# Estimates for the Constant Mean Curvature Dirichlet Problem on Catenoidal
necks

Stephen J. Kleene  Department of Mathematics, University of Rochester,
Rochester, NY<EMAIL_ADDRESS>
# Estimates for the Constant Mean Curvature Dirichlet Problem on catenoids

Stephen J. Kleene  Department of Mathematics, University of Rochester,
Rochester, NY<EMAIL_ADDRESS>
###### Abstract.

In this article, we solve the constant mean curvature dirichlet problem on
catenoidal necks with small scale in $\mathbb{R}^{3}$. The solutions are found
in exponentially weighted Hölder spaces with non-integer weight and are
a-priori bounded by a uniform constant times $r^{1+\gamma}$, where $r$ denotes
the distance to the axis of the neck and where $\gamma$ belongs to the
interval $(0,1)$. By comparing the solutions with their limits on the disk, we
improve the estimate to $\gamma=1$. As a corollary, we prove differentiability
of solutions in $\tau$ down to $\tau=0$. The surfaces we construct have
applications to gluing constructions.

##  1\. Introduction

A _catenoidal neck_ in $\mathbb{R}^{3}$ is the intersection of a catenoid with
a tube about its axis of rotation. Up to scalings and rigid motions,
catenoidal necks comprise a one parameter family
$\mathcal{N}_{\tau}=\tau\mathcal{C}\cap\\{r\leq 1\\}$, where here
$\mathcal{C}$ denotes the _standard catenoid_ implicitly given by the equation
$r=\cosh(z)$ in $\mathbb{R}^{3}$ and where $r=\sqrt{x^{2}+y^{2}}$ denotes the
distance to the $z$-axis. We refer to the parameter $\tau$ as the _scale_ of
the catenoidal neck $\mathcal{N}_{\tau}$. Let $\mathcal{N}_{\tau}^{\pm}$
denote the intersection of $\mathcal{N}_{\tau}$ with the half space $\\{\pm
z\geq 0\\}$. Then $\mathcal{N}^{\pm}_{\tau}$ converges in $C^{k}$ to the disk
$\mathcal{D}$ of radius $1$ in the plane $\\{z=0\\}$ away from the origin as
$\tau\rightarrow 0$. The sheets $\mathcal{N}^{\pm}_{\tau}$ are smooth modulo
vertical translations in the sense that the translation

$\widetilde{\mathcal{N}}^{+}_{\tau}=\mathcal{N}_{\tau}^{+}-\tau\text{arccosh}(1/\tau)e_{z}$

of $\mathcal{N}_{\tau}$ converges smoothly to the disk $\mathcal{D}$ away from
the origin in $C^{k}$, and the analogous statement holds for
$\mathcal{N}^{-}$. In order to avoid introducing additional notation, we
identify $\widetilde{\mathcal{N}}^{\pm}=\mathcal{N}^{\pm}$ throughout this
article. The variation fields generated by $\mathcal{N}^{\pm}_{\tau}$ at
$\tau$ = 0 are respectively $-\log(r)$ and $\log(r)$, where $r$ is the polar
distance on $\mathbb{R}^{2}$

In this article, we study the constant mean curvature Dirichlet problem on
catenoidal necks of small scale. For $\tau$ sufficiently small, the surfaces
$\mathcal{N}_{\tau}$ are non-degenerate, and thus the CMC dirichlet problem is
solvable for small boundary data and small mean curvature. Since the family
$\mathcal{N}_{\tau}$ is divergent in the moduli space, there is a-priori no
uniform estimate on the smallness required for solvability across the family,
and in fact there are obstructions arising from lower modes–functions in the
span of $1$, $\cos(x)$ and $\sin(x)$–on the circle. We show that the Dirichlet
problem is uniformly solvable up to lower modes across the family as the scale
tends to zero. Since the boundary values of our solutions are not completely
determined, in order to uniquely characterise our solutions we demand that the
solutions satisfy a set of orthogonality conditions along the waist of the
necks, and a function satisfying these conditions is said to be _normalized_.
Our main theorem can them be stated as follows:

######  Theorem 1.1.

There is $\epsilon>0$ and $C>0$ such that: Given a function $f\in
C^{2,\alpha}(\partial\mathcal{N}_{\tau})$ and $\delta\in(-\epsilon,\epsilon)$,
there is a unique normalized function $\mathcal{h}_{\tau,\delta,f}$ in
$C^{2,\alpha}(\mathcal{N}_{\tau})$ satisfying the estimate

$|\mathcal{h}_{\tau,\delta,f}|\leq^{C}r^{2}\|f\|$

and such that the normal graph $\mathcal{N}_{\tau,\delta,f}$ over
$\mathcal{N}_{\tau}$ has constant mean curvature $\delta$ and such that the
trace of $\mathcal{h}_{\tau,\delta,f}$ agrees with $f$ up to lower modes.

Our solutions are found by solving the Poisson-Dirichlet problem for the
Jacobi operator of the catenoid, or rather its pullback to the cylinder under
its conformal parametrization, in exponentially weighted Hölder spaces.
Similar problems arise in various formulations in numerous articles ([7], [8],
[9], [10], [12], [14], [13]). A standard feature of such problems is that
rates of exponential decay must be a non-integer. Our solutions are found in
such spaces and thus a-priori satisfy the uniform estimate
$|\mathcal{h}_{\tau,\delta,f}|\leq^{C}r^{1+\gamma}\|f\|$.

The refinement offered by Theorem 1.1 lies in the improvement of the decay
estimate for the solutions to $\gamma=1$, which allows us to prove
differentiability of the solutions in $\tau$ at $\tau=0$. We obtain these
improved estimates by comparing our solutions with their limits on this disk.

The family of solutions $\mathcal{h}_{\tau,\delta,f}$ extends differentiably
in $\tau$ to $\tau=0$ in the following sense: If
$\mathcal{h}^{\pm}_{\tau,\delta,f}$ denotes the restriction of
$\mathcal{h}_{\tau,\delta,f}$ to $\mathcal{N}^{\pm}_{\tau}$, then
$\mathcal{h}^{\pm}_{\tau,\delta,f}$ converges in $C^{2,\alpha/2}$ away from
the origin to a constant mean curvature graph over the disk satisfying the
estimate

$\|\mathcal{h}^{\pm}_{\delta,f}\|\leq^{C}r^{2}\|f^{\pm}\|$

and with trace equal to $f^{+}$–the restriction of $f$ to the top boundary
component of $\mathcal{N}^{+}_{\tau}$–up to lower modes. Since such solutions
are unique, the family $\mathcal{h}_{\tau,\delta,f}$ extends continuously in
$C^{2,\alpha}$ to $\tau=0$ in the sense that the family of functions
$\mathcal{h}^{\pm}_{\tau,\delta,f}$ extend continuously in $C^{2,\alpha}$ away
from the origin to $\tau=0$. In order to show that the family extends
differentiably in $\tau$ to $\tau=0$, we fix a family of diffeomorphisms
$\phi_{\tau_{0}}^{\tau}:\mathcal{N}_{\tau_{0}}\rightarrow\mathcal{N}_{\tau}$
for $\tau$ near $\tau_{0}$, which allows us to differentiate the solutions in
$\mathcal{h}_{\tau,\delta,f}$ in $\tau$ by fixing a common domain for $\tau$
near $\tau_{0}$. The mappings $\phi_{\tau_{0}}^{\tau}$ are obtained by
representing $\mathcal{N}_{\tau}$ as a normal graph over
$\mathcal{N}_{\tau_{0}}$ after slightly adjusting the scale of
$\mathcal{N}_{\tau}$. The maps converge to a limiting map as
$\tau_{0}\rightarrow 0$ in the following sense: Let
$\phi_{\tau_{0}}^{\tau,\pm}$ denote the restriction of
$\phi_{\tau_{0}}^{\tau}$ to the graphical sheet
$\mathcal{N}^{\pm}_{\tau_{0}}$. Then $\mathcal{N}^{\pm}_{\tau_{0}}$ converges
in $C^{k}$ on away from the origin to the disk $\mathcal{D}$, and the maps
$\phi_{\tau_{0}}^{\tau,\pm}$ converge away from the origin in $C^{k}$ to the
parametrization of $\mathcal{N}^{\pm}_{\tau}$ as a graph over $\mathcal{D}$.
The functions $\mathcal{h}_{\tau,\delta,f}$ then depend smoothly on $\tau$ for
$\tau>0$ and we define the $\tau$-derivative of $\mathcal{h}_{\tau,\delta,f}$
at $\tau_{0}$ to be

$\dot{\mathcal{h}}_{\tau_{0},\delta,f}:=\left.\frac{d}{d\tau}\right|_{\tau=\tau_{0}}\left(\mathcal{h}^{\tau}_{\tau_{0},\delta,f}\right),\quad\mathcal{h}_{\tau_{0},\delta,f}^{\tau}:=\mathcal{h}_{\tau,\delta,f}\circ\phi_{\tau_{0}}^{\tau}$

The Gateaux derivative in $\tau$ at $\tau=0$ is given by

$\dot{\mathcal{h}}^{\pm}_{\delta,f}:=\lim_{\tau\rightarrow
0}\frac{\mathcal{h}^{\pm}_{\tau,\delta,f}-\mathcal{h}^{\pm}_{\delta,f}}{\tau}$

and is well defined as a limit in $C^{2,\alpha/2}$ on compact subsets
$\mathcal{D}$ of the unit disk way from the origin. We then show:

######  Theorem 1.2.

The functions $\dot{\mathcal{h}}^{\pm}_{\tau_{0},\delta,f}$ are uniformly
bounded in $C^{2,\alpha}$ away from the origin and converge as
$\tau_{0}\rightarrow 0$ to $\dot{\mathcal{h}}^{\pm}_{\delta,f}$.

As an immediate corollary, the surfaces $\mathcal{N}_{\tau,\delta,f}$ depend
differentiably on $\tau$ at $\tau=0$, in the following sense: Letting
$\mathcal{N}^{\pm}_{\tau,\delta,f}$ denote the normal graph of
$\mathcal{h}^{\pm}_{\tau,\delta,f}$ over $\mathcal{N}_{\tau}^{\pm}$, the
surface $\mathcal{N}^{\pm}_{\tau,\delta,f}$ depends smoothly on $\delta$ and
$f$, smoothly in $\tau$ for $\tau>0$ and once differentiably in $\tau$ at
$\tau=0$, in $C^{2,\alpha}$ on compact subsets away from the origin. In
particular, the family has a $C^{1}$ extension to $\tau=0$. Continuous
extensions for the derivatives in $\delta$ and $f$ are easy and most of the
technical work of the paper is devoted to showing that the derivative in
$\tau$ extends continuously.

###  1.1. General comments about notation, conventions, and the proof

####  1.1.1. Notation, terminology and other convections

We do not define within this article widely used notation and terminology that
is considered standard, such as, for example, Hölder spaces $C^{k,\alpha}(D)$
or standard objects associated with euclidean spaces. We use the standard
practice that $C$ and $\epsilon$ denote any number of constants that must be
chosen large or small subject to various consideration. Thus, for example the
reader should regard an ‘$\epsilon$’, when it is encountered, as the minimum
of the previous encountered ‘$\epsilon$’ along with any additional constraints
required. Since we are not interested in quantifying the dependence of our
solutions on any of these constants, we do not distinguish them. We use the
non-standard notation $f\leq^{C}g$ to mean $f\leq Cg$, which is convenient in
the case that the right hand side contains many terms and would otherwise need
to be enclosed in parentheses.

Frequently, the surfaces we consider in this article come with a specified
parametrization. When this is case, we do not carefully distinguish between
objects defined on the surface and the pullbacks under the parametrization.
Whether we are considering a function as defined on a surface or on the
parametrization domain will be clear from context.

####  1.1.2. General comments of the proof

The central objects we consider are geometric operators such as the mean
curvature and unit normal operators $F\mapsto H_{F}$ and $F\mapsto N_{F}$
defined on immersions $F$, and the resulting remainder terms that arise when
perturbing immersions by graphs. Rather than write these terms explicitly, we
appeal abstractly to the fact that these operators are smooth–in the sense of
Frechet differentiability–as mappings on spaces of class $C^{2,\alpha}$ Hölder
immersions, and are thus locally bounded in $C^{k}$ in their domains. Standard
integral formulae then apply for taylor remainders and most of the required
estimates follow easily. A few of the more delicate estimates require deeper
analysis, and we pause here to observe them so they are not a surprise to the
reader later. Firstly in places–such as the proof of Lemma 9.5–we will need
the fact that the hessian of the graph mean curvature operator vanishes at
affine maps. This allows for improved decay estimates for various terms since
the standard conformal parametrization of the catenoid is exponentially flat.

####  1.1.3. Arguments requiring mode preservation

The most delicate estimates in the paper arise in the proof of Propositions
8.1 and 9.5. They require a separation of the error term into the lower and
higher parts, with the higher part satisfying improved bounds relative to the
whole terms. In both these cases, the rotational symmetry of the catenoid is
exploited in the following way: Let $F$ be an immersion of a domain
$D=\mathbb{S}^{1}\times I$ with coordinates $(x,s)$ into $\mathbb{R}^{3}$ and
$u$ a function on $D$. Then the mean curvature $H_{F}(u)$ of the normal graph
over $F$ by $u$ can be computed as:

$H_{F}(u)=\Phi(\mathcal{J}_{u}(x,s),x,s)$

where $\mathcal{J}_{u}=\left(u,\nabla u,\nabla^{2}u\right)$ is the vector of
partial derivatives of $u$ of up to order $2$, and where $\Phi$ is a smooth
function defined on an open subset of a euclidean space about the origin.
Suppose now that $F$ is an immersion of rotation, so in cylindrical
coordinates, we have:

$F(x,s)=r(s)e_{r}(x)+h(z)e_{z},$

where above we have set $e_{r}(x)=\cos(x)e_{x}+\sin(x)e_{x}$. Then the
symmetry of the parametrization $F$ implies that the function $\Phi$ does not
depend on $x$, so

$\Phi(\mathcal{J},x,s)=\Phi(\mathcal{J},s).$

We then have

$\displaystyle\int_{\mathbb{S}^{1}}e^{-ikx}\left.DH_{F}\right|_{0}(u)$
$\displaystyle=\int_{\mathbb{S}^{1}}e^{ikx}\left.D\Phi\right|_{\left(0,s\right)}\left(\mathcal{J}_{u},s\right)$
$\displaystyle\left.D\Phi\right|_{\left(0,s\right)}\left(\int_{\mathbb{S}^{1}}e^{ikx}\mathcal{J}_{u},s\right)$

This allows for bounds to be estimated for modes separately. This observation
generalizes the following way, which is needed in the proof of Proposition
9.5. Assume now that $\xi=\xi(s)$ is a rotationally symmetric function on $D$.
Then, $\mathcal{J}_{\xi}$ is a function only of $s$ and we have

$\displaystyle\int_{-\pi}^{\pi}e^{ikx}\left.D^{2}H\right|_{u}\left(\xi,u\right)dx$
$\displaystyle=\int_{-\pi}^{\pi}e^{ikx}\left.D^{2}\Phi\right|_{(0,s)}\left(\mathcal{J}_{\xi},e^{ikx}\mathcal{J}_{u}\right)dx$
$\displaystyle=\left.D^{2}\Phi\right|_{(0,s)}\left(\mathcal{J}_{\xi},\int_{-\pi}^{\pi}e^{ikx}\mathcal{J}_{u}dx\right)$

We refer this property of the operators $\left.DH_{F}\right|_{0}(0)$ and
$\left.D^{2}H_{F}\right|_{0}(\xi,-)$ as _mode preserving_. The mode preserving
property is critical for our required estimates.

####  1.1.4. Other comments

In order to keep this article as self-contained as possible, we have included
here the proofs of several results that are by this point fairly standard in
singular perturbation literature. In particular, our treatment of the mapping
properties of the jacobi operator on the catenoid in exponentially weighted
Hölder spaces closely follows similar proofs found in [12]. The only external
machinery we rely upon is what is frequently referred to as ‘standard theory’,
specifically, the existence and regularity theory for second order elliptic
operators on bounded domains (c.f. [5]), the Implicit Function Theorem, and
some basic properties of differentiability in Banach spaces. Specifically, our
argument relies on the fact a mapping which is continuously Gateaux
differentiable on a domain is Frechet differentiable (see, for example [1]
Proposition A.3).

Finally, we remark that we are considering Dirichlet-Poission problems on the
catenoid in which no symmetry is presumed. In presence of enough
symmetries–for example, if we require invariance under reflections through the
$x$ and $y$ axis, then the challenging estimates required in our proof become
trivial and go through automatically. This is because such symmetry
requirements kill the problematic lower modes, and the remaining ones
necessarily satisfy improved decay rates.

