# Axially Symmetric Willmore Minimizers with Prescribed Isoperimetric Ratio

Jan-Henrik Metsch Corresponding author: J.-H. Metsch, Department of
Mathematics, University of Freiburg, Germany<EMAIL_ADDRESS>
###### Abstract

We establish the existence and smoothness of minimizers of the Willmore energy
among axially symmetric surfaces of spherical type with prescribed
isoperimetric ratio. Afterwards, we study the behavior of these minimizers as
the isoperimetric ratio tends to zero.

00footnotetext: 2020 Mathematics Subject Classification: Primary: 49Q10,
35B07, Secondary: 35A15, 35B6500footnotetext: Keywords: Willmore Functional,
Isoperimetric Constraint, Direct Method, Singular Limit

##  1 Introduction

The isoperimetric ratio $\mathcal{I}(f)$ of a smooth embedding
$f:\mathbb{S}^{2}\rightarrow\mathbb{R}^{3}$ is defined as

$\mathcal{I}[f]:=6\sqrt{\pi}\frac{V[f]}{A[f]^{\frac{3}{2}}},$

where $A[f]$ and $V[f]$ denote the surface area of $f$ and the volume enclosed
by $f$, respectively. By the classical isoperimetric inequality

$V[f]\leq\frac{1}{6\sqrt{\pi}}A[f]^{\frac{3}{2}}$

we have $\mathcal{I}(f)\in(0,1]$ with $\mathcal{I}(f)=1$ occurring only for
round spheres. For $\sigma\in(0,1]$, Schygulla [18] introduces the problem of
minimizing the Willmore energy

$\mathcal{W}[f]:=\frac{1}{4}\int_{\mathbb{S}^{2}}H[f]^{2}d\mu_{f}$

under all embeddings $f\in C^{\infty}(\mathbb{S}^{2},\mathbb{R}^{3})$ with
$\mathcal{I}[f]=\sigma$. This problem is motivated by the models for the shape
of blood cells by Canham [2] and Helfrich [8], who independently proposed to
model the bending energy of a blood cell by a quadratic expression in its
curvature. Helfrich’s article [8] also motivates the isoperimetric constraint:
He prescribes the area and the volume. However, as the Willmore energy is
invariant under scaling, these two constraints reduce to prescribing the
isoperimetric ratio. By Hopf’s theorem, the constraint of prescribed
isoperimetric ratio $\mathcal{I}[f]=\sigma$ is nondegenerate for $\sigma<1$ so
that by Lagrange’s theorem, the minimizers satisfy the Euler-Lagrange equation

$\Delta_{g}H+\frac{1}{2}H(H^{2}-4K)=\Lambda\sigma\left(\frac{1}{V[f]}-\frac{3H}{2A[f]}\right),$
(1.1)

where $H$ and $K$ denote the mean and the Gauß curvature of $f$ respectively.
In this article, we investigate the axially symmetric case. For
$\sigma\in(0,1]$, we introduce

$\mathcal{S}_{\sigma}:=\set{f\in
C^{\infty}(\mathbb{S}^{2},\mathbb{R}^{3})\textrm{ embedding }}{\
\mathcal{I}[f]=\sigma\textrm{ and }f\circ T=f\textrm{ for all
}T\in\operatorname{SO}(2)}.$

Additionally, we let
$\beta(\sigma):=\inf_{f\in\mathcal{S}_{\sigma}}\mathcal{W}[f]$. A convenient
characterization of the surfaces in $\mathcal{S}_{\sigma}$ is achieved using
their _profile curves_. Indeed, after translation, every
$f\in\mathcal{S}_{\sigma}$ is of the form

$f_{\gamma}:[0,1]\times\mathbb{S}^{1}\rightarrow\mathbb{R}^{3},\
f_{\gamma}(s,\omega):=\begin{bmatrix}\gamma_{1}(s)\omega\\\
\gamma_{2}(s)\end{bmatrix}$ (1.2)

where $\gamma\in\mathcal{F}_{\sigma}:=\set{\gamma\in\mathcal{P}\ }{\
\mathcal{I}[f_{\gamma}]=\sigma}$ and $\mathcal{P}$ denotes the set of
admissible profile curves

$\mathcal{P}:=\left\\{\gamma\in C^{\infty}([0,1],\mathbb{R}^{2})\ \big{|}\
\begin{array}[]{l}\gamma_{1}(0)=\gamma_{1}(1)=0\textrm{ and
}\gamma_{1}(t)>0\textrm{ for all }t\in(0,1)\end{array}\right\\}.$

Conversely, using Equation (1.2), any $\gamma\in\mathcal{P}$ can be assigned a
surface of spherical type in $\mathbb{R}^{3}$ so that we can identify
$f_{\gamma}$ and $\gamma$.

######  Theorem 1.1.

For all $\sigma\in(0,1]$, there exists $f\in\mathcal{S}_{\sigma}$ such that
$\mathcal{W}[f]=\beta(\sigma)$. Moreover, $\beta(\sigma)<8\pi$ for all
$\sigma\in(0,1]$ and $\beta(\sigma)\rightarrow 8\pi$ as $\sigma\rightarrow
0^{+}$.

The proof of 1.1 is split into three steps. First, the existence of a weak
minimizing profile $\gamma\in W^{1,2}((0,1))\cap
W^{2,2}_{\operatorname{loc}}((0,1))$ is established using the direct method of
the calculus of variations. The arguments used here follow those executed by
Choski and Venernoni in [4]. Next, the smoothness of $\gamma$ on $(0,1)$ –
away from the axis of rotation – is established using elliptic regularity
arguments. To study regularity at the axis of rotation, a careful analysis of
the local graph representation

$\set{(r\omega,u(r))\ }{\ r\in[0,r_{0}),\ \omega\in\mathbb{S}^{1}}$

of $f_{\gamma}$ near the axis of rotation is required. For the unconstrained
Willmore equation, such analysis has been carried out in [3] by Chen and Li.
We adapt their analysis to prove

$u(r)=\lambda r^{2}\ln(r)+C^{2}([0,r_{0})).$ (1.3)

The parameter $\lambda$ is a residue, as has been studied by Kuwert and
Schätzle in [12] and cannot be eliminated by ODE arguments alone, as the
Euler-Lagrange equation (1.1) allows for singular solutions of the form in
Equation (1.3).

After proving the existence of smooth minimizers, we investigate their
behavior in the limit $\sigma\rightarrow 0^{+}$. This has already been done by
Schygulla [18]. He proved that as $\sigma\rightarrow 0^{+}$, his minimizers
converge to a double sphere in the sense of measures. In [10], this result was
later refined by Kuwert and Li, who prove that asymptotically, Schygulla’s
minimizers look like two spheres joined by a catenoidal neck. We prove that
our axially symmetric minimizers display the same behavior. Restricting to the
class

$\mathcal{F}^{0+}_{\sigma}:=\left\\{\gamma\in\mathcal{F}_{\sigma},\ \bigg{|}\
\gamma(0)=(0,0),\ A[f_{\gamma}]=1\textrm{ and }\int_{0}^{1}\gamma_{2}(t)dt\geq
0\right\\},$

we eliminate the invariance of the problem under vertical translation, scaling
and reflection – thereby out ruling the trivial obstructions to convergence.
Characterizing the solutions to the Euler-Lagrange equation in the limit
$\sigma\rightarrow 0^{+}$ we then establish:

######  Theorem 1.2 (Convergence to a Double Sphere).

Let $(\sigma_{k})\subset(0,1]$ be a sequence of isoperimetric ratios
$\sigma_{k}\rightarrow 0^{+}$ and $(\gamma^{(k)})_{k}$ be a sequence of
profile curves $\gamma^{(k)}\in\mathcal{F}^{0+}_{\sigma_{k}}$ satisfying
$\beta(\sigma_{k})=\mathcal{W}[f_{\gamma^{(k)}}]$. Then as
$k\rightarrow\infty$,

$\gamma^{(k)}(t)\rightarrow\kappa(t):=(0,R)+R(|\sin(2\pi t)|,-\cos(2\pi
t))\hskip 14.22636pt\textrm{with }R:=\frac{1}{\sqrt{8\pi}}$

in $C^{\infty}([0,1]\backslash\\{\frac{1}{2}\\})$ and in $W^{1,2}((0,1))$.

Since $\kappa$ is not smooth at $t=\frac{1}{2}$, it is clear that the curves
$\gamma^{(k)}$ have to develop some form of singularity near $t=\frac{1}{2}$.
We prove that near $t=\frac{1}{2}$, the profile curves solve approximately the
equation $H=0$. Using a suitable blow-up argument, we then establish the
following theorem:

######  Theorem 1.3 (Formation of a Catenoidal Neck).

Let $\sigma_{k}\rightarrow 0^{+}$ and
$\gamma^{(k)}\in\mathcal{F}^{0+}_{\sigma_{k}}$ satisfy
$\mathcal{W}[f_{\gamma^{(k)}}]=\beta(\sigma_{k})$. There exist
$k_{0}\in\mathbb{N}$ and $\rho>0$ such that the following statements hold:

  1. (1)

For $k\geq k_{0}$ there exists a unique $\tau_{k}\in[\frac{1}{8},\frac{7}{8}]$
such that
$\gamma_{1}^{(k)}(\tau_{k})=\epsilon_{k}:=\inf_{[\frac{1}{8},\frac{7}{8}]}\gamma_{1}^{(k)}$.

  2. (2)

$\dot{\gamma}_{1}^{(k)}<0$ on $[\tau_{k}-\rho,\tau_{k})$ and
$\dot{\gamma}_{1}^{(k)}>0$ on $(\tau_{k},\tau_{k}+\rho]$.

  3. (3)

The parameters from $(1)$ satisfy $\epsilon_{k}\rightarrow 0^{+}$ and
$\tau_{k}\rightarrow\frac{1}{2}$.

  4. (4)

After potentially putting
$\hat{\gamma}^{(k)}(t):=\left(\gamma_{1}^{(k)}(1-t),\
\gamma_{2}^{(k)}(1-t)-\gamma_{2}^{(k)}(1)\right)$, it can be assumed that
$\dot{\gamma}_{2}^{(k)}(\tau_{k})\leq 0$ for all $k\geq k_{0}$ and the
following convergence is in $C^{\infty}_{\operatorname{loc}}(\mathbb{R})$:

$\frac{1}{\epsilon_{k}}\left(\gamma^{(k)}(\tau_{k}+\epsilon_{k}t)-(0,\gamma_{2}^{(k)}(\tau_{k}))\right)\rightarrow\left(\sqrt{1+L_{*}^{2}t^{2}},-\operatorname{asinh}(L_{*}t)\right)\hskip
14.22636pt\textrm{with }L_{*}:=\sqrt{\frac{\pi}{2}}$

The results from Theorems 1.2 and 1.3 are in agreement with the numerically
computed minimizers in [19] (see Figure 9). Using numerical methods, several
authors have computed candidates for the minimizers. For an overview, we refer
to [19]. However, we mention that Helfrich has already investigated axially
symmetric minimizers in the aforementioned [8] and in a joint work with
Deuling in [5].

The first rigorous treatment of Willmore minimizers with isoperimetric
constraint was perhaps carried out in [15], where a one-parameter family of
critical points bifurcating from the sphere is constructed.

Regarding the axisymmetric case, Choski and Veneroni [4] study systems of
axisymmetric surfaces $S=(\Sigma_{1},...,\Sigma_{N})$ where each $\Sigma_{i}$
is either an embedded sphere or an embedded torus that minimize the Helfrich
energy under the generalized isoperimetric constraint

$6\sqrt{\pi}\frac{V[S]}{A[S]^{\frac{3}{2}}}=\sigma\in(0,1]\hskip
14.22636pt\textrm{where }A[S]=\sum_{i=1}^{N}A[\Sigma_{i}]\textrm{ and
}V[S]=\sum_{i=1}^{N}V[\Sigma_{i}].$

[17] studies axisymmetric tori minimizing the Willmore energy with an
isoperimetric constraint.

This article is structured as follows. Section 2 collects the notation and
formulas we require throughout this article. In Section 3, the existence of a
weak minimizer is proven, and in Section 4, its smoothness is established.
Sections 5 and 6 contain the proofs of Theorems 1.2 and 1.3. Technical
arguments are moved to the appendix.

##  2 Preliminaries

#### General Notation

For $k\in\mathbb{N}_{0}\cup\set{\infty}$ and $U\subset\mathbb{R}^{n}$ open, we
denote the set of $C^{k}$-functions with compact support in $U$ by
$C^{k}_{0}(U)$. The right half plane in $\mathbb{R}^{2}$ is denoted by

$\mathcal{H}^{2}:=\set{(x_{1},x_{2})\in\mathbb{R}^{2}\ }{\ x_{1}\geq 0}.$

If $Q$ is some functional, we put

$\delta Q[x]v:=\frac{d}{dt}\bigg{|}_{t=0}Q[x+tv].$

#### Immersions

Let $f:\mathbb{S}^{2}\rightarrow\mathbb{R}^{3}$ be a sufficiently regular
immersion and $n$ denote a regular unit normal vector field along $f$. The
first and second fundamental forms are respectively denoted by

$g_{ij}:=\langle\partial_{i}f,\partial_{j}f\rangle\hskip
14.22636pt\textrm{and}\hskip
14.22636pth_{ij}:=\langle\partial_{ij}f,n\rangle.$

The surfaces measure on $\mathbb{S}^{2}$ induced by $f$ is denoted by
$\mu_{f}$. In local coordinates $d\mu_{f}=\sqrt{\det g}dx$. Let $k_{1}$ and
$k_{2}$ denote the principal curvatures of $f$ – that is, the eigenvalues of
the Weingarten operator $g^{-1}h$. The mean curvature $H$ and the Gauß
curvature $K$ are given by

$H=k_{1}+k_{2}\hskip 14.22636pt\textrm{and}\hskip 14.22636ptK=k_{1}k_{2}.$

For example, considering the inclusion
$\imath:\mathbb{S}^{2}\hookrightarrow\mathbb{R}^{3}$ with the interior normal
field $n=-\imath$, we get $H[f]=2$ and $K[f]=1$.

#### Functionals

Let $f:\mathbb{S}^{2}\rightarrow\mathbb{R}^{3}$ be a sufficiently regular
immersion. The Willmore energy of $f$ is defined as

$\mathcal{W}[f]:=\frac{1}{4}\int_{\mathbb{S}^{2}}H[f]^{2}d\mu_{f}.$ (2.1)

For any of the two choices of continuous normal vector field $n$ along $f$,
the surface area, the volume and the isoperimetric ratio of $f$ are
respectively defined as

$A[f]:=\int_{\mathbb{S}^{2}}d\mu_{f},\hskip
14.22636ptV[f]:=\left|\int_{\mathbb{S}^{2}}f_{2}\langle n,e_{2}\rangle
d\mu_{f}\right|\hskip 14.22636pt\textrm{and}\hskip
14.22636pt\mathcal{I}[f]:=6\sqrt{\pi}\frac{V[f]}{A[f]^{\frac{3}{2}}}.$ (2.2)

To justify the definition of volume, we have the following lemma:

######  Lemma 2.1.

Let $f\in C^{1}(\mathbb{S}^{2},\mathbb{R}^{3})$ be an embedding. Then there
exists a bounded $C^{1}$-domain $\Omega\subset\mathbb{R}^{3}$ with
$\partial\Omega=f(\mathbb{S}^{2})$ and which satisfies $|\Omega|=V[f]$.

###### Proof.

The existence of $\Omega$ is due to the Jordan-Brouwer separation theorem (see
e.g. [14], Corollary 5.24). Let $n_{\operatorname{int}}$ denote the interior
unit normal along $\partial\Omega$. Since $\Omega\in C^{1}$, we can use Gauß’s
theorem to compute

$|\Omega|=-\int_{\partial\Omega}x_{2}\langle
n_{\operatorname{int}},e_{2}\rangle dS=-\int_{\mathbb{S}^{2}}f_{2}\langle
n_{\operatorname{int}},e_{2}\rangle d\mu_{f}.$

This establishes Equation (2.2) when $n$ in the interior normal and thereby
also for the exterior normal $n_{\operatorname{ext}}=-n_{\operatorname{int}}$.
∎

#### First Variation of the Willmore Energy

Let $f:\Sigma\rightarrow\mathbb{R}^{3}$ be a smoothly immersed surface and
$\Phi:(-\epsilon_{0},\epsilon_{0})\rightarrow\mathbb{R}^{3}$ be a smooth
variation of $f$. Denote by $n$ the unit normal field used to define the
scalar mean curvature $H=\langle\vec{H},n\rangle$. The following formula is,
for example, derived in [1], Theorem 1.

