $U^{(0)}\in\mathcal{A}_{\delta}$. Indeed,
define $U^{(m)}:=\Psi(U^{m-1}),\,m=1,2,\ldots$. Each $U^{(m)}$ is the unique
solution to the linear Cauchy problem (48)-(51) with a initial configuration
depending on $\eta_{0}$ and some constants
$\beta_{i}^{(m-1)},\,\gamma_{ij}^{(m-1)}$. By Theorem 2, the latter are
defined uniquely as functions of $U^{(m-1)}$. It follows that the limit $U$ is
the unique solution to the nonlinear problem (41), (43)-(45) with initial
condition (50)-(51), where the constants $\beta_{i},\,\gamma_{ij}$ are defined
to be the limits of $\beta_{i}^{(m)},\,\gamma_{ij}^{(m)}$ as $m\to\infty$,
respectively. Indeed, if $U^{(m)}$ is a Cauchy sequence in
$\mathcal{A}_{\delta}$, then these limits exist by (65)-(66). Moreover, the
constant $\beta_{i},\,\gamma_{ij}$ now depend on $\eta_{0}$ only. By the
uniform estimates (63)-(64), we conclude that
$\beta_{i},\,\gamma_{ij}=O(\delta^{2})$. This completes the proof.

∎

##  6 A Sable Manifold Theorem for the Rescaled MCF

In this section we consider Theorem 1 as a stable manifold theorem for the
rescaled MCF, (41).

Define a set

$M:=\left\\{\eta+\Phi(\eta):\eta\in\mathcal{B}_{\delta}\cap\mathcal{S}\right\\}\quad(0<\delta\ll
1).$ (119)

Recall that the map $\Phi$, given in Definition 5, maps each
$\eta\in\mathcal{B}_{\delta}\cap\mathcal{S}$ in the parameter space to the
fixed point of the map $\Psi(\cdot,\eta)$ in the space $\mathcal{A}_{\delta}$.

In what follows, we show the effect of $\Phi$ amounts to a second order
perturbation of $\mathcal{B}_{\delta}\cap\mathcal{S}$, and consequently the
manifold $M$ is non-degenerate. We conjecture that in fact $\Phi$ is a $C^{1}$
immersion.

We also show a Lipschitz estimate for $\Phi$, This shows $M$ is a Lipschitz
graph over $\mathcal{B}_{\delta}\cap\mathcal{S}$. By Rademacher’s theorem,
this justifies the treatment of $M$ as a manifold.

######  Theorem 4.

The map $\Phi=\Phi$ in Definition 5 satisfies

$\displaystyle\left\lVert\Phi(\eta_{0})\right\rVert_{s}$
$\displaystyle\lesssim\left\lVert\eta_{0}\right\rVert_{s}^{2}\quad(\eta_{0}\in\mathcal{B}_{\delta}\cap\mathcal{S}),$
(120) $\displaystyle\left\lVert\Phi(\eta_{0})-\Phi(\eta_{1})\right\rVert$
$\displaystyle\lesssim\delta\left\lVert\eta_{0}-\eta_{1}\right\rVert\quad(\eta_{0},\,\eta_{1}\in\mathcal{B}_{\delta}\cap\mathcal{S}).$
(121)

###### Proof.

1\. We first prove (120). Take
$\eta_{0}\in\mathcal{B}_{\delta}\cap\mathcal{S}$ with
$\left\lVert\eta_{0}\right\rVert\sim\delta$. Fix some $0<\epsilon\ll 1$ to be
determined. As in the last part of the proof of Theorem 3, for sufficiently
large $M$ and all $m\geq M$, one can iterate the map
$\Psi=\Psi(\cdot,\eta_{0})$ for $m$ times to get some
$U^{(m)}\in\mathcal{A}_{\delta}$ (this space does not depend on the choice of
$\eta_{0}$) and $\beta_{i}^{(m)},\,\gamma_{ij}^{(m)}$ s.th. the following
holds:

  1. 1.

$\left\lvert\beta_{i}^{(m)}-\beta_{i}\right\rvert+\left\lvert\gamma_{ij}^{(m)}-\gamma_{ij}\right\rvert<\epsilon;$

  2. 2.

$\left\lVert\xi^{(m+1)}(\tau)-\xi^{(m)}(\tau)\right\rVert<\epsilon\left\langle\tau\right\rangle^{-2}$
for all $\tau\geq 0$;

  3. 3.

$\beta_{i}^{(m)}=-\int_{0}^{\infty}f_{i}(U^{(m)})\lambda^{-1}$ and
$\gamma_{ij}^{(m)}=-\int_{0}^{\infty}h_{ij}(U^{(m)})$, where the functionals
$f_{i},\,h_{ij}:\mathcal{A}_{\delta}\to\mathbb{R}$ are as in (71)-(72).

Consider the path $U^{(M+1)}$ with $M\gg 1$. This solves the Cauchy problem

$\displaystyle\dot{\xi}^{(M+1)}+L(a^{(M)})\xi^{(M+1)}$
$\displaystyle=-\mathcal{W}(\sigma^{(M)})\xi^{(M+1)}-\tilde{N}(a^{(M)},\xi^{(M)}),$
(122) $\displaystyle\xi^{(M+1)}(0)$
$\displaystyle=\eta_{0}+\beta_{i}^{(M)}\Sigma^{(1,0)(i,0,0)}(a_{0})+\gamma_{ij}^{(M)}\Sigma^{(2,0),(i,j,0)}(a_{0}).$
(123)

For $\left\lVert\eta_{0}\right\rVert_{s}\sim\delta$ and $\epsilon,\,\delta\ll
1$, by (63)-(64), we have
$\left\lVert\xi^{(M+1)}(0)\right\rVert_{s}\lesssim\left\lVert\eta_{0}\right\rVert_{s}+\epsilon$.
On the other hand, by the decay condition (54), we find
$\left\lVert\xi^{(M+1)}(\tau)\right\rVert\lesssim\left\lVert\xi^{(M+1)}(0)\right\rVert_{s}\left\langle\tau\right\rangle^{-2}$.
Since $\left\lvert f_{i}(U^{(M)})\right\rvert,\,\left\lvert
h_{ij}(U^{(M)})\right\rvert\lesssim\left\lVert\xi^{(M)}\right\rVert_{s}^{2}$
by the nonlinear estimate (156) (recall the definition from (71)-(72)), with
Items 1-3 above, we find

