# Adiabatic theory for the area constrained Willmore flow

Jingxuan Zhang

###### Abstract

In this paper, we develop an adiabatic theory for the evolution of large
closed surfaces under the area-constrained Willmore (ACW) flow in a three-
dimensional asymptotically Schwarzschild manifold. We construct explicitly a
map, defined on a certain four-dimensional manifold of barycenters, which
characterizes key static and dynamical properties of the ACW flow.

In particular, using this map, we find an explicit four-dimensional effective
dynamics of barycenters, which serves as a uniform asymptotic approximation
for the (infinite-dimensional) ACW flow, so long as the initial surface
satisfies certain mild geometric constraints (which determine the validity
interval).

Conversely, given any prescribed flow of barycenters evolving according this
effective dynamics, we construct a family of surfaces evolving by the ACW
flow, whose barycenters are uniformly close to the prescribed ones on a large
time interval (whose size depends on the geometric constraints of initial
configurations).

##  1 Introduction

Let $(M,g)$ be a $3$-dimensional, complete, oriented Riemannian manifold with
non-negative curvature. Consider the area-constrained Willmore (ACW) flow,

$\partial_{t}x^{N}=-W(x)-\lambda H(x).$ (1)

Here, for $t\geq 0$, $x=x_{t}:\mathbb{S}\to M$ is a family of embeddings of
spheres (with orientation compatible with that on $M$). $\partial_{t}x^{N}$
denotes the normal velocity at $x$, given by
$\partial_{t}x^{N}:=g(\partial_{t}x,\nu)$, where $\nu=\nu(x)$ is the unit
normal vector to $\Sigma$ at $x$. $H(x)$ denotes the mean curvature scalar at
$x$. $W(x):=\Delta
H(x)+H(x)(\operatorname{Ric}_{M}(\nu,\nu)+\left\lvert\mathring{A}\right\rvert^{2}(x))$
is the Willmore operator. $\lambda$ is the Lagrange multiplier, arising due to
the area constraint.

Physically, the ACW flow (1) is the gradient flow of Hawking mass of an
embedded $2$-sphere, first introduced in [MR3960907]. In mathematical biology,
building upon Helfrich’s seminal work [Helfrich], the flow (1) and its
variants are used to model the motion of cell membranes. More recently, (1)
finds applications to nonlinear elasticity [FGJ], computer vision [MR3011864],
among various other fields in applied science.

In this paper, we develop an adiabatic (or slow-motion approximation) theory
for (1) using a map $\Phi$ from a $4$-dimensional manifold $M^{\prime}$ to the
configuration space of (1). We call $\Phi$ the Lyapunov-Schmidt map, as it
provides a dynamical analogue of the well-known Lyapunov-Schmidt reduction for
static nonlinear equations. This map $\Phi$ is constructed explicitly in
Section 3.

On the analytical ground, roughly speaking, we show that the pullback by
$\Phi$ to $M^{\prime}$ essentially reduces key static and dynamical properties
of the infinite dimensional dynamical system (1) to finite dimensional ones.

For applications to various physical models, our results in this paper provide
an explicit $4$-dimensional dynamical system, (14), whose behavior
approximates that of (1) uniformly for all sufficiently large time, and
captures key qualitative behaviours of (1). This can potentially reduces
computational complexity for this complicated forth order PDE (1).

We rigorously formulate the main results of this paper in Theorem 3.

###  1.1 Backgrounds

The problem we study here is motivated by a recent work [eichmair2021large],
in which the authors study stationary solutions to (1) using Lyapunov-Schmidt
reduction. Here we derive some dynamical analogues of the static existence
results in [eichmair2021large], with, however, rather different focus. Indeed,
the main point of our results is that we have 1. explicit information about
the adiabatic parts of a flow of surfaces evolving according to (1), with 2.
uniformly small errors in time, and 3. we can construct solution (1) with
prescribe adiabatic behavior. See the precise statements of these points in
Theorem 3.

The method of adiabatic approximation has a long history in statistical
physics. See [MR2360179] for an excellent review in this context. Our work
here is inspired by a series of papers by I.M. Sigal with several co-authors,
in which the adiabatic theory is adapted to geometric problems
[sigal2012stability, MR2465296, MR3803553, MR3824945]. Other results along
this line which we have referred to include [MR1369419, MR3251832, MR4160305,
chodosh2017global, MR4303943, MR3397388], which cover a range of static and
dynamical problems of geometric equations using Lyapunov-Schmidt reduction.

Among the papers above, we single out a recent paper [MR3824945], in which the
authors study a formally similar problem (namely, the volume-preserving mean
curvature flow with initial configurations close to small geodesics spheres),
from which we draw much inspiration. In particular, it appears that the notion
of Lyapunov-Schmidt map is first mentioned in this paper.

It seems to us that our results is the first rigorous adiabatic theory for
Willmore flows. We expect these results to be robust, in the sense that they
can be easily adapted to problems related to (1), for instance, using the
generalized Willmore energy developed in [Friedrich2020] (which covers, among
others, the applications to biomembranes). In a separate paper, we will treat
the abstract properties of the Lyapunov-Schmidt map defined in Appendix A.

##  2 Setup of the problem

###  2.1 Asymptotically Schwarzschild manifold

A $3$-dimensional complete Riemannian manifold $(M,g)$ is said to be
$C^{k}$-close to Schwarzschild if the following holds:

  1. 1.

$M\setminus K$ is diffeomorphic to
$\mathbb{R}^{3}\setminus\overline{B_{1}(0)}$ for some compact subset $K\subset
M$.

  2. 2.

The metric $g$ splits as $g_{S}+h$, where
$g_{S}:=\left(1+\frac{m}{2\left\lvert x\right\rvert}\right)^{4}\delta_{ij}$ is
the Schwarzschild metric with ADM mass $m>0$, and $h\in C^{k}$ is a small
perturbation satisfying

$h_{ij}=h_{ji},\quad\partial^{\alpha}h_{ij}\leq\eta\left\lvert
x\right\rvert^{-(2+\left\lvert\alpha\right\rvert)}\quad(\left\lvert\alpha\right\rvert\leq
k,\;\left\lvert x\right\rvert\gg 1),$ (2)

for some fixed small decay coefficient $\eta\ll 1$. Here $x\in\mathbb{R}^{3}$
denotes the coordinate in the asymptotic chart on $M$.

Physically, for applications to general relativity, such manifold $M$ is a
perturbation of the static Schwarzschild black hole
$(\mathbb{R}^{3}\setminus\overline{B_{m/2}(0)},g_{S})$.

To simplify notations, throughout the paper we normalize ADM mass to be $m=2$.
We assume that the ambiance space $M$ is $C^{k}$-close to Schwarzschild for
sufficiently large $k$, and that in (2) the decay coefficient $\eta\ll 1$.
Thus in what follows we take

$(M,g)=(\mathbb{R}^{3}\setminus\overline{B_{1}(0)},g_{S}+h)$

where $h$ is as in (2).

To use results in [MR2785762, MR4236532, eichmair2021large], we assume the
scalar curvature $\operatorname{Sc}$ on $M$ satisfies the following decay
properties:

$\displaystyle x^{j}\partial_{x^{j}}\left(\left\lvert
x\right\rvert^{2}\operatorname{Sc}\right)=o(\left\lvert x\right\rvert^{2}),$
(3) $\displaystyle\operatorname{Sc}(x)-\operatorname{Sc}(-x)=o(\left\lvert
x\right\rvert^{4}).$ (4)

The asymptotically-flatness condition (3) is satisfied if $g$ is $C^{k}$-close
to Schwarzschild with $k\geq 4$, and $\operatorname{Sc}=o(\left\lvert
x\right\rvert^{4})$, in which case $\nabla\operatorname{Sc}=o(\left\lvert
x\right\rvert^{-5})$. (4) means the scalar curvature on $M$ is asymptotically
even. Geometrically, condition (3) provides quantitative control for various
estimates involving extrinsic geometric quantities. Condition (4) provides
qualitative control of the effective action in Secs. 4, B.

###  2.2 Configuration spaces and the geometric structure of ACW

In this subsection, we layout the geometric structure of ACW flow (1). This
structure is understood in the subsequent developments in Secs. 4-5.

Let $c\gg 1,k\geq 4$ be given. Define the configuration space

$X^{k}:=H^{k}(\mathbb{S},M),\quad X^{k}_{c}:=\left\\{x\in X^{k}:\left\lvert
x(\mathbb{S})\right\rvert=c\right\\}.$ (5)

Here, for a surface $\Sigma:=x(\mathbb{S})\subset M$, we denote
$\left\lvert\Sigma\right\rvert:=\int_{\Sigma}\,d\mu_{\Sigma}^{g}$ the area of
$\Sigma$, where $\mu_{\Sigma}^{g}$ is the area form induced by the embedding
$x$ and $g$. One can check easily that (1) is well-defined in $X^{k}_{c}$.

Let $x\in X^{k}$ and write $\Sigma=x(\mathbb{S})$. The tangent spaces to $x$
at $X^{k}$ and $X^{k}_{c}$ are respectively given by

$\displaystyle T_{x}X^{k}=H^{k}(\mathbb{S},T\Sigma),$ (6) $\displaystyle
T_{x}X^{k}_{c}=\left\\{\phi\in
T_{x}X^{k}:\int_{\Sigma}Hg(\phi,\nu)=0\right\\}.$ (7)

Here, (7) is due to the well-known first variation formula of the area
functional.

On the space $T_{x}X^{k}$, we can define an inner product

$\left\langle\phi,\,\phi^{\prime}\right\rangle_{x}:=\int_{\Sigma}g(\phi,\phi^{\prime}).$
(8)

With this, we have a formal Riemannian manifold structure on the configuration
spaces $X^{k}$ and $X^{k}_{c}$.

Moreover, with this geometric structure of $X^{k}$, one can view the equation
(1) as the $L^{2}$-gradient flow, restricted to $X^{k}_{c}$, of the Willmore
energy

$\mathcal{W}(\Sigma)=\frac{1}{4}\int_{\Sigma}H^{2}\,d\mu_{\Sigma}^{g}.$ (9)

We call (the images of) static solutions to (1) surfaces of Willmore type,
following the nomenclature in [MR2785762]. Using Sobolev inequalities, one can
show that for $k\geq 4$, the functional $\mathcal{W}$ is well-defined and
$C^{2}$ (in the sense of Fréchet derivatives) on $X^{k}_{c}$.