$\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}\mathcal{W}[\Phi(\epsilon)]=\int_{\Sigma}W[f]\langle\Phi^{\prime}(0),n\rangle
d\mu_{f}+\int_{\partial\Sigma}\omega(\eta)dS_{f}.$ (2.3)

In this formula, $\eta$ is the interior unit normal along $\partial\Sigma$ and
splitting the variational vector field $\Phi^{\prime}(0)$ into a normal and a
tangential part by writing $\Phi^{\prime}(0)=\varphi n+DfX$, the Willmore
operator $W[f]$ and the one-form appearing in Equation (2.3) are given by

$\displaystyle W[f]$
$\displaystyle=\frac{1}{2}\left(\Delta_{g}H+\frac{1}{2}H(H^{2}-4K)\right),$
(2.4) $\displaystyle\omega(\eta)$
$\displaystyle=\frac{1}{2}\left(\varphi\frac{\partial
H}{\partial\eta}-H\frac{\partial\varphi}{\partial\eta}-\frac{1}{2}H^{2}\langle
Df\eta,DfX\rangle\right).$ (2.5)

###  2.1 Graphical Surfaces

Let $r_{0}>0$ and $u:[0,r_{0})\rightarrow\mathbb{R}$ be sufficiently regular.
We consider the axisymmetric surface $\Sigma_{u}$ generated by rotating the
graph of $u$ around the $x_{3}$-axis:

$f_{u}:[0,r_{0})\times[0,2\pi)\rightarrow\mathbb{R}^{3},\
f_{u}(r,\theta):=(r\cos\theta,r\sin\theta,u(r))$

We collect the geometric data of $\Sigma_{u}$. The first fundamental form is
given by

$g(r,\theta)=\begin{bmatrix}1+u^{\prime}(r)^{2}&\\\ &r^{2}\end{bmatrix}\hskip
14.22636pt\textrm{and}\hskip 14.22636pt\sqrt{\det
g}=r\sqrt{1+u^{\prime}(r)^{2}}.$ (2.6)

Next, we introduce a normal field $n$, which we choose to coincide with the
interior unit normal when $u(r)=\sqrt{1-r^{2}}$ – that is when $\Sigma_{u}$ is
a sphere.

$n(r,\theta):=\frac{1}{\sqrt{1+u^{\prime}(r)^{2}}}\begin{bmatrix}u^{\prime}(r)\cos\theta\\\
u^{\prime}(r)\sin\theta\\\ -1\end{bmatrix}$

Using this normal, the scalar second fundamental form is given by

$h=-\frac{1}{\sqrt{1+u^{\prime}(r)^{2}}}\begin{bmatrix}u^{\prime\prime}(r)&\\\
&ru^{\prime}(r)\end{bmatrix}.$ (2.7)

Combining Equations (2.6) and (2.7), we obtain the following formulas for the
two principal curvatures:

$k_{1}=-\frac{u^{\prime\prime}(r)}{\sqrt{1+u^{\prime}(r)^{2}}^{3}}\hskip
14.22636pt\textrm{and}\hskip
14.22636ptk_{2}=-\frac{u^{\prime}(r)}{r\sqrt{1+u^{\prime}(r)^{2}}}$ (2.8)

Using these, it is readily checked that the mean curvature $H$ of $\Sigma_{u}$
is given by

$H=k_{1}+k_{2}=-\frac{1}{r}\frac{d}{dr}\left[r\frac{u^{\prime}(r)}{\sqrt{1+u^{\prime}(r)^{2}}}\right].$
(2.9)

###  2.2 Profile Curves

Let $\gamma:[0,1]\rightarrow\mathcal{H}^{2}$ be a sufficiently regular curve.
We begin by recalling that the arc length of $\gamma$ is defined as

$L[\gamma]:=\int_{0}^{1}|\dot{\gamma}(t)|dt.$ (2.10)

We consider the surface $\Sigma_{\gamma}$ generated by rotating the graph of
$\gamma$ around the $x_{2}$-axis. Concretely, it is given by the
parameterization

$f_{\gamma}:[0,1]\times[0,2\pi)\rightarrow\mathbb{R}^{3},\
f_{\gamma}(t,\theta):=\begin{bmatrix}\gamma_{1}(t)\cos\theta\\\
\gamma_{1}(t)\sin\theta\\\ \gamma_{2}(t)\end{bmatrix}.$ (2.11)

In the following, we collect the geometric data of $\Sigma_{\gamma}$. The
first fundamental form is given by

$g(t,\theta)=\begin{bmatrix}|\dot{\gamma}(t)|^{2}&\\\
&\gamma_{1}(t)^{2}\end{bmatrix}\hskip 14.22636pt\textrm{and}\hskip
14.22636pt\sqrt{\det g}=\gamma_{1}(t)|\dot{\gamma}(t)|.$ (2.12)

We denote the surface element on $\Sigma_{\gamma}$ by
$d\mu_{\Sigma_{\gamma}}:=\gamma_{1}|\dot{\gamma}_{1}|dtd\theta$. Next, we
define normal fields $n$ along $\Sigma_{\gamma}$ and $\nu$ along $\gamma$.

$n(t,\theta)=\frac{1}{|\dot{\gamma}(t)|}\begin{bmatrix}-\dot{\gamma}_{2}(t)\cos\theta\\\
-\dot{\gamma}_{2}(t)\sin\theta\\\ \dot{\gamma}_{1}(t)\end{bmatrix}\hskip
14.22636pt\textrm{and}\hskip
14.22636pt\nu(t)=\frac{1}{|\dot{\gamma}(t)|}\begin{bmatrix}-\dot{\gamma}_{2}(t)\\\
\dot{\gamma}_{1}(t)\end{bmatrix}$ (2.13)

This normal is chosen, such that it agrees with the interior normal along
$\Sigma_{\gamma}$ when $\gamma(t)=(\cos(\pi t),\sin(\pi t))$ – that is when
$\gamma$ parameterizes a circle run through counterclockwise. Using this
normal, we get the following formula for the scalar second fundamental form of
$\Sigma_{\gamma}$:

$h(t,\theta):=\frac{1}{|\dot{\gamma}(t)|}\begin{bmatrix}\ddot{\gamma}_{2}\dot{\gamma}_{1}-\ddot{\gamma}_{1}\dot{\gamma}_{2}&\\\
&\gamma_{1}\dot{\gamma}_{2}\end{bmatrix}$ (2.14)

Combining Equations (2.12) and (2.14) gives the following formulas for the two
principal curvatures:

$k_{1}=\frac{\ddot{\gamma}_{2}\dot{\gamma}_{1}-\ddot{\gamma}_{1}\dot{\gamma}_{2}}{|\dot{\gamma}|^{3}}\hskip
14.22636pt\textrm{and}\hskip
14.22636ptk_{2}=\frac{\dot{\gamma}_{2}}{\gamma_{1}|\dot{\gamma}|}$ (2.15)

#### Functionals

Using Equations (2.1), (2.2), (2.12) and (2.13), we get the following formula
for the Willmore energy, the area and the volume of $\Sigma_{\gamma}$:

$\displaystyle\mathcal{W}[\gamma]$
$\displaystyle:=\mathcal{W}[\Sigma_{\gamma}]=\frac{\pi}{2}\int_{0}^{1}(k_{1}(t)+k_{2}(t))^{2}\gamma_{1}(t)|\dot{\gamma}(t)|dt$
(2.16) $\displaystyle A[\gamma]$
$\displaystyle:=A[\Sigma_{\gamma}]=2\pi\int_{0}^{1}\gamma_{1}(t)|\dot{\gamma}(t)|dt$
(2.17) $\displaystyle V[\gamma]$
$\displaystyle:=V[\Sigma_{\gamma}]=\pi\left|\int_{0}^{1}\gamma_{1}(t)^{2}\dot{\gamma}_{2}(t)dt\right|$
(2.18)

We also put $\mathcal{I}[\gamma]:=\mathcal{I}[\Sigma_{\gamma}]$. In view of
Lemma 2.1, we have the following result:

######  Lemma 2.2.

Let $\gamma\in C^{1}([0,1],\mathcal{H}^{2})$ be injective,
$\dot{\gamma}(t)\neq 0$ for all $t\in[0,1]$ and
$\gamma_{1}(0)=\gamma_{1}(1)=0$ as well as
$\dot{\gamma}_{2}(0)=\dot{\gamma}_{2}(1)=0$. Then $\Sigma_{\gamma}$ is the
boundary of a bounded $C^{1}$ domain $\Omega$ and $|\Omega|=V[\gamma]$.

#### Arc Length Parameterization

A regular curve $\gamma:[0,1]\rightarrow\mathcal{H}^{2}$ is _parameterized
proportional to arc length_ if $|\dot{\gamma}(t)|=L[\gamma]=:L$ for all
$t\in[0,1]$. These curves enjoy several beneficial properties. First

$0=\frac{1}{2}\frac{d}{dt}|\dot{\gamma}(t)|^{2}=\langle\dot{\gamma}(t),\ddot{\gamma}(t)\rangle=\dot{\gamma}_{1}(t)\ddot{\gamma}_{1}(t)+\dot{\gamma}_{2}(t)\ddot{\gamma}_{2}(t).$
(2.19)

Considering Equation (2.15), it is then straightforward to check that

$\dot{\gamma}_{1}H=\frac{\ddot{\gamma}_{2}}{L}+\frac{\dot{\gamma}_{1}\dot{\gamma}_{2}}{\gamma_{1}L}\hskip
14.22636pt\textrm{and}\hskip
14.22636pt\dot{\gamma}_{2}H=-\frac{\ddot{\gamma}_{1}}{L}+\frac{\dot{\gamma}_{2}^{2}}{\gamma_{1}L}.$
(2.20)

Additionally, it is readily checked that

$k_{1}^{2}=\frac{|\ddot{\gamma}|^{2}}{L^{4}}\hskip
14.22636pt\textrm{and}\hskip
14.22636ptK=k_{1}k_{2}=-\frac{\ddot{\gamma}_{1}}{\gamma_{1}L^{2}}.$ (2.21)

##  3 Direct Method of the Calculus of Variations

In this section, we employ the direct method of the calculus of variations to
obtain an axially symmetric minimizer of the Willmore energy with prescribed
isoperimetric ratio $\sigma$. The analysis follows with only minor
modifications, the arguments in [4].

###  3.1 Functional Space

We first define a suitable space of profile curves.

######  Definition 3.1 (Admissible Profile Curves).

A curve $\gamma=(\gamma_{1},\gamma_{2}):[0,1]\rightarrow\mathcal{H}^{2}$ is
called an _admissible profile curve_ if:

  1. (1)

$\gamma\in C^{1}([0,1])\cap W^{2,2}_{\operatorname{loc}}((0,1))$.

  2. (2)

$\gamma_{1}(0)=\gamma_{1}(1)=0$ and $\gamma_{1}(t)>0$ for all $t\in(0,1)$.

  3. (3)

$\gamma$ is parameterized proportional to arc length.

  4. (4)

$\Sigma_{\gamma}$ has curvature bounded in $L^{2}(\mu_{\Sigma_{\gamma}})$.
That is

$\int_{\mathbb{S}^{2}}k_{1}^{2}+k_{2}^{2}d\mu_{\Sigma_{\gamma}}=\int_{0}^{1}(k_{1}^{2}+k_{2}^{2})2\pi|\dot{\gamma}|\gamma_{1}dt<\infty.$

We denote the set of all admissible profile curves by $\mathcal{P}$.

We wish to deduce the existence of a curve $\gamma\in\mathcal{P}$ that
minimizes the Willmore energy under all curves in $\mathcal{P}$ with
prescribed isoperimetric ratio. Given $\sigma\in(0,1)$ we put

$\mathcal{F}_{\sigma}:=\left\\{\gamma\in\mathcal{P}\ \bigg{|}\
A[\gamma]=1\hskip 5.69046pt\textrm{and}\hskip
5.69046ptV[\gamma]=\frac{\sigma}{6\sqrt{\pi}}\right\\}\hskip
14.22636pt\textrm{and}\hskip
14.22636pt\beta(\sigma):=\inf\set{\mathcal{W}[\gamma]\ }{\
\gamma\in\mathcal{F}_{\sigma}}.$

Note that in the definition of $\mathcal{F}_{\sigma}$, we prescribe the area
and the volume of $\gamma$. However, as both the Willmore energy $\mathcal{W}$
and the isoperimetric ratio $\mathcal{I}$ are invariant under scaling, we have

$\beta(\sigma)=\inf\set{\mathcal{W}[\gamma]\ }{\
\gamma\in\mathcal{F}_{\sigma}}=\inf\set{\mathcal{W}[\gamma]\ }{\
\gamma\in\mathcal{P}\textrm{ with }\mathcal{I}[\gamma]=\sigma}.$ (3.1)

Unfortunately, the space $\mathcal{P}$ does not have sufficient compactness
properties and we require a weakened notation of admissible profile curves.

######  Definition 3.2 (Weak Profile Curves).

We say that $\gamma:[0,1]\rightarrow\mathcal{H}^{2}$ is a _weak profile curve_
if:

  1. (1)

$\gamma\in W^{1,2}((0,1))$ (so in particular $\gamma\in C^{0}([0,1])$).
Additionally $\gamma_{1}(0)=\gamma_{1}(1)=0$ and $\gamma_{1}(t)>0$ for almost
all $t\in[0,1]$.

  2. (2)

$|\dot{\gamma}|=L[\gamma]$ almost everywhere.

  3. (3)

$\gamma\in W^{2,2}_{\operatorname{loc}}(U)$ for all $U\subset[0,1]$ satisfying
$\gamma_{1}>0$ on $U$.

By (1) and (3), $\ddot{\gamma}$, and thus also $k_{1}$, is defined almost
everywhere even though $\dot{\gamma}$ is not weakly differentiable on $(0,1)$.
This allows the formulation of:

  1. (4)

$\Sigma_{\gamma}$ has curvature bounded in $L^{2}(\mu_{\Sigma_{\gamma}})$.
That is

$\int_{0}^{1}(k_{1}^{2}+k_{2}^{2})2\pi|\dot{\gamma}|\gamma_{1}dt<\infty.$

The set of weak profile curves is denoted by $\mathcal{P}^{w}$.

Clearly $\mathcal{P}\subsetneq\mathcal{P}^{w}$. However, the next theorem
establishes that weak profile curves correspond exactly to chaining several
admissible profile curves from $\mathcal{P}$ together. For a proof, we refer
to Lemma 6 in [4].

######  Theorem 3.3 (Regularity of Weak Profile Curves).

Let $\gamma\in\mathcal{P}^{w}$, put
$M:=\int_{0}^{1}(k_{1}^{2}+k_{2}^{2})2\pi|\dot{\gamma}|\gamma_{1}dt$ and let
$n_{0}:=\lfloor\frac{M}{8\pi}\rfloor$. Then there exists $m\leq n_{0}$ and
points $0=\tau_{0}<\tau_{1}...<\tau_{m}=1$ such that (up to linear
reparamterization) $\gamma|_{[\tau_{i},\tau_{i+1}]}$ belongs to $\mathcal{P}$.
Additionally, when $t\rightarrow\tau_{i}^{\pm}$ we have

$\lim_{t\rightarrow\tau_{i}^{\pm}}\dot{\gamma}_{2}(t)=0\hskip
14.22636pt\textrm{and}\hskip
14.22636pt\lim_{t\rightarrow\tau_{i}^{\pm}}\dot{\gamma}_{1}(t)=\pm L[\gamma].$

As an immediate consequence, we obtain the following corollary:

######  Corollary 3.4.

Let $\gamma\in\mathcal{P}$. Then

$\dot{\gamma}_{1}(0)=L[\gamma],\ \dot{\gamma}_{1}(1)=-L[\gamma]\hskip
14.22636pt\textrm{and}\hskip
14.22636pt\dot{\gamma}_{2}(0)=\dot{\gamma}_{2}(1)=0.$ (3.2)

Consequently $\Sigma_{\gamma}$ is a $C^{1}$ surface.

#### Properties of Profile Curves

We collect several properties of (weak) profile curves. The following theorem
is Lemma 2 from [4].