##  2\. Preliminaries and Background

###  2.1. Geometric objects as smooth mappings

The basic objects we consider in this article are spaces of class
$C^{k,\alpha}$ immersions into $\mathbb{R}^{3}$ as well as geometric
quantities associated to oriented immersions such as the unit normal and mean
curvature. These geometric quantities can be regarded as smooth mappings in
the sense of Frechet differentiability on space of class $C^{k,\alpha}$
immersions as follows: Given a smooth compact domain $D$ in $\mathbb{R}^{2}$,
let $R^{k,\alpha}(D)\subset C^{k,\alpha}(D,\mathbb{R}^{3})$ denote the space
of class $C^{k,\alpha}$ immersions of $D$ into $\mathbb{R}^{3}$. We require
immersions to extend as immersions to an open set containing $D$, so that
$R^{k,\alpha}(D)$ is an open subset of $C^{k,\alpha}(D,\mathbb{R}^{3})$. The
mean curvature mapping $H:F\mapsto H_{F}$, taking an immersion $F$ to its mean
curvature $H_{F}$ is then smooth as a map from the space
$R^{k+2,\alpha}(D,\mathbb{R}^{3})$ into $C^{k,\alpha}(D)$. Similarly, the unit
normal mapping $N:F\mapsto N_{F}$ taking an immersion to its unit normal is
smooth as a mapping from $R^{k+1,\alpha}(D)$ into
$C^{k,\alpha}(D,\mathbb{R}^{3})$. We have omitted the proof of these facts,
but they are easy to establish by, for example, computing the mean curvature
as a smooth function of components of the derivative of an immersion, so

$H_{F}=\overline{H}(\nabla F,\nabla^{2}F)$

where $\overline{H}=\overline{H}(q_{i},r_{i,j})$ is defined and smooth on the
open subset of the euclidean space $\left(\mathbb{R}^{3}\right)^{\times
2}\times\left(\mathbb{R}^{3}\right)^{\times 4}$ with $\det(q_{i}\cdot
q_{j})\neq 0$. Smoothness in this sense then works as it does in the standard
euclidean setting. In particular, the usual integral expressions for Taylor
remainders holds on locally convex domains in $R^{k,\alpha}(D)$.

We let $H_{F}$ and $N_{F}$ denote the mean curvature of a $C^{2}$ immersion of
$F:D\rightarrow\mathbb{R}^{3}$, which we regard as functions on $D$. Given a
function $u:D\rightarrow\mathbb{R}$, the normal graph over $F$ by $u$ is the
mapping

$F_{u}:=F+uN_{F}$

Provided $F_{u}$ is a $C^{2}$ immersion again, we let $H_{F}(u)$ denote its
mean curvature. When $D$ is compact, $F_{u}$ is an immersion whenever $u$ is
sufficiently small in $C^{2}$ and we refer to the mapping $u\mapsto H_{F}(u)$
as the mean curvature operator at $F$. The variation of the mean curvature of
$F$ induced by a vector field $V$ is given by

(2.1) $\displaystyle\left.DH\right|_{F}(V):=\nabla_{V^{\top}}H+L_{F}V^{\perp}$

wheret $V^{\perp}=V\cdot N_{F}$ denotes the normal part of $V$ and
$V^{\top}=V-V^{\perp}N_{F}$ the tangential part, and where $L_{F}$ is the
stability, or jacobi, operator of $F$ and is given by

$L_{F}=\Delta_{F}+\left|A_{F}\right|^{2}.$

where above $\Delta_{F}$ denotes the Laplace operator of the immersion $F$ and
$\left|A_{F}\right|^{2}$ the square length of the second fundamental form of
$F$.

###  2.2. The Poisson-Dirichlet Problem for the mean curvature operator

If $\mathcal{Q}$ is a second order elliptic operator, a solution to the
$(E,f)$-Poisson-Dirichlet problem for and operator $\mathcal{Q}$ over the
domain $D$, where $E$ is a function on $D$ and $f$ is a function on $\partial
D$, is a function $u$ on $D$ satisfying

$\mathcal{Q}_{F}(u)=E,\quad\mathcal{tr}(u)=f$

where above $\mathcal{tr}(u)$ denotes the trace of $u$. We refer to $E$ as the
Poission data and to $f$ as the Dirichlet data of the problem. We assume that
$E$ is class $C^{0,\alpha}$ and $f$ is class $C^{2,\alpha}$. Standard elliptic
regularity theory then implies that any solution $u$ is class $C^{2,\alpha}$.
The solvability of this equation for small $E$ and $f$ is determined studying
the Poisson Dirichlet problem for the linearization $\dot{\mathcal{Q}}$ of
$\mathcal{Q}$ at $u=0$:

$\dot{\mathcal{Q}}_{F}(u)=E,\quad\mathcal{tr}(u)=f$

We take $\mathcal{Q}=H_{F}$ for some fixed immersion $F\in R^{2,\alpha}(D)$,
so that $\dot{\mathcal{Q}}=L_{F}$. $L_{F}$ is a self-adjoint elliptic operator
and the linearized Possion-Dirichlet problem for $L_{F}$ is solvable if and
only if the immersion $F$ is non-degenerate, meaning that the kernel of
$L_{F}$ is trivial.

###  2.3. Conformal parametrization of the catenoid

The standard catenoid $\mathcal{C}$ is conformally parametrized by the mapping
$\mathcal{F}:\Omega\rightarrow\mathbb{R}^{3}$, where
$\Omega:=\mathbb{S}^{1}\times\mathbb{R}$ is the cylinder and where

(2.2) $\displaystyle\mathcal{F}(x,s):=\cosh(s)e_{r}(x)+se_{z},\quad
e_{r}(x)=\cos(x)e_{x}+\sin(x)e_{y}.$

The first and second fundamental forms of $\mathcal{F}$ are given by

$\mathcal{g}_{\mathcal{F}}:=\begin{pmatrix}\cosh^{2}(s)&0\\\ \\\
0&\cosh^{2}(s)\end{pmatrix},\quad A_{\mathcal{F}}:=\begin{pmatrix}1&0\\\ \\\
0&1\end{pmatrix}.$

The unit normal is

(2.3) $\displaystyle N_{\mathcal{F}}=\cosh^{-1}(s)e_{r}(x)-\tanh(s)e_{z}.$

and the stability operator is

$L_{\mathcal{F}}=\cosh^{-2}(s)\Delta_{\Omega}+2\cosh^{-4}(s).$

###  2.4. Catenoidal necks

For each $\tau>0$, let $\mathcal{G}_{\tau}:\Omega\rightarrow\mathbb{R}^{3}$
denote the mapping given by

$\mathcal{G}_{\tau}:=\tau\mathcal{F}.$

The neck $\mathcal{N}_{\tau}$ is parametrized by the restriction of
$\mathcal{G}_{\tau}$ to the domain

$\mathcal{N}_{\tau}^{*}:=\Omega_{l_{\tau}}:=\mathbb{S}^{1}\times[-l_{\tau},l_{\tau}],\quad
l_{\tau}=\text{Arccosh}\left(\frac{1}{\tau}\right).$

Given constants $\tau_{0}$ and $\tau$, we define a family of diffeomorphisms
$\phi_{\tau_{0}}^{\tau}:\mathcal{N}_{\tau_{0}}\rightarrow\mathcal{N}_{\tau}$
as follows: First, observe that there is a scalar $\lambda$ such that
$\lambda\mathcal{N}_{\tau}$ is a normal graph over $\mathcal{N}_{\tau_{0}}$.
The scalar $\lambda$ can be determined as follows: The boundary of the surface
$\mathcal{N}_{\tau}$ in cylindrical coordinates is given by

$(r,z)=\left(1,\tau l\right)$

and outward conormal is $\left(\sqrt{1-\tau^{2}},\tau\right)$. We choose the
scalar $\lambda$ so that the difference vector between the boundaries is
orthogonal to the outward conormal, which by rotational symmetry implies that
it is a multiple of the unit normal. This means that

$\left\\{\lambda(1,\tau
l)-(1,\tau_{0}l_{0})\right\\}\cdot\left(\sqrt{1-\tau^{2}},\tau\right)=0.$

Solving for $\lambda$ gives

$\lambda=\lambda_{\tau}=\frac{\tau^{2}_{0}l_{0}+\sqrt{1-\tau_{0}^{2}}}{\sqrt{1-\tau_{0}^{2}}+\tau_{0}\tau
l}$

Observe that $\lambda=1$ when $\tau=\tau_{0}$ and

$\dot{\lambda}:=\left.\frac{d}{d\tau}\right|_{\tau=\tau_{0}}\lambda\rightarrow
0$

as $\tau_{0}\rightarrow 0$. The diffeomorphisms
$\phi_{\tau_{0}}^{\tau}:\mathcal{N}_{\tau_{0}}\rightarrow\mathcal{N}_{\tau}$
induce diffeomorphims on the domains
$\phi_{\tau_{0}}^{\tau}:\mathcal{N}^{*}_{\tau_{0}}\rightarrow\mathcal{N}^{*}_{\tau}$,
which we still denote by $\phi_{\tau_{0}}^{\tau}$, which satisfy

$\lambda\mathcal{G}_{\tau_{0}}^{\tau}-\mathcal{G}_{\tau_{0}}=fN_{\mathcal{G}_{\tau}}$

for a function $f$, where above we have set
$\mathcal{G}_{\tau_{0}}^{\tau}=\mathcal{G}_{\tau}\circ\phi_{\tau_{0}}^{\tau}$.
Differentiating in $\tau$ at $\tau=\tau_{0}$ and using that
$\phi_{\tau_{0}}^{\tau}$ is a function of $s$ only gives:

(2.4)
$\displaystyle\left(\dot{\lambda}\tau+1\right)\mathcal{F}-\frac{\partial\mathcal{G}_{\tau}}{\partial
s}\dot{\phi}=\dot{f}N_{\mathcal{G}_{\tau}}.$

Thus, the normal variation field generated by the family of immersion
$\mathcal{G}_{\tau_{0}}^{\tau}$ is

$\left(\dot{\mathcal{G}}_{\tau_{0}}\right)^{\perp}=\left(\tau\dot{\lambda}+1\right)\mathcal{F}^{\perp}=\left(\tau\dot{\lambda}+1\right)\xi,$

where above $\xi=s\tanh(s)-1$. Here, we have used that
$\mathcal{N}_{\mathcal{G}_{\tau}}=\mathcal{N}_{\mathcal{F}}$ and (2.2) and
(2.3). Taking the dot product of (2.4) with
$\frac{\partial\mathcal{G}_{\tau_{0}}}{\partial s}$ gives

(2.5)
$\displaystyle\dot{\phi}_{\tau_{0}}=\left(\tau\dot{\lambda}+1\right)\frac{\mathcal{F}\cdot\frac{\partial\mathcal{G}_{\tau_{0}}}{\partial
s}}{\left|\frac{\partial\mathcal{G}_{\tau_{0}}}{\partial
s}\right|^{2}}=\left(\tau\dot{\lambda}+1\right)\frac{\tanh(s)+s/\cosh^{2}(s)}{\tau_{0}}.$

The total variation field is

(2.6)
$\displaystyle\dot{\mathcal{G}}_{\tau_{0}}=\left(\tau\dot{\lambda}+1\right)\xi
N_{\mathcal{F}}-\dot{\lambda}\tau\mathcal{F}.$

###  2.5. The modified Dirichlet-Poisson Problem on catenoidal necks

As mentioned above, there are obstructions to solving the Poission-Dirichlet
problem for $H$ on catenoidal necks with uniform estimates for small $\tau$.
We instead impose the additional condition that the solutions are _normalized_
–defined below–and in exchange we give up control on the lower modes of the
boundary values. A lower mode on the circle is a function in the span of $1$,
$\cos(\theta)$ and $\sin(\theta)$. If $f$ is a function on the circle
$\mathbb{S}^{1}$, we let $\overline{f}$ denote the projection of $f$ onto the
space of lower modes and $\mathring{f}$ denote the projection of $f$ into the
orthogonal compliment. We refer to $\overline{f}$ as the lower part of $f$ and
to $\overline{f}$ as the higher part. More generally, if $E$ is a function on
a cylindrical domain $\Omega_{a,b}$ for some $a,b$, we define the lower part
of $E$ to be the function $\overline{E}(x,s)$, where for each $s$
$\overline{E}(x,s)$ is the lower part of the function $x\mapsto E(x,s)$. The
higher part of $E$ is then

$\mathring{E}:=E-\overline{E}$

and we refer to $\mathring{E}$ as the _higher part_ of $E$.

A solution to the _modified_ $(E,f)$-Poisson Dirichlet problem on $\Omega_{l}$
for $H_{F}$ is then a normalized function $u$ satisfying

$H_{F}(u)=E,\quad\mathring{\mathcal{tr}}(u)=\mathring{f}.$

Observe here that only the higher part $\mathring{\mathcal{tr}}(u)$ is equal
to $\mathring{f}$ and thus

$\mathcal{tr}(u)=f+\ell$

for some lower mode $\ell$. The boundary $\partial\Omega_{l}$ of $\Omega_{l}$
comprises the two components
$\partial_{\pm}\Omega_{l}:=\mathbb{S}^{1}\times\\{\pm l\\}$, each of which is
canonically isomorphic to $\mathbb{S}^{1}$. A function $f$ on $\partial\Omega$
may then be identified with a pair of functions on $\mathbb{S}^{1}$ under the
identification $f\mapsto(f_{-},f_{+})$, where $f_{\pm}(x)=f(x,\pm l)$ denotes
the restriction of $f$ to $\partial_{\pm}\Omega$. A function on
$\partial\Omega_{l}$ is then a _lower mode_ if its restrictions $f_{\pm}(x)$
are lower modes on $\mathbb{S}^{1}$. Thus, the space of lower modes on
$\Omega_{l}$ is six dimensional. We will denote by
$\mathring{C}^{k,\alpha}(\mathbb{S}^{1})$ and
$\mathring{C}^{k,\alpha}(\partial\Omega_{l})$ the space of $C^{k,\alpha}$
functions on $\mathbb{S}^{1}$ and $\Omega_{l}$ orthogonal to the lower modes.

###  2.6. Geometric jacobi fields and normalized functions

The space of lower modes on $\partial\mathcal{N}_{\tau}$ is generated by the
traces of the six dimensional family of jacobi fields generated by
translations, rotatations, and dilations of $\mathbb{R}^{3}$ which we call
geometric jacobi fields. Since $\mathcal{F}$ is minimal, we have
$H_{\mathcal{F}}=0$ and thus by (2.1) we have

$\left.DH\right|_{\mathcal{F}}(V)=L_{\mathcal{F}}V^{\perp}$

for any vector field $V$. Since $H$ is invariant under translations and
rotations, the left hand side above vanishes for vector fields in the span of
the translations $e_{x}$, $e_{y}$, and $e_{z}$, as well as the rotational
vector field $xe_{y}-ye_{x}$, $xe_{z}-ze_{x}$ and $ye_{z}-ze_{y}$. Finally,
since $\mathcal{F}$ is minimal again, the mean curvature of $\mathcal{F}$ is
invariant under dilations and thus the left hand side above vanishes
additionally when $V$ is the dialtion field $xe_{x}+ye_{y}+ze_{z}$. The normal
parts of these vector fields are smooth jacobi fields. They can be computed
explicitly from the expressions for $\mathcal{F}$ and its unit normal in (2.2)
(2.3) as $\tanh(s)$, $\cos(x)\cosh^{-1}(s)$, $\sin(x)\cosh^{-1}(s)$,
$s\tanh(s)-1$, $\cos(x)\left(\sinh(s)+s\cosh^{-1}(s)\right)$ and
$\sin(x)\left(\sinh(s)+s\cosh^{-1}(s)\right)$. They are precisely the six
dimensional space of functions such that $L_{\mathcal{F}}\mathcal{j}=0$ and
$\mathring{\mathcal{j}}=0$ on $\Omega$. Observe that, although there are seven
vector fields, the normal part of the rotational field $xe_{y}-ye_{x}$
vanishes on $\mathcal{F}$, and so the space of geometric jacobi fields is six
dimensional. Moreover, it can be verified directly that the trace map
restricts to an isomorphism from the space of geometric jacobi fields onto the
space of lower modes on $\mathcal{N}_{\tau}$. Geometric jacobi fields are
uniquely determined by their Cauchy data along the curve $\\{s=0\\}$, which we
call the _signature_. More precisely, if $u$ is a function on $\Omega_{l}$, we
define its signature to be the pair
$\mathcal{s}(u):=(\overline{u},\overline{u}_{s})$, where $\overline{u}$ and
$\overline{u}_{s}$ denote the projection of $u(x,0)$ and $\frac{\partial
u}{\partial s}(x,0)$ onto the space of lower modes. The function $u$ is said
to be _normalized_ if it has vanishing signature. It is straightforward to
verify that the mapping $u\mapsto\mathcal{s}(u)$ is an isomorphism of the
space of geometric jacobi fields onto the space of signatures–i.e. pairs of
lower modes. Thus, there is for each function $u$ on $\Omega_{l}$ a unique
geometric jacobi field $\mathcal{j}$ such that $u+\mathcal{j}$ is normalized.