$\displaystyle\left\lvert\beta_{i}\right\rvert$
$\displaystyle<\epsilon+\left\lvert\beta_{i}^{(M)}\right\rvert$ (124)
$\displaystyle\lesssim\epsilon+\int_{0}^{\infty}\left\lVert\xi^{(M)}\right\rVert_{s}^{2}$
$\displaystyle<\epsilon+\int_{0}^{\infty}(\left\lVert\xi^{(M+1)}(\tau^{\prime})\right\rVert_{s}+\epsilon\left\langle\tau^{\prime}\right\rangle^{-2})^{2}\,d\tau^{\prime}$
$\displaystyle\lesssim\epsilon+\left\lVert\eta_{0}\right\rVert_{s}^{2}\int_{0}^{\infty}\Bigl{(}1+\epsilon\delta^{-1}\Bigr{)}^{2}\left\langle\tau^{\prime}\right\rangle^{-4}\,d\tau^{\prime},$

and a similar estimate for $\gamma_{ij}$. For fixed $\delta$, taking
$\epsilon\to 0$ in estimate (124) and the one for $\gamma_{ij}$ gives (120).

2\. Next, we prove (121). As discussed in Remark 4, the map $\Psi$ depends on
the choice of $\eta_{0}$. Now, consider the map

$\Psi:\mathcal{A}_{\delta}\times(\mathcal{B}_{\delta}\cap\mathcal{S})\to\mathcal{A}_{\delta}.$

With the parameter space $\mathcal{B}_{\delta}\cap\mathcal{S}$ in place of the
set $B\subset Y$ in Lemma 6, we see that (121) follows from (118) and (140) if
we can show the following uniform Lipschitz estimates:

$\displaystyle\left\lvert\Phi(\eta_{0})-\Phi(\eta_{1})\right\rvert$
$\displaystyle\lesssim\delta\lim_{M\to\infty}\left\lVert\Psi^{(M)}(U,\eta_{0})-\Psi^{(M)}(U,\eta_{1})\right\rVert$
(125) $\displaystyle\left\lVert\Psi(U,\eta_{0})-\Psi(U,\eta_{1})\right\rVert$
$\displaystyle\lesssim\left\lVert\eta_{0}-\eta_{1}\right\rVert_{s},$ (126)

for all
$\eta_{0},\,\eta_{1}\in\mathcal{B}_{\delta}\cap\mathcal{S},\,U\in\mathcal{A}_{\delta}.$
Here the notation $\Psi^{(M)}$ means iterating the map $M$ times. Notice that
the limit in (125) exists, since by Theorem 3, the sequences
$\Psi^{(M)}(U,\eta_{m}),\,m=0,1$ both converge as $M\to\infty$ in the space
$\mathcal{A}_{\delta}$.

To get (125), we use a similar argument as in Step 1. Here we only show
$\left\lvert\beta_{i}(\eta_{1})-\beta_{i}(\eta_{0})\right\rvert\lesssim\delta\left\lVert\eta_{1}-\eta_{0}\right\rVert_{s},$
as the argument to bound the $\gamma_{ij}$-difference is identical.

Indeed, define $\beta_{i}(\eta_{m}),\,\beta_{i}^{(M)}(\eta_{m})$ by adapting
the construction from Step 1 for $m=0,1$. Then we get the estimate

$\displaystyle\left\lvert\beta_{i}(\eta_{1})-\beta_{i}(\eta_{0})\right\rvert$
$\displaystyle<2\epsilon+\left\lvert\beta_{i}^{(M)}(\eta_{1})-\beta_{i}^{(M)}(\eta_{0})\right\rvert$
(127) $\displaystyle\lesssim\epsilon+\int_{0}^{\infty}\left\lvert
f_{i}(U^{(M)}(\eta_{1}))-f_{i}(U^{(M)}(\eta_{0}))\right\rvert,$

for any fix $\epsilon>0$ and all sufficiently large $M$. Here the paths
$U^{(M)}(\eta_{m}),\,m=0,1$ are obtained by iterating the map
$\Psi(\cdot,\eta_{m})$ on some fixed $U\in\mathcal{A}_{\delta}$.

The claim now is that there holds the difference estimate

$\left\lvert
f_{i}(U^{(M)}(\eta_{1}))-f_{i}(U^{(M)}(\eta_{0}))\right\rvert\lesssim\delta\left\langle\tau\right\rangle^{-2}\left\lVert
U^{(M)}(\eta_{1})-U^{(M)}(\eta_{0})\right\rVert,$ (128)

for all sufficiently large $M$. Indeed, if (128) holds, then plugging this
into (127) and taking $\epsilon\to 0,\,M\to\infty$ gives (125).

In view of the definition of $f_{i}$ from (71) and the Lipschitz estimate
(159), to get (128), it suffices to show that the paths
$U^{(M)}(\eta_{1}),\,U^{(M)}(\eta_{0})$ remains uniformly close in
$\mathcal{A}_{\delta}$ for all large $M$. This follows from the definition
$U^{(M)}(\eta_{m})=\Psi^{(M)}(U,\eta_{m})$ and the uniform bound (126),
provided $\eta_{0}$ and $\eta_{1}$ are close.

3\. It remains to prove (126). To this end, write
$(\sigma^{m},\xi^{m}):=\Psi(U,\eta_{m}),\,m=0,1$ for some fixed path
$U\in\mathcal{A}_{\delta}$. Notice the difference from Theorem 3: Here we fix
some $U$ but take different $\eta_{m}$, whereas in Theorem 3 it is the other
way around.