######  Theorem 3.5 (Choski and Veneroni).

Let $\gamma\in\mathcal{P}$. Then

$\frac{A[\gamma]}{2\pi\operatorname{diam}(\Sigma_{\gamma})}\leq
L[\gamma]\leq\frac{\sqrt{A[\gamma]}}{2\pi}\left(\left(\int_{\mathbb{S}^{2}}k_{1}^{2}d\mu_{\Sigma_{\gamma}}\right)^{\frac{1}{2}}+\left(\int_{\mathbb{S}^{2}}k_{2}^{2}d\mu_{\Sigma_{\gamma}}\right)^{\frac{1}{2}}\right).$
(3.3)

As an immediate consequence of this theorem, we can also bound the inverse arc
length. Indeed, using $\gamma_{1}(0)=0$, we can bound the diameter of
$\Sigma_{\gamma}$ by estimating

$|f_{\gamma}(t,\theta)-f_{\gamma}(t^{\prime},\theta^{\prime})|\leq|\gamma(t)-\gamma(t^{\prime})|+|\gamma_{1}(t^{\prime})|\
\left|\begin{pmatrix}\cos\theta-\cos\theta^{\prime}\\\
\sin\theta-\sin\theta^{\prime}\end{pmatrix}\right|\leq 3L[\gamma].$

Consequently $\operatorname{diam}(\Sigma_{\gamma})\leq 3L[\gamma]$ and thus

$\frac{1}{L[\gamma]^{2}}\leq\frac{2\pi\operatorname{diam}(\Sigma_{\gamma})}{A[\gamma]L[\gamma]}\leq\frac{6\pi}{A[\gamma]}.$
(3.4)

Additionally, many facts about smooth surfaces remain true for surfaces
$\Sigma_{\gamma}$ with $\gamma\in\mathcal{P}$. We collect these below and
refer to Appendix A.1 for their proofs.

######  Lemma 3.6 (Li-Yau Inequality).

Let $\gamma\in\mathcal{P}$. Then $\mathcal{W}[\Sigma_{\gamma}]\geq 4\pi$ and
equality is only true if $\Sigma_{\gamma}$ is a sphere and
$f_{\gamma}:\mathbb{S}^{2}\rightarrow\Sigma_{\gamma}$ is a diffeomorphism.

A direct consequence from Lemma 3.6 is $\beta(\sigma)\geq 4\pi$ for all
$\sigma\in(0,1]$. Additionally, since $\mathcal{W}[\mathbb{S}^{2}]=4\pi$ we
deduce $\beta(1)=4\pi$ and the minimizers are precisely rounds spheres.

######  Lemma 3.7 (Gauß-Bonnet Theorem).

Let $\gamma\in\mathcal{P}$. Then $\Sigma_{\gamma}$ satisfies the Gauß-Bonnet
theorem. That is

$\int_{\mathbb{S}^{2}}k_{1}k_{2}d\mu_{\Sigma_{\gamma}}=4\pi.$ (3.5)

######  Lemma 3.8 (Hopf Theorem).

Let $\gamma\in\mathcal{P}$ and assume that $\Sigma_{\gamma}$ has constant mean
curvature $H_{0}$. Then $\Sigma_{\gamma}$ is a sphere.

######  Lemma 3.9.

Let $\gamma\in\mathcal{P}^{w}$, $L:=L[\gamma]$ and $U\subset[0,1]$ be
measurable such that $\inf_{U}\gamma_{1}>0$. Then
$\gamma_{1}|\ddot{\gamma}|^{2}\in L^{1}((0,1))$, $|\ddot{\gamma}|^{2}\in
L^{1}(U)$ and

$\int_{0}^{1}\gamma_{1}(t)|\ddot{\gamma}(t)|^{2}dt\leq\frac{L^{3}}{2\pi}\int_{0}^{1}k_{1}^{2}2\pi
L\gamma_{1}(t)dt\hskip 14.22636pt\textrm{and}\hskip
14.22636pt\int_{U}|\ddot{\gamma}(t)|^{2}dt\leq\frac{2L^{3}}{\pi\inf_{U}\gamma_{1}}\mathcal{W}[\gamma].$

###### Proof.

In view of Theorem 3.3, we can assume that $\gamma\in\mathcal{P}$ to estblish
the first esimtate. By Definition 3.1 we have $\gamma\in
W^{2,2}(\epsilon,1-\epsilon)$ for all small $\epsilon>0$. We use the
definition of $\mathcal{P}$ and Equation (2.21) to deduce that for some $C$
independent of $\epsilon$

$C\geq\int_{\epsilon}^{1-\epsilon}k_{1}^{2}2\pi|\dot{\gamma}|\gamma_{1}dt=\frac{2\pi}{L^{3}}\int_{\epsilon}^{1-\epsilon}|\ddot{\gamma}|^{2}\gamma_{1}dt.$

The first part of the lemma follows by letting $\epsilon\rightarrow 0^{+}$ and
using the theorem of Beppo-Levi. Let $\gamma\in\mathcal{P}^{w}$. To establish
the second estimate, it suffices to bound $\int_{0}^{1}k_{1}^{2}2\pi
L\gamma_{1}dt$. By Theorem 3.3, there exists $m\in\mathbb{N}$ such that
$\gamma$ is built from chaining together $m$ many $\mathcal{P}$-curves. We use
the Gauß-Bonnet theorem (Lemma 3.5) to obtain
$\int_{0}^{1}k_{1}k_{2}2\pi|\dot{\gamma}|\gamma_{1}dt=4n_{0}\pi\geq 0$ and
estiamte

$\int_{0}^{1}k_{1}^{2}2\pi|\dot{\gamma}|\gamma_{1}dt\leq
4\mathcal{W}[\gamma]-2\int_{0}^{1}k_{1}k_{2}2\pi|\dot{\gamma}|\gamma_{1}dt\leq
4\mathcal{W}[\gamma].$

∎

###  3.2 Topology

We now introduce a notion of convergence in $\mathcal{P}^{w}$. The following
is essentially the same definition as Definition 6 in [4]. Their definition,
however, allows for systems of curves, which we exclude here. Additionally, we
have eliminated the concept of measure-function couples.

######  Definition 3.10.

We say that a sequence $(\gamma^{(n)})\subset\mathcal{P}^{w}$ _converges to
$\gamma^{*}\in\mathcal{P}^{w}$ in $\mathcal{P}^{w}$_ if:

  1. (1)

For $i=1,2$ we have $\gamma^{(n)}_{i}\rightarrow\gamma^{*}_{i}$ in
$C^{0}([0,1])$.

  2. (2)

For $i=1,2$ we have $\dot{\gamma}^{(n)}_{i}\rightarrow\dot{\gamma}^{*}_{i}$ in
$L^{2}((0,1))$.

  3. (3)

For $i=1,2$ we have
$\ddot{\gamma}^{(n)}_{i}\gamma_{1}^{(n)}\rightarrow\ddot{\gamma}^{*}_{i}\gamma_{1}^{*}$
weakly in $L^{1}((0,1))$.

We note that the third condition is well-defined by Lemma 3.9.

#### Lower Semicontinuity

We show that the notation of convergence in Definition 3.10 is strong enough
to imply lower semicontinuity of our problem. Let
$\gamma^{(n)}\rightarrow\gamma^{*}$ in $\mathcal{P}^{w}$. Then
$\gamma^{(n)}\rightarrow\gamma^{*}$ in $C^{0}([0,1])$ and
$\dot{\gamma}^{(n)}\rightarrow\dot{\gamma}^{*}$ in $L^{2}((0,1))$. Hence

$\displaystyle A[\gamma^{(n)}]$
$\displaystyle=2\pi\int_{0}^{1}|\dot{\gamma}^{(n)}(t)|\gamma_{1}^{(n)}(t)dt\rightarrow
A[\gamma^{*}],$ (3.6) $\displaystyle V[\gamma^{(n)}]$
$\displaystyle=\pi\left|\int_{0}^{1}\dot{\gamma}^{(n)}_{2}(t)\left(\gamma_{1}^{(n)}(t)\right)^{2}dt\right|\rightarrow
V[\gamma^{*}].$ (3.7)

In particular $\mathcal{I}[\gamma^{(n)}]\rightarrow\mathcal{I}[\gamma^{*}]$.
So, the area, volume and isoperimetric ratio are actually continuous with
respect to the convergence from Definition 3.10. Moreover, in Lemmas B.1 and
B.2 in the appendix we prove

$\displaystyle\mathcal{W}[\gamma^{*}]\leq\liminf_{n\rightarrow\infty}\mathcal{W}[\gamma^{(n)}],$
(3.8)
$\displaystyle\int_{0}^{1}\left((k_{1}^{*})^{2}+(k_{2}^{*})^{2}\right)2\pi|\dot{\gamma}^{*}|\gamma_{1}^{*}dt\leq\liminf_{n\rightarrow\infty}\int_{0}^{1}\left((k_{1}^{(n)})^{2}+(k_{2}^{(n)})^{2}\right)2\pi|\dot{\gamma}^{(n)}|\gamma_{1}^{(n)}dt.$
(3.9)

###  3.3 Compactness

The following compactness theorem is essentially due to Choski’s and
Veneroni’s paper [4] (see Subsection 3.4). However, as we have slightly
changed the definition of convergence in $\mathcal{P}^{w}$, their proof must
also be slightly modified. For the reader’s convenience, we provide the proof
in Appendix B.2.

######  Theorem 3.11 (Compactness Theorem).

Let $(\gamma^{(n)})\subset\mathcal{P}^{w}$ such that:

  1. (1)

There exists $R>0$ such that $\gamma^{(n)}(t)\in B_{R}(0)$ for all $t\in[0,1]$
and $n\in\mathbb{N}$.

  2. (2)

There exists $C<\infty$ such that

$\sup_{n}\int_{0}^{1}\left[\left(k_{1}^{(n)}\right)^{2}+\left(k_{2}^{(n)}\right)^{2}\right]2\pi\gamma_{1}^{(n)}|\dot{\gamma}^{(n)}|dt\leq
C.$

  3. (3)

There exist $\theta>0$ and $C<\infty$ such that $0<\theta\leq
A[\gamma^{(n)}]\leq C$ for all $n\in\mathbb{N}$.

Then there exists $\gamma^{*}\in\mathcal{P}^{w}$ and a subsequence
$\gamma^{(n_{k})}$ such that $\gamma^{(n_{k})}\rightarrow\gamma^{*}$ in
$\mathcal{P}^{w}$.

###  3.4 Existence of a Weak Minimizer

To prove the existence of a minimizer $\gamma\in\mathcal{F}_{\sigma}$
satisfying $\mathcal{W}[\gamma]=\beta(\sigma)$, we require the following
theorem, that is essentially due to Schygulla [18].

######  Theorem 3.12 (Schygulla).

Let $\beta$ be as in Equation (3.1). Then $\beta(\sigma)<8\pi$ for all
$\sigma\in(0,1]$.

Schygulla proves this for surfaces without requiring axial symmetry. Therefore
Theorem 3.12 cannot be cited directly from his paper. However, inspecting his
proof, it is straightforward to see that it can easily be adapted for our
purposes. We comment on the necessary changes in Appendix A.4.

Using Theorem 3.12, we can now prove the existence of a weak minimizer.

######  Theorem 3.13 (Existence of Weak Minimizer).

For all $\sigma\in(0,1]$ there exists $\gamma\in\mathcal{F}_{\sigma}$ such
that $\beta(\sigma)=\mathcal{W}[\gamma]$.

###### Proof.

For $\sigma=1$, any round sphere is a minimizer. For $\sigma\in(0,1)$ let
$\gamma^{(n)}\subset\mathcal{P}$ such that
$\mathcal{W}[\gamma^{(n)}]\rightarrow\beta(\sigma)$. As $\beta(\sigma)<8\pi$,
we can assume without loss of generality that $\mathcal{W}[\gamma^{(n)}]\leq
8\pi-\rho_{0}$ for all $n\in\mathbb{N}$ and some $\rho_{0}>0$. The Willmore
energy $\mathcal{W}$ and the isoperimetric ratio $\mathcal{I}$ are invariant
under translations. So, by defining

$\tilde{\gamma}^{(n)}(t):=\gamma^{(n)}(t)-(0,\gamma^{(n)}_{2}(0)),$

we can further assume that $\gamma^{(n)}(0)=0$ for all $n\in\mathbb{N}$. Let
$L_{n}:=L[\gamma^{(n)}]$. For every individual $\gamma^{(n)}$, the Gauß-Bonnet
theorem (see Lemma 3.7) holds and hence

$\displaystyle\int_{0}^{1}\left(\left(k_{1}^{(n)}\right)^{2}+\left(k_{2}^{(n)}\right)^{2}\right)2\pi
L_{n}\gamma_{1}^{(n)}(t)dt$ $\displaystyle=$ $\displaystyle
4\mathcal{W}[\gamma^{(n)}]-2\int_{0}^{1}k_{1}^{(n)}k_{2}^{(n)}2\pi
L_{n}\gamma_{1}^{(n)}(t)dt$ $\displaystyle\leq$ $\displaystyle
4\cdot(8\pi-\rho_{0})-8\pi.$ (3.10)

In view of Theorem 3.5, we get $L_{n}\leq C$. For any $t\in[0,1]$ and
$n\in\mathbb{N}$ we can now estimate

$|\gamma^{(n)}(t)|=|\gamma^{(n)}(t)-\gamma^{(n)}(0)|\leq\int_{0}^{1}|\dot{\gamma}^{(n)}(t)|dt\leq
C.$

Finally $A[\gamma^{(n)}]\equiv 1$. Thus, all assumptions from the compactness
theorem (Theorem 3.11) are satisfied and we can extract a subsequence, again
denoted by $\gamma^{(n)}$, that converges to a limit curve
$\gamma^{*}\in\mathcal{P}^{w}$. Let $L_{*}:=L[\gamma^{*}]$. We will now prove
that $\gamma^{*}\in\mathcal{P}$. Using the lower semicontinuity results from
Equations (3.8) and (3.9) and Estimate (3.10), we get

$\displaystyle\mathcal{W}[\gamma^{*}]\leq\beta(\sigma)\hskip
14.22636pt\textrm{and}\hskip
14.22636pt\int_{0}^{1}\left(\left(k_{1}^{*}\right)^{2}+\left(k_{2}^{*}\right)^{2}\right)2\pi
L_{*}\gamma_{1}^{*}(t)dt$ $\displaystyle<3\cdot 8\pi.$ (3.11)

So, by Theorem 3.3, there is at most one $\tau\in(0,1)$ such that
$\gamma^{*}_{1}(\tau)=0$. Suppose that such a $\tau$ exists. Then, by Theorem
3.3, $\gamma^{*}$ restricted to $[0,\tau]$ and $[\tau,1]$, up to
reparameterization, belong to $\mathcal{P}$. Applying the Li-Yau-inequality
(Lemma 3.6) produces the contradiction

$8\pi-\rho_{0}\geq\mathcal{W}[\gamma^{*}]=\mathcal{W}[\gamma^{*}|_{[0,\tau]}]+\mathcal{W}[\gamma^{*}|_{[\tau,1]}]\geq
2\cdot 4\pi.$ (3.12)

This shows $\gamma_{1}^{*}(t)>0$ for all $t\in(0,1)$ and hence
$\gamma^{*}\in\mathcal{P}$. Using the continuity of the area and volume
functional from Equations (3.6) and (3.7), we deduce
$\gamma^{*}\in\mathcal{F}_{\sigma}$ and therefore
$\mathcal{W}[\gamma^{*}]\geq\beta(\sigma)$. In view of Estimate (3.11) we get
$\mathcal{W}[\gamma^{*}]=\beta(\sigma)$ and hence $\gamma^{*}$ is a minimizer.
∎

##  4 Regularity of Minimizers

In this section, we prove that the weak minimizers from Theorem 3.13 are
smooth.

######  Theorem 4.1.

Let $\sigma\in(0,1]$ and $\gamma\in\mathcal{F}_{\sigma}$ such that
$\beta(\sigma)=\mathcal{W}[\gamma]$. Then $\Sigma_{\gamma}$ is a smooth
surface.

For $\sigma=1$, we have already shown that the minimizers in $\mathcal{P}$ are
round spheres. So it suffices to consider the case $\sigma\in(0,1)$. The proof
is split into two parts. In the first step, we prove that the profile curve
$\gamma$ is smooth on $(0,1)$. This shows that $\Sigma_{\gamma}$ is smooth
away from the axis of rotation. In the second step, we consider the two
intersection points $\gamma_{2}(0)e_{3}$ and $\gamma_{2}(1)e_{3}$ of
$\Sigma_{\gamma}$ with the axis of rotation and show that here locally
$\Sigma_{\gamma}$ is a smooth graph.

###  4.1 Euler-Lagrange Equation

Our main tool for improving the regularity of $\Sigma_{\gamma}$ is a suitable
Euler-Lagrange equation. However, deriving this is a little involved when we
restrict ourselves to the class $\mathcal{P}$ due to the many restrictions
imposed on curves $\gamma\in\mathcal{P}$.

######  Definition 4.2 (Generalized Profile Curves).

We say that $\gamma:[0,1]\rightarrow\mathcal{H}^{2}$ is a _generalised profile
curve_ if

  1. (1)

$\gamma\in C^{1}([0,1])\cap W^{2,2}_{\operatorname{loc}}((0,1))$,

  2. (2)

$\gamma_{1}(0)=\gamma_{1}(1)=0$ and $\gamma_{1}(t)>0$ for all $t\in(0,1)$,

  3. (3)

$\Sigma_{\gamma}$ has curvature bounded in $L^{2}(\Sigma_{\gamma})$:
$\int_{\mathbb{S}^{2}}k_{1}^{2}+k_{2}^{2}d\mu_{\Sigma_{\gamma}}<\infty$.