###  2.7. Local compactness of the catenoid modulo scalings

The parametrization $\mathcal{F}$ induces an atlas on the catenoid which is
compact modulo translations and scalings, which allows us to prove uniform
estimates for the mean curvature operator at $\mathcal{F}$ in the weighted
holder spaces $W^{k,\alpha,\gamma}_{\tau}$. Namely, if
$u:\Omega_{a,b}\rightarrow\mathbb{R}^{k}$ is a function defined on the finite
cylinder $\Omega_{a,b}:=\mathbb{S}^{1}\times[a,b]$, we define the
_localization of $u$ at $s_{0}$_ for $a+1/2\leq s_{0}\leq b-1/2$ to be the
mapping $\left.u\right|_{s_{0}}:\Omega_{0}\rightarrow\mathbb{R}^{3}$,
$\Omega_{0}:=\Omega_{-1/2,1/2}$ given by

$\left.u\right|_{s_{0}}(x,s):=u(x,s+s_{0}).$

The localizations $\left.\mathcal{F}\right|_{s_{0}}$ of $\mathcal{F}$ are
compact in the moduli space–i.e. up to scalings and rigid motions. In
particular, if we define the normalization of
$\left.\mathcal{F}\right|_{s_{0}}$ to be the mapping
$\left.\widetilde{\mathcal{F}}\right|_{s_{0}}:\Omega_{0}\rightarrow\mathbb{R}^{3}$
given by

$\left.\widetilde{\mathcal{F}}\right|_{s_{0}}=\left(\left.\mathcal{F}\right|_{s_{0}}-s_{0}e_{z}\right)/\cosh(s_{0}),$

which differs from $\left.\mathcal{F}\right|_{s_{0}}$ by a translation and
scaling, then explicitly we have

$\left.\widetilde{\mathcal{F}}\right|_{s_{0}}=\mathcal{c}_{s_{0}}(s)e_{r}(x)+\frac{s}{\cosh(s_{0})}e_{z},$

where above we have set

$\mathcal{c}_{s_{0}}(s)=\frac{\cosh(s+s_{0})}{\cosh(s_{0})}=\cosh(s)+\tanh(s_{0})\sinh(s).$

Thus, as $s_{0}\rightarrow\infty$, the normalized localizations
$\left.\widetilde{\mathcal{F}}\right|_{s_{0}}$ converge in
$C^{k}(\Omega_{0},\mathbb{R}^{3})$ to the limiting map

$\left.\widetilde{\mathcal{F}}\right|_{\infty}(x,s):=e^{s}e_{r}(x)$

conformally parametrizing a flat annulus in the plane.

###  2.8. The weighted mean curvature operator

We exploit the compactness of the normalized charts
$\left.\widetilde{\mathcal{F}}\right|_{s_{0}}$ by working with the weighted
mean curvature operator $H_{\mathcal{F}}^{\prime}$ at $\mathcal{F}$ defined
below:

$H^{\prime}_{\mathcal{F}}(u):=\cosh(s)H_{F}(\cosh(s)u)$

Localizing $H^{\prime}_{\mathcal{F}}(u)$ at $s_{0}$ and using that the mean
curvature operator is homogeneous degree $-1$ gives

(2.7) $\displaystyle\left.H^{\prime}_{\mathcal{F}}(u)\right|_{s_{0}}$
$\displaystyle=\left.\cosh(s)H_{\mathcal{F}}(\cosh(s)u)\right|_{s_{0}}$
$\displaystyle=\mathcal{c}_{s_{0}}H_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}\left(\mathcal{c}_{s_{0}}\left.u\right|_{s_{0}}\right)$

A function $u$ is a solution to modified $(E,f)$-Poission-Dirichlet problem
for $H_{\mathcal{F}}$ if and only if $u/\cosh(s)$ is a solution to the
normalized $(E\cosh(s),f)$ Dirichlet-Possion problem for
$H^{\prime}_{\mathcal{F}}$. The linearization
$\mathcal{L}^{\prime}_{\mathcal{F}}$ of $H^{\prime}_{\mathcal{F}}$ at 0 is
then

(2.8) $\displaystyle L^{\prime}_{\mathcal{F}}u$
$\displaystyle=\cosh^{-1}(s)\Delta\left(\cosh(s)u\right)+2\cosh^{-2}(s)u.$

Observe that $L^{\prime}_{\mathcal{F}}$ is conjugate the operator

(2.9) $\displaystyle
L^{{}^{\prime\prime}}_{\mathcal{F}}=\Delta+2\cosh^{-2}(s)$

###  2.9. Weighted Holder spaces

Given $\tau>0$, set

$\omega_{\tau}(s)=\tau\cosh(s),\quad s\in\mathbb{R}.$

Then, given $\gamma\in\mathbb{R}$, $\alpha\in(0,1)$, a non-negative integer
$k$ and a locally $C^{2,\alpha}$ function on $\Omega_{l}$ we set

$\|u\|^{k,\alpha,\gamma}_{\tau}=\sup_{s_{0}}\left(\omega_{\tau}(s_{0})\right)^{-\gamma}\left\|\left.u\right|_{s_{0}}\right\|$

where above the supremum is taken over $s_{0}\in(-l+1/2,l-1/2)$. We let
$W^{k,\alpha,\gamma}_{\tau}$ denote the space
$C^{k,\alpha,\gamma}(\Omega_{l_{\tau}})$ equipped with the norm
$\|-\|^{k,\alpha,\gamma}_{\tau}$.

It follows easily from the definition that mapping
$\|-\|^{k,\alpha,\gamma}_{\tau}$ is indeed a norm and that
$W^{k,\alpha,\gamma}_{\tau}$ is a Banach space. Observe that
$\omega_{\tau}(s)\leq 1$ for $|s|\leq l_{\tau}$ and $\gamma>0$. It then
follows that

(2.10)
$\displaystyle\left\|\left.u\right|_{s_{0}}\right\|^{k,\alpha}\leq\left\|u\right\|^{k,\alpha,\gamma}_{\tau}\omega_{\tau}(s_{0})\leq\left\|u\right\|^{k,\alpha,\gamma}_{\tau}.$

We will use (2.10), along with the compactness of the normalized mappings
$\left.\widetilde{\mathcal{F}}\right|_{s_{0}}$, in establishing uniform bounds
for $H^{\prime}_{\mathcal{F}}$ as map on the spaces
$W^{k,\alpha,\gamma}_{\tau}$. We let $W^{k,\alpha,\gamma}_{\tau,n}\subset
W^{k,\alpha,\gamma}_{\tau}$ denote the subspace of normalized functions in
$W^{k,\alpha,\gamma}_{\tau}$, and we set

$\widehat{W}^{k,\alpha,\gamma}_{\tau}:=W^{k,\alpha,\gamma}_{\tau}\times\mathring{C}^{2,\alpha}\left(\partial\Omega_{l_{\tau}}\right)$

##  3\. Outline of the argument

We find solutions to the modified Poisson-Dirichlet problem by applying the
implicit function theorem. We do this in in two parts. First, we show in
Proposition 4.1 that the operator $H^{\prime}_{\mathcal{F}}$ is defined on a
ball
$\mathcal{B}=\mathcal{B}_{\tau}=B_{\epsilon}\left(0;W^{2,\alpha,\gamma}_{\tau}\right)$
of fixed radius independent of $\tau$ in the weighted spaces
$W^{2,\alpha,\gamma}_{\tau}$ and that the corresponding Possion-Dirichlet
mapping

$\widehat{H}^{\prime}_{\mathcal{F}}:u\mapsto\left(H^{\prime}_{\mathcal{F}}(u),\mathring{\mathcal{tr}}(u)\right)$

is uniformly bounded in $C^{2}$ as a map from $\mathcal{B}_{\tau}$ into
$\widehat{W}^{0,\alpha,\gamma}_{\tau}$. Then, in Proposition 5.3, we prove
that the lineraized mapping

$\widehat{L}^{\prime}_{\mathcal{F}}:u\mapsto\left(L^{\prime}_{\mathcal{F}}(u),\mathring{\mathcal{tr}}(u)\right)$

is an isomorphism from $W^{2,\alpha,\gamma}_{\tau,n}$ onto
$\widehat{W}^{0,\alpha}_{\tau}$ for small $\tau$ and with a uniform
invertibility constant independent of $\tau$. This allows us to directly apply
the implicit function theorem to construct solutions
$\mathcal{h}^{\prime}_{\tau,E,f}$ to the modified $(E,f)$-Possion-Dirichlet
for $H^{\prime}_{\mathcal{F}}$ for data small in
$\widehat{W}^{0,\alpha,\gamma}_{\tau}$. This is done Proposition 6.1.
Corollary 6.2 then uses Proposition 6.1 to obtain solutions
$\mathcal{h}_{\tau,\delta,f}$ to the modified constant mean curvature
Dirichlet problem for the operator $H_{\mathcal{F}}$. As a function on
$\mathcal{N}_{\tau}$, $\mathcal{h}_{\tau,\delta,f}$ satisfies the estimate

$\|\mathcal{h}_{\tau,\delta,f}\|\leq^{C}\epsilon r^{1+\gamma}$

for a universal constant $C>0$. Thus, the restrictions
$\mathcal{h}_{\tau,\delta,f}^{\pm}$ of $\mathcal{h}_{\tau,\delta,f}$ to
$\mathcal{N}^{\pm}_{\tau}$ converge on compact subsets of the unit disk
$\mathcal{D}$ way from the origin to a limit $\mathcal{h}_{\delta,f_{\pm}}$ at
$\tau=0$, satisfying

$H_{\mathcal{D}}\left(\mathcal{h}\right)=0,\quad\mathring{\mathcal{tr}}(\mathcal{h})=f_{\pm}.$

and the weighted estimate

$\left|\mathcal{h}(r,\theta)\right|\leq^{C}\epsilon r^{2}.$

Here, the improved estimate for the solution on disk follows from fact that it
extends smoothly to the origin. Such solutions are unique (Corollary 7.2), and
thus the solutions $\mathcal{h}_{\tau,\delta,f}$ extend continuously to
$\tau=0$.

The remainder of the article is aimed at establishing the $C^{1}$ extension of
the family of solutions to $\tau=0$. This is done in several steps. First, in
Lemma 9.2 we show that the functions $\dot{\mathcal{h}}^{+}_{\delta,f}$ are
well defined in $C^{2}$ and solve a modified Possion-Dirichlet problem on the
unit disk. Then in Proposition 9.1, we show that
$\dot{\mathcal{h}}^{+}_{\tau_{0},\delta,f}\rightarrow\dot{\mathcal{h}}^{+}_{\delta,f}$
as $\tau_{0}\rightarrow 0$. The estimates required for this are delicate and
require optimal bounds for the solutions in the weighted spaces. We obtain
optimal bounds by comparing the solutions $\mathcal{h}^{\pm}_{\tau,\delta,f}$
with their limits $\mathcal{h}^{\pm}_{\delta,f}$ at $\tau=0$, which extend
smoothly to the unit disk and vanish to second order at the origin. Since the
catenoid has rapidly decaying curvature, the limiting solutions serve as good
approximate solutions on the necks. The error term is then small enough in the
weighted space to obtain the improved estimate. This is done in Proposition
8.1. In Lemma 9.4, we show that the $\tau$-variation
$\dot{\mathcal{h}}_{\tau_{0},\delta,f}$ satisfies a normalized Poisson-
Dirichlet equation with vanishing Dirichlet data. The Poisson data decomposes
as a sum of two terms $\mathcal{E}_{1}+\mathcal{E}_{2}$ satisfying uniform
estimates in different weighted norms; one large but exponentially decaying
and the second small but exponentially growing. The mapping properties for the
linearized operator $L^{\prime}_{\mathcal{F}}$ for exponentially decaying
source terms is similar to the case of exponentially growing terms belonging
to the spaces $W^{k,\alpha,\gamma}_{\tau}$ for positive $\gamma$. However
there are some complications caused by the decay related to the presence of
small eigenvalues for $L^{\prime}_{\mathcal{F}}$ on long domains $\Omega_{l}$.
We handle this by assuming an orthogonality condition of the source term to
the lower modes, and directly integrate the low modes. The invertibility
theorem for source terms with exponential decay is recorded in Proposition
5.4. We prove uniform estimates in Lemma 9.6 for
$\dot{\mathcal{h}}_{\tau,\delta,f}$. The variations
$\dot{\mathcal{h}}^{\pm}_{\tau,\delta,f}$ then converge to a limit on the
disk. To show that the limit agrees with $\dot{\mathcal{h}}^{+}_{0,\delta,f}$,
we appeal to the uniqueness of solutions to the modified Poission-Dirichlet
problem on the disk. To apply this theorem, we need improved estimates for the
lower modes of term $\mathcal{E}_{1}$. This is proved in Lemma 9.5 and is done
by exploiting the rotational symmetry of the catenoid. The solutions then
satisfy the uniform estimate

$\left|\dot{\mathcal{h}}_{\tau,\delta,f}\right|\leq^{C\epsilon}\tau/r^{\gamma}+r^{1+\gamma}.$

The limit is then normalized and solves the same modified Poisson-Dirichlet
problem as $\dot{\mathcal{h}}^{\pm}_{\delta,f}$ and thus the two solutions
agree by Lemma 7.1. Thus the $\tau$-variations extend in continuously to
$\tau=0$.

##  4\. The mean curvature operator in weighted holder spaces

In this section we record Proposition 4.1, which states that the operator
$H^{\prime}_{\mathcal{F}}$ is defined and smooth on a ball of fixed radius
independent of $\tau$:

######  Proposition 4.1.

There are constants $\epsilon>0$ and $C>0$ such that: Given
$\tau\in(0,\epsilon)$, the mapping $H^{\prime}_{\mathcal{F}}$ is defined on
the ball $\mathcal{B}:=B_{\epsilon}(0:W^{2,\alpha,\gamma}_{\tau})$ and it
holds that

$\left\|H^{\prime}_{\mathcal{F}}:C^{2}\left(\mathcal{B},W^{0,\alpha,\gamma}_{\tau}\right)\right\|\leq
C_{0}.$

###### Proof.

Fix a precompact open set $\mathcal{O}_{0}\subset R^{3,\alpha}(\Omega_{0})$
such that the normalized localizations
$\left.\widetilde{\mathcal{F}}\right|_{s_{0}}$ of $\mathcal{F}$ belong to
$\mathcal{O}_{0}$ for all $s_{0}\in\mathbb{R}$. There are then constants
$\epsilon_{0}>0$ and $C_{0}>0$ such that the mapping $H_{-}(-):(F,u)\mapsto
H_{F}(u)$ is defined on $\mathcal{O}_{0}\times\mathcal{B}_{0}$ and satisfies
the bound:

(4.1)
$\displaystyle\left\|H_{-}(-):C^{2}\left(\mathcal{O}\times\mathcal{B}_{0},C^{0,\alpha}(\Omega_{0})\right)\right\|\leq
C,$

where above we have set
$\mathcal{B}_{0}=B_{\epsilon_{0}}(0:C^{2,\alpha}(\Omega_{0}))$. Pick $u\in
W^{2,\alpha,\gamma}_{\tau}$ and assume that $\|u\|\leq\epsilon$ for
$\epsilon>0$ to be determined. By (2.7) we have

$\displaystyle\left.\left.DH^{\prime}_{\mathcal{F}}\right|_{u}(\dot{u})\right|_{s_{0}}$
$\displaystyle=\mathcal{c}_{s_{0}}\left.DH_{\left.\tilde{\mathcal{F}}\right|_{s_{0}}}\right|_{\mathcal{c}_{s_{0}}\left.u\right|_{s_{0}}}\left(\left.\mathcal{c}_{s_{0}}\dot{u}\right|_{s_{0}}\right)$

Observe that

$\displaystyle\left\|\mathcal{c}_{s_{0}}\left.u\right|_{s_{0}}\right\|^{2,\alpha}\leq^{C}\left\|\left.u\right|_{s_{0}}\right\|^{2,\alpha}\leq
C\epsilon.$

Taking $\epsilon$ small and using (2.10) and Equation (4.1) gives

$\displaystyle\left\|\left.\left.DH^{\prime}_{\mathcal{F}}\right|_{u}(\dot{u})\right|_{s_{0}}\right\|^{0,\alpha}$
$\displaystyle\leq^{C}\left\|\mathcal{c}_{s_{0}}\left.\dot{u}\right|_{s_{0}}\right\|^{2,\alpha}$
$\displaystyle\leq^{C}\left\|\left.\dot{u}\right|_{s_{0}}\right\|^{2,\alpha}$
$\displaystyle\leq^{C}\left\|\left.\dot{u}\right|_{s_{0}}\right\|^{2,\alpha,\gamma}_{\tau}\omega_{\tau}^{\gamma}(s_{0}).$

Thus, taking the supremum over $s_{0}$ in $[-l_{\tau}+1/2,l_{\tau}-1/2]$ gives

$\left\|\left.DH^{\prime}_{\mathcal{F}}\right|_{u}(\dot{u})\right\|^{0,\alpha,\gamma}_{\tau}\leq^{C}\|\dot{u}\|^{2,\alpha,\gamma}_{\tau}$

Estimates for the remaining derivatives are similar and we omit a detailed
proof. ∎

##  5\. The stability operator in the spaces $W^{k,\alpha,\gamma}_{\tau}$

In this section we study the mapping properties of the operator
$L^{\prime}_{\mathcal{F}}$ as a map between the weighted holder spaces
$W^{k,\alpha,\gamma}_{\tau}$. The results we present here are standard, and
similar statements, along with methods of proof, can be found in [12] (c.f.
Proposition 21, [12]). The basic idea is to reduce the problem to a
classification of Jacobi fields on cylinders with various limitations on their
boundary conditions and growth rates. This is done below in Lemma 5.1.