Let $d\mathcal{W}(x):T_{x}X^{k}_{c}\to T_{x}X^{k-4}_{c}$ be the Fréchet
derivative of $\mathcal{W}$ at an embedding $x$ in the class $X^{k}_{c}$.
Define the normal $L^{2}$-gradient
$\nabla^{N}\mathcal{W}(x)\phi:=d\mathcal{W}(x)\phi$ for every normal, area-
preserving variation $\phi$ on the surface $\Sigma=x(\mathbb{S})$. Then by the
first variation formula of the Willmore energy (see e.g. [MR2780248]*Sec. 3),
this $\nabla^{N}\mathcal{W}(x)$ is given by the r.h.s. of (1). This allows us
to rewrite (1) as

$\partial_{t}x^{N}=\nabla^{N}\mathcal{W}(x)\quad(x\in X^{k}_{c}).$

Equivalently, (1) is the (negative) gradient flow of the Hawking mass,

$m_{\text{Haw}}(\Sigma):=\frac{\left\lvert\Sigma\right\rvert^{1/2}}{(16\pi)^{3/2}}\left(16\pi-\frac{1}{2}\int_{\Sigma}H^{2}\,d\mu_{\Sigma}^{g}\right),$

in the sense that a flow of surfaces evolving according to (1) increases the
mass $m_{\text{Haw}}$. For interests from physics related to this problem,
especially in general relativity, see [MR645761].

###  2.3 Preliminary results

Let $R\gg 1$ be given. Let $K\subset M$ be a large compact set. As explained
in the last subsection, for asymptotically Schwarzschild manifold $M$, we can
identify the $M\setminus K$ with its coordinate space
$\mathbb{R}^{3}\setminus\overline{B_{R}(0)}$.

Let $\delta>0$ be given.

######  Definition 1 (Admissible surfaces).

For a closed surface $\Sigma\subset M$, define the inner and outer radii
$\rho(\Sigma),\,\lambda(\Sigma)$ as

$\displaystyle\rho(\Sigma)=\min_{x\in\Sigma}\left\lvert x\right\rvert,$ (10)
$\displaystyle\lambda(\Sigma):=\sqrt{\left\lvert\Sigma\right\rvert/4\pi}.$
(11)

We say $\Sigma$ is admissible if the interior of $\Sigma$ contains $K$, and

$\rho(\Sigma)>R,\quad\left\lvert\frac{\rho(\Sigma)}{\lambda(\Sigma)}-1\right\rvert+\int_{\Sigma}|\mathring{A}|^{2}<\delta.$
(12)

Here $\mathring{A}$ denotes the traceless part of the second fundamental form
on $\Sigma$.

######  Remark 1.

Geometrically, a surface $\Sigma$ is admissible if the origin lies
sufficiently deep inside the interior of $\Sigma$ (this property is called
centering in [eichmair2021large]), and at the same time the surface does not
wiggle too much. Obviously we have
$\lambda(\Sigma)\leq\max_{x\in\Sigma}\left\lvert x\right\rvert$. Using the
terminology in [eichmair2021large], every admissible surface $\Sigma$
satisfying (12) with $R,\delta^{-1}\gg 1$ is on-center

For the class of admissible surfaces, we have the following well-posedness
result for (1):

######  Theorem 1 ([MR4236532]*Thm. 5.3).

Assume $M$ is $C^{4}$-close to Schwarzschild and satisfies (3)-(4). Then for
$R\gg 1,\,\delta\ll 1$ and every admissible surface $\Sigma_{*}$ satisfying
(12), there exists a global solution to (1) with initial configuration
$\Sigma|_{t=0}=\Sigma_{*}$.

Stationary solution to (1) are called surfaces of Willmore type. The existence
and stability of such surfaces are studied in [eichmair2021large, MR4236532].

######  Theorem 2 ([MR2785762]*Thm. 1, [MR4236532]*Thm. 5.3).

Assume $M$ is $C^{4}$-close to Schwarzschild and satisfies (3)-(4). Then there
exists a compact subset $K\subset M$, s.th. $M\setminus K$ is foliated by
surfaces of Willmore type.

Moreover, for $R\gg 1,\,\delta\ll 1$ and every admissible surface $\Sigma_{*}$
satisfying (12), the flow generated by $\Sigma_{*}$ under (1) converges
smoothly to one of the leave of this foliation.

###  2.4 Main results and organization of the paper

The main result of this paper is the following:

######  Theorem 3 (Main).

Let $k\geq 4,\,c\gg 1$. Let $X^{k}$ be the configuration space defined in (5).
Fix $R\gg 1,\,\delta\ll 1$ in Definition 1.

Then there exists a map

$\tilde{\Phi}:M^{\prime}:=\mathbb{R}_{>R}\times
B_{1}(0)\subset\mathbb{R}\times\mathbb{R}^{3}\to X^{k}$

with the following property: Let $x_{r,z}:=\tilde{\Phi}(r,z)$. There hold:

  1. 1.

(Critical point) $x_{z}$ parametrizes a surface of Willmore type if and only
if $z$ is a critical point of the function
$\mathcal{W}\circ\tilde{\Phi}:\mathbb{R}^{4}\to\mathbb{R}$, restricted to the
submanifold $\left\\{(r,z)\in
M^{\prime}:\left\lvert\tilde{\Phi}(r,z)(\mathbb{S})\right\rvert=c\right\\}$.

  2. 2.

(Stability) Suppose $x_{z}$ parametrize an admissible surface of Willmore
type. Then $x_{z}$ is uniformly stable with small area-preserving
$H^{k}$-perturbation 111This means that if $y$ is another admissible surface
that is $H^{k}$-close to $x_{z}$ in the topology given in Definition 3, then
for every $\epsilon>0$ there exists $T>0$ s.th. $\lVert
x_{t}-y_{t}\rVert_{X^{k}}<\epsilon$ for all $t\geq T$, where $x_{t},\,y_{t}$
are respectively the flows generated by $x_{z},\,y$ under (1). if $z$ is a
strict local minimum of the function $\mathcal{W}\circ\tilde{\Phi}$ restricted
to the submanifold $\left\\{(r,z)\in
M^{\prime}:\left\lvert\tilde{\Phi}(r,z)(\mathbb{S})\right\rvert=c\right\\}$.

  3. 3.

(Effective dynamics) Let $\Sigma_{*}=x_{*}(\mathbb{S})$ be an admissible
surface. Let $\Sigma_{t}=x_{t}(\mathbb{S})$ be the global solution to (1) with
initial configuration $\Sigma|_{t=0}=\Sigma_{*}$ as in Theorem 1.

Then there exist $\alpha>0$, $T=O(R^{-\alpha})$, and a path $(r_{t},z_{t})\in
M^{\prime}$, s.th. for every $t\geq T$,

$\lVert\tilde{\Phi}(r_{t},z_{t})-x_{t}\rVert_{X^{k}}=O(R^{-3}).$ (13)

Moreover, the path $(r_{t},z_{t})$ evolves according to

$\displaystyle\dot{z}=\nabla_{z}\mathcal{W}\circ\tilde{\Phi}(r,z)+O(R^{-3}),$
(14) $\displaystyle\dot{r}=O(R^{-3}).$ (15)

In (14) the leading term is of the order $O(R^{-2})$.

  4. 4.

Conversely, if $(r_{t},z_{t})\in M^{\prime}$ is a flow evolving according to
(14)-(15), then there exists a global solution $x_{t}$ to (1) s.th. (13) holds
for this choice of $(r_{t},z_{t})$ and every $T\leq t\leq T+R$.

######  Remark 2.

The map $\tilde{\Phi}$ in Theorem 3 is defined by
$\tilde{\Phi}(r,z)=\theta(\Phi(r,z),r,z)$, where $\theta,\,\Phi$ are given
respectively in (17) and Definition 4.

The content of Theorem 3 says that the static and dynamical properties of (5)
are captured by the effective action $\mathcal{W}\circ\tilde{\Phi}$, on the
$3$-manifold given by the level set $\left\\{(r,z)\in
M^{\prime}:\left\lvert\Phi(r,z)(\mathbb{S})\right\rvert=c\right\\}$.

(13)-(15) constitute the crucial part of the theorem, which says that the
infinite dimensional dynamical system (1) reduces uniformly in time to the
finite system of ODEs, (14)-(15).

We organize this paper as follows: In Section 2, we define the important map
$\Phi$, which arises by reconceptualizing the Lyapunov-Schmidt reduction. This
allows us to identify the adiabatic part of a flow evolving according to (1).
This adiabatic part accounts for much of the motion along the flow (modulo
some uniformly small fluctuation), and is finite-dimensional. In Section 3, we
discuss the static property of the effective action
$\mathcal{W}\circ\tilde{\Phi}$, and prove the first part of Theorem 3. A
similar but different function defined on a domain in $\mathbb{R}^{3}$ is used
in [eichmair2021large] (denoted by $G$ in that paper). In Section 4, we prove
the remaining part of Theorem 3 by deriving the effective dynamics (14) of
(1). Here we exploit the spectral property of certain linearized operators, in
order to bound a Lyapunov-type functional that controls fluctuations.

### Notation

######  Remark 3.

Throughout the paper, the notation $A\lesssim B$ means that there is a
constant $C>0$ depends only on $c,k$ in (5) and $R,\delta$ in (12), s.th.
$A\leq CB$. For two vectors $A,\,B$ in Banach spaces $X,\,Y$, the notation
$A=O_{Y}(B)$ means $\lVert A\rVert_{X}\lesssim\lVert B\rVert_{Y}$. For a
vector $A$ in some Sobolev space $H^{k}$, the notation $A=O_{Y}(B)$ means
$\lVert A\rVert_{H^{k}}=O_{Y}(B)$.

##  3 The Lyapunov-Schmidt map

Let $k\geq 4,c\gg 1$. Let $K\subset M$, $R\gg 0$ to be determined, and let
$M^{\prime}:=\mathbb{R}_{>R}\times
B_{1}(0)\subset\mathbb{R}\times\mathbb{R}^{3}$. In this section we construct
the map $\tilde{\Phi}:M^{\prime}\to X^{k}$ as in Theorem 3.

###  3.1 Graphs over sphere

Denote $H^{k}=H^{k}(\mathbb{S},\mathbb{R})$. This space is equipped with the
$L^{2}$-inner product $\left\langle u,\,v\right\rangle=\int_{\mathbb{S}}uv$.
Define the configuration space

$Y^{k}:=H^{k}\times M^{\prime}.$ (16)

Define a map

$\begin{array}[]{ccrcl}{\theta}&\colon&{Y^{k}}&\longrightarrow&{X^{k}}\\\
\mbox{}&\mbox{}&{(\phi,r,z)}&\longmapsto&{r(1+\phi(v))v+z}\end{array}.$ (17)

Here $v\in\mathbb{S}\subset\mathbb{R}^{3}$ is the spherical coordinate, and
recall we identify the asymptotic part $(M\setminus
K)\cong(\mathbb{R}^{3}\setminus B_{R}(0))$.