We denote the set of all generalized profile curves by $\mathcal{P}^{\prime}$.

Clearly $\mathcal{P}\subset\mathcal{P}^{\prime}$. We consider the
isoperimetric ratio functional

$\mathcal{I}:\mathcal{P}^{\prime}\rightarrow\mathbb{R}_{0}^{+},\
\mathcal{I}[\gamma]:=6\sqrt{\pi}\frac{V[\gamma]}{A[\gamma]^{\frac{3}{2}}}\hskip
14.22636pt\textrm{ and put }\hskip 14.22636pt\
\mathcal{F}^{\prime}_{\sigma}:=\set{\gamma\in\mathcal{P}^{\prime}\ }{\
\mathcal{I}[\gamma]=\sigma}.$

As both the Willmore energy $\mathcal{W}$ and the isoperimetric ratio
$\mathcal{I}$ are invariant under scaling and reparameterization, we have

$\beta(\sigma)=\inf\left\\{\mathcal{W}[\gamma]\ |\
\gamma\in\mathcal{F}_{\sigma}\right\\}=\inf\left\\{\mathcal{W}[\gamma]\ |\
\gamma\in\mathcal{F}^{\prime}_{\sigma}\right\\}.$

Consequently, given a minimizer $\gamma\in\mathcal{F}_{\sigma}$ and a
variation
$\Phi:(-\epsilon_{0},\epsilon_{0})\rightarrow\mathcal{F}^{\prime}_{\sigma}$
satisfying $\Phi(0)=\gamma$ we have

$0=\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}\mathcal{W}[\Phi(\epsilon)]=\delta\mathcal{W}[\gamma]\Phi^{\prime}(0).$
(4.1)

#### Variation Formulas

In the following, we require the variation of the isoperimetric ratio
functional. Let $\gamma\in\mathcal{P}^{\prime}$ with $V[\gamma]>0$ and
$\phi\in C^{\infty}([0,1])$, not necessarily vanishing at $t=0,1$. Using the
definition of $\mathcal{I}$ as well as Equations (2.17) and (2.18) it is easy
to see that

$\displaystyle\delta\mathcal{I}[\gamma]\phi$
$\displaystyle=\frac{6\sqrt{\pi}}{A[\gamma]^{\frac{3}{2}}}\left[\delta
V[\gamma]\phi-\frac{3}{2}\frac{V[\gamma]}{A[\gamma]}\delta
A[\gamma]\phi\right],$ (4.2) $\displaystyle\delta V[\gamma]\phi$
$\displaystyle=\pm_{\gamma}\pi\int_{0}^{1}\gamma_{1}^{2}(t)\dot{\phi}_{2}+2\gamma_{1}(t)\dot{\gamma}_{2}(t)\phi_{1}(t)dt,$
(4.3) $\displaystyle\delta A[\gamma]\phi$
$\displaystyle=2\pi\int_{0}^{1}|\dot{\gamma}(t)|\phi_{1}(t)+\frac{\gamma_{1}(t)\langle\dot{\phi}(t),\dot{\gamma}(t)\rangle}{|\dot{\gamma}(t)|}dt.$
(4.4)

The $\pm_{\gamma}$ in Equation (4.3) is the sign of the integral in Equation
(2.18). Now suppose that $\phi\in C^{\infty}_{0}((0,1))$. Then, as $\gamma\in
W^{2,2}_{\operatorname{loc}}((0,1))$, we can integrate the integrals in
Equations (4.3) and (4.4) by parts. Using $\nu$ from Equation (2.13) we get

$\displaystyle\delta V[\gamma]\phi$
$\displaystyle=\mp_{\gamma}2\pi\int_{0}^{1}\langle\nu,\phi\rangle|\dot{\gamma}|\gamma_{1}dt,$
(4.5) $\displaystyle\delta A[\gamma]\phi$
$\displaystyle=-2\pi\int_{0}^{1}H[\gamma]\langle\nu,\phi\rangle|\dot{\gamma}|\gamma_{1}dt.$
(4.6)

The following lemma proves that the constraint of prescribed isoperimetric
ration $\sigma\neq 1$ is non-degenerate.

######  Lemma 4.3.

Let $\gamma\in\mathcal{P}$. Either $\Sigma_{\gamma}$ is a sphere or there
exists $\psi_{0}\in C_{0}^{\infty}((0,1))$ such that

$\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}\mathcal{I}[\gamma+\epsilon\psi_{0}]=1.$

###### Proof.

If no $\phi\in C_{0}^{\infty}((0,1))$ satisfying
$\delta\mathcal{I}[\gamma]\phi\neq 0$ exists, then we can use Equations (4.2),
(4.5) and (4.6) to get

$\pm_{\gamma}2\pi\int_{0}^{1}\langle\nu,\phi\rangle|\dot{\gamma}|\gamma_{1}dt=2\pi\frac{3V[\gamma]}{2A[\gamma]}\int_{0}^{1}H[\gamma]\langle\nu,\phi\rangle|\dot{\gamma}|\gamma_{1}dt\hskip
14.22636pt\textrm{for all }\phi\in C^{\infty}_{0}((0,1)).$

By the fundamental lemma of the calculus of variations analysis
$H\equiv\pm_{\gamma}\frac{2A[\gamma]}{2V[\gamma]}$ and by Lemma 3.8 – the Hopf
theorem – $\Sigma_{\gamma}$ is a sphere. ∎

###  4.2 Regularity Away from the Axis of Rotation

Let $\sigma\in(0,1)$ and $\gamma\in\mathcal{F}_{\sigma}$ such that
$\mathcal{W}[\gamma]=\beta(\sigma)$. We derive a suitable Euler-Lagrange
equation for $\gamma$ on $(0,1)$. Using Lemma 4.3, we choose $\psi_{0}\in
C_{0}^{\infty}((0,1))$ such that $\delta\mathcal{I}[\gamma]\psi_{0}=1$. For
any smooth $\eta\in C_{0}^{\infty}((0,1))$ we consider the vector field

$\tilde{\eta}:=\eta-\alpha(\eta)\psi_{0}\hskip 14.22636pt\textrm{with}\hskip
14.22636pt\alpha(\eta):=\delta\mathcal{I}[\gamma]\eta.$

In Lemma A.1 we prove that for any such $\eta$ there exists a variation
$\Phi_{\eta}:(-\epsilon_{0},\epsilon_{0})\rightarrow\mathcal{F}_{\sigma}^{\prime}$
such that $\Phi_{\eta}^{\prime}(0)=\tilde{\eta}$. So, by Equation (4.1) we get

$0=\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}\mathcal{W}[\Phi_{\eta}(\epsilon)]=\delta\mathcal{W}[\gamma]\tilde{\eta}\hskip
14.22636pt\textrm{and hence}\hskip
14.22636pt\delta\mathcal{W}[\gamma]\eta=\left(\delta\mathcal{W}[\gamma]\psi_{0}\right)\delta\mathcal{I}[\gamma]\eta.$
(4.7)

Using this Euler-Lagrange equation, we can now derive regularity of $\gamma$
as long as we _‘stay away from the axis of rotation’_. To do so, we require a
general regularity result, which we have moved to Appendix C.

######  Corollary 4.4.

Let $\sigma\in(0,1)$, $\gamma\in\mathcal{F}_{\sigma}$ such that
$\mathcal{W}[\gamma]=\beta(\sigma)$, $0<\tau_{1}<\tau_{2}<1$ and
$U:=(\tau_{1},\tau_{2})$. Assume that
$\kappa_{1}(\gamma;U)=\inf_{U}\gamma_{1}>0$ and that there exists $\psi_{0}\in
C^{2}_{0}(U)$ such that $\delta\mathcal{I}[\gamma]\psi_{0}=1$. Then $\gamma\in
C^{\infty}(U)$ and for all $m\in\mathbb{N}$

$\|\gamma-\gamma(0)\|_{C^{m}((\tau_{1},\tau_{2}))}\leq
C(m,\kappa(\gamma;U),\|\psi_{0}\|_{C^{2}(U)}).$

The constant is decreasing in the second slot.

###### Proof.

Without loss of generality, we assume $\gamma(0)=0$. We wish to apply Theorem
C.1. To do so, we first establish an appropriate differential equation. Using
Equation (4.7), we get

$\delta\mathcal{W}[\gamma]=\left(\delta\mathcal{W}[\gamma]\psi_{0}\right)\delta\mathcal{I}[\gamma](\phi)=I(\phi)\hskip
14.22636pt\textrm{for all }\phi\in C^{\infty}_{0}(U,\mathbb{R}^{2}).$

Here $I(\phi)$ is an in Equation (C.1) with parameters $\alpha$ and $\beta$
that, in view of Equations (4.2), (4.3) and (4.4), are given as

$\alpha=\pm_{\gamma}\frac{6\sqrt{\pi}^{3}}{A[\gamma]^{\frac{3}{2}}}\left(\delta\mathcal{W}[\gamma]\psi_{0}\right)\hskip
14.22636pt\textrm{and}\hskip
14.22636pt\beta=-\frac{3\pi}{A[\gamma]}\frac{6\sqrt{\pi}V[\gamma]}{A[\gamma]^{\frac{3}{2}}}\left(\delta\mathcal{W}[\gamma]\psi_{0}\right).$

Next, we collect the appropriate bounds to satisfy assumption (C.3). We have
$\mathcal{W}[\gamma]=\beta(\sigma)\leq 8\pi$. Using $A[\gamma]=1$, Theorem 3.5
and Estimate (3.4), we get $L[\gamma]+L[\gamma]^{-1}\leq C$. Combining the
bound for $L[\gamma]$ with Lemma 3.9, we have
$\|\ddot{\gamma}\|_{L^{2}(U)}\leq C(\kappa(\gamma;U),U)$. Since
$|\dot{\gamma}|=L[\gamma]\leq C$ and we assume $\gamma(0)=0$, we obtain
$\|\gamma\|_{C^{0}}\leq C$.

To estimate $\alpha$ and $\beta$ we recall $A[\gamma]=1$ and
$V[\gamma]=\frac{\sigma}{6\sqrt{\pi}}$. It remains to estimate
$|\delta\mathcal{W}[\gamma]\psi_{0}|$. Considering Equations (2.15) and (2.16)
it is easy to see that

$|\delta\mathcal{W}[\gamma]\psi_{0}|\leq
C(\|\gamma\|_{W^{1,\infty}(U)},\|\ddot{\gamma}\|_{L^{2}(U)},\kappa(\gamma;U))\|\psi_{0}\|_{C^{2}(U)}.$

In fact, we essentially establish this estimate in Step 2 in the proof of
Lemma C.2.

Summarising the above, we see that we can satisfy Estimate (C.3) with a
parameter $M$ that depends only on $U$, $\kappa(\gamma;U)$ and
$\|\psi_{0}\|_{C^{2}(U)}$. So, Theorem C.1 implies the corollary. ∎

Combining Lemma 4.3, $\gamma_{1}(t)>0$ for all $t\in(0,1)$ and Corollary 4.4,
we immediately get:

######  Corollary 4.5.

Let $\sigma\in(0,1)$ and $\gamma\in\mathcal{F}_{\sigma}$ such that
$\mathcal{W}[\gamma]=\beta(\sigma)$. Then $\gamma\in C^{\infty}((0,1))$.

###  4.3 Regularity At the Axis of Rotation

We continue to fix $\sigma\in(0,1)$ and $\gamma\in\mathcal{F}_{\sigma}$ such
that $\mathcal{W}[\gamma]=\beta(\sigma)$. By Definition 3.1 and Corollary 4.5
we have $\gamma\in C^{\infty}((0,1))\cap C^{1}([0,1])$. The main ingredient in
the discussion of the regularity of $\Sigma_{\gamma}$ near the axis of
rotation is the study of an appropriate Euler-Lagrange equation:

######  Lemma 4.6 (Euler-Lagrange Equation).

There exists $\Lambda\in\mathbb{R}$ such that $\gamma$ satisfies the following
Euler-Lagrange equation on $(0,1)$:

$\Delta_{g}H+\frac{1}{2}H(H^{2}-4K)=\Lambda(\pm_{\gamma}4\sqrt{\pi}-\sigma H)$

###### Proof.

Let $\phi\in C^{\infty}_{0}((0,1))$ and $\psi_{0}$ be as in Lemma 4.3. Then
$\delta\mathcal{W}[\gamma](\phi-\alpha(\phi)\psi_{0})=0$ by Equation (4.7) .
As $\phi$ is supported in $(0,1)$ where $\gamma$ is smooth, we can use
Equation (A.9) for the variation of Willmore energy from the appendix to get

$\delta\mathcal{W}[\gamma]\phi=2\pi\int_{0}^{1}W[\gamma]\langle\phi,\nu\rangle\gamma_{1}|\dot{\gamma}|dt.$

We put $\Lambda:=-3(\delta\mathcal{W}[\gamma]\psi_{0})$ and recall from
Equations (4.2), (4.5) and (4.6) that

$\alpha(\phi)=\delta\mathcal{I}[\gamma]\phi=-2\pi\sigma\int_{0}^{1}\left(\pm_{\gamma}\frac{1}{V[\gamma]}-\frac{3}{2}\frac{H}{A[\gamma]}\right)\langle\phi,\nu\rangle\gamma_{1}|\dot{\gamma}|ds.$

Using $A[f]=1$, $V[f]=\frac{\sigma}{6\sqrt{\pi}}$, the explicit formula for
the Willmore operator as well as the fundamental lemma of the calculus of
variations, we get

$\frac{1}{2}\left(\Delta_{g}H+\frac{1}{2}H(H^{2}-4K)\right)=W[\gamma]=\frac{1}{3}\Lambda\sigma\left(\pm_{\gamma}\frac{6\sqrt{\pi}}{\sigma}-\frac{3}{2}H\right).$

∎

To continue the investigation of the regularity of $\Sigma_{\gamma}$, we will
now introduce a more suitable description of the surface $\Sigma_{\gamma}$
close to $(0,0,\gamma_{2}(0))$.

#### Graph Representation near $(0,0,\gamma_{2}(0))$

Near $(0,0,\gamma_{2}(0))$ we can write the surface $\Sigma_{\gamma}$ as a
graph

$(r\cos\theta,r\sin\theta,u(r))\hskip 14.22636pt\textrm{where}\hskip
14.22636ptr\in[0,r_{0}),\ \theta\in[0,2\pi).$

Indeed, putting $r(t):=\gamma_{1}(t)$ we have $r\in C^{1}([0,1])$ and
$r^{\prime}(0)=L[\gamma]\neq 0$. By the inverse function theorem, there exists
$t_{0}>0$ such that $\gamma_{1}:[0,t_{0}]\rightarrow\gamma_{1}([0,t_{0}])$ is
a $C^{1}$-diffeomorphism and we may define

$u:[0,r_{0})\rightarrow\mathbb{R},\ u(r):=\gamma_{2}(\gamma_{1}^{-1}(r))\hskip
14.22636pt\textrm{where }r_{0}=\gamma_{1}(t_{0}).$

Since $\gamma\in C^{\infty}((0,t_{0}])$, we get $u\in C^{1}([0,r_{0}])\cap
C^{\infty}((0,r_{0}])$. Additionally, $\dot{\gamma}_{2}(0)=0$ implies
$u^{\prime}(0)=0$.

Throughout the rest of this subsection, we fix $u$ as above.