By the definition of $\Psi$ from (56), we find that the difference
$\xi^{1}-\xi^{0}$ satisfies the Cauchy problem

$\displaystyle\partial_{\tau}(\xi^{1}-\xi^{0})+L(a)(\xi^{1}-\xi^{0})=-\mathcal{W}(\sigma)(\xi^{1}-\xi^{0}),$
(129)
$\displaystyle\partial_{\tau}(\sigma^{1}-\sigma^{0})=\vec{F}(\sigma)(\xi^{1}-\xi^{0}),$
(130)
$\displaystyle(\sigma^{1}-\sigma^{0},\xi^{1}-\xi^{0})|_{\tau=0}=(0,\eta_{1}-\eta_{0}).$
(131)

Here $\sigma$ and $a$ are components of the fixed path $U$, and the r.h.s. of
(129) is bounded by (148).

Now we study (129)-(131) as in the proof of Theorem 3. Indeed, if we decompose
$\xi^{1}(\tau)-\xi^{0}(\tau)$ in terms of the eigenfunctions of $L(a(\tau))$,
then the projections of this difference along the stable modes decay
exponentially by the evolution (129), and those along the zero-unstable modes
decay at the rate of $\left\langle\tau\right\rangle^{-2}$ by the construction
from Theorem 2. The conclusion is the estimate
$\left\lVert\xi^{1}(\tau)-\xi^{0}(\tau)\right\rVert_{s}\lesssim\left\langle\tau\right\rangle^{-2}\left\lVert\xi^{1}(0)-\xi^{0}(0)\right\rVert=\left\langle\tau\right\rangle^{-2}\left\lVert\eta_{1}-\eta_{0}\right\rVert$.
Plugging this into (130) and integrating, we get the estimate
$\left\lvert\sigma^{1}-\sigma^{0}\right\rvert\lesssim\left\langle\tau\right\rangle^{-1}\left\lVert\eta_{1}-\eta_{0}\right\rVert$
(here we use the zero initial condition from (131)). Thus (126) follows.

∎

## Acknowledgment

The Author is supported by Danish National Research Foundation grant CPH-
GEOTOP-DNRF151. The Author thanks NM Møller and JMS Ma for helpful
discussions.

## Declarations

  * •

Conflict of interest: The Author has no conflicts of interest to declare that
are relevant to the content of this article.

##  Appendix A Elementary Lemmas

In this section we prove some abstract and elementary lemmas. Despite their
simple appearcance, they play crucial roles in Sects. 4-5.

Recall that the weighted Sobolev space $X^{s}(a)$ is defined in Section 2. As
discussed in that section, for two numbers $b\leq 1/2\leq a$ we have the
trivial embedding

$X^{s}(b)\leq X^{s}\equiv X^{s}(1/2)\subset X^{s}(a)\quad(s\geq 0).$

In Sect. 4, we are concerned with the $X^{s}$-estimate for a family of
functions $\phi(t)\in X^{s}(b)$, knowing some estimate of the varying norm
$\left\lVert\phi(t)\right\rVert_{s,a(t)}$ for a function $a(t)\geq 1/2$. Lemma
4 below allows us to deal with this issue.

######  Definition 6 (pivot condition).

Fix some $0<\delta<1/8$. A family of functions
$\phi(\cdot,t):\mathbb{R}^{n-k}\times\mathbb{S}^{k}\to\mathbb{R},\,t\geq 0$
satisfied the pivot condition if there is constant $c>0$ independent of $t$
s.th.

$\int\sum_{\left\lvert\alpha\right\rvert\leq
s}\left\lvert\partial^{\alpha}\phi(t)\right\rvert^{2}\rho_{b}\leq c,\quad
b:=\frac{1}{2}-4\delta>0.$ (132)

Equivalently, this means $\phi(t)\in X^{s}(b)$ and is uniformly bounded for
all $t$ in the $X^{s}(b)$-norm.

######  Lemma 4.

Let $a(t)$ be a function satisfying $0\leq a(t)-1/2\leq 2\delta$. Suppose
$\phi(t)$ satisfies the pivot condition (132), and
$\left\lVert\phi(t)\right\rVert_{s,a(t)}\leq g(t)$ for some positive function
$g$. Then there holds

$\left\lVert\phi(t)\right\rVert_{s}\leq c^{1/12}g(t)^{2/3}.$ (133)

###### Proof.

Let $u:=\sum_{\left\lvert\alpha\right\rvert\leq
s}\left\lvert\partial^{\alpha}\phi\right\rvert^{2}$. Then
$u\rho_{1/2}=(u^{2/3}\rho_{2a/3})(u^{1/3}\rho_{\frac{1}{2}-\frac{2a}{3}}).$ By
Hölder’s inequality with $p=3/2$ and $q=3$, we find

$\left\lVert\phi\right\rVert_{s}^{2}=\int u\rho_{1/2}\leq\left(\int
u\rho_{a}\right)^{2/3}\left(\int
u\rho_{\frac{3}{2}-2a}\right)^{1/3}=\left\lVert\phi\right\rVert_{s,a}^{4/3}\left(\int
u\rho_{\frac{3}{2}-2a}\right)^{1/3}.$ (134)

The assumption $0\leq a-1/2\leq 2\delta<1/4$ implies $\tfrac{3}{2}-2a\geq
b>0$, and therefore the second factor in the r.h.s. of (134) is bounded by the
l.h.s. of (132). Hence (133) follows from the assumption
$\left\lVert\phi\right\rVert_{s,a}\leq g$ and taking square root on both sides
of (134). ∎

######  Remark 7.

More generally, for all $1<p<\infty$, one can vary the pivot condition to
obtain similar estimate as (133) with higher power in the r.h.s.. For our
need, any power $1<p<2$ will do.

By Lemma 4, every dissipative estimate in terms of the $X^{s}(a)$-norm with
possibly changing $a\geq 1/2$ gives a (weaker) dissipative estimate in the
fixed $X^{s}$-norm. This is important for many estimates in Sect.4.

The following two lemmas are the mechanism behind Theorem 2 and Theorem 3,
respectively. These results are obtained in [MR2480603].

The first lemma concerns with Cauchy problem for an inhomogeneous ODE.