######  Lemma 5.1.

The following statements hold:

  1. (1)

Let $u$ be a smooth solution to the equation

$L_{\mathcal{F}}^{{}^{\prime\prime}}u=\left(\Delta_{\Omega}+2\cosh^{-2}(s)\right)u=0$

on the domain $\Omega$ and assume that

$\left|u(x,s)\right|\leq^{C}\cosh^{1+\gamma}(s)$

for some $\gamma\in(0,1)$. Then $u$ is a geometric jacobi field.

  2. (2)

Let $u$ be a smooth solution to the equation

$\Delta_{\Omega}u=0$

on $\Omega$ and assume that

$\left|u(x,s)\right|\leq^{C}\cosh^{1+\gamma}(s)$

Then $u\equiv 0$.

  3. (3)

Let $u$ be a smooth solution to the equation

$\Delta_{\Omega}u=0,\quad\mathring{\mathcal{tr}}(u)=0.$

on $\Omega_{-}:=\mathbb{S}^{1}\times(-\infty,0]$ and assume that

$\left|u(x,s)\right|\leq^{C}e^{1+\gamma}(s)$

Then $u\equiv 0$.

.

###### Proof.

Let $u$ be as in statement (1), and set

(5.1) $\displaystyle u_{k}(s):=\int_{-\pi}^{\pi}u(x,s)e^{ikx}dx.$

Then $u_{k}$ satisfies the equation

(5.2) $\displaystyle
u_{k}^{\prime\prime}=\left(k^{2}-2\cosh^{-2}(s)\right)u_{k},$

and by assumption we have

(5.3) $\displaystyle|u_{k}(s)|\leq^{C}\cosh^{1+\gamma}(s).$

For $k\geq 2$ there is $K>0$ such that

$u^{\prime\prime}_{k}\geq(1+\gamma)^{2}u_{k}$

for $s>K$ and for any $\gamma\in(0,1)$. Write

$f=ae^{(1+\gamma)s}+be^{-(1+\gamma)s}$

for constants $a$ and $b$ to be determined. Clearly it holds that

$f^{\prime\prime}=(1+\gamma)^{2}f.$

Since the matrix

$\begin{pmatrix}e^{(1+\gamma)K}&e^{-(1+\gamma)K}\\\ \\\
(1+\gamma)e^{(1+\gamma)K}&-(1+\gamma)e^{(1+\gamma)K}\end{pmatrix}$

is non-singular, the values of $f$ and $f^{\prime}$ at $s=K$ can be prescribed
by choosing $a$ and $b$. In particular, with $r=1+\gamma$ we have

$\displaystyle a$
$\displaystyle=-\frac{1}{2r}\left(f(K)re^{rK}+f^{\prime}(K)e^{-rK}\right)$
$\displaystyle b$
$\displaystyle=\frac{1}{2r}\left(-re^{rK}f(K)+e^{rK}f^{\prime}(K)\right)$

Taking $f(K)=u(K)$, we can pick $f^{\prime}(K)$ sufficiently negative such
that $a\geq 0$ . The function $w=u-f$ then satisfies $w(0)=0$ and
$w^{\prime}(0)>0$ and the inequality

$w^{\prime\prime}\geq(1+\gamma)^{2}w.$

We claim that $w>0$ on the interval $[K,\infty)$. If not, there is a smallest
time $s_{0}$ such that $w(s_{0})=0$. This implies that $w$ has a positive
maximum on $[K,s_{0}]$ which contradicts the maximum principle. Thus, $w$ is
positive on $[K,\infty)$ and

$w(s)\geq\frac{a}{2}e^{rs}$

which contradicts (5.3). This is (1)

Now, let $u$ be as in Proposition 5.1 (2) and let $u_{k}$ be as in the proof
of Proposition 5.1 (1). Then in this case $u_{k}$ satisfies

$u^{\prime\prime}_{k}=k^{2}u_{k}$

and thus

(5.4) $\displaystyle u_{k}=ae^{ks}+be^{-ks}$

for some constants $a$ and $b$, and thus $a$ and $b$ must vanish for $k\geq
0$. This gives (2).

Now, let $u$ be as in Proposition 5.1 (3). Then, as in the previous case, we
have that $u_{k}$ is of the form (5.4). Since
$|u_{k}(x,s)|\leq^{C}e^{(1+\gamma)s}$, we must have that $b=0$ for all $k$ and
$a=0$ for $k=1,2$. Also, since $\mathring{\mathcal{tr}}(u)=0$, we must have
$a=0$ for $k\geq 2$. This proves (3). ∎

######  Lemma 5.2.

The operator $\Delta_{\Omega}+2\cosh^{-2}(s)$ has trivial dirichlet kernel on
$\Omega_{l}$ for $l\neq\sqrt{2}$

###### Proof.

Let $u$ be a smooth solution to the equation

$\Delta u+2\cosh^{-2}(s)u=0,\quad\mathcal{tr}(u)=0$

on $\Omega_{l}$. Let $u_{k}$ denote the $k^{th}$ mode of $u$ as in (5.1). Then
$u_{k}$ satisfies (5.2). For $k\geq 2$, it then follows from the fact that
$u(-l)=0$ that $u_{k}^{\prime\prime}(s)>0$ for $s>-\ell$ if $u_{k}$ does not
vanish, which is a contradiction. Thus $u_{k}\equiv 0$ for all $k\geq 2$. For
the $0$ mode $u_{0}$, we have

$u_{0}=a\tanh(s)+b\left(s\tanh(s)-1\right)$

for some constants $a$ and $b$. Since $\tanh(s)$ is even and $s\tanh(s)-1$ is
odd and since neither vanishes for $s\neq\sqrt{2}$, it follows that $a=b=0$.
The case of $u_{1}$ is similar. We have

$u_{1}=a\left(\cosh^{-1}(s)\right)+b\left(\sinh(s)+s\cosh^{-1}(s)\right)$

Since the first of these functions is odd and the second is even, and since
niether vanishes, it again follows that $a=b=0$. ∎

We are now ready to prove the invertibility results for the stability operator
in the weighted spaces.

######  Proposition 5.3.

There are constants $\epsilon>0$ and $L>0$ such that: Given
$\tau\in(0,\epsilon)$, the mapping
$u\mapsto(L^{\prime}_{\mathcal{F}}u,\mathring{\mathcal{tr}}(u))$ is an
isomorphism from $W^{2,\alpha,\gamma}_{\tau,n}$ onto
$\widehat{W}^{0,\alpha,\gamma}_{\tau}$ with invertibility constant $L$.

We also record here an invertibility theorem for source terms with vanishing
lower modes, which is needed in the proof of Proposition 9.1.

######  Proposition 5.4.

There is $\epsilon>0$ such that: Given $\tau\in(0,\epsilon)$,
$\gamma\in(-1,1)$ and a function $E\in C^{0,\alpha}(\mathcal{N}_{\tau}^{*})$
with $\overline{E}=0$, there is a unique function $u\in
C^{2,\alpha}_{0}(\mathcal{N}_{\tau})$ solving the $E$-Possion problem for the
operator $L^{\prime}_{\mathcal{F}}$ on $\mathcal{N}_{\tau}$. Moreover, there
is a uniform constant $C>0$ such that

$\left\|u\right\|^{2,\alpha,\gamma}_{1}\leq^{C}\left\|E\right\|^{0,\alpha,\gamma}_{1}$

We prove Propositions 5.3 and 5.4 together.

###### Proof of Propositions 5.3 and 5.4.

We claim first that there is a constant $C>0$ such that

$\left\|u\right\|^{2,\alpha,\gamma}_{\tau}\leq^{C}\left\|L^{\prime}_{\mathcal{F}}(u)\right\|^{0,\alpha,\gamma}_{\tau}+\|\mathring{\mathcal{tr}}(u)\|^{2,\alpha}$

for all $\tau$ sufficiently small. If not, there is a sequence
$\tau_{j}\rightarrow 0$ and $u_{j}\in W^{2,\alpha,\gamma}_{\tau_{j}}$ with
$\|u_{j}\|=1$ and
$\left\|L^{\prime}_{\mathcal{F}}(u_{j})\right\|+\|\mathring{\mathcal{tr}}(u_{j})\|\rightarrow
0$. Pick $s_{j}$ such that

$1=\|u_{j}\|=\omega_{\tau_{j}}^{-\gamma}(s_{j})\left\|\left.u_{j}\right|_{s_{j}}\right\|,$

and set

(5.5) $\displaystyle\tilde{u}_{j}:=\frac{u_{j}}{\omega_{\tau_{j}}(s_{j})}$

Then it holds that

$\displaystyle\left\|\left.L_{j}\tilde{u}_{j}\right|_{s_{0}}\right\|$
$\displaystyle\leq^{C}\left(\frac{\omega_{\tau_{j}}(s_{0}+s_{j})}{\omega_{\tau_{j}}(s_{j})}\right)^{\gamma}$
$\displaystyle\leq^{C}\cosh_{s_{j}}(s_{0}).$

where we have set

$L_{j}u=\left.L^{\prime}_{\mathcal{F}}\right|_{s_{j}}u:=\Delta
u+u+2\tanh(s+s_{j})\frac{\partial u}{\partial s}+2\cosh^{-2}(s+s_{j})u.$

We may assume without loss of generality that $s_{j}\geq 0$. Assume first that
$s_{j}$ is convergent to $s_{\infty}\in\mathbb{R}$. Then the sequence $u_{j}$
converges in $C^{2,\alpha/2}$ to a non-zero limiting function $u$ satisfying

$L^{\prime}_{\mathcal{F}}u=0$

on $\Omega$ and the estimate

$\left|u(x,s)\right|\leq\cosh^{\gamma}(s)$

Equivalently, setting $v=\cosh(s)u$, then $v$ satisfies

$L^{{}^{\prime\prime}}_{\mathcal{F}}v=0$

on $\Omega$ and the estimate

$\left|u(x,s)\right|\leq\cosh^{1+\gamma}(s)$

By Lemma 5.1 (1), the only such functions are the geometric jacobi fields.
However, since $u$ is the limit of normalized functions, it is also
normalized, which is a contradiction. Now assume that both $s_{j}$ and
$l_{\tau_{j}}-s_{j}$ are properly divergent. Then the sequence $\tilde{u}_{j}$
converges to limiting function $u$ satisfying

$\left.\mathcal{L}^{\prime}_{\mathcal{F}}\right|_{\infty}u=\Delta
u+u+2u_{s}=e^{-s}\Delta e^{s}(u)=0.$

on $\Omega$ and the growth estimate

(5.6) $\displaystyle\sup|u(x,s)|\leq^{C}e^{\gamma s}.$

Arguing as in the previous case we obtain a contradiction to Lemma 5.1 (2).
Finally if the sequence $l_{\tau_{j}}-s_{j}$ remains bounded, we obtain a non-
trivial limit function $u$ satifying

$\Delta u=0,\quad\mathring{\mathcal{tr}}(u)=0$

on the half cylinder $\Omega_{-}$ and the growth estimate (5.6), which is a
contradiction to Lemma 5.1 (3). This completes the proof of the a-priori
estimate for solutions. To prove existence, first observe that for
$l\neq\sqrt{2}$, using the fact that $L^{\prime}_{\mathcal{F}}$ and
$L^{\prime\prime}_{\mathcal{F}}$ are conjugate, there is by Lemma 5.2 a unique
solution $u_{0}\in C^{2,\alpha}(\Omega_{l})$ to the $(E,f)$-Poisson Dirichlet
problem for $L^{\prime}_{F}$ on $\Omega_{l}$:

$L^{\prime}_{\mathcal{F}}u_{0}=E,\quad\mathcal{tr}(u_{0})=f.$

Recalling the discussion in Section 2.3, there is a unique geometric jacobi
field $\mathcal{j}$ such that $u:=u_{0}+\mathcal{j}$ is normalized. Clearly
$u$ is normalized and solves the modified $(E,f)$-Dirichlet-Problem for
$L^{\prime}_{\mathcal{F}}$ on $\Omega_{l}$. Combing the existence with the
a-priori estimate completes the proof of Proposition 5.4.

The proof Proposition 5.4 follows similarly. The existence of a solution $u$
to the problem follows again from Lemma 5.2. It follows easily that $u$
inherits the orthogonality imposed on $E$. To see this, observe that the
Fourier expansion of $u$ along meridians satisfies a homogeneous second order
ODE with vanishing boundary conditions. We then have

$\displaystyle\int_{-\pi}^{\pi}L^{\prime}_{\mathcal{F}}$ $\displaystyle
u=\int_{-\pi}^{\pi}\left(u_{xx}+u_{ss}+2(\tanh(s)+\cosh^{-2}(s))u\right)$
$\displaystyle=\left(\int
udx\right)_{ss}+2(\tanh(s)+\cosh^{-2}(s))\left(\int_{-\pi}^{\pi}udx\right)$
$\displaystyle=L^{\prime}_{\mathcal{F}}\bar{u}$

where we have set $\bar{u}=\int_{-\pi}^{\pi}udx$. The orthogonality on $E$
then implies that $\overline{u}$ satifisfies the homogenous equation
$L^{\prime}_{\mathcal{F}}\overline{u}=0$ with vanishing boundary conditions.
By Lemma 5.2 it follows that $\overline{u}\equiv 0$. Thus, the modes $u_{0}$
and $u_{1}$ vanish. Now, as in the proof of of Proposition 5.3, assume that
there is a sequence $l_{j}\rightarrow\infty$ and functions $u_{j}$ on
$\Omega_{l_{j}}$ with

$\left\|u_{j}\right\|^{2,\alpha,\gamma}_{1}=1,\quad\left\|L^{\prime}_{F}u_{j}\right\|^{0,\alpha,\gamma}_{1}\rightarrow
0.$

As above, pick $s_{j}$ such that

$1=\left\|u_{j}\right\|^{2,\alpha,\gamma}_{1}=\cosh^{\gamma}(s_{j})\left\|\left.u_{j}\right|_{s_{j}}\right\|$

and define $\tilde{u}_{j}$ as in (5.5). We consider the same three
possibilities as before. In the case that $s_{j}$ is convergent, we obtain a
non-trivial limiting function $u$ satisfying

$\Delta_{\Omega}u+2\cosh^{-2}(s)u=0,\quad\left|u\right|\leq\cosh^{1+\gamma}(s_{0}).$

Thus, by Lemma 5.1 (1) is a geometric jacobi field. However, since $u$ is
orthogonal to lower modes along each meridian, this is a contradiction.
Assuming now that $l_{j}-s_{j}$ and $s_{j}$ are both divergent, we obtain as a
nontrivial limit a function $u$ satisfying

$\Delta_{\Omega}u=0,\quad\left|u(x,s)\right|\leq\cosh^{1+\gamma}(s)$

which again by Lemma 5.1 (2) is a contradiction. Finally, when $l_{j}-s_{j}$
is convergent, we obtain as a limit a function $u$ satisfying

$\Delta u=0,\quad\quad\left|u(x,s)\right|\leq\cosh^{1+\gamma}(s)$

on $\Omega_{-}$ and the boundary condition $u(x,0)=0$. $u$ is then a multiple
of the coordinate function $s$. However, this again contradicts the fact that
$u$ is orthogonal to lower modes along the meridians. This completes the proof
of Proposition 5.4. ∎

##  6\. Solutions to the non-linear Poisson-Dirichlet problem

As a direct consequence of Propositions 4.1, 5.3 and the Implicit Function
Theorem, we can solve the non-linear modified Poission-Dirichlet Problem for
$H_{\mathcal{F}}$ for small data.