We now wish to write the Euler-Lagrange equation from Lemma 4.6 in the graph
parameterization. Using a result from [6], the authors of [3] derive an
elegant formula for the Willmore operator. Letting $w:=u^{\prime}(r)$ and
$v:=\sqrt{1+w^{2}}$ we deduce that $w$ solves the following ODE on
$(0,r_{0})$:

$\frac{1}{r}\frac{d}{dr}\left[r\frac{1}{v^{5}}\left(w^{\prime\prime}+\left(\frac{w}{r}\right)^{\prime}-\varphi(w)\right)\right]=W[\gamma]=a-bH$
(4.8)

In view of Lemma 4.6 $a$ and $b$ are explicitly given by
$a=\pm_{\gamma}4\sqrt{\pi}\Lambda$ and $b=\sigma\Lambda$. Furthermore,
$\varphi(w)$ is computed in Lemma 2.2 in [3] and given by

$\varphi(w):=\frac{5w}{2(1+w^{2})}(w^{\prime})^{2}+\frac{w^{3}}{2r^{2}}(3+w^{2}).$
(4.9)

Note from Equation (2.9), that $H$ is also of the form
$r^{-1}\partial_{r}(...)$. This allows us to reduce the order of the ODE in
Equation (4.8). Indeed, multiplying Equation (4.8) by $r$, inserting Equation
(2.9) for $H$, integrating and finally multiplying by $\frac{v^{5}}{r}$ shows
that there exists $\lambda\in\mathbb{R}$ such that

$\left\\{\begin{aligned}
&w^{\prime\prime}+\left(\frac{w}{r}\right)^{\prime}-\varphi(w)=\frac{1}{2}v^{5}ar+bv^{4}w+\frac{\lambda}{r}v^{5},\\\
&w(0)=0.\end{aligned}\right.$ (4.10)

#### Regularity of $u$

$u$ satisfies Equation (4.10) on $(0,r_{0})$. We note that Equation (4.10)
allows for nonsmooth solutions. For example, the first remark in Section 3 in
[12] discusses that the inverted catenoid solves (4.10) on $(0,r_{0})$ with
$a=b=0$. It is also shown that the graph function $u_{ic}$ describing the
inverted catenoid satisfies $u_{ic}\in C^{1,\alpha}([0,r_{0}])$ for all
$\alpha\in(0,1)$ and $u_{ic}^{\prime\prime}\in L^{p}((0,r_{0}))$ for all
$p\in[1,\infty)$ but $u_{ic}\not\in C^{1,1}((0,r_{0}))$. In the next lemma, we
prove that our solution $u$ possesses at least the same regularity.

The following lemma is essentially the same as Lemma 3.3 in [3]. Here the same
analysis is carried out for the unconstrained Willmore equation. We show that
the same argument can be applied when including the Lagrange multiplier in
Equation (4.10).

######  Lemma 4.7.

There exists $\xi\in C^{1}([0,r_{0}])$ such that
$u^{\prime}(r)=\frac{1}{2}\lambda r\ln(r)+\xi(r)$. In particular,
$u^{\prime}\in C^{0,\alpha}([0,r_{0}])$ and $u^{\prime\prime}\in
L^{p}((0,r_{0}))$ for all $\alpha\in(0,1)$ and $p\in[1,\infty)$.

###### Proof.

The proof is separated into six steps.

Step 1
Note $w\varphi(w)\geq 0$ by the definition of $\varphi(w)$ in Equation (4.9).
We multiply Equation (4.10) by $w$, choose any $\rho\in(0,r_{0})$ and
integrate:

$\int_{r}^{\rho}w^{\prime\prime}w+\left(\frac{w}{t}\right)^{\prime}wdt\geq\frac{a}{2}\int_{r}^{\rho}v^{5}twdt+b\int_{r}^{\rho}v^{4}w^{2}dt+\lambda\int_{r}^{\rho}\frac{v^{5}}{t}wdt$

We already know that $u$ is $C^{1}([0,r_{0}))$ and satisfies
$u^{\prime}(0)=0$. Hence $v$ and $w$ are bounded and we get the estimate

$\int_{r}^{\rho}w^{\prime\prime}w+\left(\frac{w}{t}\right)^{\prime}wdt\geq-C\left(1+\int_{r}^{\rho}\frac{|w|}{t}dt\right).$

$C$ depends on $\sigma$, $\lambda$ and on the Lagrange multiplier. We will not
make this dependence explicit in the notation. Integrating by parts gives

$-w^{\prime}(r)w(r)-\frac{w(r)^{2}}{r}-\int_{r}^{\rho}(w^{\prime}(t))^{2}+\frac{ww^{\prime}}{t}dt\geq-C(\rho)\left(1+\int_{r}^{\rho}\frac{|w|}{t}dt\right).$

We have included the boundary terms at $t=\rho$ in the constant $C(\rho)$.
Note that $C(\rho)<\infty$ for all $\rho\in(0,r_{0})$ as $w\in
C^{\infty}((0,r_{0}))$. Next, we integrate by parts again and get

$-w^{\prime}(r)w(r)-\frac{w(r)^{2}}{r}+\frac{w(r)^{2}}{2r}-\int_{r}^{\rho}(w^{\prime}(t))^{2}+\frac{w(t)^{2}}{2t^{2}}dt\geq-C(\rho)\left(1+\int_{r}^{\rho}\frac{|w|}{t}dt\right).$

Rearranging this estimate gives

$\displaystyle\int_{r}^{\rho}(w^{\prime}(t))^{2}+\frac{w(t)^{2}}{2t^{2}}dt$
$\displaystyle\leq
C(\rho)\left(1+\int_{r}^{\rho}\frac{|w|}{t}dt\right)+\frac{w(r)^{2}}{2r}-w(r)\frac{w(r)+rw^{\prime}(r)}{r}$
$\displaystyle=C(\rho)\left(1+\int_{r}^{\rho}\frac{|w|}{t}dt\right)+\frac{w(r)^{2}}{2r}-w(r)\frac{(rw(r))^{\prime}}{r}.$
(4.11)

Step 2
Let again $\rho\in(0,r_{0})$. This time we integrate Equation (4.10) directly
and use $w^{\prime}+\frac{w}{r}=\frac{1}{r}(wr)^{\prime}$ to get

$\frac{(wt)^{\prime}}{t}\bigg{|}_{t=\rho}-\frac{(wr)^{\prime}}{r}=\int_{r}^{\rho}\varphi(w)+\lambda\frac{v^{5}-1}{t}+\frac{1}{2}av^{5}t+bv^{4}wdt-\lambda\ln\frac{r}{\rho}.$
(4.12)

Let $\epsilon>0$. Since $w=u^{\prime}\in C^{0}([0,r_{0}))$ and $w(0)=0$, we
can choose $\rho(\epsilon)$ small enough such that $|w(r)|\leq\epsilon$ for
all $r\in[0,\rho(\epsilon)]$. For $\rho\leq\rho(\epsilon)$ we get

$\displaystyle\left|\frac{(rw(r))^{\prime}}{r}\right|$ $\displaystyle\leq
C(\epsilon)+\left|\lambda\ln\frac{r}{\rho}\right|+\int_{r}^{\rho}\frac{5|w(t)|w^{\prime}(t)^{2}}{2(1+w(t)^{2})}+w(t)^{2}\frac{3|w(t)|+|w(t)|^{3}}{2t^{2}}+|\lambda|\left|\frac{v^{5}-1}{t}\right|dt$
$\displaystyle\hskip 14.22636pt+C\int_{r}^{\rho}tv^{5}+v^{4}|w|dt.$

The constant $C(\epsilon)$ is due to the boundary terms at $\rho(\epsilon)$.
The second constant $C$ does not depend on $\epsilon$. Using
$u^{\prime}(0)=w(0)=0$, we can shrink $\rho(\epsilon)$ to achieve

$\displaystyle\left|\frac{(rw(r))^{\prime}}{r}\right|$ $\displaystyle\leq
C(\epsilon)\ln\frac{1}{r}+\epsilon\int_{r}^{\rho}w^{\prime}(t)^{2}+\frac{w(t)^{2}}{2t^{2}}dt\hskip
14.22636pt\textrm{for all }r\in(0,\rho(\epsilon)).$ (4.13)

Step 3
Let $\epsilon>0$ be arbitrary. By taking $\rho=\rho(\epsilon)$ small enough in
Estimate (4.11) we may insert Estimate (4.13) to get

$\displaystyle\int_{r}^{\rho}w^{\prime}(t)^{2}+\frac{w(t)^{2}}{2t^{2}}dt$
$\displaystyle\leq
C(\epsilon)\left(1+\int_{r}^{\rho}\frac{|w|}{t}dt\right)+\frac{w(r)^{2}}{2r}$
$\displaystyle\hskip
28.45274pt+|w(r)|\left(C(\epsilon)\ln\frac{1}{r}+\epsilon\int_{r}^{\rho}w^{\prime}(t)^{2}+\frac{w(t)^{2}}{2t^{2}}dt\right).$

By potentially shrinking $\rho$ we can assume without loss of generality that
$|w|\leq\epsilon$ on $[0,\rho]$ and so, in particular $|w(r)|\leq\epsilon$.
Choosing $\epsilon$ small enough, we can absorb the second integral on the
right to the left and obtain that for some small enough $\rho$ and
$r\in(0,\rho)$

$\int_{r}^{\rho}w^{\prime}(t)^{2}+\frac{w(t)^{2}}{2t^{2}}dt\leq
C(\rho)\left(\frac{w(r)^{2}}{2r}+|w(r)|\ln\frac{1}{r}+\int_{r}^{\rho}\frac{|w(t)|}{t}dt+1\right).$
(4.14)

Step 4
Let $\rho$ be small enough so that Estimate (4.14) holds. Inserting Estimate
(4.14) into Estimate (4.13) with $\epsilon=1$ and using that $|w|$ is bounded,
we get

$\displaystyle\left|\frac{(rw(r))^{\prime}}{r}\right|\leq$ $\displaystyle
C\ln\frac{1}{r}+C\left(\frac{w(r)^{2}}{2r}+|w(r)|\ln\frac{1}{r}+\int_{r}^{\rho}\frac{|w(t)|}{t}dt+1\right)$
$\displaystyle\leq$ $\displaystyle
C\ln\frac{1}{r}+C\left(\frac{w(r)^{2}}{2r}+\int_{r}^{\rho}\frac{1}{t}dt\right).$

We multiply by $r$. Clearly $|\ln(\rho)-\ln(r)|\leq C\ln(\frac{1}{r})$ as
$r\leq\rho$. Let $\mu\in(0,1)$. By potentially shrinking $\rho$ we can assume
that $|w|\leq C^{-1}\mu$ on $[0,\rho]$. Hence

$|(rw)^{\prime}|\leq C(w(r)^{2}+r\ln\frac{1}{r})\leq\mu|w|+Cr\ln\frac{1}{r}.$

In general, for any weakly differentiable $g$ we have
$|g|^{\prime}\leq|g^{\prime}|$. As $|rw|=r|w|$ we get

$r|w|^{\prime}+|w|=(r|w|)^{\prime}=|rw|^{\prime}\leq|(rw)^{\prime}|\leq\mu|w|+Cr\ln\frac{1}{r}\hskip
14.22636pt\textrm{for all $r\in(0,\rho]$}.$

Multiplying with $r^{1-\mu}$ we get

$(r^{1-\mu}|w|)^{\prime}=r^{1-\mu}|w|^{\prime}+(1-\mu)r^{-\mu}|w|\leq
Cr^{1-\mu}\ln\frac{1}{r}.$

We integrate this inequality from $r_{1}<r$ to $r$ and subsequently send
$r_{1}\rightarrow 0^{+}$. Since $w\in C^{0}([0,r_{0}))$, we get

$r^{1-\mu}|w(r)|\leq C\int_{0}^{r}t^{1-\mu}\ln\frac{1}{t}dt\leq
C(\mu)r^{2-\mu}\ln\frac{1}{r}.$

Taking e.g. $\mu=\frac{1}{2}$ shows that there exists $\rho_{0}>0$ such that

$|w(r)|\leq Cr\ln\frac{1}{r}\hskip 14.22636pt\textrm{for all }0<r<\rho_{0}.$
(4.15)

Step 5
Using Estimate (4.15) we deduce the following three estimates for
$r\in(0,\rho_{0})$:

$\displaystyle\frac{w(r)^{2}}{r}$ $\displaystyle\leq
C\frac{r^{2}\ln^{2}\frac{1}{r}}{r}\leq Cr\ln^{2}\frac{1}{r}$ (4.16)
$\displaystyle|w(r)|\ln\frac{1}{r}$ $\displaystyle\leq Cr\ln^{2}\frac{1}{r}$
(4.17) $\displaystyle\int_{0}^{r}\frac{|w(t)|}{t}dt$ $\displaystyle\leq
C\int_{0}^{r}\ln\frac{1}{t}dt\leq C$ (4.18)

After potentially shrinking $\rho_{0}$, we may use Estimate (4.14). Inserting
Estimates (4.16)-(4.18) into Estimate (4.14), we get

$\int_{r}^{\rho_{0}}w^{\prime}(t)^{2}+\frac{w(t)^{2}}{2t^{2}}dt\leq C.$ (4.19)

Potentially shrinking $\rho_{0}$ even further, we may assume
$|w|\leq\frac{1}{2}$ and, recalling that $v=\sqrt{1+w^{2}}$, we get

$\left|\frac{5w}{2(1+w^{2})}(w^{\prime})^{2}\right|\leq
C|w^{\prime}|^{2},\hskip
14.22636pt\left|\frac{3w^{3}+w^{5}}{2t^{2}}\right|\leq
C\frac{w^{2}}{t^{2}}\hskip 14.22636pt\textrm{and}\hskip
14.22636pt\left|\frac{v^{5}-1}{t}\right|\leq C\frac{w^{2}}{t}.$ (4.20)

Combining Estimates (4.19) and (4.20), we obtain

$\int_{0}^{\rho_{0}}\left|\frac{5w}{2(1+w^{2})}(w^{\prime})^{2}+\frac{3w^{3}+w^{5}}{2t^{2}}+\lambda\frac{v^{5}-1}{t}\right|dt<\infty.$
(4.21)

Step 6
We choose $\rho$ small enough, such that the results from all previous steps
are valid. Considering the integral appearing in Equation (4.12) from Step 2
we introduce

$\psi(r):=-\int_{r}^{\rho}\varphi(w)+\lambda\frac{v^{5}-1}{t}+\frac{1}{2}av^{5}t+bv^{4}wdt.$

Recalling the definition of $\varphi(w)$ from Equation (4.9) and using
Estimate (4.21) as well as $v\in C^{0}([0,\rho])$ we deduce $\psi\in
C^{0}([0,\rho])$. Rewriting Equation (4.12) we have shown that there exists a
constant $\tilde{c}_{0}\in\mathbb{R}$ such that

$(w(r)r)^{\prime}=r\psi(r)+\tilde{c}_{0}+\lambda r\ln(r).$

This implies that there exist constants $c_{0},c_{1}\in\mathbb{R}$ such that

$w(r)r=\int_{0}^{r}t\psi(t)dt+c_{0}r+c_{1}+\frac{1}{2}\lambda
r^{2}\ln(r)-\frac{1}{4}\lambda r^{2}.$

Letting $r\rightarrow 0^{+}$ we get that $c_{1}=0$ so that

$w(r)-\frac{1}{2}\lambda r\ln(r)-c_{0}+\frac{1}{4}\lambda
r=\frac{1}{r}\int_{0}^{r}t\psi(t)dt=:h(r).$

It remains to prove that $h\in C^{1}([0,\rho])$. First note, that $\psi\in
C^{0}([0,\rho])$ is bounded. Hence $h$ is continuous at $r=0$ with $h(0)=0$.
Indeed

$|h(r)|\leq\frac{C}{r}\int_{0}^{r}tdt\leq Cr\rightarrow 0,\hskip
14.22636pt\textrm{as $r\rightarrow 0^{+}$}.$

On $(0,\rho]$ we can differentiate the definition of $h$ to derive the
equation $\frac{(rh(r))^{\prime}}{r}=\psi(r)$. As $\psi\in C^{0}([0,\rho])$ we
may use Lemma D.1 from the appendix and get $h\in C^{1}([0,\rho])$. Finally,
as $u$ is smooth on $[\rho,r_{0}]$, the lemma follows. ∎

The next step is to derive an expansion of the mean curvature $H$.

######  Lemma 4.8.

There exists a function $H_{0}(r)\in C^{1,\alpha}([0,r_{0}])$ for all
$\alpha\in(0,1)$ such that $H(r)=-\lambda\ln(r)+H_{0}(r)$.