For $\lVert\phi\rVert_{H^{k}}\ll 1$, the map $\theta(\phi,r,z)$ is a well-
defined graph over the coordinate sphere $\theta(0,r,z)(\mathbb{S})=:S_{r,z}$.
Thus we can also identify $\theta(\phi,r,z)$ as a function from
$S_{r,z}\subset M\to\mathbb{R}$. Note also that for sufficiently large $c\gg
1$ and every $z\in B_{1}(0)\subset\mathbb{R}^{3}$, there is a coordinate
sphere with area $c$ around $z$. Thus the map $\theta$ is surjective onto
$X^{k}_{c}$.

######  Definition 2 (topology on graphs).

We say two graphs
$\theta(\phi,r,z),\,\theta(\phi^{\prime},r^{\prime},z^{\prime})$ are
$H^{k}$-close if $\lVert\phi-\phi^{\prime}\rVert_{H^{k}}+\left\lvert
r-r^{\prime}\right\rvert+\left\lvert z-z^{\prime}\right\rvert\ll 1$.

###  3.2 Lyapunov-Schmidt reduction

Denote $\bar{W}(\phi,r,z),\,\Omega(\phi,r,z)$ the pullbacks of the r.h.s. of
(1) and the Willmore energy (9) to $Y^{k}$ through $\theta$, respectively.
Explicitly, we have

$\displaystyle\bar{W}(\phi,r,z):=-W(\theta(\phi,r,z))-\lambda
H(\theta(\phi,r,z)),$ (18)
$\displaystyle\Omega(\phi,r,z):=\mathcal{W}(\theta(\phi,r,z)).$ (19)

Since $\mathcal{W}$ is $C^{2}$ on $X^{k}$ with $k\geq 4$ and $\theta$ is
smooth, the pullback energy $\Omega$ is $C^{2}$ on $Y^{k},\,k\geq 4$. Using
Sobolev inequalities, one can check that the partial Fréchet derivative
$\bar{W}$ is $C^{1}$ in $\phi$ and smooth in $r,\,z$. This $\bar{W}$ is the
$L^{2}$-gradient of $\Omega(\cdot,r,z)$, and satisfies the mapping property
$\bar{W}:Y^{k}\to H^{k-4}$.

######  Remark 4.

Notice that (18)-(19) both depend implicitly on the background metric $g$.

######  Lemma 1.

The linearized operator $L_{r,z}$ of $\bar{W}$ at $(0,r,z)$ with background
metric $g$ is given by

$\begin{split}L_{r,z}^{g}:&=\partial_{\phi}\bar{W}(\phi,r,z)|_{\phi=0}\\\
&=(\Delta^{2}+2r^{-2}\Delta+O(r^{-4}))\partial_{\phi}\theta(0,r,z):H^{k}\to
H^{k-4}.\end{split}$ (20)

Here $\Delta:X^{k}\to X^{k-2}$ denotes the Laplace-Beltrami operator on the
coordinate sphere $S_{r,z}\subset M\setminus K$, with center $z$ and radius
$r$. The partial Fréchet derivative $\partial_{\phi}\theta(0,r,z):H^{k}\to
X^{k}$ is given by $\xi(v)\mapsto\xi(v)rv$.

Moreover, the operator $L_{r,z}$ is self-adjoint on $H^{k}$. The spectrum of
$L_{r,z}$ is purely discrete. The operator $\partial_{\phi}\theta(0,r,z)$ is
invertible and satisfies

$\lVert\partial_{\phi}\theta(0,r,z)\rVert_{H^{k}\to
H^{k}}=\lVert\partial_{\phi}\theta(0,r,z)^{-1}\rVert^{-1}_{H^{k}\to H^{k}}=r.$
(21)

###### Proof.

The operator $L_{r,z}^{g}$ is explicitly calculated in [MR2785762]*Sec. 3. The
spectral properties of $L_{r,z}$ are studied in [MR2785762]*Sec. 7. The
mapping properties of $\partial_{\phi}\theta$ is obvious. ∎

######  Remark 5.

The linearized operator (20) depends on (the curvature of ) the background
metric $g$ on $M$. In the special case when the ambient manifold $M$ is flat,
i.e. $g=\delta_{ij}$, the linearized operator $L_{r,z}^{0}$ has eigenvalue
$0$, and $\ker L_{r,z}^{0}$ is spanned by the constant function $y^{0}\equiv
1$, together with the spherical harmonics $y^{1},y^{2},y^{3}$. Thus, so long
as $(M,g)$ is asymptotically flat and $r\gg 1$ in (20) (such as in our
setting), one can view $L_{r,z}^{g}$ as a perturbation of $L_{r,z}^{0}$. This
motivates the following definition.

######  Definition 3.

Define $P:H^{k}\to H^{k}$ to be the $L^{2}$-orthogonal projection onto
$\operatorname{span}\left\\{y^{0},\ldots,y^{4}\right\\}=\ker{L_{r,z}^{0}}$.
Define $\bar{P}:=1-P:H^{k}\to H^{k}$ be the complement of $P$.

Let $\mathcal{S}$ be the set of all smooth symmetric two tensors on $M$.
Define a map

$\begin{array}[]{ccrcl}{F}&\colon&{Y^{k}\times\mathcal{S}}&\longrightarrow&{H^{k-4}}\\\
\mbox{}&\mbox{}&{(\phi,r,z,h)}&\longmapsto&{\bar{P}\bar{W}(\phi,r,z)}\end{array},$
(22)

where $\bar{W}$ is computed with background metric $g=g_{S}+h$ (see Section
2.1).

######  Proposition 1.

Assume the ambient manifold $(M,g)$ is $C^{k}$-closed to Schwarzschild.

  1. 1.

For every $z\in\mathbb{R}^{3}$ with $\left\lvert z\right\rvert<1$ and
sufficiently large $r\geq R\gg 1$, there is a unique solution
$\phi=\phi_{r,z}\in\bar{P}H^{k}$ to the equation

$F(\phi,r,z,h)=0,$ (23)

where $F$ is defined in (22), and $g=g_{S}+h$.

  2. 2.

Moreover, the map $(r,z)\mapsto\phi_{r,z}$ is $C^{2}$, and satisfies the
estimate

$\lVert\partial_{r}^{m}\partial^{\alpha}_{z}\phi_{r,z}\rVert_{H^{k}}\lesssim
r^{-(2+2m)}.$ (24)

for every $m+\left\lvert\alpha\right\rvert\leq 2$.

  3. 3.

Moreover, the surface $\theta(\phi_{r,z},r,z)$ lies in the class of admissible
surfaces in Definition 1

###### Proof.

1\. By the Implicit Function Theorem, it suffices to check that the map $F$
defined in (23) satisfies the following properties:

  1. 1.

$F$ is $C^{1}$ in $\phi$.

  2. 2.

$F(0,r,z,0)=0$ for every $r,z$.

  3. 3.

$\partial_{\phi}F(0,r,z,0)=L_{r,z}^{0}$ is invertible on $\bar{P}H^{k}$.

The first claim follows from the regularity of $\bar{W}$ on $Y^{k}$ and its
smooth dependence on the background metric.

If the background metric is Schwarzschild, i.e. $h=0$, then it is well-known
that by conformal invariance the coordinate sphere $\theta(0,r,z)$ is the
global minimizer of the Willmore energy $\mathcal{W}$. Since
$\bar{W}=\partial_{\phi}\Omega$ (see (19)), the second claim follows.

The spectrum of $L_{r,z}^{0}$ can be calculated explicitly. See for instance
[eichmair2021large]*Cor. 33. In particular, $0$ is an isolated eigenvalue with
finite multiplicity. By elementary spectral theory, this implies the
restriction $\bar{L}_{r,z}^{0}:=L_{r,z}^{0}|_{\bar{P}}$ is invertible as a map
from $\bar{P}H^{k}\to\bar{P}H^{k}$. Thus the third claim follows.

2\. For the estimate (24), we expand

$L_{r,z}^{g}=L_{r,z}^{0}+V_{r,z},$ (25)

where $V_{r,z}$ is defined by this expression. As we discuss in Remark 5, this
$V_{r,z}$ is bounded from $H^{k}\to H^{k-4}$, and satisfies $\lVert
V_{r,z}\rVert_{H^{k}\to H^{k-4}}=O(r^{-4})$. The restriction
$\bar{L}_{r,z}^{0}$ can bounded from below by $Cr^{-2}$ for some $C>0$ only
depending on $k$. It follows that

$\lVert(\bar{L}_{r,z}^{0})^{-1}V_{r,z}\rVert_{H^{k-4}\to H^{k}}=O(r^{-2}).$

For sufficiently large $r$, this together with the expansion (25) implies that
the restriction $\bar{L}_{r,z}^{g}:\bar{P}H^{k}\to\bar{P}H^{k-4}$ is
invertible, given explicitly as the Neumann series

$(\bar{L}_{r,z}^{g})^{-1}=\sum_{n=0}^{\infty}(\bar{L}_{r,z}^{0})^{-1}(-V_{r,z}(\bar{L}_{r,z}^{0})^{-1})^{n}.$

From here one can also read off the estimate

$\lVert(\bar{L}_{r,z}^{g})^{-1}\rVert_{\bar{P}H^{k-4}\to\bar{P}H^{k}}=O(r^{2}).$
(26)

Expand $F(\phi,r,z,h)=F(0,r,z,h)+\bar{L}_{r,z}^{g}\phi+N_{r,z}(\phi)$, where
the nonlinearity $N_{r,z}$ is defined by this expression. This $N_{r,z}$ is
calculated explicitly in (83). For every $\phi$ satisfying (23), we can
rearrange to get

$\phi=-(\bar{L}_{r,z}^{g})^{-1}(F(0,r,z,h)+N_{r,z}(\phi)).$ (27)

In the r.h.s. we have $F(0,r,z,h)=O(r^{-4})$ by [eichmair2021large]*Cor. 45.
Thus, for sufficiently small $\phi$, we have by (26)-(27) that
$\lVert\phi\rVert_{H^{k}}=O(r^{-2})$.