######  Proposition 6.1.

There are constants $\epsilon>0$ and $C>0$ such that:

  1. (1)

Given $(E,f)\in\widehat{W}^{0,\alpha,\gamma}_{\tau}$ with
$\left\|(E,f)\right\|\leq\epsilon$, there is a unique function
$\mathcal{h}=\mathcal{h}^{\prime}_{\tau,E,f}\in W^{2,\alpha,\gamma}_{\tau,n}$
with

$\|\mathcal{h}^{\prime}_{\tau,E,f}\|^{2,\alpha,\gamma}_{\tau}\leq^{C}\|f\|+\|E\|$

and such that

$H^{\prime}_{\mathcal{F}}(\mathcal{h}^{\prime}_{E,f})=E,\quad\mathring{\mathcal{tr}}(\mathcal{h}^{\prime}_{E,f})=f.$

  2. (2)

The mapping $u_{\tau,\cdot}(E,f)\mapsto u_{\tau,(E,f)}$ is smooth from
$\mathcal{B}:=B_{\epsilon}\left(0,\widehat{W}^{0,\alpha,\gamma}_{\tau}\right)$
into $W^{2,\alpha,\gamma}_{\tau,n}$ and it holds that

$\left\|u_{\tau,\cdot}:C^{k}\left(\mathcal{B},W^{2,\alpha,\gamma}_{\tau,n}\right)\right\|\leq
C.$

###### Proof.

This is a direct consequence of the Implicit Function Theorem. ∎

The following Proposition reinterprets the solutions constructed in
Proposition 6.1 as solutions for the constant mean curvature Dirichlet problem
for $H_{\mathcal{F}}$.

######  Corollary 6.2.

There are constants $\epsilon>0$ and $C>0$ such that: Given
$\delta\in(-\epsilon,\epsilon)$, and $f$ with $\|f\|<\epsilon$, there is a
unique function $\mathcal{h}^{\prime}_{\tau,\delta,f}\in
W^{2,\alpha,\gamma}_{\tau}$ with

$H^{\prime}_{\mathcal{F}}(\mathcal{h}^{\prime}_{\tau,\delta,f})=\delta\cosh(s),\quad\mathring{\mathcal{tr}}(\mathcal{h}^{\prime}_{\tau,\delta,f})=f.$

and satisfying the estimate
$\|\mathcal{h}^{\prime}_{f,\delta,\tau}\|\leq^{C}\delta+\|f\|$.

###### Proof.

Given $\delta>0$ and $\tau>0$, set

$E_{\tau,\delta}=\tau\delta\cosh(s),\quad
l_{\tau}:=\text{Arcsosh}\left(\frac{1}{\tau}\right).$

The mapping $\delta\mapsto E_{\tau,\delta}=\delta\omega_{\tau}$ is linear and
uniformly bounded from $\mathbb{R}$ into $W^{0,\alpha,\gamma}_{\tau}$
independent of $\tau$. Thus, by Proposition 6.1, there is $\epsilon>0$ such
that
$\mathcal{h}^{\prime}_{\tau,\delta,f}:=\mathcal{h}^{\prime}_{\tau,E_{\tau,\delta},f}\in
W^{2,\alpha,\gamma}_{\tau}$ is defined for $\delta\in(-\epsilon,\epsilon)$ and
$\|f\|\leq\epsilon$. By Proposition 6.1 (2), the mapping
$\mathcal{h}^{\prime}_{\tau}:(\delta,f)\mapsto\mathcal{h}^{\prime}_{\tau,\delta,f}$
is smooth for $\tau$ fixed and uniformly bounded in $C^{k}$ independent of
$\tau$ in $(0,\epsilon).$ Moreover, by construction we have

$H_{\mathcal{F}}^{\prime}(\mathcal{h}^{\prime}_{\tau,\delta,f})=\tau\delta\cosh(s).$

∎

The functions
$\mathcal{h}_{\tau,\delta,f}:=\omega_{\tau}\mathcal{h}^{\prime}_{\tau,\delta,f}$
then satisfy

$H_{\mathcal{G}_{\tau}}(\mathcal{h}_{\tau,\delta,f})=\delta,\quad\mathring{\mathcal{tr}}=f.$

In the following section, we show that the solutions extend continuously to
$\tau=0$.

##  7\. $C^{0}$ extension to $\tau=0$

We will need the following lemma which establishes the uniqueness of the limit
solutions on the disk. We define solutions to the modified Poission-Dirichlet
problem on the disk as follows: If $\mathcal{Q}$ is a differential operator on
the disk $\mathcal{D}$, a function $u$ on $\mathcal{D}$ is a solution to the
modified $(E,f)$ Poisson-Dirichlet Problem for $\mathcal{Q}$ on $\mathcal{D}$
if

$\mathcal{Q}u=E,\quad\mathring{\mathcal{tr}}(u)=\mathring{f}$

and if

(7.1) $\displaystyle u(0)=\nabla u(0)=0$

A function $u$ on the unit disk satisfying the conditions (7.1) is said to be
_normalized_. When $\mathcal{Q}$ agrees with the Laplace operator
$\Delta_{\mathcal{D}}$, clearly the modified Poission Dirichlet problem is
uniquely solvable for $E\in C^{0,\alpha}(\mathcal{D})$ and $f\in
C^{2,\alpha}(\partial\mathcal{D})$. In the following, we let
$C_{n}^{k,\alpha}(\mathcal{D})$ denote the space of normalized class
$C^{k,\alpha}$ functions on $\mathcal{D}$ and we set
$\widehat{C}^{0,\alpha}(\mathcal{D}):=C^{0,\alpha}(\mathcal{D})\times\mathring{C}^{2,\alpha}(\mathbb{S}^{1})$.
Observe that here we are identifying $\partial\mathcal{D}=\mathbb{S}^{1}$.

######  Lemma 7.1.

The mapping $u\mapsto\left(\Delta u,\mathring{\mathcal{tr}}(u)\right)$ is an
isomorphism from $C^{2,\alpha}_{n}\left(\mathcal{D}\right)$ onto
$\widehat{C}^{0,\alpha}(\mathcal{D})$

###### Proof.

Since the disk is Dirichlet non-degenerate, there is a unique function
$u_{0}\in C^{2,\alpha}(\mathcal{D})$ with

$\Delta_{\mathcal{D}}u_{0}=E,\quad\mathcal{tr}(u_{0})=f.$

The function $\ell$ given by

$\ell(x,y)=u_{0}(0)+\left.\nabla u_{0}\right|_{0}\cdot(x,y)$

is harmonic and $\mathring{\mathcal{tr}}(u)=0$. The function $u=u_{0}-\ell$
then solves the modified $(E,f)$ Poission-Dirichlet problem on $\mathcal{D}$
for $\Delta_{\mathcal{D}}$. Clearly, $u$ is unique. This completes the proof.
∎

######  Corollary 7.2.

There are constants $\epsilon>0$ and $C>0$ such that: Given
$f\in\mathring{C}^{2,\alpha}(\mathbb{S}^{1})$ and $E\in
C^{0,\alpha}(\mathcal{D})$ with $\|f\|,\|E\|\leq\epsilon$, there is a unique
function $\mathcal{h}=\mathcal{h}_{f,E}\in C^{2,\alpha}(\mathcal{D})$ such
that

$H_{\mathcal{D}}(\mathcal{h})=E,\quad\mathring{\mathcal{tr}}(\mathcal{h})=f$

and such that $\mathcal{h}(0)=\left.\nabla\mathcal{h}\right|_{0}=0$.

###### Proof.

The mapping
$\widehat{H}_{\mathcal{D}}:u\mapsto(H_{\mathcal{D}}(u),\mathring{\mathcal{tr}}(u))$
is smooth and defined on a ball near the origin in
$C_{n}^{2,\alpha}(\mathcal{D})$ and maps into
$\widehat{C}^{0,\alpha}(\mathcal{D})$. By Lemma 7.1, its linearization at
$u=0$ is an isomorphism. It is then a diffeomorphism near zero, which is
equivalent to the statement of the corollary ∎

######  Corollary 7.3.

The functions $\mathcal{h}^{\pm}_{\tau,\delta,f}$ converge in $C^{2,\alpha/2}$
on $\mathcal{D}$ away from the origin to the function
$\mathcal{h}_{\delta,f^{+}}$ on $\mathcal{D}$, where $f^{+}$ denotes the
restriction of $f$ to $\partial_{+}\Omega_{l_{\tau}}$.

###### Proof.

By Proposition 6.2, the functions
$\mathcal{h}^{\pm}_{\tau,\delta,f}:\mathcal{N}^{\pm}_{\tau}\rightarrow\mathbb{R}$
are uniformly bounded in $C^{2,\alpha}$ and satisfy the bound

(7.2)
$\displaystyle\left|\mathcal{h}^{\pm}_{\tau,\delta,f}\right|\leq^{C}r^{1+\gamma},$

where $\gamma\in(0,1)$ and are locally bounded in $C^{2,\alpha}$. Taking a
sequence $\tau_{j}\rightarrow 0$, the solutions
$\mathcal{h}_{j}:=\mathcal{h}^{\pm}_{\tau_{j},\delta,f}$, after possibly
passing to a subsequence, converge on the disk $\mathcal{D}$ in
$C^{2,\alpha/2}$ away from the origin to a function $\mathcal{h}$ satisfying

$H_{\mathcal{D}}(\mathcal{h})=0,\quad\mathring{\mathcal{tr}}(\mathcal{h})=f.$

The uniform estimate (7.2) then implies that $u$ is normalized. By Corollary
7.2, we then have $\mathcal{h}=\mathcal{h}_{\delta,f^{+}}$. This completes the
proof. ∎

##  8\. Improved decay estimates for solutions

In order to establish uniform estimates for the $\tau$-derivative of the
solutions, we need optimal decay estimates for the solutions
$\mathcal{h}^{\prime}_{\tau,\delta,f}$, which a-priori lie in the spaces
$W^{k,\alpha,\gamma}_{\tau}$. By comparing these solutions with their limits
$\mathcal{h}^{\pm}_{\delta,f}$ on the disk, which extend smoothly to the
origin and vanish to first order, we obtain improved estimates for the decay.
For later applications, we will also need to separate out the dominant mode,
which is higher, from the rest of $\mathcal{h}_{\tau,\delta,f}$. This allows
improved estimate for the lower part of $\mathcal{h}_{\tau,\delta,f}$ which is
needed in the proof of Proposition 9.1 and Lemma 9.2.

######  Proposition 8.1.

It holds that

$\left\|\mathcal{h}^{\prime}_{\tau,\delta,f}\right\|^{2,\alpha,1}_{\tau}\leq^{C}\epsilon,\quad\left\|\overline{\mathcal{h}}^{\prime}_{\tau,\delta,f}\right\|^{2,\alpha,\gamma}_{\tau}\leq^{C}\epsilon\tau.$

###### Proof of Proposition 8.1.

Fix $f\in\mathring{C}^{2,\alpha}(\partial\mathcal{N}_{\tau})$ and
$\delta\in\mathbb{R}$ such that
$\mathcal{h}^{\prime}:=\mathcal{h}^{\prime}_{\tau,\delta,f}$ is defined, and
let $f_{\pm}(x)=f(x,\pm l_{\tau})$ denote the restriction of $f$ to the
boundary component $\partial_{\pm}\mathcal{N}_{\tau}$. By Corollary 7.2, there
are constants $\epsilon>0$ and $C>0$ such that the function
$\mathcal{h}_{+}:=\mathcal{h}_{\delta,f_{+}}$ is defined and satisfies

$\|\mathcal{h}_{+}\|^{2,\alpha}\leq^{C}\epsilon$

Since $\mathcal{h}_{+}$ vanishes to first order on the disk, the taylor
expansion of $\mathcal{h}_{+}$ at the origin gives that

$\mathcal{h}_{+}=\mathcal{f}_{+}+\mathcal{g}_{+}$

where we have set

$\mathcal{f}_{+}:=ax^{2}+by^{2}+cxy$

for constants $a$, $b$ and $c$ satisfying $|a|,|b|,|c|\leq^{C}\epsilon$ and
where $\mathcal{g}_{+}$ satisfies the estimate:

(8.1) $\displaystyle\sup\left|\mathcal{g}_{+}(r,\theta)\right|\leq^{C}\epsilon
r^{3},$

where $(r,\theta)$ denotes polar coordinates on $\mathcal{D}$. Define
$\mathcal{h}^{*}_{+}$ on $\mathcal{N}^{*}_{\tau}$ as follows: Fix a smooth
monotonic function $\psi:\mathbb{R}\rightarrow\mathbb{R}$ such that
$\psi(s)=0$ for $s\leq 10$ and $\psi(s)=1$ for $s\geq 11$ and set

$\displaystyle\mathcal{h}^{*}_{+}(x,s)$
$\displaystyle=\frac{\psi(s)\left(\mathcal{h}_{\delta,f_{+}}\circ\overline{\mathcal{G}}_{\tau}(x,s)\right)}{\omega}$
$\displaystyle=\frac{\psi(s)\left(\mathcal{f}_{+}\circ\overline{\mathcal{G}}_{\tau}(x,s)\right)}{\omega}+\frac{\psi(s)\left(\mathcal{g}_{+}\circ\overline{\mathcal{G}}_{\tau}(x,s)\right)}{\omega}$
$\displaystyle=:\mathcal{f}^{*}_{+}+\mathcal{g}^{*}_{+}$

where we have set

$\overline{\mathcal{G}}_{\tau}(x,s)=\tau\cosh(s)e_{r}(x),\quad\omega=\omega_{\tau}=\tau\cosh(s).$

Observe that $\overline{\mathcal{G}}_{\tau}$ takes the domain
$\mathcal{N}^{*,+}_{\tau}:=\Omega_{0,l_{\tau}}$ into $\mathcal{D}$ and that it
is a diffeomorphism for $s>0$. Since we have

$\left(\overline{\mathcal{G}}_{\tau}\right)^{*}(r)=r\circ\overline{\mathcal{G}}_{\tau}=\tau\cosh(s)=\omega,$

the function $\mathcal{f}_{+}^{*}$ is then explicitly given by

(8.2)
$\displaystyle\mathcal{f}_{+}^{*}=\omega\psi\left(a\cos^{2}(x)+g\sin^{2}(x)+2\cos(x)\sin(x)\right).$

Additionally, we have the estimate

(8.3) $\displaystyle\sup\left|\mathcal{g}_{+}\right|\leq^{C}\omega^{2}.$

We can similarly define a function $\mathcal{h}^{*}_{-}$ by pulling back
$\mathcal{h}_{-\delta,f_{-}}$ from the disk to the bottom half of the
catenoidal neck and we set

$\mathcal{h}^{*}:=\mathcal{h}_{+}^{*}+\mathcal{h}^{*}_{-}.$

We then have that $\mathcal{h}^{*}$ is defined on $\mathcal{N}_{\tau}^{*}$ and
it holds that

$\mathring{\mathcal{tr}}(\mathcal{h}^{*})=f=\mathring{\mathcal{tr}}(\mathcal{h}_{\tau,\delta,f}).$

We will estimate the difference

(8.4) $\displaystyle\mathcal{E}:=\
H^{\prime}_{F}\left(\mathcal{h}^{*}\right)-\delta\tau\cosh(s)$

We consider only that case that $s_{0}\in\mathbb{R}$ is positive, as the case
$s_{0}\leq 0$ is treated identically. The estimate then follows from
considering two cases separately: In the first, we assume $s_{0}$ to be near
the support of $\nabla\psi$, in which case the function $\mathcal{h}^{*}$ is
sufficiently small and the structure of the error term is unimportant. In the
second case, $s_{0}$ is far from the support of $\nabla\phi$ and we use the
fact that here $\mathcal{h}^{*}$ is exactly the pullback of the corresponding
solution on the disk. In this region, $\mathcal{h}^{*}$ is no longer small,
but the distance from the catenoidal neck to the flat disk is, which we
exploit. We consider now the first case. For $s_{0}$ belonging to the region
$[8,13]$, we have

$\left\|\left.\mathcal{h}^{*}\right|_{s_{0}}\right\|^{2,\alpha}\leq^{C}\omega_{0}\leq^{C}\epsilon\tau$

and thus trivially

$\left\|\left.L_{\mathcal{F}}^{\prime}\left(\mathcal{h}^{*}\right)\right|_{s_{0}}\right\|^{0,\alpha}\leq^{C}\epsilon\tau.$

For the second case, we localize at $s_{0}\geq 12$–that is, away from the
support of $\nabla\psi$. Observe that