###### Proof.

Writing out the Willmore operator in Equation (4.8) in terms of $H$, we get

$\Delta_{g}H+\frac{1}{2}H(H^{2}-4K)=a-bH.$

We put $f(r):=a-bH-\frac{1}{2}H(H^{2}-4K)$. Using Equations (2.6) and (2.9) we
may write

$\frac{1}{r\sqrt{1+u^{\prime}(r)^{2}}}\frac{d}{dr}\left[\frac{r}{\sqrt{1+u^{\prime}(r)^{2}}}H^{\prime}(r)\right]=\Delta_{g}H(r)=f(r).$

Lemma 4.7 gives $u^{\prime}(r)=\frac{1}{2}\lambda r\ln(r)+\xi$ with $\xi\in
C^{1}([0,r_{0}])$. As $u^{\prime}(0)=0$ we have $\xi(0)=0$. Using Equations
(2.6) and (2.8) allows us to estimate both $|H(r)|\leq C\ln\frac{1}{r}$ and
$|K(r)|\leq C\ln^{2}\frac{1}{r}$. Therefore $|f(r)|\leq C\ln^{3}\frac{1}{r}$
and for all $p\in[1,\infty)$

$\frac{d}{dr}\left[\frac{r}{\sqrt{1+u^{\prime}(r)^{2}}}H^{\prime}(r)\right]=r\sqrt{1+u^{\prime}(r)^{2}}f(r)=:z(r)\in
L^{p}((0,r_{0})).$ (4.22)

Since $u\in C^{\infty}((0,r_{0}])\cap C^{1}([0,r_{0}])$, $|H|\leq
C\ln\frac{1}{r}$ and $|K|\leq C\ln^{2}\frac{1}{r}$ we get $z\in
C^{\infty}((0,r_{0}])\cap C^{0}([0,r_{0}])$. Integrating Equation (4.22)
implies that there exists $c_{0}\in\mathbb{R}$ such that

$H^{\prime}(r)=\frac{c_{0}}{r}\sqrt{1+u^{\prime}(r)^{2}}+\frac{\sqrt{1+u^{\prime}(r)^{2}}}{r}\int_{0}^{r}z(s)ds.$

We put $H_{0}(r):=H(r)-c_{0}\ln(r)$. Clearly $H_{0}\in C^{\infty}((0,r_{0}])$
and we can compute

$\displaystyle H_{0}^{\prime\prime}(r)=$
$\displaystyle\frac{d}{dr}\left[c_{0}\frac{\sqrt{1+u^{\prime}(r)^{2}}-1}{r}+\frac{\sqrt{1+u^{\prime}(r)^{2}}}{r}\int_{0}^{r}z(s)ds\right]$
$\displaystyle=$ $\displaystyle-
c_{0}\frac{\sqrt{1+u^{\prime}(r)^{2}}-1}{r^{2}}+c_{0}\frac{u^{\prime}(r)u^{\prime\prime}(r)}{r\sqrt{1+u^{\prime}(r)^{2}}}-\frac{\sqrt{1+u^{\prime}(r)^{2}}}{r^{2}}\int_{0}^{r}z(s)ds$
$\displaystyle+\frac{u^{\prime}(r)u^{\prime\prime}(r)}{r\sqrt{1+u^{\prime}(r)^{2}}}\int_{0}^{r}z(s)ds+\frac{\sqrt{1+u^{\prime}(r)^{2}}}{r}z(r).$

We have $|u^{\prime}(r)|\leq Cr\ln\frac{1}{r}$ and $|u^{\prime\prime}(r)|\leq
C\ln\frac{1}{r}$. Also $|z(r)|\leq Cr\ln^{3}\frac{1}{r}$. This gives

$|H_{0}^{\prime\prime}(r)|\leq
C\left[\ln^{2}\frac{1}{r}+\frac{1}{r^{2}}\int_{0}^{r}s\ln^{3}\frac{1}{s}ds+\ln^{2}\frac{1}{r}\int_{0}^{r}s\ln^{3}\frac{1}{s}ds+\ln^{3}\frac{1}{r}\right]\leq
C\ln^{3}\frac{1}{r}.$

Hence $|H_{0}^{\prime\prime}(r)|\leq C\ln^{3}\frac{1}{r}\in L^{p}((0,r_{0}))$
for all $p\in[1,\infty)$. Using Lemma D.3 we get $H_{0}\in C^{1}([0,r_{0}])$
and thus $H_{0}\in W^{2,p}((0,r_{0}))$ for all $p<\infty$. By the Sobolev
embedding theorem we deduce $H_{0}\in C^{1,\alpha}([0,r_{0}])$ for all
$\alpha\in(0,1)$. Recalling the definition of $H_{0}$, this gives

$H(r)=c_{0}\ln(r)+H_{0}(r)\hskip 14.22636pt\textrm{with $H_{0}\in
C^{1,\alpha}([0,r_{0}])$}.$ (4.23)

It remains to check that $c_{0}=-\lambda$. We recall
$u^{\prime}(r)=\frac{\lambda}{2}r\ln(r)+\xi(r)$ where $\xi\in
C^{1}([0,r_{0}])$. In particular $u^{\prime}\in C^{0}([0,r_{0}])$. Using
$u^{\prime}(0)=0$ and Equation (2.8) we compute

$\frac{H(r)}{\ln(r)}=-\frac{u^{\prime\prime}(r)}{\ln(r)}\frac{1}{\sqrt{1+u^{\prime}(r)^{2}}^{3}}-\frac{u^{\prime}(r)}{r\ln(r)}\frac{1}{\sqrt{1+u^{\prime}(r)^{2}}}\rightarrow-\lambda\hskip
14.22636pt\textrm{as }r\rightarrow 0^{+}.$

In view of Equation (4.23), we obtain $c_{0}=-\lambda$. ∎

#### Higher Regularity

In the next lemma, we prove that the parameter $\lambda$ from Equation (4.10)
vanishes. In view of Lemma 4.7 we then get $u\in C^{2}([0,r_{0}])$. The key
ingredient that allows this further improvement of regularity is the fact that
the surface $\Sigma_{\gamma}$ minimizes the Willmore energy inside the class
$\mathcal{F}_{\sigma}$.

Recall that $\gamma_{1}$ is a $C^{1}$-diffeomorphism from $[0,t_{0}]$ onto
$[0,r_{0}]$ and that $u:[0,r_{0}]\rightarrow\mathbb{R},\
u(r):=\gamma_{2}(\gamma_{1}^{-1}(r))$. We put $r_{1}:=\frac{r_{0}}{2}$,
$t_{1}:=\gamma_{1}^{-1}(r_{1})\in(0,t_{0})$ and choose $\chi\in
C^{\infty}([0,1])$ such that $\chi\equiv 1$ on $[0,t_{1}]$ and $\chi\equiv 0$
on $[\frac{t_{0}+t_{1}}{2},1]$. For $\varphi\in C^{\infty}_{0}([0,r_{1}))$
satisfying $\varphi^{\prime}(0)=0$ we consider the variation

$\Phi:(-\epsilon_{0},\epsilon_{0})\times[0,1]\rightarrow\mathcal{H}^{2},\
\Phi(\epsilon,t):=\gamma(t)+\epsilon\chi(t)\varphi(\gamma_{1}(t))e_{2}\hskip
14.22636pt\textrm{and put}\hskip
14.22636pt\phi(t):=\frac{\partial\Phi}{\partial\epsilon}\bigg{|}_{\epsilon=0}.$

We note that $\Phi(\epsilon,t)\equiv\gamma(t)$ for $t\geq t_{1}$ and
$\Phi(\epsilon,\cdot)\in\mathcal{P}^{\prime}$ for all small $\epsilon$.
However, the variation $\Phi$ does not preserve the isoperimetric ratio.
Employing the function $\psi_{0}\in C^{\infty}_{0}((0,1))$ from Lemma 4.3 we
may define an admissible variational vector field by putting

$\eta:=\phi-\alpha(\phi)\psi_{0}\hskip 14.22636pt\textrm{with}\hskip
14.22636pt\alpha(\phi)=\delta\mathcal{I}[\gamma]\phi.$ (4.24)

We require the following lemma,

######  Lemma 4.9.

There exists a variation
$\tilde{\Phi}:(-\epsilon_{0},\epsilon_{0})\rightarrow\mathcal{F}_{\sigma}^{\prime}$
such that $\tilde{\Phi}^{\prime}(0)=\eta$. In particular
$\delta\mathcal{W}[\gamma]\eta=0$ by Equation (4.7). Additionally

$0=\delta\mathcal{I}[\gamma]\eta=-3\pi\lim_{\tau\rightarrow
0^{+}}\int_{\tau}^{1}\left(\pm_{\gamma}4\sqrt{\pi}-\sigma
H[\gamma]\right)\langle\nu,\eta\rangle L\gamma_{1}dt.$

###### Proof.

The existence of $\tilde{\Phi}$ is established in Appendix A.3. Here we only
establish the formula. Note that
$\eta=\chi\cdot(\varphi\circ\gamma_{1})e_{2}-\alpha(\phi)\psi_{0}\in
C^{1}([0,1])$ since $\gamma\in C^{1}([0,1])$. Using Equation (4.4),
$\gamma_{1}(0)=0$ and the fact that $\eta$ vanishes at $1$, we compute

$\displaystyle\delta A[\gamma]\eta=$ $\displaystyle 2\pi\lim_{\tau\rightarrow
0^{+}}\int_{\tau}^{1}L\eta_{1}+\frac{\gamma_{1}(t)\langle\dot{\eta}(t),\dot{\gamma}(t)\rangle}{L}dt$
$\displaystyle=$ $\displaystyle-2\pi\lim_{\tau\rightarrow
0^{+}}\left[\frac{\gamma_{1}(\tau)\langle\eta(\tau),\dot{\gamma}(\tau)\rangle}{L}+\int_{\tau}^{1}H[\gamma]\langle
n,\eta\rangle L\gamma_{1}(t)dt\right]$ $\displaystyle=$
$\displaystyle-2\pi\lim_{\tau\rightarrow
0^{+}}\int_{\tau}^{1}H[\gamma]\langle\nu,\eta\rangle L\gamma_{1}(t)dt.$ (4.25)

Using Equation (4.3), $\gamma_{1}(0)=0$, $\eta\equiv 0$ near $1$ and the
regularity of $\gamma$ we write

$\displaystyle\delta V[\gamma]\eta=$
$\displaystyle\mp_{\gamma}2\pi\int_{0}^{1}\langle\nu,\eta\rangle
L\gamma_{1}(t)dt\mp_{\gamma}\pi\gamma_{1}^{2}(0)\eta_{2}(0)$ $\displaystyle=$
$\displaystyle\mp_{\gamma}2\pi\lim_{\tau\rightarrow
0^{+}}\int_{\tau}^{1}\langle\nu,\eta\rangle L\gamma_{1}(t)dt.$ (4.26)

Inserting Equations (4.25) and (4.26) into Equation (4.2) and using
$A[\gamma]=1$ and $V[\gamma]=\frac{\sigma}{6\sqrt{\pi}}$, we get

$\displaystyle\delta\mathcal{I}[\gamma]\eta=$
$\displaystyle\frac{6\sqrt{\pi}}{A[\gamma]^{\frac{3}{2}}}\left[\delta
V[\gamma]\eta-\frac{3V[\gamma]}{2A[\gamma]}\delta A[\gamma]\eta\right]$
$\displaystyle=$ $\displaystyle-6\sqrt{\pi}\lim_{\tau\rightarrow
0^{+}}2\pi\int_{\tau}^{1}\left(\pm_{\gamma}1-\frac{\sigma}{4\sqrt{\pi}}H[\gamma]\right)\langle\nu,\eta\rangle
L\gamma_{1}dt.$ (4.27)

∎

######  Lemma 4.10.

$\lambda=0$ and consequently by Lemmas 4.7 and 4.8, $u\in C^{2}([0,r_{0}])$
and $H\in C^{1}([0,r_{0}])$.

###### Proof.

Throughout the proof, we use both the profile curve and the graph function
parameterization. To shorten the notation, we denote the mean curvature by
$\tilde{H}$ when working in the graph function parameterization and by $H$
when using the profile curve parameterization. That is:
$\tilde{H}(r)=\tilde{H}(\gamma_{1}(s))=H(s)$.

Let $\varphi\in C^{\infty}_{0}([0,r_{1}))$ such that $\varphi^{\prime}(0)=0$
and define $\eta$ as in Equation (4.24). By Lemma 4.9 there exists a variation
$\tilde{\Phi}:(-\epsilon_{0},\epsilon_{0})\rightarrow\mathcal{F}_{\sigma}^{\prime}$
such that
$\frac{\partial\tilde{\Phi}(\epsilon)}{\partial\epsilon}\big{|}_{\epsilon=0}=\eta$
and we get the Euler-Lagrange equation

$0=\delta\mathcal{W}[\gamma]\eta=\delta\mathcal{W}[\gamma]\phi-\alpha(\phi)\delta\mathcal{W}[\gamma]\psi_{0}.$
(4.28)

#### The first term

$\phi$ is supported on $[0,t_{1}]$. Here $\chi\equiv 1$ and hence

$\displaystyle\delta\mathcal{W}[\gamma]\phi$
$\displaystyle=\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}\mathcal{W}[\gamma+\epsilon\phi]=\frac{2\pi}{4}\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}\int_{0}^{r_{1}}\tilde{H}[\gamma+\epsilon\phi]^{2}\sqrt{1+(u^{\prime}+\epsilon\varphi^{\prime})^{2}}rdr$
$\displaystyle=$
$\displaystyle\frac{2\pi}{4}\int_{0}^{r_{1}}\left(2\tilde{H}(r)\delta\tilde{H}[\gamma]\phi\sqrt{1+u^{\prime}(r)^{2}}+\tilde{H}(r)^{2}\frac{u^{\prime}(r)\varphi^{\prime}(r)}{\sqrt{1+u^{\prime}(r)^{2}}}\right)rdr.$

Let us consider $\delta\tilde{H}[\gamma]\phi$. By Lemma 4.7 there exists
$\xi\in C^{1}([0,r_{1}])$ such that
$u^{\prime}(r)=\frac{\lambda}{2}r\ln(r)+\xi(r)$. Since $u^{\prime}(0)=0$ we
have $\xi(0)=0$. Also
$u^{\prime\prime}(r)=\frac{\lambda}{2}\ln(r)+\tilde{\xi}(r)$ where
$\tilde{\xi}=\xi^{\prime}+\frac{\lambda}{2}$ is continuous on $[0,r_{0}]$.
Using Equation (2.8) shows that for all $p\in[1,\infty)$

$\delta\tilde{H}[\gamma]\phi=-\frac{\varphi^{\prime\prime}(r)}{\sqrt{1+u^{\prime}(r)^{2}}^{3}}+3\frac{u^{\prime\prime}(r)u^{\prime}(r)\varphi^{\prime}(r)}{\sqrt{1+u^{\prime}(r)^{2}}^{5}}-\frac{\varphi^{\prime}(r)}{r\sqrt{1+u^{\prime}(r)^{2}}}+\frac{u^{\prime}(r)^{2}\varphi^{\prime}(r)}{r\sqrt{1+u^{\prime}(r)^{2}}^{3}}\in
L^{p}((0,r_{1})).$

Additionally, $\tilde{H}\in L^{p}((0,r_{0}))$ for all $p\in[1,\infty)$ by
Lemma 4.8. Let $\rho>0$ be a small parameter. We put
$\gamma_{\rho}:=\gamma|_{[\gamma_{1}^{-1}(\rho),\gamma_{1}^{-1}(r_{1})]}$.
Using Lebesgue’s theorem, we get

$\displaystyle\delta\mathcal{W}[\gamma]\phi$
$\displaystyle=\lim_{\rho\rightarrow
0^{+}}\frac{2\pi}{4}\int_{\rho}^{r_{1}}\left(2\tilde{H}(r)\delta\tilde{H}[\gamma]\phi\sqrt{1+u^{\prime}(r)^{2}}+\tilde{H}(r)^{2}\frac{u^{\prime}(r)\varphi^{\prime}(r)}{\sqrt{1+u^{\prime}(r)^{2}}}\right)rdr$
$\displaystyle=\lim_{\rho\rightarrow
0^{+}}\delta\mathcal{W}[\gamma_{\rho}]\phi.$ (4.29)

Since $\gamma\in C^{\infty}((0,1))$ we have $\gamma_{\rho}\in
C^{\infty}([\gamma_{1}^{-1}(\rho),\gamma_{1}^{-1}(r_{1})])$ for all small
$\rho>0$. Using Equation (A.9) from the appendix and using that $\phi$ is
compactly supported in $[0,\gamma_{1}^{-1}(r_{1}))$, we get

$\displaystyle\delta\mathcal{W}[\gamma_{\rho}]\phi=2\pi\int_{\gamma_{1}^{-1}(\rho)}^{\gamma_{1}^{-1}(r_{1})}W[\gamma]\langle\phi,\nu\rangle\gamma_{1}|\dot{\gamma}|dt+B(\rho)$
$\displaystyle=2\pi\int_{\gamma_{1}^{-1}(\rho)}^{1}W[\gamma]\langle\phi,\nu\rangle\gamma_{1}|\dot{\gamma}|dt+B(\rho).$
(4.30)

Here we have introduced the boundary term

$B(\rho)=\pi\left[\frac{\varphi(\gamma_{1}(s))H^{\prime}(s)-(\varphi\circ\gamma_{1})^{\prime}(s)H(s)-\frac{1}{2}H^{2}(s)\langle\dot{\gamma}(s),\phi(s)\rangle}{|\dot{\gamma}(s)|}\right]\gamma_{1}(s)\bigg{|}_{s=\gamma_{1}^{-1}(\rho)}.$
(4.31)

#### The second term

As $\psi_{0}\in C^{\infty}_{0}((0,1))$, its support lies in an interval
$(a,b)$ with $0<a<b<1$ where $\gamma\in C^{\infty}([a,b])$. We use Equation
(A.9) from the appendix. As $\psi_{0}$ has compact support in $(a,b)$ we get