We now claim for $\phi\in H^{k}$ and $m+\left\lvert\alpha\right\rvert\leq 2$,
there hold

$\displaystyle\lVert\partial_{r}^{m}\partial^{\alpha}_{z}(\bar{L}_{r,z}^{g})^{-1}\phi\rVert_{H^{k}}\lesssim\lVert\phi\rVert_{H^{k-4}},$
(28)
$\displaystyle\lVert\partial_{r}^{m}\partial^{\alpha}_{z}F(0,r,z,h)\rVert_{H^{k-4}}\lesssim
r^{-(4+m)},$ (29)
$\displaystyle\lVert\partial_{r}^{m}\partial^{\alpha}_{z}N_{r,z}(\phi)\rVert_{H^{k-4}}\lesssim\lVert\phi\rVert_{H^{k}}^{2}.$
(30)

For (28), one uses the identity
$\partial^{\beta}(\bar{L}_{r,z}^{g})^{-1}=-(\bar{L}_{r,z}^{g})^{-1}\bar{\partial}^{\beta}L_{r,z}^{g}(\bar{L}_{r,z}^{g})^{-1}$,
where $\left\lvert\beta\right\rvert\leq 2$ is a multi-index in both $r$ and
$z$. This, together with the fact that $\partial^{\beta}L_{r,z}$ is uniformly
bounded (see (81)), implies (28). The rest follows from the expansion in
Proposition 4. Using (28)-(30) and differentiating both sides of (27), we
conclude the estimates (24).

3\. For sufficiently large $R$ and every $r\geq R$, we find using (24) with
$m=0,\alpha=0$ that the surface $\theta(\phi_{r,z},r,z)$ is $H^{k}$-close to
the coordinate sphere $S_{r,z}$. This implies $\theta(\phi_{r,z},r,z)$ is an
admissible surface.

∎

From now on we write $\zeta=\zeta^{\alpha},\,\alpha=0,\ldots,4$, for a point
in $(r,z)\in M^{\prime}$. Thus, $\zeta^{0}=r$ and $\zeta^{j}=z^{j}$ for
$j=1,2,3$.

######  Definition 4 (The Lyapunov-Schmidt map $\Phi$).

Let $K\subset M$ be the compact set as in Theorem 2. Let $R\gg 1,\,\delta\ll
1$ be given as in Theorem 1. Let $M^{\prime}:=\mathbb{R}_{>R}\times
B_{1}(0)\subset\mathbb{R}\times\mathbb{R}^{3}$.

Define the Lyapunov-Schmidt map $\Phi:M^{\prime}\to H^{k}$ by
$\zeta\mapsto\phi_{\zeta}$, where $\phi_{\zeta}$ is the solution to (23) given
in Proposition 1.

######  Remark 6.

This $\Phi$ is equivalent to the map $\tilde{\Phi}$ in Theorem 3, through the
diffeomorphism $\Phi(\zeta)\mapsto\theta(\Phi(\zeta),\zeta)$.

In the next proposition, we describe the geometric structure induced by the
map $\Phi$.

######  Proposition 2.

The set

$E:=\left\\{\theta(\phi,\zeta):\phi=\Phi(\zeta),\zeta\in M^{\prime}\right\\}$

forms an immersed $C^{1}$ submanifold in $X^{k}$. The tangent space
$T_{\theta(\Phi(\zeta),\zeta)}E$ consists of vector fields over the surface
$\theta(\Phi(\zeta),\zeta)(\mathbb{S})$. A basis of
$T_{\theta(\Phi(\zeta),\zeta)}E$ is given by
$\partial_{\zeta^{\alpha}}\theta(\Phi(\zeta),\zeta)$.

######  Remark 7.

Using the projection constructed in Lemma 2, one can view this manifold $E$ as
consisting of the adiabatic parts of low (Willmore) energy surfaces in
$X^{k}$.

###### Proof.

The manifold structure of $E$ follows from Definition 4, where
$\Phi:M^{\prime}\to E$ is a $C^{1}$ parametrization. We check the tangent
space is non-degenerate. Compute

$\displaystyle\partial_{\zeta^{0}}\theta(\Phi(\zeta),\zeta)(v)=(1+\Phi(\zeta)+\zeta^{0}\partial_{\zeta^{0}}\Phi(\zeta))v,$
(31)
$\displaystyle\partial_{\zeta^{j}}\theta(\Phi(\zeta),\zeta)(v)=\zeta^{0}\partial_{\zeta^{j}}\Phi(\zeta)v+e^{j},$
(32)

where $e^{j}$ is the $j$-th unit vector in $\mathbb{R}^{3}$. By the estimate
(24), we find
$\left\langle\partial_{\zeta^{\alpha}}\theta(\Phi(\zeta),\zeta),\,\partial_{\zeta^{\beta}}\theta(\Phi(\zeta),\zeta)\right\rangle_{\theta}=4\pi\delta_{\alpha\beta}+O(R^{-2})$,
where the inner product is defined in (8). This implies the claim if $R$ is
sufficiently large. ∎

In Appendix, we introduce the general concepts of the Lyapunov-Schmidt map,
and relate it to our setting above.

###  3.3 Barycenter

In this subsection, we develop a new concept of barycenter for a certain class
of closed surfaces in $X^{k}$.

######  Definition 5 (Barycenter).

Let $x_{*}$ be an embedding of sphere that is $H^{k}$-close to the manifold
$E\subset X^{k}$ constructed in Definition 4, w.r.t. the topology on graphs
introduced in Definition 2. Then we can write
$x_{*}=\theta(\Phi(\zeta)+\xi,\zeta)$ for some $\zeta\in
M^{\prime},\lVert\xi\rVert_{H^{k}}\ll 1$. (There can in general be many such
choice of $\zeta$ and $\xi$.) Expand $x_{*}$ in $\xi$ around
$\theta(\Phi(\zeta),\zeta)$ as

$x_{*}=\theta(\Phi(\zeta),\zeta)+\partial_{\phi}\theta(\Phi(\zeta),\zeta)\xi+O(\lVert\xi\rVert_{H^{k}}^{2}).$
(33)

Define $f_{\alpha}\in H^{k}$ as

$\begin{split}f_{\alpha}(\zeta)(v)&=\partial_{\zeta^{\alpha}}\theta(\Phi(\zeta),\zeta)(v)^{N}\\\
&=g(\partial_{\zeta^{\alpha}}\theta(\Phi(\zeta),\zeta)),\nu(\theta(\Phi(\zeta),\zeta))\quad(\alpha=0,\ldots,3).\end{split}$
(34)

We say a point $\zeta_{*}\in M^{\prime}$ is the barycenter of $x_{*}$ if
$\zeta_{*}$ solves the following algebraic system:

$\left\langle\xi,\,f_{\alpha}\right\rangle_{L^{2}}=0\quad(\alpha=0,\ldots,3),$
(35)

where $\xi$ is defined by the relation
$x_{*}=\theta(\Phi(\zeta_{*})+\xi,\zeta_{*})$.

######  Remark 8.

The four vectors $f_{\alpha}$ span the tangent space at
$\theta(\Phi(\zeta_{t}),\zeta_{t})^{N}$ to $E^{N}\subset H^{k}$, where $E^{N}$
consists of the normal components of the elements in the manifold $E$ defined
in Definition 4. Geometrically, the defining condition (35) for barycenter
means that the Gâteaux derivative of the map $\theta(\cdot,\zeta_{*})^{N}$ at
$\Phi(\zeta_{*})$ along $\xi$-direction is perpendicular to the tangent space
$T_{\theta(\Phi(\zeta_{*}),z_{*})^{N}}E^{N}$. In terms of the expansion (33),
this means the second term in the r.h.s. is $L^{2}$-orthogonal to the tangent
space at the first term to $E$. In this sense the choice of barycenter is
optimal.

######  Remark 9.

Our definition of barycenter differs from the classical one, given by
averaging over $\Sigma$ w.r.t. Euclidean background metric, namely
$\left\lvert\Sigma\right\rvert_{g}^{-1}\int_{\Sigma}xd\mu_{\Sigma}^{\delta_{ij}}$.
See [MR4236532] and the references therein. Our version of barycenter retains
the key decay property as [MR4236532]*Sec. 5. Namely, the motion of barycenter
is controlled by a differential inequality using a Lyapunov functional,
defined in Section 5.

Moreover, our definition allows us to retain explicit and uniform control of a
flow evolving according to (1), as we show in Sec. 5.

In the next lemma, we define a nonlinear projection (or coordinate map) that
associates barycenters to low energy configurations in $X^{k}$.

######  Lemma 2 (nonlinear projection).

There exists $\delta>0$ s.th. on the space

$U_{\delta}:=\left\\{x=\theta(\Phi(\zeta)+\xi,\zeta):\zeta\in
M^{\prime},\,\lVert\xi\rVert_{H^{k}}<\delta\right\\},$ (36)

there exists a $C^{1}$ map $S:U_{\delta}\to M^{\prime}$ s.th. $S(x)$ is the
barycenter of $x$ as in Definition 5.

Moreover, we have uniform estimate on $S$ and its derivative.

######  Remark 10.

Essentially, the existence of such projection depends on the non-degeneracy
shown in Proposition 2. Later, we see that the barycenter $\zeta=S(x)$
determines the adiabatic (or slowly-varying) part of $x$.

We call the remainder $\xi$ that satisfies $x=\theta(\Phi(S(x))+\xi,S(x))$ the
fluctuation of $x$.

###### Proof.

Define a map

$\begin{array}[]{ccrcl}{\Gamma}&\colon&{M^{\prime}\times
U_{\delta}\subset\mathbb{R}^{4}\times
H^{k}}&\longrightarrow&{\mathbb{R}^{4}}\\\
\mbox{}&\mbox{}&{(\zeta,x)}&\longmapsto&{\left\langle\xi,\,f_{\alpha}\right\rangle_{L^{2}}}\end{array},$
(37)

where $\xi$ is defined by the relation $x=\theta(\Phi(\zeta)+\xi,\zeta)$. It
suffices to find a map $S$ s.th. $\zeta=S(x)$ solves

$\Gamma(\zeta,x)=0.$ (38)

By the Implicit Function Theorem, it suffices to check that the map $\Gamma$
satisfies the following properties:

  1. 1.

$\Gamma$ is $C^{1}$ in $\zeta$.

  2. 2.

$\Gamma(\zeta,\theta(\Phi(\zeta),\zeta))=0$.

  3. 3.

The matrix
$\nabla_{\zeta}\Gamma(\zeta,\theta(\Phi(\zeta),\zeta)):\mathbb{R}^{4}\to\mathbb{R}^{4}$
is invertible.

The first claim follows from the regularity of $\Phi$ as in Proposition 1. The
second claim is trivial because in this case $\xi=0$.

We now claim the rescaled matrix

$A_{\alpha\beta}:=\zeta^{0}\partial_{\zeta^{\alpha}}\Gamma_{\beta}|_{(\theta(\Phi(\zeta),\zeta))}$

is invertible, which implies the third claim above.