(8.5)
$\displaystyle\left\|\mathcal{h}^{*}\right\|^{2,\alpha,1}_{\tau}\leq^{C}\epsilon.$

Given $s_{0}\geq 12$, we have

$\displaystyle\left.\widetilde{\mathcal{F}}\right|_{s_{0}}(x,s)$
$\displaystyle=\mathcal{c}_{s_{0}}(s)e_{r}(x)+\frac{s}{\cosh(s_{0})}e_{z}$
$\displaystyle=:\left.\widetilde{\overline{\mathcal{G}}}\right|_{s_{0}}(x,s)+\frac{s}{\cosh(s_{0})}e_{z}$

where $\left.\widetilde{\overline{\mathcal{G}}}\right|_{s_{0}}$ is defined
implicitly above and thus

(8.6)
$\displaystyle\left\|\left.\widetilde{\mathcal{F}}\right|_{s_{0}}-\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}}\right\|^{2,\alpha}\leq^{C}\cosh^{-1}(s_{0}).$

Since

$\cosh(s)H_{\overline{\mathcal{G}}_{\tau}}(\tau\cosh(s)\mathcal{h}^{*})=\delta\cosh(s),$

we have

(8.7)
$\displaystyle\left.\mathcal{E}\right|_{s_{0}}:=\mathcal{c}_{s_{0}}\left\\{H_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}-H_{\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}}}\right\\}\left(\mathcal{c}_{s_{0}}\left.\mathcal{h}^{*}\right|_{s_{0}}\right)$

In order to get the estimates we want we will have to decompose the error term
into its lower and higher parts. Given immersions $F,G\in
R^{3,\alpha}(\Omega_{0})$ and a function $u\in C^{2,\alpha}(D)$, set

$\Phi(F,G,u):=H_{F}(u)-H_{G}(u)$

so that above we have

$\left.\mathcal{E}\right|_{s_{0}}=\Phi\left(\left.\widetilde{\mathcal{F}}\right|_{s_{0}},\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}},\left.\mathcal{h}^{*}\right|_{s_{0}}\right)$

Then $\Phi$ is smooth as a map into $C^{0,\alpha}(\Omega_{0})$ and defined
near
$(F,G,u)=\left(\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}},\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}},0\right)$
and vanishes whenever $F=G$. We then have

(8.8) $\displaystyle\left\|\left.\frac{\partial^{k}\Phi}{\partial
u^{k}}\right|_{(F,G,u)}\right\|\leq^{C}\left\|F-G\right\|^{3,\alpha}\|u\|^{2,\alpha}$

Since $\left.\widetilde{\mathcal{F}}\right|_{s_{0}}$ and
$\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}}$ are both
minimal we have
$\Phi\left(\left.\widetilde{\mathcal{F}}\right|_{s_{0}},\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}},0\right)=0$
and thus

$\displaystyle\left.\mathcal{E}\right|_{s_{0}}$
$\displaystyle=\left(\left.\frac{\partial\Phi}{\partial
u}\right|_{\left(\left.\widetilde{\mathcal{F}}\right|_{s_{0}},\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}},0\right)}\right)\left(\mathcal{h}^{*}\right)+R$
$\displaystyle=\mathcal{E}^{\prime}+R$

where $R$ is the first order remainder term. Directly from (8.6) and (8.8) we
get:

$\left\|R\right\|^{0,\alpha}\leq^{C}\left(\cosh^{-1}(s_{0})\right)\left(\left\|\mathcal{h}^{*}\right\|^{2,\alpha}\right)^{2}\leq^{C\epsilon}\tau\omega.$

and

$\left\|\mathcal{E}^{\prime}\right\|^{0,\alpha}\leq^{C}\left(\cosh^{-1}(s_{0})\right)\left\|\mathcal{h}^{*}\right\|^{2,\alpha}\leq^{C\epsilon}\tau.$

We now estimate $\overline{\mathcal{E}^{\prime}}$. To do this, write

$\left(\left.\frac{\partial\Phi}{\partial
u}\right|_{\left(\left.\widetilde{\mathcal{F}}\right|_{s_{0}},\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}},0\right)}\right)\left(\mathcal{h}^{*}\right)=\left.DH_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}\right|_{0}\left(\mathcal{h}^{*}\right)-\left.DH_{\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}}}\right|_{0}\left(\mathcal{h}^{*}\right)$

Since $\left.\widetilde{\mathcal{F}}\right|_{s_{0}}$ and
$\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}}$ are
rotationally symmetric, the linearizations
$\left.DH_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}\right|_{0}$ and
$\left.DH_{\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}}}\right|_{0}$
are then mode separating (recall Section 1.1.3) and thus

$\overline{\left(\left.\frac{\partial\Phi}{\partial
u}\right|_{\left(\left.\widetilde{\mathcal{F}}\right|_{s_{0}},\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}},0\right)}\right)\left(\mathcal{h}^{*}\right)}=\left(\left.\frac{\partial\Phi}{\partial
u}\right|_{\left(\left.\widetilde{\mathcal{F}}\right|_{s_{0}},\left.\widetilde{\overline{\mathcal{G}}}_{\tau}\right|_{s_{0}},0\right)}\right)\left(\overline{\mathcal{h}^{*}}\right).$

This then gives:

$\left\|\overline{\mathcal{E}^{\prime}}\right\|^{0,\alpha}\leq^{C}\left(\cosh^{-1}(s_{0})\right)\left(\|\overline{\mathcal{h}^{*}}\|^{2,\alpha}\right)\leq^{C}\tau\omega.$

Thus we can split $\mathcal{E}$ into the sum of the higher and lower parts
$\mathcal{E}=\overline{\mathcal{E}}+\mathring{\mathcal{E}}$ where

$\left\|\mathring{\mathcal{E}}\right\|^{0,\alpha,0}_{1}\leq^{C}\tau,\quad\left\|\overline{\mathcal{E}}\right\|^{0,\alpha,1}_{\tau}\leq^{C}\tau$

Observe that

$\displaystyle
H^{\prime}_{\mathcal{F}}(\mathcal{h}^{\prime})-H^{\prime}_{\mathcal{F}}(\mathcal{h}^{*})$
$\displaystyle=\int_{0}^{1}\left.L^{\prime}_{\mathcal{F}}\right|_{\mathcal{h}(t)}\left(\mathcal{h}^{\prime}-\mathcal{h}^{*}\right)dt$
$\displaystyle=:\mathcal{L}\left(\mathcal{h}^{\prime}-\mathcal{h}^{*}\right)$

where above we have set
$\mathcal{h}(t)=(1-t)\mathcal{h}^{\prime}+t\mathcal{h}^{*}$ and where the
operator $\mathcal{L}$ is defined implicitly above. Combining with (8.7) we
have

$\mathcal{L}\left(\mathcal{h}^{\prime}-\mathcal{h}^{*}\right)=\mathcal{E}.$

By Proposition 5.4, there is a unique function $u$ such that

$L_{F}^{\prime}(u)=\mathring{\mathcal{E}},\quad\mathcal{tr}(u)=0$

and satisfying the estimate
$\left\|u\right\|^{2,\alpha,0}_{1}\leq^{C}\epsilon\tau$. Then

$\displaystyle\mathcal{L}\left(\mathcal{h}^{\prime}-\mathcal{h}^{*}-u\right)$
$\displaystyle=\overline{\mathcal{E}}+\left(\mathcal{L}-L^{\prime}_{\mathcal{F}}\right)(u).$

We then have

$\displaystyle\left\|\left.\left(\mathcal{L}-L^{\prime}_{\mathcal{F}}\right)(u)\right|_{s_{0}}\right\|$
$\displaystyle\leq^{C}\sup\left\|\left.\mathcal{h}(t)\right|_{s_{0}}\right\|\left\|\left.u\right|_{s_{0}}\right\|$
$\displaystyle\leq^{C}\epsilon\tau\omega^{\gamma}.$

and thus

$\left\|\mathcal{L}\left(\mathcal{h}^{\prime}-\mathcal{h}^{*}-u\right)\right\|^{0,\alpha,\gamma}_{\tau}\leq^{C}\epsilon\tau.$

The operator $\mathcal{L}$ is a easily seen to be perturbation of
$L_{F}^{\prime}$ in the weighted spaces $W^{k,\alpha,\gamma}_{\tau}$, and thus
by Proposition 5.3, there is a unique $v\in W^{2,\alpha,\gamma}_{\tau,n}$ such
that

$\mathcal{L}\left(\mathcal{h}^{\prime}-\mathcal{h}^{*}-u-v\right)=0,\quad\mathring{\mathcal{tr}}(v)=0,$

and such that $\left\|v\right\|^{2,\alpha,\gamma}_{\tau}\leq^{C}\tau$. In
particular, this gives the local estimate

(8.9)
$\displaystyle\left\|\left.\left(\mathcal{h}^{\prime}-\mathcal{h}^{*}\right)\right|_{s_{0}}\right\|$
$\displaystyle\leq^{C\epsilon}\tau+\tau\omega_{0}^{\gamma}$
$\displaystyle\leq^{C\epsilon}\tau$

We then have

$\displaystyle\left\|\left.\mathcal{h}^{\prime}\right|_{s_{0}}\right\|$
$\displaystyle\leq\left\|\left.\mathcal{h}^{*}\right|_{s_{0}}\right\|+\left\|\left.\mathcal{h}^{*}\right|_{s_{0}}-\left.\mathcal{h}^{\prime}\right|_{s_{0}}\right\|$
$\displaystyle\leq^{C\epsilon}\omega_{0}+\tau$
$\displaystyle\leq^{C\epsilon}\omega_{0}.$

Taking the supremum over $s_{0}$ gives

$\|\mathcal{h}^{{}^{\prime}}\|^{2,\alpha,\gamma}_{1}\leq^{C}\epsilon.$

Finally, the second estimate follow from the fact that $\overline{u}=0$ and
thus

(8.10)
$\displaystyle\left\|\overline{\mathcal{h}^{\prime}}-\overline{\mathcal{h}^{*}}\right\|^{2,\alpha,\gamma}_{\tau}=\left\|v\right\|^{2,\alpha,\gamma}_{\tau}\leq^{C}\tau.$

This completes the proof. ∎

##  9\. $C^{1}$-extension of the Dirichlet Solutions to scale zero

In this section, we prove that the $\tau$-derivative of the solutions
$\mathcal{h}_{\tau,\delta,f}$ extend continuously to $\tau=0$. Recall that
$\mathcal{h}_{\tau,\delta,f}$ solves the modified $(\delta,f)$-Poisson
Dirichlet problem for $\mathcal{G}_{\tau}$ on the domain
$\mathcal{N}_{\tau}^{*}=\Omega_{l_{\tau}}$, so:

$H_{\mathcal{G}_{\tau}}\left(\mathcal{h}_{\tau,\delta,f}\right)=\delta,\quad\mathring{\mathcal{tr}}(\mathcal{h}_{\tau,\delta,f})=f.$

######  Proposition 9.1.

Let $\dot{\mathcal{h}}_{\tau_{0},\delta,f}^{\pm}$ denote the restrictions of
$\dot{\mathcal{h}}_{\tau_{0},\delta,f}$ to $\mathcal{N}^{\pm}_{\tau_{0}}$.
Then the functions
$\dot{\mathcal{h}}^{\pm}=\dot{\mathcal{h}}^{\pm}_{\tau_{0}}$ converge in
$C^{2,\alpha}$ away from the origin to $\dot{\mathcal{h}}^{+}_{\delta,f}$

The proof of Proposition 9.1 is long and we thus split it apart into the
following steps. First in Lemma 9.2, we show that the Gateaux derivative
$\dot{\mathcal{h}^{\pm}_{\delta,f}}$ at $\tau=0$ is well-defined in
$C^{2,\alpha}$ and solves a modified Poission-Dirichlet problem on the disk
with vanishing Dirichlet data. Next, in Lemma 9.4, we show that for positive
$\tau$, the derivatives $\dot{\mathcal{h}}_{\tau,\delta,f}$ solves a modified
Poission-Dirichlet problem on $\mathcal{N}_{\tau}$ with vanishing Dirichlet
data. The Poisson data of problem splits into two parts $\mathcal{E}_{1}$ and
$\mathcal{E}_{2}$, both bounded in different weighted holder spaces. The
first, $\mathcal{E}_{1}$, is large but decaying, and we can apply the
invertibility result Proposition 5.4 to obtain a normalized solution to the
Poisson problem for the higher part $\mathring{\mathcal{E}}_{1}$ of
$\mathcal{E}_{1}$. The term $\mathcal{E}_{2}$ is small but growing and can be
solve for using Proposition 5.3. Solving for the higher part
$\overline{\mathcal{E}}_{1}$ of $\mathcal{E}_{1}$ can then be done using
separation of variables. This then allows us to show, in Lemma 9.6, that the
functions $\dot{\mathcal{h}}_{\tau,\delta,f}$ are uniformly bounded in
$C^{2,\alpha}$ away from the origin and thus the restrictions
$\dot{\mathcal{h}}^{\pm}_{\tau,\delta,f}$ converge to a limit on the disk. The
limit then satisfies the same Poisson problem as
$\dot{\mathcal{h}}^{\pm}_{\delta,f}$, with boundary values agreeing up to
lower modes. In order to show that the two functions agree, we need to know
that that the limit is normalized. To do this, in Lemma 9.5 we show that
higher part $\overline{\mathcal{E}}_{1}$ satisfies an improved bound relative
to the whole term.

######  Lemma 9.2.

The function $\dot{\mathcal{h}}^{\pm}_{\delta,f}$ is well-defined away from
the origin on $\mathcal{D}$ and satisfies

$\left.DH_{\mathcal{D}}\right|_{\mathcal{h}^{+}_{\delta,f}}\left(\dot{\mathcal{h}}^{\pm}_{\delta,f}\right)=\pm\left.DH\right|_{\mathcal{N}^{+}_{\delta,f}}\left(\log(r)e_{z}-\mathcal{h}_{\delta,f}^{\pm}\frac{e_{r}}{r}\right),\quad\mathring{\mathcal{tr}}\left(\dot{\mathcal{h}}^{\pm}_{\delta,f}\right)=0$

and the estimate

$\sup\left|\dot{\mathcal{h}}^{\pm}_{\delta,f}(r,\theta)\right|\leq^{C\epsilon}r^{1+\gamma}$

######  Remark 9.3.

It is not immediately obvious that the terms on the right hand side of the
equation satisfied by $\dot{\mathcal{h}}^{\pm}_{\delta,f}$ are uniformly
bounded over the whole disk. It is possible to compute directly that it is,
bounded, using the uniform bound $|\mathcal{h}|\leq^{c}r^{2}$.