$\delta\mathcal{W}[\gamma]\psi_{0}=2\pi\int_{a}^{b}W[\gamma]\langle\psi_{0},\nu\rangle\gamma_{1}|\dot{\gamma}|ds=2\pi\lim_{\rho\rightarrow
0^{+}}\int_{\gamma_{1}^{-1}(\rho)}^{1}W[\gamma]\langle\psi_{0},\nu\rangle\gamma_{1}|\dot{\gamma}|ds.$
(4.32)

#### Combining the results

By Lemma 4.6, $\gamma$ satisfies the Euler-Lagrange equation
$W[\gamma]=\Lambda(\pm_{\gamma}4\sqrt{\pi}-\sigma H[\gamma])$ on $(0,1)$.
Using Lemma 4.9, we get

$\displaystyle\lim_{\rho\rightarrow
0^{+}}\left[2\pi\int_{\gamma_{1}^{-1}(\rho)}^{1}W[\gamma]\langle\eta,\nu\rangle\gamma_{1}|\dot{\gamma}|ds\right]$
$\displaystyle=$ $\displaystyle\lim_{\rho\rightarrow
0^{+}}\left[2\pi\Lambda\int_{\gamma_{1}^{-1}(\rho)}^{1}(\pm_{\gamma}4\sqrt{\pi}-\sigma
H[\gamma])\langle\eta,\nu\rangle\gamma_{1}|\dot{\gamma}|ds\right]$
$\displaystyle=$
$\displaystyle-\frac{2\Lambda}{3}\delta\mathcal{I}[\gamma]\eta=0.$ (4.33)

Combining Equations (4.28), (4.29), (4.30), (4.32) and (4.33) we deduce
$\lim_{\rho\rightarrow 0^{+}}B(\rho)=~{}0.$ We now rewrite the boundary term
in terms of the graph function $u$. Note

$r=\gamma_{1}(s)\hskip 14.22636pt\textrm{and consequently}\hskip
14.22636pt\frac{\partial}{\partial s}=\frac{\partial r}{\partial
s}\frac{\partial}{\partial r}=\dot{\gamma}_{1}(s)\frac{\partial}{\partial r}.$

Hence
$\dot{\gamma}_{2}(s)=\dot{\gamma}_{1}(s)(\gamma_{2}\circ\gamma_{1}^{-1})^{\prime}(r)=\dot{\gamma}_{1}(s)u^{\prime}(r)$
and therefore
$|\dot{\gamma}(s)|=\dot{\gamma}_{1}(s)\sqrt{1+u^{\prime}(r)^{2}}$. Inserting
into Equation (4.31), we get

$B(\rho)=\pi
r\frac{\varphi(r)\tilde{H}^{\prime}(r)-\varphi^{\prime}(r)\tilde{H}(r)-\frac{1}{2}\tilde{H}^{2}(r)u^{\prime}(r)\varphi(r)}{\sqrt{1+u^{\prime}(r)^{2}}}\bigg{|}_{r=\rho}.$

Using the formulas $u^{\prime}(r)=\frac{\lambda}{2}r\ln(r)+\xi(r)$ where
$\xi\in C^{1}([0,r_{0}])$ satisfies $\xi(0)=0$ from Lemma 4.7,
$\tilde{H}(r)=-\lambda\ln(r)+\tilde{H}_{0}(r)$ with $\tilde{H}_{0}\in
C^{1,\alpha}([0,r_{0}))$ from Lemma 4.8, the smoothness of $\varphi$ and
$\varphi^{\prime}(0)=0$, we get

$0=\lim_{\rho\rightarrow 0^{+}}B(\rho)=\lim_{\rho\rightarrow
0^{+}}\left[\pi\rho\frac{\varphi(\rho)}{\sqrt{1+u^{\prime}(\rho)^{2}}}\left(-\frac{\lambda}{\rho}+\tilde{H}_{0}^{\prime}(\rho)\right)\right]=-\pi\lambda\varphi(0).$

As $\varphi(0)$ is arbitrary we deduce $\lambda=0$. ∎

Having improved the regularity of $u$ to $C^{2}$, we can now easily deduce
that $u$ is, in fact, smooth.

######  Lemma 4.11.

$u\in C^{\infty}([0,r_{0}))$.

###### Proof.

The proof is inductive. Let $k\geq 0$ and suppose that we know $H\in
C^{k}([0,r_{0}))$. By Equation (2.9)

$-\frac{1}{r}\frac{d}{dr}\left[r\frac{u^{\prime}(r)}{\sqrt{1+u^{\prime}(r)^{2}}}\right]=H\in
C^{k}([0,r_{0})).$

Lemma D.1 implies $u^{\prime}\in C^{k+1}([0,r_{0}))$ and thus $u\in
C^{k+2}([0,r_{0}))$. In view of Equation (2.8) we get $k_{1}\in
C^{k}([0,r_{0}))$ and since $u^{\prime}(0)=0$

$k_{2}(r)=-\frac{u^{\prime}(r)}{r\sqrt{1+u^{\prime}(r)^{2}}}=-\frac{1}{\sqrt{1+u^{\prime}(r)^{2}}}\int_{0}^{1}u^{\prime\prime}(sr)ds\hskip
14.22636pt\textrm{is also of class }C^{k}([0,r_{0})).$

Thus $K=k_{1}k_{2}\in C^{k}([0,r_{0}))$. Writing out the Euler-Lagrange
equation from Lemma 4.6, we get

$\frac{1}{r\sqrt{1+u^{\prime}(r)^{2}}}\frac{d}{dr}\left[\frac{r}{\sqrt{1+u^{\prime}(r)^{2}}}\frac{dH}{dr}\right]=\Delta_{g}H=-\frac{1}{2}H(H^{2}-4K)+a-bH\in
C^{k}([0,r_{0})).$

Using $u^{\prime}\in C^{k+1}([0,r_{0}))$ and putting
$\alpha(r):=\frac{1}{\sqrt{1+u^{\prime}(r)^{2}}}\frac{dH}{dr}$, we get
$\frac{(r\alpha)^{\prime}}{r}\in C^{k}([0,r_{0}))$. Lemma D.1 implies
$\alpha\in C^{k+1}([0,r_{0}))$. By definition of $\alpha$ and using
$u^{\prime}\in C^{k+1}([0,r_{0}))$ we deduce $H^{\prime}(r)\in
C^{k+1}([0,r_{0}))$ and consequently $H\in C^{k+2}([0,r_{0}))$.

We have shown the following two implications: $H\in C^{k}([0,r_{0})\Rightarrow
H\in C^{k+2}([0,r_{0}))$ and $H\in C^{k}([0,r_{0}))\Rightarrow u\in
C^{k+2}([0,r_{0}))$. Since we have shown that $H\in C^{1}([0,r_{0}))$ in Lemma
4.10, we deduce $u\in C^{\infty}([0,r_{0}))$. ∎

In Lemma 4.11, we have shown that the graph function $u$ describing $\gamma$
near $t=0$ is smooth. By the same arguments, the graph function $\tilde{u}$
describing $\gamma$ near $t=1$ is also smooth. As an easy corollary, we obtain
that $\gamma$ is smooth up to $t=0$ and $t=1$.

######  Corollary 4.12.

$\gamma\in C^{\infty}([0,1])$

###### Proof.

In view of Corollary 4.5 we already have $\gamma\in C^{\infty}((0,1))$. Recall
that we chose $t_{0}>0$ and $r_{0}:=\gamma_{1}(t_{0})$ such that
$\gamma_{1}:[0,t_{0})\rightarrow[0,r_{0})$ is a $C^{1}$-diffeomorphism. In
particular, as $\dot{\gamma}_{1}(0)=L:=L[\gamma]>0$, we have
$\dot{\gamma}_{1}(t)>0$ for all $t\in[0,t_{0})$. From Lemma 4.11, we know that
$u:[0,r_{0})\rightarrow\mathbb{R},\ u(r):=\gamma_{2}(\gamma_{1}^{-1}(r))$ is
in $C^{\infty}([0,r_{0}))$. Clearly $\gamma_{2}(t)=u(\gamma_{1}(t))$.
Differentiating and using $|\dot{\gamma}(t)|^{2}=L^{2}$ yields

$L^{2}-\dot{\gamma}_{1}(t)^{2}=\dot{\gamma}_{2}(t)^{2}=u^{\prime}(\gamma_{1}(t))^{2}\dot{\gamma}_{1}(t)^{2}\hskip
14.22636pt\textrm{and so}\hskip
14.22636pt\dot{\gamma}_{1}(t)^{2}=\frac{L^{2}}{1+u^{\prime}(\gamma_{1}(t))^{2}}.$

Since $\dot{\gamma}_{1}(t)\geq 0$ for all $t\in[0,t_{0})$ we can take the
square root and obtain

$\dot{\gamma}_{1}(t)=\frac{L}{\sqrt{1+u^{\prime}(\gamma_{1}(t))^{2}}}.$

As $\gamma_{1}\in C^{0}([0,t_{0}))$ and $u^{\prime}\in C^{\infty}([0,r_{0}))$,
we deduce that $\gamma_{1}\in C^{\infty}([0,t_{0}))$ and consequently
$\gamma_{2}=u\circ\gamma_{1}\in C^{\infty}([0,t_{0}))$. This establishes the
smoothness of $\gamma$ near $t=0$. The smoothness near $t=1$ can be
established by the same method. ∎

We now prove Theorem 4.1 by demonstrating that $u^{(2k+1)}(0)=0$ for all
integers $k\geq 0$. Indeed the same proof shows the analog result for the
graph function $\tilde{u}$ describing $\gamma$ near $t=1$.

######  Lemma 4.13 (Proof of Theorem 1.1 (apart from $\beta(\sigma)\rightarrow
8\pi$ for $\sigma\rightarrow 0^{+}$)).

$u^{(2k+1)}(0)=0$ for all $k\in\mathbb{N}_{0}$ and consequently
$\Sigma_{\gamma}$ is smooth.

###### Proof.

The proof is inductive. By Equation (4.10) we have $w(0)=u^{\prime}(0)=0$. We
show that if there exists $n\geq 0$ such that $u^{(2k+1)}(0)=0$ for all $0\leq
k\leq n$, then $u^{(2n+3)}(0)=0$. By Lemma 4.10 and Equation (4.10) we have

$u^{\prime\prime\prime}+\left(\frac{u^{\prime}}{r}\right)^{\prime}=\frac{1}{2}\sqrt{1+u^{\prime}(r)^{2}}^{5}ar+b(1+u^{\prime}(r)^{2})^{2}u^{\prime}(r)+\varphi(u^{\prime}).$
(4.34)

Since $u\in C^{\infty}([0,r_{0})$ and $u^{(2k+1)}(0)=0$ for $0\leq k\leq n$,
the function

$U:(-r_{0},r_{0})\rightarrow\mathbb{R},\ U(r):=\left\\{\begin{aligned}
u(r)&\hskip 14.22636pt\textrm{ if }r\geq 0,\\\ u(-r)&\hskip 14.22636pt\textrm{
if }r<0\end{aligned}\right.$

is smooth on $[0,r_{0})$, $(-r_{0},0]$ and $(-r_{0},r_{0})\backslash\set{0}$.
However, the derivatives at $0$ of order$\geq 2n+3$ might not match up so that
only $U\in C^{2n+2}((-r_{0},r_{0}))$ is guaranteed. Consequently, the function

$F(r):=\frac{1}{2}\sqrt{1+U^{\prime}(r)^{2}}^{5}ar+b(1+U^{\prime}(r)^{2})^{2}U^{\prime}(r)+\varphi(U^{\prime})$

is of class $C^{2n}((-r_{0},r_{0}))$. Indeed, considering the definition of
$\varphi(U^{\prime})$ from Equation (4.9), we see that only the term
$\frac{(U^{\prime}(r))^{3}}{2r^{2}}(3+(U^{\prime}(r))^{2})$ is of concern.
However, using that $u^{\prime}(0)=0$, we get

$\frac{U^{\prime}(r)^{2}}{r^{2}}=\left(\int_{0}^{1}U^{\prime\prime}(sr)ds\right)^{2},\hskip
14.22636pt\textrm{which is of class }C^{2n}(-r_{0},r_{0}).$

Additionally, it is readily checked that $F(-r)=-F(r)$. Thus $F^{(2n)}(0)=0$.
Differentiating Equation (4.34) $2n$ times, we get

$u^{(2n+3)}+\left(\frac{u^{\prime}}{r}\right)^{(2n+1)}=F^{(2n)}(r)\hskip
14.22636pt\textrm{on }[0,r_{0}).$ (4.35)

Since $u\in C^{\infty}([0,r_{0}))$ we may write

$\left(\frac{u^{\prime}}{r}\right)^{(2n+1)}=\left(\int_{0}^{1}u^{\prime\prime}(rs)ds\right)^{(2n+1)}=\int_{0}^{1}u^{(2n+3)}(rs)s^{2n+1}ds.$

Now we consider Equation (4.35) the limit $r\rightarrow 0^{+}$. Since
$F^{(2n)}(0)=0$ we get

$0=\lim_{r\rightarrow
0^{+}}\left(u^{(2n+3)}(r)+\int_{0}^{1}u^{(2n+3)}(rs)s^{2n+1}ds\right)=\left(1+\frac{1}{2n+2}\right)u^{(2n+3)}(0)$
(4.36)

and hence $u^{(2n+3)}(0)=0$. ∎

##  5 Convergence to a Double Sphere

Let $\sigma_{k}\rightarrow 0^{+}$ and
$\gamma^{(k)}\in\mathcal{F}_{\sigma_{k}}$ satisfy
$\mathcal{W}[\gamma^{(k)}]=\beta(\sigma_{k})$. Since vertical translations
$(\gamma_{1},\gamma_{2})\mapsto(\gamma_{1},\gamma_{2}+c)$ and the reflection
$(\gamma_{1},\gamma_{2})\mapsto(\gamma_{1},-\gamma_{2})$ leave the Willmore
energy and the isoperimetric ratio invariant, we cannot expect the profile
curves $\gamma^{(k)}$ to converge. To rule out both obstructions, we introduce

$\mathcal{F}^{0+}_{\sigma}:=\set{\gamma\in\mathcal{F}_{\sigma}\ }{\
\gamma(0)=0\textrm{ and }\int_{0}^{1}\gamma_{2}(t)dt\geq 0},$

and stress that $\beta(\sigma)=\inf\set{\mathcal{W}[\gamma]\ }{\
\gamma\in\mathcal{F}_{\sigma}}=\inf\set{\mathcal{W}[\gamma]\ }{\
\gamma\in\mathcal{F}_{\sigma}^{0+}}$. Now, considering a sequence
$\gamma^{(k)}\in\mathcal{F}_{\sigma_{k}}^{0+}$ convergence to a limit is
possible. Indeed, letting $R:=\frac{1}{\sqrt{8\pi}}$ and

$\kappa:[0,1]\rightarrow\mathbb{R}^{2},\ \kappa(t):=(0,R)+R(|\sin(2\pi
t)|,-\cos(2\pi t)),$

we prove the following:

######  Theorem 5.1 (Convergence to a Double Sphere).

Let $(\sigma_{k})\subset(0,1)$ such that $\sigma_{k}\rightarrow 0^{+}$ and let
$\gamma^{(k)}\in\mathcal{F}_{\sigma_{k}}^{0+}$ such that
$\mathcal{W}[\gamma^{(k)}]=\beta(\sigma_{k})$. Then
$\beta(\sigma_{k})\rightarrow 8\pi$ and $\gamma^{(k)}\rightarrow\kappa$ in
$W^{1,2}((0,1))$ and in
$C^{\infty}_{\operatorname{loc}}([0,1]\backslash\\{\frac{1}{2}\\})$.

In particular, we note that Theorem 5.1 implies Theorem 1.2 and concludes the
proof of Theorem 1.1.

For the entirety of this section we fix a sequence $\sigma_{k}\rightarrow
0^{+}$ as well as $\gamma^{(k)}\in\mathcal{F}^{0+}_{\sigma_{k}}$ such that
$\beta(\sigma_{k})=\mathcal{W}[\gamma^{(k)}]$. We will prove that there exists
a subsequence $\gamma^{(k_{l})}$ that converges to $\kappa$ in
$W^{1,2}((0,1))$ and in
$C^{\infty}_{\operatorname{loc}}([0,1]\backslash\\{\frac{1}{2}\\})$. Once we
have shown this, Theorem 5.1 follows by the usual subsequence argument.

######  Lemma 5.2.

There exists a subsequence $\gamma^{(k_{l})}$ and a curve
$\gamma^{*}\in\mathcal{P}^{w}$ such that
$\gamma^{(k_{l})}\rightarrow\gamma^{*}$ in the sense of Definition 3.10.
Moreover, either $\gamma^{*}\in\mathcal{P}$ or there exists $\tau\in(0,1)$
such that, after reparameterization, $\gamma^{*}|_{[0,\tau]}$ and
$\gamma^{*}|_{[\tau,1]}$ both belong to $\mathcal{P}$.