Using (31)-(32), the definition (34), and the assumption on the background
metric $g_{ij}=\delta_{ij}+O((\zeta^{0})^{-1})$, we compute

$\displaystyle
f_{0}=(1+O((\zeta^{0})^{-1}))y^{0}+\Phi(\zeta)+\zeta^{0}\partial_{\zeta^{0}}\Phi(\zeta)+\Phi(\zeta)O((\zeta^{0})^{-1})+O((\zeta^{0})^{-4}),$
(39) $\displaystyle
f_{j}=(1+O((\zeta^{0})^{-1}))y^{i}+\zeta^{0}\partial_{\zeta^{j}}\Phi(\zeta)+O((\zeta^{0})^{-4}),$
(40)

where the vectors $y^{\alpha}$ span $\operatorname{ran}P$ as in Definition 3.

For $\xi=\xi(\zeta,x)$, we find

$\displaystyle\xi(v)=\left\langle\frac{x(v)-\zeta}{\zeta^{0}},\,v\right\rangle-1-\Phi(\zeta)(v),$
(41)
$\displaystyle\partial_{\zeta^{0}}\xi(v)=-\left\langle\frac{x(v)-\zeta}{(\zeta^{0})^{2}},\,v\right\rangle-\partial_{\zeta^{0}}\Phi(\zeta),$
(42)
$\displaystyle\partial_{\zeta^{j}}\xi(v)=-\frac{e^{j}}{\zeta^{0}}-\partial_{\zeta^{j}}\Phi(\zeta)(v).$
(43)

Now, since $x\in U_{\delta}$, we have
$A_{\alpha\beta}=\zeta^{0}\left\langle\partial_{\zeta^{\alpha}}\xi,\,f_{\beta}\right\rangle+\zeta^{0}\left\langle\xi,\,\partial_{\zeta^{\alpha}}f_{\beta}\right\rangle=\left\langle\partial_{\zeta^{\alpha}}\xi,\,f_{\beta}\right\rangle+O(\zeta^{0}\delta)$.
By this, and the formula (39)-(43) above, we find that
$A_{\alpha\beta}=O(1)\delta_{\alpha\beta}+O((\zeta^{0})^{-2})+O(\zeta^{0}\delta)$.
For sufficiently large $R\gg 1$, $\delta=o(R^{-1})$, and every $\zeta^{0}\geq
R$, we can conclude from here that $A_{\alpha\beta}$ is invertible. This
proves the existence of the $C^{1}$ map $S$. The uniform estimates for $S$ and
its Fréchet derivative are implicit in the arguments above.

∎

##  4 Effective action

In this section we prove Part 1 of Theorem 3. We formulate this as follows:

######  Theorem 4.

The embedding $\theta(\Phi(\zeta),\zeta)$ parametrizes a surface of Willmore
type (i.e. static solution to (1)) if and only if $\zeta$ is a critical point
of the function $G:=\Omega(\Phi(\cdot),\cdot):\mathbb{R}^{4}\to\mathbb{R}$,
where $\Omega$ is defined in (19).

######  Remark 11.

Similar results are obtained in [eichmair2021large]*Thms. 5,8.

###### Proof.

For the forward direction, we use the chain rule to get ,

$\partial_{\zeta^{\alpha}}G(\zeta)=\mathcal{W}^{\prime}(\theta(\Phi(\zeta),\zeta))\partial_{\zeta^{\alpha}}\theta(\Phi(\zeta),\zeta)^{N}.$
(44)

If $\theta(\Phi(\zeta),\zeta)$ is a critical point of $\mathcal{W}$, i.e.
$\mathcal{W}^{\prime}(\theta(\Phi(\zeta),z))=0$, then the first factor of
r.h.s. of (44) vanishes, and therefore $z$ is a critical point of $G_{r}$.

Now, suppose $\partial_{\zeta^{\alpha}}G(\zeta)=0$. For the backward
direction, it suffices to show the pullback $\bar{W}(\Phi(\zeta),\zeta)=0$.

Let $Q_{\zeta}$ to be the projection onto the tangent space at
$T_{\theta(\Phi(\zeta_{t}),\zeta_{t})^{N}}E^{N}$, as in Remark 8. Explicitly,
this map $Q_{\zeta}$ is given by

$Q_{\zeta}\phi=V^{\alpha\beta}\left\langle
f_{\alpha},\,\phi\right\rangle_{L^{2}}f_{\beta}\quad(\phi\in H^{k}),$ (45)

where $V_{\alpha\beta}:=\left\langle f_{\alpha},\,f_{\beta}\right\rangle$,
$V^{\alpha\beta}$ is its inverse, and $f_{\alpha}$ are given in (39)-(40).
Notice that this matrix $V_{\alpha\beta}$ is indeed invertible because the
tangent space $T_{\theta(\Phi(\zeta_{t}),\zeta_{t})^{N}}E^{N}$ is non-
degenerate, c.f. Proposition 2. From the definition (45) and the formula
(39)-(40), we also get an uniform estimate $\lVert Q_{\zeta}\rVert\lesssim 1$.

The claim now is that $Q_{\zeta}:H^{k}\to H^{k}$ is uniformly close in
operator norm to the projection $P$ defined in Definition 3. Geometrically,
this is because the manifold $E^{N}$ is a uniformly small normal perturbation
of $\operatorname{ran}P$. Analytically, this claim follows by comparing
(39)-(40) to the basis $y^{\alpha}$ of $\operatorname{ran}P$, and using the
estimate (24) for the derivatives $\partial_{\zeta^{\alpha}}\Phi$. As a result
we find

$\lVert Q_{\zeta}-P\rVert_{H^{k}\to H^{k}}=O(R^{-2}).$ (46)

Now, by the construction in Definition 4, we know
$P\bar{W}(\Phi(\zeta),\zeta)=\bar{W}(\Phi(\zeta),\zeta)$. Thus we can write

$\bar{W}(\Phi(\zeta),\zeta)=P\bar{W}(\Phi(\zeta),\zeta)=(P-Q_{\zeta})\bar{W}(\Phi(\zeta),\zeta)+Q_{\zeta}\bar{W}(\Phi(\zeta),\zeta).$
(47)

From (44) one can see that $\partial_{\zeta^{\alpha}}G=0$ is equivalent to
$Q_{\zeta}\bar{W}=0$. Thus by assumption the last term in (47) vanishes. Using
this fact and the estimate (46), we find

$\displaystyle\lVert\bar{W}(\Phi(\zeta),\zeta)\rVert_{H^{k}}$
$\displaystyle\lesssim\lVert(P-Q_{\zeta})\rVert_{H^{k}\to
H^{k}}\lVert\bar{W}(\Phi(\zeta),\zeta)\rVert_{H^{k}}$ $\displaystyle\lesssim
R^{-1}\lVert\bar{W}(\Phi(\zeta),\zeta)\rVert_{H^{k}}.$

For sufficiently large $R$, this is impossible unless
$\bar{W}(\Phi(\zeta),\zeta)=0$.

∎

##  5 Effective dynamics

In this section we prove the rest of Theorem 3. We first derive the effective
dynamics (13)-(15), and then use this to derive Parts 2 and 4 of Theorem 3.

######  Theorem 5.

Let $\Phi:M^{\prime}\to H^{k}$ be the map defined in Definition 4. Let
$\Sigma_{*}=x_{*}(\mathbb{S})$ be an admissible surface. Let
$\Sigma_{t}=x_{t}(\mathbb{S})$ be the global solution to (1) with initial
configuration $\Sigma|_{t=0}=\Sigma_{*}$ as in Theorem 1.

Then there exists $\alpha>0$, $T=O(R^{-\alpha})$, and a path $\zeta_{t}\in
M^{\prime}$, such that

$\lVert\Phi(\zeta_{t})-x_{t}^{N}\rVert_{H^{k}}=O(R^{-3})\quad(t\geq T).$ (48)

Moreover, the path $(\zeta_{t})$ evolves according to

$\dot{\zeta}=\nabla G(\zeta)+O(R^{-3}),$ (49)

where the leading term in the r.h.s. is of the order $O(R^{-2})$.

###### Proof.

Step 1. Fix $0<\delta\ll 1$ to be determined. To begin with, denote $T>0$ the
first time $u_{t}$ enters the manifold $U_{\delta}$ defined in (36). By the
stability result Theorem 2, this $T$ is finite, and there exists $\alpha>0$
s.th. $T=O(\delta^{-\alpha})$.

Now we prove the claims (48)-(49), assuming the a priori estimate

$x_{t}\in U_{\delta}\text{ for all $t>T$}.$ (50)

By Lemma 2, so long as (50) is satisfied, we can associate a path of
barycenters $\zeta_{t}\in M^{\prime}$ to the full flow $x_{t}$. Later we show
this Ansatz is satisfied with $\delta=O(R^{-3})$ by proving an a priori
estimate for the fluctuation of $x_{t}$ around its adiabatic part.

Write $x=\theta(\Phi(\zeta)+\xi,\zeta)$, where $\zeta=S(x)$ is the barycenter
as in Lemma 2, and $\xi$ is the fluctuation. Then we can rewrite the l.h.s. of
(1) as

$\displaystyle\partial_{t}x^{N}$
$\displaystyle=\partial_{\xi}\theta(\Phi+\xi,\zeta)^{N}\partial_{t}\xi+\partial_{\zeta^{\alpha}}\theta(\Phi+\xi,\zeta)^{N}\dot{\zeta^{\alpha}}$
(51)
$\displaystyle=A(\xi)\partial_{\phi}\theta(\Phi,\zeta)^{N}\partial_{t}\xi+B(\xi)\partial_{\zeta^{\alpha}}\theta(\Phi,\zeta)^{N}\dot{\zeta^{\alpha}}$
$\displaystyle=A(\xi)(\zeta^{0}+O_{H^{k}}(1))\partial_{t}\xi+B(\xi)f_{\alpha}\dot{\zeta}^{\alpha},$

where we define

$A(\xi):=\frac{\partial_{\xi}\theta(\Phi+\xi,\zeta)^{N}}{\partial_{\phi}\theta(\Phi,\zeta)^{N}},\quad
B(\xi):=\frac{\partial_{\zeta^{\alpha}}\theta(\Phi+\xi,\zeta)^{N}}{\partial_{\zeta^{\alpha}}\theta(\Phi,\zeta)^{N}}.$

For simplicity, here and below we omit the dependence of $f_{\alpha}$ and
$\Phi$ on $\zeta$.

The expansion (51) follows from the chain rule, (34), and the assumption on
the background metric, $g_{ij}=\delta_{ij}+O((\zeta^{0})^{-1})$. From (17),
one can see that the prefactors $A(\xi),B(\xi)$ are of the order
$1+O_{H^{k}}(\xi)$.