###### Proof.

Fix $\delta>0$. Then for $\tau<\delta$, $\mathcal{N}^{+}_{\tau}\cap K$ is a
normal graph over the domain $\mathcal{D}^{\prime}:=\mathcal{D}\setminus
B_{\delta}(0)$. and thus the mapping
$\phi_{0}^{\tau,+}:\mathcal{D}^{\prime}\rightarrow\mathcal{N}^{+}_{\tau}$ is
defined. Recalling the notation used in the proof of Proposition 8.1 we have

$\displaystyle\mathcal{h}^{\prime}$
$\displaystyle:=\frac{\mathcal{h}_{0,\delta,f}^{\tau}\circ\overline{\mathcal{G}}_{\tau}}{\omega}$
$\displaystyle=\frac{\mathcal{h}_{\tau,\delta,f}\circ\mathcal{G}_{\tau}}{\omega}$

and

$\displaystyle\mathcal{h}^{*}:=\frac{\mathcal{h}_{\delta,f}\circ\overline{\mathcal{G}}_{\tau}}{\omega}.$

Recalling the estimate (8.9) we have

$\left\|\left(\left.\mathcal{h}^{\prime}-\mathcal{h}^{*}\right)\right|_{s_{0}}\right\|^{2,\alpha}\leq^{C\epsilon}\tau$

Thus, as functions on $\mathcal{D}$ we have the uniform bound

$\left|\left.\left(\mathcal{h}^{\tau,+}_{0,\delta,f}-\mathcal{h}^{+}_{\delta,f}\right)\right|_{(r,\theta)}\right|\leq^{C\epsilon}\tau$

as well as local uniform bounds in $C^{2,\alpha}$. Thus, the difference
quotients
$\frac{\mathcal{h}^{\tau,+}_{0,\delta,f}-\mathcal{h}_{\delta,f+}}{\tau}$ and
converge locally in $C^{2,\alpha/2}$ as $\tau\rightarrow 0$ to a limiting
function $\dot{\mathcal{h}}^{+}_{\delta,f}$ satisfying the estimate:

$\left\|\dot{\mathcal{h}}^{+}_{\delta,f}\right\|\leq^{C\epsilon}r^{1+\gamma}$

Moreover since $\mathcal{N}_{\tau}^{+}$ is a smooth family with variation
field $\dot{\mathcal{N}}^{+}$ at $\tau=0$ given by $-\log(r)e_{z}$, the
variation in the unit normal at $\tau=0$ is
$\dot{N}_{\mathcal{N}}=\frac{1}{r}e_{r}$. We then have

$\displaystyle 0$
$\displaystyle=\frac{H(\mathcal{N}^{+}_{\tau,\delta,f})-H(\mathcal{D}^{+}_{\delta,f})}{\tau}$
$\displaystyle=\int_{0}^{1}\left.DH\right|_{\mathcal{N}(t)}\left(\frac{\mathcal{N}^{+}_{\tau,\delta,f}-\mathcal{D}^{+}_{\delta,f}}{\tau}\right)$
$\displaystyle=\int_{0}^{1}\left.DH\right|_{\mathcal{N}(t)}\left(\frac{\mathcal{N}_{\tau}-\mathcal{D}}{\tau}+\left(\frac{\mathcal{h}^{+}_{\tau,\delta,f}-\mathcal{h}^{+}_{\delta,f}}{\tau}\right)e_{z}+\mathcal{h}^{+}_{\tau,\delta,f}\frac{N_{\mathcal{N}_{\tau}}-e_{z}}{\tau}\right)$

where we have set
$\mathcal{N}(t)=(1-t)\mathcal{D}_{\delta,f}+t\mathcal{N}_{\tau,\delta,f}$. As
$\tau=0$ we have
$\frac{\mathcal{N}_{\tau}-\mathcal{D}}{\tau}\rightarrow-\log(r)e_{z}$,
$\left(\frac{\mathcal{h}^{+}_{\tau,\delta,f}-\mathcal{h}^{+}_{\delta,f}}{\tau}\right)\rightarrow\dot{\mathcal{h}}^{+}_{\delta,f}$
and $\frac{N_{\mathcal{N}_{\tau}}-e_{z}}{\tau}\rightarrow\frac{1}{r}e_{r}$,
and the operator $\int_{0}^{1}\left.DH\right|_{\mathcal{N}}(t)dt$ converges to
$DH_{\mathcal{D}}(-)$. Thus, in the limit as $\tau=0$, we have

$\left.DH_{\mathcal{D}}\right|_{\mathcal{h}^{+}_{\delta,f}}(\dot{\mathcal{h}}^{+}_{\delta,f})+\left.DH_{\mathcal{D}}\right|_{\mathcal{h}^{+}_{\delta,f}}\left(-\log(r)e_{z}+\mathcal{h}^{+}_{\delta,f}\frac{1}{r}e_{r}\right)=0,\quad\mathring{\mathcal{tr}}(\dot{\mathcal{h}}^{+}_{\delta,f})=0.$

Finally, we observe that by (8.10), the higher part of
$\dot{\mathcal{h}}^{+}_{\delta,f}$ satisfies the improved bound

$\left|\overline{\dot{\mathcal{h}}^{+}_{\delta,f}}\right|\leq^{C\epsilon}r^{1+\gamma}$

The function $\dot{\mathcal{h}}^{+}_{\delta,f}$ then extends to the disk in
$C^{2}$ and the higher part satisfies
$\mathring{\mathcal{h}}(0)=\nabla\mathring{\mathcal{h}}(0)=0$ and thus

$\left|\mathring{\mathcal{h}}(r,\theta)\right|\leq^{C\epsilon}r^{2}.$

It then follows that:

$\left|\dot{\mathcal{h}}^{+}_{\delta,f}\right|\leq^{C\epsilon}r^{1+\gamma}.$

This completes the proof. ∎

######  Lemma 9.4.

It holds that

$\left.L^{\prime}_{\mathcal{F}}\right|_{\mathcal{h}^{\prime}}\left(\frac{\dot{\mathcal{h}}}{\omega}\right)=\mathcal{E}_{1}+\mathcal{E}_{2}$

where

$\left\|\mathcal{E}_{2}\right\|^{0,\alpha,\gamma}\leq^{C}\epsilon$

and where

$\mathcal{E}_{1}:=\left(\dot{\lambda}+1\right)\left.D^{2}H^{\prime}\right|_{\mathcal{F}}\left(\frac{\xi}{\tau_{0}},\mathcal{h}^{\prime}_{\tau,\delta,f}\right)$

and where above $\xi=s\tanh(s)-1$.

###### Proof.

Set

$\mathcal{G}=\mathcal{G}_{\tau_{0},\delta,f}^{\tau}=\mathcal{G}_{\tau,\delta,f}\circ\phi_{\tau_{0}}^{\tau},\quad\mathcal{G}_{0}:=\mathcal{G}_{\tau_{0},0,0}^{\tau},\quad\mathcal{h}:=\mathcal{h}_{\tau_{0},\delta,f}^{\tau},\quad\mathcal{h}^{\prime}=\mathcal{h}/\omega,$

and for $s_{0}\in\mathbb{R}$ and with $\omega_{0}=\omega_{\tau}(s_{0})$ set

$\left.\tilde{\mathcal{G}}\right|_{s_{0}}:=\left.\widetilde{\mathcal{F}}\right|_{s_{0}}+\mathcal{c}_{s_{0}}\mathcal{h}^{\prime}\left.N_{\mathcal{F}}\right|_{s_{0}}.$

When $\tau=\tau_{0}$ we have $\mathcal{G}=\mathcal{G}_{\tau_{0},\delta,f}$,
$\mathcal{h}=\mathcal{h}_{\tau_{0},\delta,f}$ and
$\mathcal{G}_{0}=\mathcal{G}_{\tau_{0}}=\tau_{0}\mathcal{F}$. We also have the
estimate

$\left\|\left.\widetilde{\mathcal{G}}\right|_{s_{0}}-\left.\tilde{\mathcal{F}}\right|_{s_{0}}\right\|^{2,\alpha}\leq^{C}\left\|\left.\mathcal{h}^{\prime}\right|_{s_{0}}\right\|\leq^{C}\omega_{0}.$

Observe that by construction we have

$H\left(\mathcal{G}\right)=\delta.$

Differentiating gives

$\displaystyle\left.DH\right|_{\mathcal{G}}\left(\dot{\mathcal{G}}\right)=\frac{1}{\tau_{0}}\left.DH\right|_{\mathcal{G}/\tau_{0}}\left(\dot{\mathcal{G}}/\tau_{0}\right)=0,$

where above
$\dot{\mathcal{G}}:=\left.\frac{d}{d\tau}\mathcal{G}\right|_{\tau=\tau_{0}}$.
From (2.6) we have

$\dot{\mathcal{G}}=\left(\dot{\lambda}\tau+1\right)\xi
N_{\mathcal{G}}+\mathcal{h}\dot{N_{\mathcal{G}}}+\dot{\mathcal{h}}N_{\mathcal{G}_{0}}-\dot{\lambda}\tau\mathcal{F}$

At $\tau=\tau_{0}$ we have
$\mathcal{G}/\tau_{0}=\mathcal{F}+\frac{\mathcal{h}}{\tau_{0}}N_{\mathcal{F}}$.
This then gives

$\displaystyle\left.L^{\prime}_{\mathcal{F}}\right|_{\mathcal{h}^{\prime}}\left(\frac{\dot{\mathcal{h}}}{\omega}\right)$
$\displaystyle=\cosh(s)\left.L_{\mathcal{F}}\right|_{\frac{\mathcal{h}}{\tau_{0}}}\left(\frac{\dot{\mathcal{h}}}{\tau_{0}}\right)$
$\displaystyle=\cosh(s)\left.DH\right|_{\mathcal{G}/\tau_{0}}\left(\frac{\dot{\mathcal{h}}}{\tau_{0}}N_{\mathcal{F}}\right).$

We thus have

$\displaystyle\left.L^{\prime}_{\mathcal{F}}\right|_{\mathcal{h}^{\prime}}\left(\frac{\dot{\mathcal{h}}}{\omega}\right)$
$\displaystyle=-\cosh(s)\dot{\lambda}\left.DH\right|_{\mathcal{G}/\tau_{0}}\left(\mathcal{F}\right)-\left(\tau\dot{\lambda}+1\right)\cosh(s)\left.DH\right|_{\mathcal{G}/\tau_{0}}\left(\frac{\xi}{\tau_{0}}N\right)$
$\displaystyle\quad-\cosh(s)\left.DH\right|_{\mathcal{G}/\tau_{0}}\left(\frac{\mathcal{h}}{\tau_{0}}\dot{N}\right)$
$\displaystyle=T_{1}+T_{2}+T_{3}.$

We will estimate the terms $T_{1}$, $T_{2}$, and $T_{3}$ above separately.
First, observe that

$\left.DH\right|_{\mathcal{F}}(\xi
N_{\mathcal{F}})=\left.DH_{\mathcal{F}}\right|_{0}\left(\xi\right)=0.$

This then gives

$\displaystyle\cosh(s)\left.DH\right|_{\mathcal{G}/\tau_{0}}\left(\frac{\xi}{\tau_{0}}N\right)$
$\displaystyle=\cosh(s)\left.D^{2}H\right|_{\mathcal{F}}\left(\frac{\xi}{\tau_{0}}N,\frac{\mathcal{h}}{\tau_{0}}N\right)+R$
$\displaystyle:=\mathcal{E}_{1}+R,$

where $\mathcal{E}_{1}$ and $R$ are defined implicitly above. By definition we
have

$\displaystyle\mathcal{E}_{1}$
$\displaystyle=\cosh(s)\left.D^{2}H\right|_{\mathcal{F}}\left(\frac{\xi}{\tau_{0}}N,\frac{\mathcal{h}}{\tau_{0}}N\right)$
$\displaystyle=D^{2}H^{\prime}_{\mathcal{F}}\left(\frac{\xi}{\omega},\mathcal{h}^{\prime}\right).$

With

$\mathcal{E}_{2}=T_{1}+T_{3}+R,$

we then have

$\left.L^{\prime}_{\mathcal{F}}\right|_{\mathcal{h}^{\prime}}\left(\frac{\dot{\mathcal{h}}}{\omega}\right)=\mathcal{E}_{1}+\mathcal{E}_{2}.$

It remains to estimate $\mathcal{E}_{2}$. We will estimate the terms $T_{1}$,
$T_{3}$ and $R$ above separately. Fix $s_{0}\in\mathbb{R}$. To estimate
$T_{1}$, observe that since $\mathcal{F}$ is minimal we have
$DH_{\mathcal{F}}(\mathcal{F})=0$. Then

$\displaystyle\cosh(s)\left.\left.DH\right|_{\mathcal{G}/\tau_{0}}\left(\mathcal{F}\right)\right|_{s_{0}}$
$\displaystyle=\mathcal{c}_{s_{0}}\left.DH\right|_{\frac{\left.\mathcal{G}\right|_{s_{0}}}{\omega_{0}}}\left(\frac{\left.\mathcal{F}\right|_{s_{0}}}{\cosh(s_{0})}\right)$
$\displaystyle=\mathcal{c}_{s_{0}}\left.DH\right|_{\left.\widetilde{\mathcal{G}}\right|_{s_{0}}}\left(\left.\widetilde{\mathcal{F}}\right|_{s_{0}}\right)$

and thus

$\left\|\cosh(s)\left.\left.DH\right|_{\mathcal{G}/\tau_{0}}\left(\mathcal{F}\right)\right|_{s_{0}}\right\|^{0,\alpha}\leq^{C}\epsilon.$

Since $\dot{\lambda}\rightarrow 0$ as $\tau_{0}\rightarrow 0$, we have:

(9.1) $\displaystyle\left\|T_{1}\right\|^{0,\alpha,1}_{\tau_{0}}\rightarrow 0$

as $\tau_{0}\rightarrow 0$. Next, we estimate the term $T_{3}$. Observe that
since $\dot{N}=\dot{N}_{\mathcal{G}_{0}}$ is tangential to $\mathcal{F}$ and
$\mathcal{F}$ is minimal we have

$\left.DH\right|_{\mathcal{F}}\left(\mathcal{h}\dot{N}\right)=\mathcal{h}\nabla_{\dot{N}}H_{\mathcal{F}}=0.$

with $\mathcal{c}_{0}:=\mathcal{c}_{s_{0}}$ this gives:

$\displaystyle\cosh(s)\left.\left.DH\right|_{\mathcal{G}/\tau_{0}}\left(\frac{\mathcal{h}}{\tau_{0}}\dot{N}\right)\right|_{s_{0}}$
$\displaystyle=\mathcal{c}_{s_{0}}\left.DH\right|_{\left.\widetilde{\mathcal{G}}\right|_{s_{0}}}\left(\mathcal{c}_{0}\left.\mathcal{h}^{\prime}\right|_{s_{0}}\left.\dot{N}\right|_{s_{0}}\right)$
$\displaystyle=\mathcal{c}_{s_{0}}\left.DH\right|_{\left.\widetilde{\mathcal{G}}\right|_{s_{0}}}\left(\mathcal{c}_{0}\left.\mathcal{h}^{\prime}\right|_{s_{0}}\left.\dot{N}\right|_{s_{0}}\right)-\mathcal{c}_{s_{0}}\left.DH\right|_{\left.\tilde{\mathcal{F}}\right|_{s_{0}}}\left(\mathcal{c}_{0}\left.\mathcal{h}^{\prime}\right|_{s_{0}}\left.\dot{N}\right|_{s_{0}}\right)$

and thus

(9.2)
$\displaystyle\left\|\left.T_{3}\right|_{s_{0}}\right\|^{0,\alpha}\leq^{C}\left\|\left.\mathcal{h}^{\prime}\right|_{s_{0}}\right\|^{2}\left\|\left.\dot{N}\right|_{s_{0}}\right\|\leq^{C}\epsilon^{2}\omega_{0}.$

Here we have used the expression for $\dot{\phi}_{\tau}$ in (2.5) to conclude
that $\dot{N}_{\mathcal{G}}=\frac{\partial N_{\mathcal{F}}}{\partial
s}\dot{\phi}_{\tau_{0}}$ satisfies the bound

$\left\|\left.\dot{N}_{\mathcal{G}}\right|_{s_{0}}\right\|\leq^{C}\frac{1}{\omega_{0}}.$

To estimate the term $R$, using the integral form of the Taylor remainder and
setting $\Xi=\xi N$ we can write explicitly that

$\displaystyle R$
$\displaystyle:=\cosh(s)\int^{1}_{0}\frac{(1-t)^{2}}{2}\left.D^{3}H\right|_{F_{t}}\left(\frac{\Xi}{\tau_{0}},\mathcal{h}N_{\mathcal{F}},\mathcal{h}N_{\mathcal{F}}\right)dt$

where above we have set
$\mathcal{F}_{t}:=\mathcal{F}+t\mathcal{h}N_{\mathcal{F}}$. Thus, with
$\widetilde{\mathcal{F}}_{t}:=\left.\widetilde{\mathcal{F}}\right|_{s_{0}}+t\mathcal{c}_{0}\mathcal{h}^{\prime}N_{\mathcal{F}}$,
we get

$\left.R\right|_{s_{0}}=\mathcal{c}_{0}\int^{1}_{0}\frac{(1-t)^{2}}{2}\left.D^{3}H\right|_{\widetilde{F}_{t}}\left(\frac{\Xi}{\omega_{0}},\mathcal{c}_{0}\left.\mathcal{h}^{\prime}\right|_{s_{0}}\left.N_{\mathcal{F}}\right|_{s_{0}},\mathcal{c}_{0}\left.\mathcal{h^{\prime}}\right|_{s_{0}}\left.N_{\mathcal{F}}\right|_{s_{0}}\right)$

In order estimate this term, we recall the basic observation made in Section
2.1 that the mean curvature operator depends only on first and second
derivatives of an immersion or; equivalently, that it is invariant under
translations. This is needed since, although the vector field $\Xi$ is large,
we have

(9.3)
$\displaystyle\left\|\left.\nabla\Xi\right|_{s_{0}}\right\|^{1,\alpha}\leq C$

for a universal constant $C$. To see this, observe that from the expression
for the unit normal $N_{\mathcal{F}}$ to $\mathcal{F}$ in (2.3) we see

$\left\|\left.|\nabla
N_{\mathcal{F}}\right|_{s_{0}}\right\|\leq^{C}\cosh^{-1}(s_{0}).$

Since $\nabla\Xi=\nabla\xi N_{\mathcal{F}}+\xi\nabla N_{\mathcal{F}}$ and
since $\|\nabla\xi\|^{1,\alpha}\leq C$, we get (9.3). We thus have

(9.4) $\displaystyle\left\|\left.R\right|_{s_{0}}\right\|^{0,\alpha}$
$\displaystyle\leq^{C}\left\|\nabla\left(\left.\frac{\Xi}{\omega_{0}}\right|_{s_{0}}\right)\right\|^{1,\alpha}\left(\|\left.\mathcal{h}^{\prime}\right|_{s_{0}}\|^{2,\alpha}\right)^{2}$
$\displaystyle\leq^{C}\left(\frac{1}{\omega_{0}}\right)\left(\epsilon^{2}\omega^{2}_{0}\right)$
$\displaystyle\leq^{C}\epsilon^{2}\omega_{0}$

Combining this with the estimates for $T_{1}$ and $T_{3}$ in (9.1) and (9.2)
gives.