######  Lemma 5 (c.f. [MR2480603]*Lem. 23).

Fix two functions $a(t)\geq c>0$ and $f\in L^{1}([0,\infty),\mathbb{R})$.
Consider the Cauchy problem for $x:[0,\infty)\to\mathbb{R}$:

$\displaystyle\dot{x}-a(t)x=f,$ (135) $\displaystyle x(0)=x_{0}\in\mathbb{R}.$
(136)

There exists a unique bounded solution if and only if

$x_{0}=-\int_{0}^{\infty}f_{i}(t^{\prime})e^{-\int_{0}^{t^{\prime}}a}\,dt^{\prime}\text{
in \eqref{B.2}}.$ (137)

Moreover, if (137) holds, then the solution to (135)-(136) is given by

$x(t)=-\int_{t}^{\infty}f(t^{\prime})e^{-\int_{t}^{t^{\prime}}a},\,d\tau^{\prime}.$
(138)

###### Proof.

The variation of parameter formula gives the general solution to (135)-(136)
as

$x(t)=e^{\int_{0}^{t}a}(x_{0}+\int_{0}^{t}fe^{-\int_{0}^{t^{\prime}}a}\,dt^{\prime}).$
(139)

Multiplying both sides of (139) by $e^{-\int_{0}^{t}a}$, and then taking
$t\to\infty$, we find that if $x(t)\leq C$, then $x(t)e^{-\int_{0}^{t}a}\to 0$
as $t\to\infty$ because of the uniform bound $a(t)\geq c>0$, and therefore
(137) holds.

Conversely, if (137) holds, then the general formula (139) simplifies to
(138). For $a(t)>0$ and $f\in L^{1}$ we can conclude from here that
$x(t)\leq\left\lVert f\right\rVert_{L^{1}}$.

∎

For the next lemma, we adopt similar notations as in Sects. 4-6.

######  Lemma 6 (c.f. [MR2480603]*Lem. 28).

Let $A\subset X$ be a closed subset of a Banach space
$(X,\left\lVert\cdot\right\rVert)$. Let $B\subset Y$ be a subset of a normed
vector space $(Y,\left\lVert\cdot\right\rVert_{Y})$. Let $\Psi:X\times Y\to X$
be a map s.th.

  1. 1.

$\Psi(A\times B)\subset A$;

  2. 2.

There exists $0<\rho<1$ s.th.

$\sup_{\eta\in
B}\left\lVert\Psi(u^{(1)},\eta)-\Psi(u^{(0)},\eta)\right\rVert\leq\rho\left\lVert
u^{(1)}-u^{(0)}\right\rVert$

for two vectors $u^{(0)},u^{(1)}\in A$;

  3. 3.

There exists $\alpha,\delta>0$ s.th.

$\sup_{u\in
A}\left\lVert\Psi(u,\eta_{1})-\Psi(u,\eta_{0})\right\rVert\leq\delta\left\lVert\eta_{1}-\eta_{0}\right\rVert_{Y}^{\alpha}$

for two vectors $\eta_{1},\eta_{0}\in B$.

Then for every $\eta\in B$, there exists a unique fixed point $u(\eta)$ s.th.
$\Psi(u(\eta),\eta)=u(\eta)$. Moreover, there holds

$\left\lVert
u(\eta_{1})-u(\eta_{0})\right\rVert\leq\frac{\delta}{1-\rho}\left\lVert\eta_{1}-\eta_{0}\right\rVert_{Y}^{\alpha}.$
(140)

###### Proof.

Fix $u_{*}\in A$. For every $\eta\in B$, let $u^{(0)}(\eta):=u_{*}$,
$u^{(m)}(\eta)=\Psi(u^{(m-1)}(\eta),\eta)$ for $m=1,2,\ldots,$ and
$u(\eta):=\lim_{m\to\infty}\Psi(u^{(m)}(\eta),\eta)$. This limit exists and is
the unique fixed point of $\Psi$ in $A$, by Item 1-2 in the assumptions and
the standard contraction mapping theorem.

For the estimate (140), compute for $m=1,2,\ldots$

$\displaystyle\left\lVert u^{(m)}(\eta_{1})-u^{(m)}(\eta_{0})\right\rVert\leq$
$\displaystyle\left\lVert\Psi(u^{(m-1)}(\eta_{1}),\eta_{1})-\Psi(u^{(m-1)}(\eta_{0}),\eta_{1})\right\rVert$
$\displaystyle+\left\lVert\Psi(u^{(m-1)}(\eta_{0}),\eta_{1})-\Psi(u^{(m-1)}(\eta_{0}),\eta_{0})\right\rVert$
$\displaystyle\leq$ $\displaystyle\rho\left\lVert
u^{(m-1)}(\eta_{1})-u^{(m-1)}(\eta_{0})\right\rVert+\delta\left\lVert\eta_{1}-\eta_{0}\right\rVert_{Y}^{\alpha}$
$\displaystyle\leq$
$\displaystyle\delta\sum_{n=0}^{m}\rho^{n}\left\lVert\eta_{1}-\eta_{0}\right\rVert_{Y}^{\alpha}.$

In the last step we use induction on $m$ and Item 3 in the assumptions. Taking
$m\to\infty$ gives (140). ∎

##  Appendix B Global well-posedness of a linear Cauchy problem

In this section we study the global well-posedness of the problem (48)-(51).
In particular, we show the key map $\Psi$ introduced in Section 3.1 is well-
defined.

Indeed, for fixed $(\sigma^{(0)},\xi^{(0)})$, (52), which is equivalent to the
system (48)-(49) as we show below, is a linear parabolic equation w.r.t. the
drift Laplacian on the space $X^{s}(a)$, together with a source term as a
function of $(\sigma^{(0)},\xi^{(0)})$. In view of this, the well-posedness
results in this section is rather standard. The small twist in our situation
is that the drift Laplacian in (52) is slowly changing, as the drift parameter
$a\sim 1/2$ is a function of time.

For the main existence and regularity result in this section, we need the
following propagator estimate for the linearized operator $L(a)$.