Let $Q=Q_{\zeta(t)}$ be the projection onto the tangent space
$T_{\theta(\Phi(\zeta_{t}),\zeta_{t})^{N}}E^{N}$, as given explicitly in (45).
The space $\operatorname{ran}Q$ is spanned by the four vectors $f_{\alpha}$,
which we computed in (39)-(40). Applying $Q$ to (51), we find

$Q\partial_{t}x^{N}=(\zeta^{0}+O_{H^{k}}(1))Q\partial_{t}\xi+f_{\alpha}\dot{\zeta}^{\alpha}+O_{H^{k}}(\xi(\left\lvert\dot{\zeta}\right\rvert+\partial_{t}\xi)).$
(52)

By the definition of barycenter, (35), we know $Q\xi=0$. Differentiating this,
we find

$Q(\partial_{t}\xi)=-(\partial_{t}Q)\xi=-(\partial_{\zeta^{\alpha}}Q\dot{\zeta}^{\alpha})\xi.$
(53)

Here, the partial Fréchet derivative $\partial_{\zeta^{\alpha}}Q$ maps a real
number to a linear operator from $H^{k}\to\operatorname{ran}Q\subset H^{k}$.
From the explicit formula (45) for $Q$ and the estimate (24), we get the
uniform bound $\lVert\partial_{\zeta^{\alpha}}Q\rVert_{\mathbb{R}\to
L(H^{k},H^{k})}\lesssim(\zeta^{0})^{-2}$. Thus plugging (53) to (52) gives

$Q\partial_{t}x^{N}=f_{\alpha}\dot{\zeta}^{\alpha}+O_{H^{k}}(\xi(\left\lvert\dot{\zeta}\right\rvert+\partial_{t}\xi))+R^{-1}O_{H^{k}}(\xi\left\lvert\dot{\zeta}\right\rvert)$
(54)

Step 2. Next, expanding the r.h.s. of (1) at $\theta(\Phi(\zeta),\zeta)$, we
find

$\begin{split}W(x)+\lambda H(x)&=\bar{W}(\Phi(\zeta)+\xi,\zeta)\\\
&=\bar{W}(\Phi(\zeta),\zeta))+L_{\zeta}\xi+N_{\zeta}(\xi).\end{split}$ (55)

Here, as in Section 3, $\bar{W}:Y^{k}\to H^{k-4}$ is the pullback of
$W+\lambda H$, and $L_{\zeta}$ is the partial Fréchet derivative
$\partial_{\phi}\bar{W}$ evaluated at $\theta(\Phi(\zeta),\zeta)$. The
nonlinear term $N_{\zeta}(\xi)$ is defined by collecting the rest terms in
this expansion.

Applying $Q$ to (55), we find

$Q\bar{W}(\Phi(\zeta)+\xi,\zeta)=Q\bar{W}(\Phi(\zeta),\zeta)+QL_{\zeta}\xi+QN_{\zeta}(\xi).$
(56)

By the chain rule and the definition (45), the first term in (56) can be
written as

$Q\bar{W}(\Phi(\zeta),\zeta)=V^{\alpha\beta}\left\langle\mathcal{W}^{\prime}(\theta(\Phi(\zeta),\zeta)),\,f_{\alpha}\right\rangle
f_{\beta}.$ (57)

By the uniform estimates $\lVert Q\rVert_{H^{k}\to H^{k}}\lesssim 1$ and (87),
we can bound the nonlinearity as

$\lVert QN_{\zeta}(\xi)\rVert_{H^{k-4}}=O(\lVert\xi\rVert^{2}_{H^{k}}).$ (58)

Lastly, using the approximate zero modes property of the elements in
$\operatorname{ran}Q$, which we show in Lemma 3, one sees that the restriction
of $L_{\zeta}$ is small on $\operatorname{ran}Q$. In particular, this implies
for $\xi,\xi^{\prime}\in H^{k}$,

$\left\langle
QL_{\zeta}\xi,\,\xi^{\prime}\right\rangle=\left\langle\xi,\,L_{\zeta}Q\xi^{\prime}\right\rangle=O(R^{-2})\lVert\xi\rVert_{H^{k}}\lVert\xi^{\prime}\rVert_{H^{k-4}}.$

Plugging $\xi^{\prime}=QL_{\zeta}\xi$ into this expression , we find

$\lVert QL_{\zeta}\xi\rVert_{H^{k-4}}=O(R^{-2})\lVert\xi\rVert_{H^{k}}.$ (59)

Recall the Ansatz (50), which implies $\lVert\xi\rVert_{H^{k}}<\delta\ll 1$.
Collecting (54), (56)-(59), plugging these back to (1), and multiplying both
sides by the matrix $V_{\alpha\beta}=O(1)\delta_{\alpha\beta}$ (see (45)), we
find

$\dot{\zeta^{\alpha}}=\frac{1}{1+O(\delta)+R^{-1}O(\delta)}\left(\partial_{\zeta^{\alpha}}G(\zeta)+R^{-2}O_{H^{k}}(\delta)+O_{H^{k}}(\delta\partial_{t}\xi))\right).$
(60)

Step 3. From (60), we see that (49) follows once we can prove an a priori
estimate of the form

$\lVert\xi\rVert_{H^{k}}+\lVert\partial_{t}\xi\rVert_{H^{k}}\lesssim\lVert\xi_{T}\rVert_{H^{k}},$
(61)

where $\xi_{T}$ is the fluctuation when the flow $x_{t}$ first enters the
manifold $U_{\delta}$, as in Ansatz (50).

To this end, define

$\Lambda(t):=\frac{1}{2}\left\langle\xi_{t},\,L_{\zeta_{t}}\xi_{t}\right\rangle.$
(62)

We show this is a Lyapunov-type functional along the flow of $\xi$. Indeed,
differentiating $\Lambda$, we find

$\dot{\Lambda}(t)=\left\langle\partial_{t}\xi,\,L_{\zeta_{t}}\xi\right\rangle+\frac{1}{2}\left\langle(\partial_{t}L_{\zeta})\xi,\,\xi\right\rangle.$
(63)

The second term is bounded as

$\left\lvert\left\langle(\partial_{t}L_{\zeta})\xi,\,\xi\right\rangle\right\rvert\leq\lVert(\partial_{\zeta}L_{\zeta}\dot{\zeta})\xi\rVert_{H^{k-4}}\lVert\xi\rVert_{L^{2}}=O(\left\lvert\dot{\zeta}\right\rvert\lVert\xi\rVert_{H^{k}}^{2}),$
(64)

where we used the estimate (86).

To bound the first term in the r.h.s. of (63), we isolate the dynamics of
$\xi$ after equating (51) to (55). We find

$\partial_{t}\xi=\frac{1}{\partial_{\phi}\theta(\Phi,\zeta)^{N}A(\xi)}(-\bar{W}(\Phi,\zeta)-L_{\zeta}\xi-
N_{\zeta}(\xi)-B(\xi)f_{\alpha}\dot{\zeta}^{\alpha}).$ (65)

Consider the r.h.s. of (64)-(65). Using (85)-(87) and the governing equation
for barycenter, (60), we find for $R\gg 1$,

$\displaystyle\partial_{\phi}\theta(\Phi,\zeta)^{N}A(\xi)=O_{H^{k}}(\zeta^{0}(1+\xi))+O_{H^{k}}(1+\xi)$
(66)
$\displaystyle\left\lvert\left\langle\bar{W}(\Phi,\zeta),\,L_{\zeta}\xi\right\rangle\right\rvert\lesssim
R^{-4}\lVert\xi\rVert_{L^{2}},$ (67) $\displaystyle\left\lvert\left\langle
N_{\zeta}(\xi),\,L_{\zeta}\xi\right\rangle\right\rvert\lesssim\lVert\xi\rVert_{H^{k}}^{3},$
(68) $\displaystyle\left\lvert\left\langle
f_{\alpha}\dot{\zeta}^{\alpha},\,L_{\zeta}\xi\right\rangle\right\rvert\lesssim
R^{-2}(1+\lVert\xi\rVert_{H^{k}})\lVert\xi\rVert_{H^{k}}.$ (69)

Plugging (64)-(69) back to (63), we find that so long as
$\lVert\xi\rVert_{H^{k}}<1/2$ and $R\gg 1$, there holds

$\displaystyle\dot{\Lambda}(t)$ $\displaystyle\leq
C_{1}R^{-3}\lVert\xi\rVert_{H^{k}}$ (70) $\displaystyle-\lVert
L_{\zeta}\xi\rVert_{H^{k-4}}^{2}+(C_{2}R^{-3}+C_{3}R^{-1}\lVert\xi\rVert_{H^{k}})\lVert\xi\rVert_{H^{k}}^{2}.$

By the coercivity of $L_{\zeta}$ shown in Lemma 4, the first term in the
second line of (70) can be bounded by $-\lVert
L\xi\rVert_{H^{k}}^{2}\leq-\beta\lVert\xi\rVert_{H^{k}}$ for some $\beta>0$
independent of $\zeta$. This, together with the upper bound in (92), implies
that there exists some $\gamma>0$ independent of $\zeta$, such that

$\dot{\Lambda}(t)+\gamma\Lambda(t)\leq
C_{1}R^{-3}\lVert\xi\rVert_{H^{k}}+(C_{2}R^{-3}+C_{3}R^{-1}\lVert\xi\rVert_{H^{k}}-\beta/2)\lVert\xi\rVert_{H^{k}}^{2}.$
(71)

Thus, so long as

$C_{2}R^{-3}+C_{3}R^{-1}\lVert\xi\rVert_{H^{k}}-\beta/2<0,$ (72)

we can drop the last term in (71) to deduce that

$\tfrac{d(\Lambda(t)e^{\gamma t})}{dt}\lesssim R^{-3}e^{\gamma t}.$ (73)

Integrating (73) on $[T,\infty)$ and using (92), we find

$\lVert\xi_{t}\rVert_{H^{k}}^{2}\lesssim\Lambda(t)\lesssim e^{-\gamma
t}\lVert\xi_{T}\rVert_{H^{k}}^{2}+R^{-3}M(t)\quad(M(t):=\sup_{T\leq
t^{\prime}\leq t}\lVert\xi_{t^{\prime}}\rVert_{H^{k}}),$ (74)

where $\xi_{T}$ is the fluctuation at $t=T$. Taking supremum of both sides of
(74) in $t$, and then dividing by $M(t)$ (which is positive for all $t$ in the
nontrivial case), we find

$M(t)\lesssim e^{-\gamma t}\lVert\xi_{T}\rVert+R^{-3}.$ (75)

This, in particular, implies that the Ansatz (72) is satisfied as long as
$R\gg 1$. Iterating this process, we get the estimate $M(t)\lesssim R^{-3}$.
Plugging this back to the dynamics (60) and (65) for $\zeta$ and $\xi$
respectively, we find that the velocity of the fluctuation satisfies
$\lVert\partial_{t}\xi\rVert_{H^{k}}\lesssim R^{-4}.$ Thus for sufficiently
large $R$, the claim (61) follows.