$\displaystyle\left\|\mathcal{E}_{2}\right\|^{0,\alpha,1}_{\tau}\leq^{C}\epsilon.$

∎

######  Lemma 9.5.

It holds that

$\left\|\mathcal{E}_{1}\right\|^{0,\alpha,-1}_{1}\leq^{C}\epsilon,\quad\left\|\overline{\mathcal{E}}_{1}\right\|^{0,\alpha,0}_{1}\leq^{C\epsilon}\tau^{\gamma}.$

###### Proof.

We will use the notation from the proof of Lemma 9.4. We have

$\displaystyle\left.\mathcal{E}_{1}\right|_{s_{0}}$
$\displaystyle=\left.D^{2}H^{\prime}_{\mathcal{F}}\left(\frac{\xi}{\tau_{0}},\mathcal{h}^{\prime}\right)\right|_{s_{0}}$
$\displaystyle=\mathcal{c}_{0}D^{2}H_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}\left(\frac{\left.\xi\right|_{s_{0}}}{\omega_{0}},\mathcal{c}_{0}\left.\mathcal{h}^{\prime}\right|_{s_{0}}\right)$
$\displaystyle=\mathcal{c}_{0}\left.D^{2}H\right|_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}\left(\left.\frac{\Xi}{\omega_{0}}\right|_{s_{0}},\mathcal{c}_{0}\left.\mathcal{h}^{\prime}\right|_{s_{0}}\left.N_{\mathcal{F}}\right|_{s_{0}}\right).$

Recall that $\left.\widetilde{\mathcal{F}}\right|_{\infty}(x,s)=e^{s}e_{r}(x)$
and thus:

$\left.D^{2}H_{\left.\widetilde{\mathcal{F}}\right|_{\infty}}\right|_{0}=0.$

Using the estimate

(9.5)
$\displaystyle\left\|\left.\widetilde{\mathcal{F}}\right|_{s_{0}}-\left.\widetilde{\mathcal{F}}\right|_{\infty}\right\|\leq^{C}\cosh^{-1}(s_{0}),$

the bound (9.3) and the smoothness of the mean curvature on
$R^{k,\alpha}(\Omega_{0})$ gives

$\displaystyle\left\|\left.\mathcal{E}_{1}\right|_{s_{0}}\right\|^{0,\alpha}$
$\displaystyle\leq^{C}\left\|\left.D^{2}H\right|_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}\right\|\left\|\nabla\left(\frac{\left.\Xi\right|_{s_{0}}}{\omega_{0}}\right)\right\|^{1,\alpha}\left\|\left.\mathcal{h}^{\prime}\right|_{s_{0}}\right\|^{2,\alpha}$
$\displaystyle\leq^{C}\left(\cosh^{-1}(s_{0})\right)\left(\frac{1}{\omega_{0}}\right)\left(\epsilon\omega_{0}\right)$
$\displaystyle\leq^{C}\epsilon\cosh^{-1}(s_{0}).$

This completes the estimate for $\mathcal{E}_{1}$. To estimate
$\overline{\mathcal{E}_{1}}$, write:

$\displaystyle\left.D^{2}H_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}\right|_{0}\left(\frac{\left.\xi\right|_{s_{0}}}{\omega_{0}},\mathcal{c}_{s_{0}}\left.\mathcal{h}^{\prime}\right|_{s_{0}}\right)$
$\displaystyle=\left.D^{2}H_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}\right|_{0}\left(\frac{\left.\xi\right|_{s_{0}}}{\omega_{0}},\mathcal{c}_{s_{0}}\left.\overline{\mathcal{h}^{\prime}}\right|_{s_{0}}\right)+\left.D^{2}H_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}\right|_{0}\left(\frac{\left.\xi\right|_{s_{0}}}{\omega_{0}},\mathcal{c}_{s_{0}}\left.\mathring{\mathcal{h}}^{\prime}\right|_{s_{0}}\right)$
$\displaystyle=T_{1}+T_{2}$

where above $\overline{\mathcal{h}^{\prime}}$ and
$\mathring{\mathcal{h}^{\prime}}$ denote the higher and lower parts of
$\mathcal{h}^{\prime}$ respectively. From the estimate for
$\overline{\mathcal{h}^{\prime}}$ in Proposition 8.1, we have that

$\displaystyle\left\|T_{1}\right\|$
$\displaystyle\leq^{C}\left\|\left.D^{2}H\right|_{\left.\widetilde{F}\right|_{s_{0}}}\right\|\left(\frac{1}{\omega_{0}}\right)\left(\tau\omega_{0}^{\gamma}\right)$
$\displaystyle\leq^{C}\tau^{\gamma}$

where above we have used (9.5) to bound
$\left\|\left.D^{2}H\right|_{\left.\widetilde{F}\right|_{s_{0}}}\right\|$. We
now claim

(9.6) $\displaystyle\overline{T}_{2}=0.$

This follows from the fact that, since $\mathcal{F}$ is rotationally
symmetric, the operator
$\left.D^{2}H_{\left.\widetilde{\mathcal{F}}\right|_{s_{0}}}\right|_{0}\left(\frac{\left.\xi\right|_{s_{0}}}{\omega_{0}},-\right)$
is mode preserving (recall Section 1.1.3) and the fact that
$\mathring{\mathcal{h}}^{\prime}$ is orthogonal to the lower modes. This
completes the proof. ∎

######  Lemma 9.6.

There are functions $\dot{\mathcal{h}}_{1}$ and $\dot{\mathcal{h}}_{2}$ such
that

$\left\|\dot{\mathcal{h}}^{\prime}_{1}\right\|^{2,\alpha,-\gamma}_{1}\leq^{C}\epsilon,\quad\left\|\dot{\mathcal{h}}^{\prime}_{2}\right\|^{2,\alpha,\gamma}_{\tau}\leq^{C}\epsilon$

and such that
$\dot{\mathcal{h}}=\omega\left(\dot{\mathcal{h}}^{\prime}_{1}+\dot{\mathcal{h}}^{\prime}_{2}\right)$.

###### Proof.

By Proposition 5.3, there is a unique function
$\dot{\mathcal{h}}^{\prime}_{2}\in W^{2,\alpha,\gamma}_{\tau,n}$ such that

$L_{F}^{\prime}(\dot{\mathcal{h}}^{\prime}_{2})=\mathcal{E}_{2},\quad\mathring{\mathcal{tr}}(\dot{\mathcal{h}}^{\prime}_{2})=0,$

and satisfying the estimate

$\|\dot{\mathcal{h}}^{\prime}_{2}\|^{2,\alpha,\gamma}_{\tau}\leq^{C}\|\mathcal{E}_{2}\|^{0,\alpha,\gamma}_{\tau}\leq^{C}\epsilon,$

where $C>0$ is a universal constant. Next, we solve for the function
$\mathcal{E}_{1}$. Write:

$\mathcal{E}_{1}:=\mathring{\mathcal{E}}_{1}+\overline{\mathcal{E}}_{1},$

where $\overline{\mathcal{E}}_{1}$ and $\mathring{\mathcal{E}}_{1}$ denote the
lower and higher parts of $\mathcal{E}_{1}$, respectively. By Proposition 5.4,
the estimate for $\mathcal{E}_{1}$ in Lemma 9.5 and since
$\left\|\mathring{\mathcal{E}}_{1}\right\|\leq\left\|\mathcal{E}_{1}\right\|$,
there is a unique function $\mathring{\mathcal{h}}^{\prime}_{1}$ such that

$L^{\prime}_{\mathcal{F}}\mathring{\mathcal{h}}^{\prime}_{1}=\mathring{\mathcal{E}}_{1},\quad\mathcal{tr}(\mathring{\mathcal{h}}^{\prime})=0$

and satisfying the estimate

$\left\|\mathring{\mathcal{h}}^{\prime}_{1}\right\|^{2,\alpha,-\gamma}_{1}\leq^{C}\epsilon.$

Solving for $\overline{\mathcal{E}}_{1}$ can be done by separating variables
and integrating as follows: We have

$\overline{\mathcal{E}}_{1}=a(s)+b(s)\cos(x)+c(s)\sin(x).$

for class $C^{0,\alpha}$ funtions $a$, $b$ and $c$. Moreover, by the estimate
for $\overline{\mathcal{E}_{1}}$ in Lemma (9.5), the coefficients $a$, $b$ and
$c$ satisfy the bounds

$|a|,|b|,|c|\leq^{C}\epsilon\tau^{\gamma}.$

Since $\frac{\tanh(s)}{\cosh(s)}$ is in the kernel of
$L_{\mathcal{F}}^{\prime}$, variation of parameters gives

$\overline{\mathcal{h}}^{\prime}_{a}(s):=\frac{\tanh{s}}{\cosh(s)}\int_{0}^{s}\tanh^{-2}(s^{\prime})\int_{0}^{s^{\prime}}\sinh(s^{\prime\prime})a(s^{\prime\prime})ds^{\prime\prime}ds^{\prime}$

as a solution to the equation

$L^{\prime}_{\mathcal{F}}\overline{\mathcal{h}}^{\prime}_{a}=a.$

Moreover, we clearly have

$\overline{\mathcal{h}}^{\prime}_{a}(0)=\left.\frac{\partial\overline{\mathcal{h}}^{\prime}_{a}}{\partial
s}\right|_{(x,0)}=0,$

and the estimate:

$\left|\overline{\mathcal{h}}^{\prime}_{a}(s)\right|\leq^{C}\|a\|/\cosh(s)\leq^{C}\tau^{\gamma}\epsilon/\cosh(s).$

Solving for the term $b(s)\cos(x)$ is similar, and variation of parameters
yields the solution

$\overline{\mathcal{h}}^{\prime}_{b}=\left(\cosh^{-2}(s)\int_{0}^{s}\cosh^{2}(s^{\prime})\int_{0}^{s^{\prime}}b(s^{\prime\prime})ds^{\prime\prime}ds^{\prime}\right)\cos(x),$

which is again normalized and satisfies the estimate

$\left\|\overline{\mathcal{h}}^{\prime}_{b}\right\|^{2,\alpha}\leq^{C}\|b\|\leq^{C}\tau^{\gamma}\epsilon.$

A function $\overline{\mathcal{h}}^{\prime}_{c}$ solving
$L_{\mathcal{F}}^{\prime}\overline{\mathcal{h}}^{\prime}_{c}=c\sin(x)$ and
given by the expression above–with $\sin(x)$ substituted for $\cos(x)$–can be
found with the same methods. The function
$\overline{\mathcal{h}}^{\prime}_{1}:=\overline{\mathcal{h}}^{\prime}_{a}+\overline{\mathcal{h}}^{\prime}_{b}+\overline{\mathcal{h}}^{\prime}_{c}$
then satisfies

(9.7) $\displaystyle
L^{\prime}_{\mathcal{F}}\overline{\mathcal{h}}^{\prime}_{1}=\overline{\mathcal{E}}_{1}.$

and the estimate

(9.8)
$\displaystyle\left\|\overline{\mathcal{h}}^{\prime}_{1}\right\|^{2,\alpha}\leq^{C}\tau^{\gamma}\epsilon$

and thus

$\displaystyle\left\|\overline{\mathcal{h}}^{\prime}_{1}\right\|^{2,\alpha,-\gamma}_{1}\leq^{C}\epsilon.$

The function
$\mathcal{h}^{\prime}_{1}:=\overline{\mathcal{h}}^{\prime}_{1}+\mathring{\mathcal{h}}^{\prime}_{1}$
then satisfies

$\mathcal{L}^{\prime}_{\mathcal{F}}\mathcal{h}^{\prime}_{1}=\mathcal{E}_{1}$

and

$\left\|\mathring{\mathcal{h}}^{\prime}_{1}\right\|^{2,\alpha,-\gamma}_{1}\leq^{C}\epsilon.$

We then have

$L^{\prime}_{\mathcal{F}}\left(\frac{\dot{\mathcal{h}}}{\omega}+\mathcal{h}^{\prime}_{1}+\mathcal{h}^{\prime}_{2}\right)=0,\quad\mathring{\mathcal{tr}}\left(\frac{\dot{\mathcal{h}}}{\omega}+\mathcal{h}^{\prime}_{1}+\mathcal{h}^{\prime}_{2}\right)=0$

Since $\dot{\mathcal{h}}$, $\mathcal{h}^{\prime}_{1}$ and
$\dot{\mathcal{h}}^{\prime}_{2}$ are normalized, by Proposition 5.3 we have

$\dot{\mathcal{h}}=-\omega\left(\mathcal{h}^{\prime}_{1}+\mathcal{h}^{\prime}_{2}\right)$

This completes the proof. ∎

###### Proof of Proposition 9.1/Theorem 1.2.

We will identify the functions
$\dot{\mathcal{h}}=\dot{\mathcal{h}}_{\tau_{0}}$ with their push-forwards to
$\mathcal{N}_{\tau_{0}}$ under $\mathcal{G}_{\tau_{0}}$. Then the bound in
Lemma 9.6 translates to

(9.9)
$\displaystyle\left|\dot{\mathcal{h}}(r,\theta)\right|\leq^{C}\frac{\tau^{\gamma}}{r^{\gamma}}+r^{1+\gamma}$

which holds on compact subsets of $\mathcal{N}_{\tau_{0}}$ away from the
origin. Consider a sequence $\tau_{j}\rightarrow 0$. Then the functions
$\dot{\mathcal{h}}^{+}_{j}:=\dot{\mathcal{h}}^{+}_{\tau_{j}}$ satisfy

$\left.DH_{\mathcal{N}_{\tau}}\right|_{\mathcal{h}_{j}^{+}}\left(\dot{\mathcal{h}}^{+}_{j}\right)=-\left.DH\right|_{\mathcal{N}^{+}_{j}}\left(\dot{\mathcal{N}}^{+}_{j}+\mathcal{h}_{j}\dot{N}_{\mathcal{N}}\right)$

The variation field generated by the family $\mathcal{N}^{+}_{\tau}$ at
$\tau=0$ is $-\log(r)e_{z}$, and thus

$\dot{N}_{\mathcal{N}^{+}_{j}}\rightarrow\frac{1}{r}e_{r}$

as $j\rightarrow 0$ smoothly away from the origin. The functions
$\dot{\mathcal{h}}^{+}_{j}$ then converge in $C^{2,\alpha/2}$ to a limiting
function $\dot{\mathcal{h}}^{+}_{0}$ on the punctured unit disk $\mathcal{D}$
satisfying the limiting equation

(9.10)
$\displaystyle\left.DH_{\mathcal{D}}\right|_{\mathcal{h}}\left(\dot{\mathcal{h}}^{\pm}_{0}\right)=\mathcal{E},\quad\mathring{\mathcal{tr}}(\dot{\mathcal{h}}^{\pm})=0,$

where

$\mathcal{E}=\pm\left.DH\right|_{\mathcal{N}^{+}_{\delta,f}}\left(+\log(r)e_{z}-\mathcal{h}^{+}\frac{e_{r}}{r}\right)$

and the estimate

$\left\|\dot{\mathcal{h}}^{+}\right\|\leq^{C}r^{1+\gamma}.$

Thus, $\dot{\mathcal{h}}^{+}$ is normalized and solves the same modified
Poission-Dirichlet problem on $\mathcal{D}$ as
$\dot{\mathcal{h}}^{+}_{\delta,f}$. Since solutions are unique (Lemma 7.1), it
follows that the family of functions $\dot{\mathcal{h}}^{\pm}_{\tau,\delta,f}$
extends continuously to $\tau=0$. ∎

###  9.1. Criteria for embeddedness

When $\delta=0$, the surfaces $\mathcal{N}_{\tau,0,f}$ are minimal surfaces,
and thus by the maximum principle they are embedded provided the boundary is.
If we orient the surface $\mathcal{N}_{\tau,\delta,f}$ by the downward
pointing unit normal, then for $\delta>0$ the surfaces
$\mathcal{N}_{\tau,\delta,0}$, which are rotationally symmetric about the
$z$-axis and under reflexions through the plane $z=0$, are embedded. Moreover,
as a direct consequence of the estimate (8.9), it follows that the surface
$\mathcal{N}_{\tau,\delta,f}$ is embedded provided $f$ is small in terms of
$\delta$.

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