######  Lemma 7.

Fix a function $a(t),\,t\geq 0$ satisfying $0\leq a(t)-1/2\leq 2\delta$. For
each $t$, define the linear propagator

$U(t):=e^{-L(a(t)}:X^{s-2}(a(t))\to X^{s}(a(t)),$ (141)

where the operator $L(a)$ is defined in (30). Then for every constant
$b\in[1/2-4\delta,1/2]$, there holds

$\displaystyle U(t)(X^{s-2}(b))\subset X^{s}(b),$ (142)
$\displaystyle\left\lVert U(t)\right\rVert_{s-2,b\to s,b}\leq C(t).$ (143)

###### Proof.

By the spectral theorem, it suffices to prove the restriction of $L(a(t))$
maps $X^{s}(b)\subset X^{s}(a(t))$ continuously into $X^{s-2}(b)$.

Using the formula (30), we have the relation

$L(a(t))=L(b)+(b-a(t))V,\quad V:\phi\mapsto(-\left\langle
y,\,\nabla_{y}(\cdot)\right\rangle+\tfrac{1}{k}\Delta_{\omega}+2).$

The operator $L(b)$ is continuous from $X^{s}(b)\to X^{s-2}(b)$. The operator
$V$, which is independent of the parameters $a(t)$ and $b$, satisfies
$\left\lVert V\phi\right\rVert_{s-2,b}\leq
c_{n,k}\left\lVert\phi\right\rVert_{s,b}$. (For this, one uses the estimate
$\left\lVert\left\langle
y,\,\nabla_{y}\phi\right\rangle\right\rVert_{s-2,b}\leq
c_{n}\left\lVert\phi\right\rVert_{s,b}$, c.f. [MR2289828]*Sec. 4.) Since
$\left\lvert b-a(t)\right\rvert\leq 6\delta$, it follows that for sufficiently
small $\delta$ there holds

$\left\lVert L(a(t))\phi\right\rVert_{s-2,b}\leq(1+\delta^{1/2})\left\lVert
L(b)\right\rVert_{s,b\to s-2,b}\left\lVert\phi\right\rVert_{s,b},\quad t\geq
0.$ (144)

This shows the mapping property (142) and the estimate (143) with
$C(t)=e^{(1+\delta^{1/2})t\left\lVert L(b)\right\rVert_{s,b\to s-2,b}}.$

∎

######  Lemma 8.

Fix $a_{0}>1/2$ in (50). Let $b=\tfrac{1}{2}-4\delta$ for some $0<\delta<1/8$.
For every $\eta_{0}\in X^{s}(b)\cap\mathcal{S}$ (see Definition 4), every
fixed path $(\sigma^{(0)},\xi^{(0)})\in\mathcal{A}_{\delta}$, and any
constants $\beta_{i},\gamma_{ij}$ in (51), the problem (48)-(51) has a unique
global solution in the space

$Lip([0,\infty),\Sigma)\times(C([0,\infty),X^{s}(b))\cap
C^{1}([0,\infty),X^{s-2}(b))),\quad b=\tfrac{1}{2}-4\delta.$

Moreover, for all $\tau\geq 0$, the solution satisfies the orthogonality
condition (c.f. (42))

$\left\langle\xi(\tau),\,\Sigma^{(m,n)}(a^{(0)}(\tau))\right\rangle_{a^{(0)}(\tau)},\quad\;(m,n)=(0,0),(0,1),(1,1).$
(145)

###### Proof.

Once the coefficients are frozen, (48)-(49) are decoupled. Thus, we first
solve for $\xi$, and then $\sigma$ can be solved by integrating (49).

Plugging (49) to (50), we find

$\dot{\xi}+L(a^{(0)})\xi=-\mathcal{W}(\sigma^{(0)})\xi-\tilde{N}(a^{(0)},\xi^{(0)}).$
(146)

Here, for $\vec{F},\,\vec{M}$ as in (49), we define
$\mathcal{W}(\sigma^{(0)}):=\partial_{\sigma}W(\sigma^{(0)})\vec{F}(\sigma^{(0)})$,
and

$\tilde{N}(a^{(0)},\xi^{(0)}):=N(a^{(0)},\xi^{(0)})+\partial_{\sigma}W(\sigma^{(0)})\vec{M}(\sigma^{(0)},\xi^{(0)}),$
(147)

which is the sum of the (frozen) nonlinearity in both equations.

Consider the two terms in the the r.h.s. of (146). The second, nonlinear term
is studied in Section C below. In the first term, the linear operator
$\mathcal{W}$ satisfies the uniform bound

$\left\lVert\mathcal{W}(\sigma^{(0)}(\tau))\right\rVert_{s,b\to s,b}\leq
c\quad(s\geq 2,b>0,\tau\geq 0),$ (148)

for some absolute constant $c>0$ (say, $500$). To get this, one uses the
explicit form of $W(\sigma)$ (see also Remark 1), the decay condition (53) of
$\sigma$, and the formulae (43)-(45), in which the linear terms in $\xi$
defines $\vec{F}$.

Fix some $T>0$ to be determined. Define the space

$A:=C([0,T],X^{s}(b)),\quad
b:=\tfrac{1}{2}-4\delta,\left\lVert\phi\right\rVert_{A}:=\sup_{0\leq\tau\leq
T}\left\lVert\phi(\tau)\right\rVert_{s,b}.$

Clearly, this $\left\lVert\cdot\right\rVert_{A}$ turns $A$ into a Banach
space.

For any initial configuration $\xi(0)$ of the form (50), together with a path
$\phi\in A$, consider the map

$F:\phi\mapsto\left(\tau\mapsto
U(\tau)\xi(0)-\int_{0}^{\tau}U(\tau-\tau^{\prime})\left(\mathcal{W}(\sigma^{(0)})\phi(\tau^{\prime})+\tilde{N}(a^{(0)},\xi^{(0)})\right)\,d\tau^{\prime}\right).$
(149)

Here $U(\tau):=e^{-L(a^{(0)})\tau}$ is the propagator associated to
$L(a^{(0)}(\tau))$. By Duhamel’s principle, any fixed point of the map $F$ is
a (mild) local solution to (146).

With the choice $a=a^{(0)}(\tau)$ satisfying (53) and $b=1/2-4\delta$, the
assumption of Lemma 7 is satisfied. By (142), we see that $F(A)\subset A$.