∎

######  Corollary 1.

Let $\zeta_{t}\in M^{\prime}$ be a flow evolving according to (49). Then there
exist some $T>0$ and a global solution $x_{t}$ to (1), s.th. (48) holds on the
time interval $T\leq t\leq T+R$ with this choice of $\zeta_{t}$.

###### Proof.

Let $T>0$ the first time $u_{t}$ enters the manifold $U_{\delta}$ defined in
(36) (this $T$ is finite by Theorem 2). Consider the flow $x_{t},\,t\geq T$
generated under (1) by $x_{*}=\Phi(\zeta_{T})$, as in Theorem 1. Let
$\zeta_{t},\,t\geq T$ be the flow of barycenter as in (48). Then using (24)
and (48), we can estimate

$\displaystyle\lVert\Phi(\tilde{\zeta}_{t})-x_{t}^{N}\rVert_{H^{k}}$
$\displaystyle\leq\lVert\Phi(\zeta_{t})-x_{t}^{N}\rVert_{H^{k}}+\lVert\Phi(\zeta_{t})-\Phi(\tilde{\zeta}_{t})\rVert_{H^{k}}$
(76) $\displaystyle\lesssim
R^{-2}\left\lvert\zeta_{t}-\tilde{\zeta}_{t}\right\rvert+R^{-3}$
$\displaystyle\lesssim
R^{-2}\int_{T}^{T+R}\left\lvert\dot{\zeta}_{t}-\dot{\tilde{\zeta_{t}}}\right\rvert+R^{-3}$
$\displaystyle\lesssim R^{-3}\quad(T\leq t\leq T+R).$

In the last step we use the effective dynamics (49).

∎

In the following corollary, we display the $t$-dependence in subscripts.

######  Corollary 2.

Suppose the embedding $\theta(\Phi(\zeta),\zeta)$ parametrizes a surface of
Willmore type (i.e. static solution to (1)). Then $\theta(\Phi(\zeta),\zeta)$
is uniformly stable with small area-preserving $H^{k}$-perturbation if $\zeta$
is a strict local minimum of the function $G$ defined in Theorem 4.

###### Proof.

Suppose $\zeta$ is a strict local minimum of $G$. Then every flow $\zeta_{t}$
starting at some $\zeta^{\prime}$ near $\zeta$ under (49) converges to
$\zeta$, i.e. $\zeta_{t}\to\zeta$ as $t\to\infty$. Now, for this $\zeta$,
consider a perturbation $x^{\prime}:=\theta(\Phi(\zeta)+\xi,\zeta)$ with
$\lVert\xi\rVert_{H^{k}}\ll 1$. By the regularity of the nonlinear projection
$S$ in Lemma 2, this perturbation has barycenter $\zeta^{\prime}$ close to
$\zeta$. It follows that the flow of barycenters $\zeta_{t}$ associated to the
flow of embeddings $x_{t}$ generated by $x^{\prime}$ under (1) satisfies
$\left\lvert\zeta_{t}-\zeta\right\rvert<\delta$ for any $\delta>0$ and all
sufficiently large $t>T$. If we choose $\delta\ll R^{-3}$, then we can
conclude from (48) that $\lVert x_{t}^{N}-\Phi(\zeta)\rVert_{H^{k-4}}\lesssim
R^{-3}$ for all large $t$.

∎

## Acknowledgments

The Author is supported by Danish National Research Foundation grant CPH-
GEOTOP-DNRF151.

## Declarations

  * •

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

##  Appendix A The abstact Lyapunov Schmidt map

Motivated by the synonymous reduction procedure, in general we can define
Lyapunov-Schmidt map as follows:

######  Definition 6.

Let $X\subset Y$ be two Hilbert spaces. Let $U\subset X$ be an open set, and
$u_{0}\in U$. Let $P:X\to X$ be an orthogonal projection onto some subspace of
$X$, and $\bar{P}:=1-P$ be the complement of $P$.

The Lyapunov-Schmidt map $\Phi=\Phi_{P,u_{0}}$ is defined on the following
set:

$\left\\{\begin{aligned} F:U\subset X\to Y:&\text{$F$ is $C^{1}$,
$\bar{P}L\bar{P}:\bar{P}X\to\bar{P}Y$ has bounded inverse, }\\\ &\text{where
$L:=dF(u)|_{u=u_{0}}:X\to Y$ is}\\\ &\text{ the linearized operator of $F$ at
$u_{0}$}.\end{aligned}\right\\}$

For such $F$, by Implicit Function Theorem, there exists an open neighbourhood
$V\subset PX$ around $v=0$ and a $C^{1}$ map $w:V\to\bar{P}X$ s.th.
$\bar{P}F(u_{0}+v+w(v))=0$ for every $v\in V$. This map $w$ parametrizes a
manifold $E$ in $X$. This manifold $E$ is a normal perturbation of
$\operatorname{ran}P$.

Let $w^{\prime}$ be the Fréchet derivative of $w$. (This is a map from $V$ to
the linear operators on $X$.) Then the tangent space at $w=w(v)$ to $E$ is
spanned by $w^{\prime}(v)(V)$, and can be trivialized as a subspace in $X$,
with dimension up to $\dim V$.

Now, the image $\Phi(F)$ of the Lyapunov-Schmidt map is given by

$\begin{array}[]{ccrcl}{\Phi(F)}&\colon&{V\subset PX}&\longrightarrow&{X}\\\
\mbox{}&\mbox{}&{v}&\longmapsto&{Q_{v}F(u_{0}+v+w(v))}\end{array},$

where $w=w(v)$ is as above, and $Q_{v}$ is the projection onto the tangent
space $T_{w(v)}E$.

This notion is first informally conceived in [MR3824945]. Here, notice that
the domain $V$ of the map $\Phi(F)$ depends on the spectrum of $L$ only.

Essentially, the map $\Phi$ identifies a subspace which can be either finite-
dimensional or otherwise more tractable. The behavior on the reduced space,
which is locally isomorphic to $PX$, of a map in the domain of $\Phi$
determines the behavior of this map in the vicinity of $0$ in the full space
$X$. We will treat the analytical and geometric properties of the abstract
Lyapunov-Schmidt map elsewhere.

In the context of this paper, the family of maps involved is
$F(\cdot,r,z):H^{k}\to H^{k-4}$, defined in Definition 3. These maps are
parametrized by the manifold $M^{\prime}$. The projection $P$ is defined as
the orthogonal projection onto the subspace
$\operatorname{span}\left\\{1,y^{1},y^{2},y^{3}\right\\}\subset H^{k}$, and
the vector $u_{0}$ is the zero function in $H^{k}$.

##  Appendix B Asymptotics of $\mathcal{W}$

In this section, we record various expansions related to the Willmore energy
$\mathcal{W}$ at large coordinate spheres. Most of the results can be read off
from [eichmair2021large]*Sec. C, [MR2785762]*Secs. 3,7.

Fix some $R\gg 1$ and $0<\delta\ll 1$. In what follows, we restore the
parameter $r,z$ and assume $r\geq R$, $z\in\mathbb{R}^{3},\,\left\lvert
z\right\rvert<1-\delta$.

######  Proposition 3.

For $\Omega$ given in (19), $G$ given in Theorem 4, there hold

$\displaystyle\Omega(0,r,z)=4\pi-16\pi r^{-1}+2\pi\left(\frac{10-6\left\lvert
z\right\rvert^{2}}{(\left\lvert
z\right\rvert^{2}-1)^{2}}+3\frac{1}{\left\lvert
z\right\rvert}\log\frac{1+\left\lvert z\right\rvert}{1-\left\lvert
z\right\rvert}\right)+O(r^{-3}),$ (77) $\displaystyle
G(r,z)=\Omega(0,r,z)+O(r^{-5}).$ (78)

###### Proof.

(77) can be found in [eichmair2021large]*Lem. 42. To derive this expansion,
one uses the assumption (4) that the scalar curvature $\operatorname{Sc}$ is
asymptotically even. (78) follows from the expansion

$G(r,z)=\Omega(0,r,z)+\left\langle\bar{W}(0,r,z),\,\Phi(r,z)\right\rangle+O(\lVert\Phi(r,z)\rVert_{H^{k}}^{2}),$

the expansion for $\bar{W}(0,r,z)$ below, and the estimate (24). ∎

The following results are used to derive the key estimate (24).

######  Proposition 4.

Let $k\geq 4$. Let $\phi=\phi_{r,z}\in H^{k}$ be a family of functions with
$\lVert\phi\rVert_{H^{k}}\ll 1$, and suppose the map $(r,z)\mapsto\phi_{r,z}$
is $C^{2}$. Let $\bar{W}$ be as in (18). Then for all
$m+\left\lvert\alpha\right\rvert\leq 2$, there hold

$\displaystyle\bar{W}(\phi,r,z)=\bar{W}(0,r,z)+L_{r,z}\phi+N_{r,z}(\phi),$
(79)
$\displaystyle\lVert\partial_{r}^{m}\partial^{\alpha}_{z}\bar{W}(0,r,z)\rVert_{H^{k-4}}=O(r^{-(4+m)}),$
(80)
$\displaystyle\lVert\partial_{r}^{m}\partial^{\alpha}_{z}L_{r,z}\phi\rVert_{H^{k-4}}\lesssim\lVert\partial_{r}^{m}\partial^{\alpha}_{z}\phi\rVert_{H^{k}},$
(81)
$\displaystyle\lVert\partial_{r}^{m}\partial^{\alpha}_{z}N_{r,z}(\phi)\rVert_{H^{k-4}}\lesssim\lVert\partial_{r}^{m}\partial^{\alpha}_{z}\phi\rVert_{H^{k}}^{2}.$
(82)

###### Proof.

The expansion (79) is valid by the regularity of $\bar{W}$ from $Y^{k}\to
H^{k-4}$. (80) follows from the formula [eichmair2021large]*Cor. 45. Since
$L_{r,z}$ is linear in $\phi$, (81) is immediate. For (82), we calculate
$N_{r,z}(\phi)$ explicitly as

$\displaystyle N_{r,z}(\phi)$ $\displaystyle=\Delta
H(\theta(\phi,r,z))+H(\theta(\phi,r,z))\Bigl{(}\operatorname{Ric}_{M}(\nu,\nu)+\left\lvert\mathring{A}\right\rvert^{2}(\theta(\phi,r,z))\Bigr{)}$
(83)
$\displaystyle-\Lambda^{2}_{r,z}\phi-\frac{1}{2}H^{2}_{r,z}\Lambda_{r,z}\phi-\lambda\Lambda_{r,z}\phi$
$\displaystyle-2H_{r,z}g(\mathring{A}_{r,z},\nabla^{2}\phi)-2H_{r,z}\operatorname{Ric}_{M}(\nu,\nabla\phi)+2\mathring{A}_{r,z}(\nabla
H_{r,z},\nabla\phi)$ $\displaystyle-V_{r,z}\phi+O(r^{-4}),$

where $\lambda$ is the Lagrange multiplier in (1),
$\Lambda=-\Delta-(\left\lvert
A\right\rvert^{2}+\operatorname{Ric}_{M}(\nu,\nu))$ is the Jacobi operator,
$V$ is some smooth function depending on the sphere $\theta(0,r,z)$ and the
background metric only, and the subscript on $r,z$ indicates evaluation at the
sphere $\theta(0,r,z)$. Formula (83) follows from [MR2780248]*Sec. 7.2.