Take $\phi_{1},\,\phi_{2}\in A$ and consider the difference

$(F(\phi_{1})-F(\phi_{2}))|_{\tau}=-\int_{0}^{\tau}U(\tau-\tau^{\prime})\mathcal{W}(\sigma^{(0)}(\tau^{\prime}))(\phi_{1}(\tau^{\prime})-\phi_{2}(\tau^{\prime})),\,d\tau^{\prime}.$

By the propagator estimate $\left\lVert U(\tau)\right\rVert_{s-2,s}\leq
C(\tau)$ in (143), together with (148), from here we can choose the length $T$
of the existence interval, s.th. the map $F$ in (149) is a contraction in $A$.
This gives a local solution $\xi(\tau),\tau\leq T$ to (146). This choice $T$
depends only on the constant $c>0$ in (148) and the constant $C(t)$ in the
propagator estimate (143). (The latter is large as $t$ increases due to the
unstable modes, and therefore is a constraint for choosing $T$.) Since this
choice of $T$ does not depend on the initial condition $\xi(0)$, we can
iterate this process starting from $\xi(T)$ to get a local solution on
$T\leq\tau\leq 2T$, etc. It follows that we have now a global solution $\xi\in
C([0,\infty),X^{s}(b))$ so long as the initial condition $\xi(0)\in X^{s}(b)$.

Moreover, by Lemma 6, in particular the estimate (140), we find the fixed
point $\xi=\xi(\tau)$ to be locally Lipschitz in $\tau$. By Rademacher’s
theorem, this implies the path $\partial_{\tau}\xi$ exists everywhere. Thus we
can differentiate the fixed point equation $\xi(\tau)=F(\xi(\tau))$ w.r.t.
$\tau$ to get (146) for a.e. $\tau$. On the other hand, by (144) and the
expression (154) below, for $(\sigma^{(0)},\xi^{(0)})\in\mathcal{A}_{\delta}$,
so long as $\xi(\tau)$ lies in $X^{s}(b)$, the path

$\tau\mapsto
L(a^{(0)}(\tau))\xi(\tau)+\mathcal{W}(\sigma^{(0)}(\tau))\xi(\tau)+\tilde{N}(a^{(0)}(\tau),\xi^{(0)}(\tau))\in
X^{s}(b)$

lies in $X^{s-2}(b)$ and is continuous. Thus we see in fact $\xi\in
C^{1}([0,\infty),X^{s-2}(b)).$

As in Section 2.3, once we have solved $\sigma$ from (49), we can conclude the
orthogonality condition (145) as long as it holds at $\tau=0$. The latter is
indeed the case, because of the choice for $\eta_{0}$ as
$X^{0}(a_{0})$-perpendicular to all the zero-unstable modes of $L(a_{0})$, and
the fact that the remaining terms in (50) already satisfy (145) at $\tau=0$,
owning to the orthogonality among various eigenfunctions of $L(a_{0})$.

∎

Next, we consider the regularity of the solution constructed above.

######  Lemma 9.

Let $(\sigma,\xi)$ be the global solution constructed in Lemma 8. Let
$P^{(m,n)}$ be the projection onto the eigenspaces spanned by the vectors
$\Sigma^{(m,n)(i,j,l)}(a)$ defined in (31)-(35) above. Then, for $s\geq 2$,
there holds the a priori estimate

$\left\lVert
P^{(m,n)}\xi\right\rVert_{s}\lesssim\left\lVert\xi\right\rVert_{0}.$ (150)

Moreover, this solution satisfies the pivot condition

$\left\lVert\xi(\tau)\right\rVert_{s,b}^{2}\leq c\quad
b:=\tfrac{1}{2}-4\delta.$ (151)

c.f. (132) and (55).

Moreover, the path $\sigma(\tau)$ satisfies

$\left\lvert\sigma(\tau)-\sigma(0)\right\rvert\lesssim\tau\sup_{0\leq\tau^{\prime}\leq\tau}\left\lVert\xi(\tau^{\prime})\right\rVert_{s}.$
(152)

where $a_{0}>1/2$ is as in (50).

###### Proof.

For the unstable parts $\xi^{(m,n)}:=P^{(m,n)}\xi$, consider the eigenvalue
equation $L(a)\xi^{(m,n)}=\lambda_{m,n}\xi^{(m,n)}$ in the space $X^{s}\subset
X^{s}(a)$. (This problem is well-posed by (144).) Following [MR3374960]*Lem.
3.11, we use Bochner’s formula for the drift Laplacian to get elliptic
estimates of the form $\left\lVert\xi^{(m,n)}\right\rVert_{s}\leq
C_{m,n}\left\lVert\xi^{(m,n)}\right\rVert_{s-2}$. These imply
$\left\lVert\sum\xi^{(m,n)}\right\rVert_{s}\leq(\sum
C_{m,n})\left\lVert\sum\xi^{(m,n)}\right\rVert_{s-2}$. Thus (150) follows.

For (151), consider the equation (146). In Lemma 8 we have shown the existence
of $X^{s}(b)$-solution $\xi$ to this linear parabolic equation. Standard semi-
group theory for such flows provides continuous dependence of the
$X^{s}(b)$-norm of the solution $\xi$ in terms of the $X^{s-2}(b)$-norm of the
data in (146). In view of (144) and (148), it suffices to show the source term
$\tilde{N}(a^{(0)},\xi^{(0)})$ lies in $X^{s-2}(b)$. (Here we are not claiming
any decay estimates in the $X^{s}(b)$-norm.)

Consider the two terms in the r.h.s. of

$\tilde{N}(a^{(0)},\xi^{(0)})=N(a^{(0)},\xi^{(0)})+\partial_{\sigma}W(\sigma^{(0)})\vec{M}(\sigma^{(0)},\xi^{(0)}).$

By the expression (154) below, the nonlinear map $N$ is continuous from
$X^{s}(b)\to X^{s-2}(b)$. The second term in $\tilde{N}$ is of the order
$\left\lVert N\right\rVert_{s-2,b}\left\lVert\xi^{(0)}\right\rVert_{0,b}$. The
conclusion is the continuous dependence

$\left\lVert\tilde{N}(a^{(0)},\xi^{(0)})\right\rVert_{s-2,b}\leq
c\left\lVert\xi^{(0)}\right\rVert_{s,b}\quad(\tau\geq 0),$ (153)

where the constant $c$ in the r.h.s. depends on that in (156) only. Estimate
(153), the assumption that $\xi^{(0)}$ satisfies the pivot condition (132),
together with the continuous dependence mentioned earlier, gives (151).