Now, inspecting each term in (83), one can see from this explicit expression
that the nonlinearity $N_{r,z}$ is a finite sum of the form
$N_{r,z}=N_{r,z}(D^{\leq 4}\phi,g)$, where $D^{\leq 4}\phi$ denotes all
derivatives of $\phi$ up to order $4$. Moreover, each term in this finite sum,
together with its derivatives, is uniformly bounded by some $C>0$ independent
of $r,z$ as a map from $H^{k}\to H^{k-4}$ . This implies
$\lVert\partial_{r}^{m}\partial^{\alpha}_{z}N_{r,z}(\phi)\rVert_{H^{k-4}}\lesssim\lVert\partial_{r}^{m}\partial^{\alpha}_{z}\phi\rVert_{H^{k}}^{M}$
for some $M=M(m,\alpha)\geq 2$. For $\lVert\phi\rVert_{H^{k}}\ll 1$, this
implies (82).

∎

By perturbing the estimates in Proposition 4, we get the following results:

######  Proposition 5.

Let $k\geq 4$. Let $\phi=\phi_{r,z}\in H^{k}$ be a family of functions with
$\lVert\phi\rVert_{H^{k}}\ll 1$. Let $\bar{W}$ be as in (18).

Suppose the map $(r,z)\to\phi_{r,z}$ is $C^{2}$, and satisfies (24). Then, for
every $\lVert\xi\rVert\ll 1$ , there hold

$\displaystyle\bar{W}(\phi+\xi,r,z)=\bar{W}(\phi,r,z)+\tilde{L}_{r,z}\xi+\tilde{N}_{r,z}(\xi),$
(84) $\displaystyle\lVert\bar{W}(\phi,r,z)\rVert_{H^{k-4}}=O(r^{-4}),$ (85)
$\displaystyle\lVert\partial_{r}^{m}\partial^{\alpha}_{z}\tilde{L}_{r,z}\xi\rVert_{H^{k-4}}\lesssim\lVert\partial_{r}^{m}\partial^{\alpha}_{z}\xi\rVert_{H^{k}}\quad(m+\left\lvert\alpha\right\rvert\leq
1),$ (86)
$\displaystyle\lVert\tilde{N}_{r,z}(\xi)\rVert_{H^{k-4}}\lesssim\lVert\xi\rVert_{H^{k}}^{2}.$
(87)

###### Proof.

The point is that, by the regularity of $\Omega(\cdot,r,z)$ on $H^{k}$ with
$k\geq 4$, the map $\bar{W}(\cdot,r,z)$ together with its derivatives varies
continuously as $\phi$ varies. Thus, if $\phi$ satisfies (24), then (85)-(87)
follow from (80)-(82) with appropriate choice of $r,\alpha$. ∎

##  Appendix C Spectral properties of $L_{\zeta}$

Let $k\geq 4$. Fix $\zeta=(r,z^{j})\in M^{\prime}\subset\mathbb{R}_{>R}\times
B_{1}(0)$ for sufficiently large $R$. In this section, we prove various
uniform estimates for the linearized operator $L_{\zeta}$ of
$\bar{W}(\cdot,\zeta)$ at $\Phi(\zeta)$. For more details, see
[MR2785762]*Sections 3, 7.

Recall $P:H^{k}\to H^{k}$ is the $L^{2}$-orthogonal projection onto the span
the constants and the three spherical coordinates. $Q=Q_{\zeta}:H^{k}\to
H^{k}$ is the $L^{2}$-orthogonal projection onto the tangent space at
$\theta(\Phi(\zeta),\zeta)^{N}$ to $E^{N}:=\left\\{\theta^{N}:\theta\in
E\right\\}\subset H^{k}$, where the manifold $E$ is given in Definition 4.

######  Lemma 3 (approximate zero modes).

For every $\phi\in H^{k}$ there holds

$\lVert L_{\zeta}Q\phi\rVert_{H^{k-4}}\lesssim
O(R^{-2})\lVert\phi\rVert_{H^{k}}.$ (88)

###### Proof.

Recall that the functions $f_{\alpha}$ are calculated in (39)-(40), and that
the projection $Q$ maps onto $\operatorname{span}\left\\{f_{\alpha}\right\\}$.
Using these facts, we have shown $\lVert Q-P\rVert_{H^{k}}\lesssim R^{-2}$ as
in (46).

Denote $L^{0}$ the linearized operator at the coordinate sphere $S_{\zeta}$
with Euclidean background metric, and $L^{g}$ the same with the metric $g$ on
$M$. These two operators satisfy

$\lVert L^{0}_{\zeta}\phi\rVert_{H^{k-4}}=\lVert
L^{g}_{\zeta}\phi\rVert_{H^{k-4}}+O(R^{-3})\lVert\phi\rVert_{H^{k}}.$ (89)

See a discussion about this in Remark 5.

By the definition of $P$ in Definition 3, $L^{0}$ vanishes on
$\operatorname{ran}P$. It follows that

$\lVert L^{0}_{\zeta}Q\phi\rVert_{H^{k}}\leq\lVert
L^{0}_{\zeta}\rVert_{H^{k}\to
H^{k-4}}\lVert(P-Q)\phi\rVert_{H^{k}}=O_{H^{k}}(R^{-2})\lVert\phi\rVert_{H^{k}}.$
(90)

Since the functional $\mathcal{W}$ is $C^{2}$ on $X^{k}$, we have by (24) that

$L_{\zeta}\lesssim L^{g}_{\zeta}+O_{H^{k}\to
H^{k-4}}(\zeta^{\alpha}\partial_{\zeta^{\alpha}}\Phi)=L^{g}_{\zeta}+O_{H^{k}\to
H^{k-4}}(R^{-2}).$ (91)

The claim follows by plugging (89)-(90) into (91). ∎

######  Lemma 4 (coercivity).

Let $k\geq 4,\,R\gg 1$. There exist $\alpha,\,\beta>0$ depending on $k,\,R$
and the background metric $g$ only, s.th.

$\alpha\lVert\xi\rVert_{H^{k}}^{2}\leq\left\langle\xi,\,L_{\zeta}\xi\right\rangle\leq\beta\lVert\xi\rVert_{H^{k}}^{2}\quad(\xi\in\ker
Q).$ (92)

###### Proof.

The upper bound follows from the fact that $\bar{W}$ is $C^{2}$ and
$L_{\zeta}=\bar{W}^{\prime\prime}$. Thus it suffices to find the lower bound.

In [MR2785762]*Thm. 10, it is shown that

$\left\langle\xi,\,L_{\zeta}\xi\right\rangle\geq\alpha\lVert\xi\rVert_{L^{2}}^{2}\quad(\xi\in\ker
P).$ (93)

where $\alpha$ depends only on the lower bound $R$ and the ambient metric $g$.
The claim now is

$\left\langle\xi,\,L_{\zeta}\xi\right\rangle\geq\frac{\alpha}{4}\lVert\xi\rVert_{L^{2}}^{2}\quad(\xi\in\ker
Q),$ (94)

provided $R$ is sufficiently large. Notice that here we have one caveat,
namely that $Q$ is not necessarily orthogonal.

Put $\bar{Q}=1-Q$. Then we can rewrite (94) as

$\left\langle L_{\zeta}\xi,\,\xi\right\rangle=\left\langle
L_{\zeta}Q\xi,\,\xi\right\rangle+\left\langle
L_{\zeta}\bar{Q}\xi,\,\xi\right\rangle.$ (95)

The first term is $O(R^{-2})\lVert\xi\rVert_{H^{k}}^{2}$ by the approximate
zero mode property (88). The second term further splits as $\left\langle
L_{\zeta}\bar{Q}\xi,\,\bar{Q}\xi\right\rangle+\left\langle
L_{\zeta}\bar{Q}\xi,\,Q\xi\right\rangle=\left\langle
L_{\zeta}\bar{Q}\xi,\,\bar{Q}\xi\right\rangle+\left\langle\bar{Q}\xi,\,LQ\xi\right\rangle=\left\langle
L\bar{Q}\xi,\,\bar{Q}\xi\right\rangle+O(R^{-2})\lVert\xi\rVert_{H^{k}}^{2}$,
again by (88). Thus it suffices to find a lower bound of the quadratic form
$\left\langle
L\bar{Q}\xi,\,\bar{Q}\xi\right\rangle\geq(\alpha/2)\lVert\xi\rVert_{H^{k}}$

Consider the operator $\bar{Q}L\bar{Q}$. By the uniform closeness (46), we
have $\bar{Q}L\bar{Q}=\bar{P}L\bar{P}+O_{H^{k}\to H^{k-4}}(R^{-2})$. This and
$k\geq 4$ implies $\lVert\bar{Q}L\bar{Q}\rVert_{H^{k}\to L^{2}}\geq\alpha/2$
for sufficiently large $R$.

Thus (92) follows once we can find a uniform estimate
$\lVert\xi\rVert_{H^{k}}\lesssim\lVert\xi\rVert_{L^{2}}.$ This can be done
using the fact that $\xi$ solves (65), which we can rewrite as a fourth order
elliptic equation as

$L_{\zeta}\xi=-\bar{W}(\Phi,\zeta)-N_{\zeta}(\xi)+O(R^{-2}).$

By elliptic regularity, this implies that there is some $C>0$ depending on
$R$, the background metric $g$, and the Sobolev order $k$ only s.th.
$\lVert\xi\rVert_{H^{k}}\leq C\lVert\xi\rVert_{L^{2}}.$

∎

## References