Lastly, (152) follows from integrating (49).

∎

##  Appendix C Properties of the Nonlinearity

In this section we prove two key estimates of the nonlinear map $N$ in (41).

For $a>0$, the nonlinear map $N(a,\cdot):X^{s}(a)\to X^{s-2}(a)$ is defined in
(41), as the remainder in the expansion
$F^{\prime}_{a}(\sqrt{k/a}+\xi)=L(a)\xi+N(a,\xi)$, which is valid in a
neighbourhood around zero in $X^{s}(a)$ due to the $C^{2}$ regularity of the
$F$-functional. By this definition, (26), and (30), we find that $N(a,\xi)$ is
given by

$\displaystyle\begin{split}N(a,\xi)=&\left(\frac{a}{k}-v^{-2}\right)\Delta_{\omega}\xi-
av+kv^{-1}+2a\xi\\\ &+N_{1}(v),\end{split}$ (154)
$\displaystyle\begin{split}N_{1}(v)=&\frac{v^{-4}\left\langle\nabla^{2}_{\omega}v\nabla_{\omega}v,\,\nabla_{\omega}v\right\rangle-v^{-3}\left\lvert\nabla_{\omega}v\right\rvert^{2}}{1+\left\lvert\nabla_{y}v\right\rvert^{2}+v^{-2}\left\lvert\nabla_{\omega}v\right\rvert^{2}}\\\
&-\frac{2v^{-2}\left\langle\nabla_{y}v,\,\left\langle\nabla_{\omega}v,\,\nabla_{\omega}\right\rangle\nabla_{y}v\right\rangle-\left\langle\nabla^{2}_{y}v\nabla_{y}v,\,\nabla_{y}v\right\rangle}{1+\left\lvert\nabla_{y}v\right\rvert^{2}+v^{-2}\left\lvert\nabla_{\omega}v\right\rvert^{2}},\end{split}$
(155)

where $v=\sqrt{k/a}+\xi$. This expansion is obtained in e.g.
[MR3397388]*Chap.1, [MR4303943]*Sect.3.

As pointed out in [MR3374960], though $N(a,\xi)$ arises as the quadratic
remainder from the expansion of the $X^{0}(a)$-gradient $F_{a}^{\prime}$, due
to the decay property of the Gaussian measure, one cannot establish a
quadratic estimate of the form $\left\lVert
N(a,\xi)\right\rVert_{s-2}\lesssim\left\lVert\xi\right\rVert_{s}^{2}$.

However, it is possible to get some quadratic estimate for the projection of
$N(a,\xi)$ into specific eigenspaces of $L(a)$, owning to the explicit forms
of the latter. As far as the decay estimates in Sect. 4-6 are concerned, it
suffices to bound the projections of $N$ onto the span of (33) and (35).

######  Lemma 10.

Fix $0<\delta\ll 1$ and suppose $1/2+\delta\leq a\leq 1/2+2\delta$. Let
$P^{(m,n)}$ be the projection onto the eigenspaces spanned by the vectors
$\Sigma^{(m,n)(i,j,l)}(a)$ defined in (31)-(35) above. Then there hold

$\left\lVert P^{(m,n)}N(a,\xi)\right\rVert_{s-2,a}\leq
c\left\lVert\xi\right\rVert_{s}^{2},$ (156)

where the constant $c>0$ on the r.h.s. depends on dimension only.

###### Proof.

In [MR3374960]*(4.19), the following pointwise estimate for $N$ is obtained:

$N(a,\xi(y,\omega))\lesssim(1+\left\lvert
y\right\rvert)(\left\lvert\xi(y,\omega)\right\rvert^{2}+\left\lvert\nabla\xi(y,\omega)\right\rvert^{2}+\left\lvert\nabla^{2}\xi(y,\omega)\right\rvert^{2}).$
(157)

By a rescaling argument, in principle the implicit constant can depend on $a$.
But for all $\left\lvert a-1/2\right\rvert\ll 1$, one can make the constant
uniform in $a$.

The claim now is that (156) holds with $(m,n)=(2,0)$. Indeed, this gives the
worst bound of all such projections, and only this projection and the one with
$(m,n)=(1,0)$ are needed in the previous sections, and all other estimates
follow the same argument.

Let
$u=\left\lvert\xi\right\rvert^{2}+\left\lvert\nabla\xi\right\rvert^{2}+\left\lvert\nabla^{2}\xi\right\rvert^{2}$.
Recall the definition of the function $\Sigma^{(2,0)(i,j,0)}$ from (35). For
$s=2$, integrating (157) over the cylinder against the Gaussian measure, we
find

$\displaystyle\left\lVert P^{(2,0)}N\right\rVert_{0,a}$
$\displaystyle=\sum_{i,j=1}^{n-k}\left\lvert\int
N\Sigma^{(2,0)(i,j,0)}(a)\rho_{a}\right\rvert$ (158)
$\displaystyle\lesssim\int(\left\lvert y\right\rvert^{2}+\left\lvert
y\right\rvert^{3})u\rho_{a}$
$\displaystyle\leq\sup_{\mathbb{R}^{n-k}\times\mathbb{S}^{k}}(\left\lvert
y\right\rvert^{2}+\left\lvert y\right\rvert^{3})\rho_{a-1/2}(y,\omega)\int
u\rho_{1/2}.$

For $a-1/2\geq\delta,$ the prefactor in the last line can be made uniform, and
one can conclude the claim from here.

For $s\geq 3$, one can differentiate the point-wise estimate (157), replace
the second line of (158) with the result, and then proceed as above. ∎

From the proof above one can see what prevents an estimate of the form
$\left\lVert N\right\rVert_{s-2,a}\lesssim\left\lVert\xi\right\rVert_{s}^{2}$,
which is already weaker than an estimate of the form $\left\lVert
N\right\rVert_{s-2}\lesssim\left\lVert\xi\right\rVert_{s}^{2}$. Indeed, take
an orthonormal basis of the Hilber space $X^{s}(a)$. Then this consists of a
sequence of functions that has increasing asymptotic growth. For instance, in
the proof of Lemma 10 above, we consider the basis of $X^{s}(a)$ given by the
eigenfunctions of the self-adjoint operator $L(a)$, whose growth are dominated
by the Hermite polynomials. The latter have growing orders of polynomial
growth. Thus, one cannot proceed from the second to the third line in (158) if
one sums over all projections.

The following lemma is needed to derive the Lipschitz estimate (99) in Section
5.

######  Lemma 11.

For $s\geq 2$, define

$V(\delta_{1}):=\left\\{(a,\xi)\in\mathbb{R}\times
X^{s}:\left\lVert\xi\right\rVert_{s}\leq\delta_{1},0<a<1\right\\}$

for every sufficiently small $0<\delta_{1}\ll a^{-1/2}$. Then for any two
paths $(a^{(m)},v^{(m)})\in U(\delta),\,m=1,2$, there holds the Lipschitz
estimate

$\left\lVert
N(a^{(1)},\xi^{(1)})-N(a^{(0)},\xi^{(0)})\right\rVert_{s-2}\lesssim\delta_{1}\left(\left\lvert
a^{(1)}-a^{(0)}\right\rvert+\left\lVert\xi^{(1)}-\xi^{(0)}\right\rVert_{s}\right).$
(159)

###### Proof.

Consider the trivial inequality

$\left\lVert N(a^{(1)},v^{(1)})-N(a^{(0)},v^{(0)})\right\rVert_{s-2}\\\
\lesssim\left\lVert
N(a^{(1)},v^{(1)})-N(a^{(1)},v^{(0)})\right\rVert_{s-2}+\left\lVert
N(a^{(1)},v^{(0)})-N(a^{(0)},v^{(0)})\right\rVert_{s-2}.$ (160)

We bound the two terms in the r.h.s. respectively.

To bound the first term, consider the explicit expression (154) above. From
here we see that for fixed $a$, the map $N(a,\cdot)$ is a rational function of
$v^{-k},\,k=0,\ldots,4$ and the derivatives
$\partial^{\alpha}_{y}\partial^{\beta}_{\omega}v,\,\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert\leq
2$. Hence, for $s\geq 2$, the map $N(a,\cdot):V(\delta_{1})\to X^{s-2}$ is a
smooth by the uniform bound (156) proven earlier. The claim now is that the
partial Fréchet derivative $\partial_{\xi}N(a,0):X^{s}\to X^{s-2}$ vanishes.
If this holds, then we have

$\left\lVert
N(a^{(1)},v^{(1)})-N(a^{(1)},v^{(0)})\right\rVert_{s-2}\lesssim\delta_{1}\left\lVert\xi^{(1)}-\xi^{(0)}\right\rVert_{s},$
(161)

which follows from the mean value theorem for Fréchet differentiable maps,
c.f. [MR1225101]*Thm. 1.8.

To see $\partial_{\xi}N(a,0)=0$, we differentiate the expansion
$N=F^{\prime}_{a}-L(a)$ w.r.t. $\xi$ to obtain
$\partial_{\xi}N(a,\cdot)=dF_{a}^{\prime}(\cdot)-L(a)$, where
$dF_{a}^{\prime}$ is the Hessian of the $F$-functional defined in (25). By the
$C^{2}$ regularity of $F_{a}$, (say, from the analysis of the Hessian in
[MR2993752]*Sect. 4), one can evaluate this expression at a vanishing sequence
of $\xi_{n}$ in $X^{s}$ to get the claim. Thus (161) follows.

To bound the second term in the r.h.s. of (160), we need to use the expression
(154). The point here is that for fixed $\xi$ with
$\left\lVert\xi\right\rVert_{2}\ll 1$, the leading order term of
$N(\cdot,\xi)$ lies in the first five terms from (154), namely

$\left(\frac{a}{k}-v^{-2}\right)\Delta_{\omega}\xi-av+kv^{-1}+2a\xi.$ (162)

The claim is that the $X^{s-2}$-norm of this expression can be bounded by some
product of a Lipschitz function in $a$ and a function of the order
$O(\left\lVert\xi\right\rVert_{s})$. If this the case, then it follows from
the assumption $\left\lVert\xi\right\rVert\leq\delta_{1}$ that

$\left\lVert
N(a^{(1)},v^{(0)})-N(a^{(0)},v^{(0)})\right\rVert_{s-2}\lesssim\delta_{1}\left\lvert
a^{(1)}-a^{(0)}\right\rvert.$ (163)

To see (162) has the claimed property, we find

$\displaystyle\left\lVert\left(\frac{a}{k}-v^{-2}\right)\Delta_{\omega}\xi\right\rVert_{s-2}$
$\displaystyle\lesssim\left\lVert\xi\right\rVert_{s}a,$
$\displaystyle\left\lVert-av+kv^{-1}+2a\xi\right\rVert_{s}$
$\displaystyle\lesssim(\left\lVert\xi\right\rVert_{s}+\left\lVert\xi\right\rVert_{s}^{2})(a+a^{3/2}).$

This follows by plugging $v=\sqrt{k/a}+\xi$ into (162) and rearranging. For
the second line above to be Lipschitz, we need the assumption $0<a<1$.

Combining (160)-(162) gives (159).

∎

###### Proof of (99).

Now, for paths $U^{m}\in\mathcal{A}_{\delta},\,m=1,2$, define two paths
$(a^{(m)},\xi^{(m)})$ in the obvious way. By (54), at each $\tau$ we find

$(a^{(m)},\xi^{(m)})\in V(\delta\left\langle\tau\right\rangle^{-2}).$

This gives (99). ∎

By the same argument, using (153), one can get a similar estimate as (99) for
the nonlinear map $\tilde{N}$ in (147).

## References