###### Proof.

The proof for the existence of $\gamma^{*}$ and the convergence is essentially
the same as the one from Theorem 3.13. However, instead of Estimate (3.11) we
only get

$\mathcal{W}[\gamma^{*}]\leq 8\pi\hskip 14.22636pt\textrm{and}\hskip
14.22636pt\int_{0}^{1}\left((k_{1}^{*})^{2}+(k_{2}^{*})^{2}\right)2\pi
L_{*}\gamma_{1}^{*}(t)\leq 3\cdot 8\pi.$

A priori Theorem 3.3 implies that there are at most 2 points
$\tau\neq\tau^{\prime}\in(0,1)$ such that
$\gamma_{1}^{*}(\tau)=\gamma_{1}^{*}(\tau^{\prime})=0$. However, following the
arguments after Equation (3.11) implies that there can be at most one such
point. ∎

Let $\delta>0$ be small. Depending on whether the limit $\gamma^{*}$ from
Lemma 5.2 satisfies $\gamma^{*}_{1}(\tau)=0$ for some $\tau\in(0,1)$ or not,
we put

$I_{\delta}:=\left\\{\begin{aligned} (\delta,1-\delta)&\ \textrm{if
$\gamma^{*}\in\mathcal{P}$},\\\
(\delta,\tau-\delta)\cup(\tau+\delta,1-\delta)&\ \textrm{if
$\gamma^{*}\not\in\mathcal{P}$}.\end{aligned}\right.$

We can now prove that along a subsequence $\gamma^{(k)}$ converges to
$\gamma^{*}$ not only in $\mathcal{P}^{w}$ but in $C^{m}(I_{\delta})$ for all
$\delta>0$ and $m\in\mathbb{N}$.

######  Lemma 5.3.

Let $\gamma^{(k_{l})}$ and $\gamma^{*}$ be as in Lemma 5.2. Then
$\gamma^{*}\in C^{\infty}(I_{\delta})$ for all $\delta>0$ and
$\gamma^{(k_{l})}\rightarrow\gamma^{*}$ in $C^{m}(I_{\delta})$ for all
$m\in\mathbb{N}$ and small $\delta>0$.

###### Proof.

We prove that for arbitrary $m\in\mathbb{N}$ the sequence $\gamma^{(k_{l})}$
is bounded in $W^{m+1,\infty}(I_{\delta})$. The Sobolev embedding
$W^{m+1,\infty}(I_{\delta})\hookrightarrow C^{m,\alpha}(\bar{I}_{\delta})$
then shows that $\gamma^{(k_{l})}$ is bounded in $C^{m,\alpha}(I_{\delta})$.
Once this is shown, we can use $\gamma^{(k_{l})}\rightarrow\gamma^{*}$ in
$\mathcal{P}^{w}$ to deduce the lemma. From now on we write $\gamma^{(k)}$
instead of $\gamma^{(k_{l})}$

Uniform Bounds in $W^{m,\infty}$
By Definition 3.10 and Lemma 5.2 we get $\gamma^{(k)}\rightarrow\gamma^{*}$ in
$C^{0}([0,1])$. So we can assume that

$\inf_{t\in I_{\delta}}\gamma_{1}^{(k)}(t)\geq\frac{1}{2}\inf_{t\in
I_{\delta}}\gamma_{1}^{*}(t)=\frac{1}{2}\kappa(\gamma^{*};I_{\delta})>0.$
(5.1)

Either $I_{\delta}=(\delta,1-\delta)$ or
$I_{\delta}=(\delta,\tau-\delta)\cup(\tau+\delta,1-\delta)$. Let
$(\tau_{1},\tau_{2})$ denote any of these intervals. We claim that there is
$\psi^{*}\in C^{\infty}_{0}((\tau_{1},\tau_{2}))$ such that

$1\overset{!}{=}6\sqrt{\pi}\delta
V[\gamma^{*}]\psi^{*}=\pm_{\gamma^{*}}6\sqrt{\pi}^{\frac{3}{2}}\int_{\tau_{1}}^{\tau_{2}}(\gamma_{1}^{*})^{2}\dot{\psi}^{*}_{2}+2\gamma_{1}^{*}\dot{\gamma}_{2}^{*}\psi^{*}_{1}dt.$
(5.2)

Indeed, otherwise, we could use partial integration to get for all
$\psi^{*}\in C^{\infty}_{0}((\tau_{1},\tau_{2}))$

$0=\int_{\tau_{1}}^{\tau_{2}}(\gamma_{1}^{*})^{2}\dot{\psi}^{*}_{2}+2\gamma_{1}^{*}\dot{\gamma}_{2}^{*}\psi^{*}_{1}dt=2\int_{\tau_{1}}^{\tau_{2}}\gamma_{1}^{*}\langle\dot{\gamma}^{*},(-\psi_{2}^{*},\psi_{1}^{*})\rangle
dt.$

Using $\gamma^{*}\in W^{1,2}((0,1))$ and $\gamma_{1}^{*}>0$ almost everywhere
we conclude $\dot{\gamma}^{*}=0$ almost everywhere in $(\tau_{1},\tau_{2})$.
This is a contradiction as $\gamma^{*}\in\mathcal{P}^{w}$ and hence
$|\dot{\gamma}^{*}|=L[\gamma^{*}]$ almost everywhere.

Now let $\psi^{*}$ be as in Equation (5.2). Using
$\gamma^{(k)}\rightarrow\gamma^{*}$ in $C^{0}([0,1])$ and $W^{1,2}((0,1))$ and
$\sigma_{k}\rightarrow 0$, we can use Equations (4.2)-(4.4) to get
$1=\lim_{k\rightarrow\infty}\delta\mathcal{I}[\gamma^{(k)}]\psi^{*}$. This
proves that, after potentially ignoring the first couple of terms in the
sequence, there exists a sequence $\zeta_{k}\subset(0,\infty)$ converging to
$1$ such that

$1=\zeta_{k}\left(\delta\mathcal{I}[\gamma^{(k)}]\psi^{*}\right)=\delta\mathcal{I}[\gamma^{(k)}](\zeta_{k}\psi^{*}).$

This allows us to use Corollary 4.4 with $U=(\tau_{1},\tau_{2})$ and
$\psi_{0}^{(k)}=\zeta_{k}\psi^{*}$ to prove the lemma. Indeed, Estimate (5.1)
provides us with a uniform lower bound for $\kappa(\gamma^{(k)};U)$ and, as
$\zeta_{k}\rightarrow 1$, the $C^{2}$-norms of $\psi^{(k)}_{0}$ are bounded
uniformly. ∎

We now prove a first part of Theorem 5.1.

######  Lemma 5.4.

Let $(\sigma_{k})\subset(0,1)$ such that $\sigma_{k}\rightarrow 0^{+}$ and let
$\gamma^{(k)}\in\mathcal{F}_{\sigma_{k}}^{0+}$ such that
$\mathcal{W}[\gamma^{(k)}]=\beta(\sigma_{k})$. Then
$\beta(\sigma_{k})\rightarrow 8\pi$ and $\gamma^{(k)}\rightarrow\kappa$ in
$W^{1,2}((0,1))$ and in
$C^{\infty}_{\operatorname{loc}}((0,1)\backslash\\{\frac{1}{2}\\})$.

###### Proof.

Combining Lemmas 5.2 and 5.3 we have $\gamma^{(k)}\rightarrow\gamma^{*}$ in
$\mathcal{P}^{w}$ and in $C^{m}(I_{\delta})$ for all $m\in\mathbb{N}$ and
$\delta>0$ along a subsequence again denoted by $\gamma^{(k)}$. Additionally,
either $\gamma^{*}\in\mathcal{P}$ or there exists $\tau\in(0,1)$ such that
$\gamma_{1}^{*}(\tau)=0$ and $\gamma^{*}|_{[0,\tau]}$,
$\gamma^{*}|_{[\tau,1]}$ both belong to $\mathcal{P}$. In either case, we can
use the weak lower semicontinuity from Equation (3.8) and deduce

$\mathcal{W}[\gamma^{*}]\leq\liminf_{k\rightarrow\infty}\mathcal{W}[\gamma^{(k)}]\leq\limsup_{k\rightarrow\infty}\mathcal{W}[\gamma^{(k)}]\leq
8\pi.$ (5.3)

Using $\mathcal{I}[\gamma^{(k)}]=\sigma_{k}\rightarrow 0^{+}$,
$A[\gamma^{(k)}]\equiv 1$ and the continuity of the area and volume with
respect to the $\mathcal{P}^{w}$ convergence (see Equations (3.6) and (3.7)),
we get

$A[\gamma^{*}]=\lim_{k\rightarrow\infty}A[\gamma^{(k)}]\equiv 1\hskip
14.22636pt\textrm{and}\hskip
14.22636ptV[\gamma^{*}]=\lim_{k\rightarrow\infty}V[\gamma^{(k)}]=0.$ (5.4)

#### The first case: $\gamma^{*}\in\mathcal{P}$

We claim that there exist $\tau<\tau^{\prime}\in[0,1]$ such that
$\gamma(\tau)=\gamma(\tau^{\prime})=:(p_{1},p_{2})$. Indeed, if $\gamma^{*}$
was injective, then $V[\gamma^{*}]>0$ by Lemma 2.2 which contradicts Equation
(5.4). By Lemma A.2 we conclude $\mathcal{W}[\gamma^{*}]\geq 8\pi$. Combining
this with Estimate (5.3), we deduce $\mathcal{W}[\gamma^{*}]=8\pi$. Putting
$p:=(p_{1},0,p_{2})$, we are in the equality case of the monotonicity formula
from Lemma A.2 and get

$\frac{1}{4}\vec{H}[f_{\gamma^{*}}]=\frac{(f_{\gamma^{*}}-p)^{\perp}}{|f_{\gamma^{*}}-p|^{2}}.$

Using Lemma A.3 and $\mathcal{W}[\Sigma_{\gamma^{*}}]=8\pi$, we deduce that
$\Sigma_{\gamma^{*}}$ is the inversion of a scaled and vertically translated
catenoid and thus encloses a positive volume, which contradicts Equation
(5.4).

#### The Second Case: $\gamma^{*}\not\in\mathcal{P}$

Let $\alpha^{(1)}:=\gamma^{*}|_{[0,\tau]}$ and
$\alpha^{(2)}:=\gamma^{*}|_{[\tau,1]}$. Lemma 3.6 gives
$\mathcal{W}[\alpha^{(i)}]\geq 4\pi$ for $i=1,2$. This shows

$8\pi\leq\mathcal{W}[\alpha^{(1)}]+\mathcal{W}[\alpha^{(2)}]=\mathcal{W}[\gamma^{*}]\leq
8\pi.$ (5.5)

Hence $\mathcal{W}[\gamma^{*}]=8\pi$ and in view of Estimate (5.3) we get
$\beta(\sigma_{k})=\mathcal{W}[\gamma^{(k)}]\rightarrow 8\pi$. Additionally,
Estimate (5.5) gives $\mathcal{W}[\alpha^{(i)}]=4\pi$ for $i=1,2$ so that by
the Li-Yau inequality (Lemma 3.6), both curves are the profile curve of a
sphere. Since $\alpha^{(i)}\in\mathcal{P}$, are parameterized proportional to
arc length, there exist $z_{i}\in\mathbb{R}$, $\epsilon_{i}\in\set{\pm 1}$ and
$\omega_{i},R_{i}\in\mathbb{R}_{0}^{+}$, such that

$\displaystyle\alpha^{(1)}:[0,\tau]\rightarrow\mathcal{H}^{2},\
\alpha^{(1)}(t)=(0,z_{1})+R_{1}(\sin(\omega_{1}t),\epsilon_{1}\cos(\omega_{1}t)),$
$\displaystyle\alpha^{(2)}:[\tau,1]\rightarrow\mathcal{H}^{2},\
\alpha^{(2)}(t)=(0,z_{2})+R_{2}(\sin(\omega_{2}(t-\tau)),\epsilon_{1}\cos(\omega_{2}(t-\tau))).$

Using Equation (5.4) we get

$\displaystyle 1$
$\displaystyle=A[\gamma^{*}]=A[\gamma^{(1)}]+A[\gamma^{(2)}]=4\pi(R_{1}^{2}+R_{2}^{2}),$
$\displaystyle 0$
$\displaystyle=V[\gamma^{*}]=V[\gamma^{(1)}]+V[\gamma^{(2)}]=\frac{4}{3}\pi|\epsilon_{1}R_{1}^{3}+\epsilon_{2}R_{2}^{3}|.$

These imply $\epsilon_{2}=-\epsilon_{1}=:-\epsilon$ and
$R_{1}=R_{2}=(8\pi)^{-\frac{1}{2}}=:R$. Since $\gamma^{*}$ is parameterized
proportional to arc length, we have
$\omega_{1}R=|\dot{\alpha}^{(1)}|=|\dot{\gamma}^{*}|=|\dot{\alpha}^{(2)}|=\omega_{2}R$
and hence $\omega_{1}=\omega_{2}=:\omega$. Note $\alpha^{(1)}_{1}(t)>0$ for
$t\in(0,\tau)$ and $\alpha^{(1)}_{1}(\tau)=0$. This proves $\omega\tau=\pi$.
Similarly $\alpha^{(2)}_{1}(t)>0$ for $t\in(\tau,1)$ and
$\alpha^{(2)}_{1}(1)=0$ give $\omega(1-\tau)=\pi$. Therefore $\omega=2\pi$ and
$\tau=\frac{1}{2}$. By continuity of $\gamma^{*}$ at $\tau$

$z_{1}+\epsilon R\cos(\omega\tau)=z_{2}-\epsilon R\cos(\omega(\tau-\tau)).$

Using $\omega=2\pi$ and $\tau=\frac{1}{2}$ we get $z_{1}=z_{2}=:z$. As
$\gamma^{*}(0)=(0,0)$, we get $\alpha_{2}^{(1)}(0)=0$ and hence $z=-\epsilon
R$. Finally, since $\gamma^{(k)}\in\mathcal{F}^{0+}_{\sigma_{k}}$ we get

$0\leq\lim_{k\rightarrow\infty}\int_{0}^{1}\gamma_{2}^{(k)}(t)dt=\int_{0}^{1}\gamma_{2}^{*}(t)dt=\frac{z_{1}+z_{2}}{2}=z=-\epsilon
R.$

So $\epsilon=-1$ and $\gamma^{*}=\kappa$. ∎

It remains to study the asymptotic behaviour of $\gamma^{(k)}$ near $t=0$ and
$t=1$ when $k\rightarrow\infty$.

######  Corollary 5.5.

We denote by $\Lambda_{k}$ the Lagrange multiplier from Lemma 4.6. Then
$\Lambda_{k}\rightarrow 0$ as $k\rightarrow\infty$

###### Proof.

By Lemma 4.6

$|\Lambda_{k}|=\left|\frac{2W[\gamma^{(k)}]}{\pm_{\gamma^{(k)}}4\sqrt{\pi}-\sigma_{k}H[\gamma^{(k)}]}\right|.$
(5.6)

Let $I:=(\frac{1}{8},\frac{3}{8})$. By Lemma 5.4 we have
$\gamma^{(k)}\rightarrow\kappa$ in $C^{r}(I)$ for all $r\in\mathbb{N}$ and
hence

$W[\gamma^{(k)}]\bigg{|}_{I}\rightarrow W[\kappa]\bigg{|}_{I}\equiv 0\hskip
14.22636pt\textrm{and}\hskip 14.22636ptH[\gamma^{(k)}]\bigg{|}_{I}\rightarrow
H[\kappa]\bigg{|}_{I}\equiv 2.$

In particular, $\sigma_{k}H[\gamma^{(k)}]\rightarrow 0$ on $I$ and
consequently $\Lambda_{k}\rightarrow 0$ by Equation (5.6). ∎

For all $k\in\mathbb{N}$ we define

$\tau_{k}:=\sup\\{t\geq 0\ |\ \dot{\gamma}_{1}^{(k)}(s)>0\hskip
5.69046pt\textrm{for all $0\leq s\leq t$}\\}.$

We prove that $\tau_{k}$ is bounded from below. In fact, we even prove the
following fact:

######  Lemma 5.6.

Let $\epsilon>0$. There exist $k_{0}(\epsilon)\in\mathbb{N}$ and
$t_{0}(\epsilon)>0$ such that

$\dot{\gamma}_{1}^{(k)}(t)\geq L[\gamma^{(k)}](1-\epsilon)\hskip
14.22636pt\textrm{for all $k\geq k_{0}(\epsilon)$ and
}t\in[0,t_{0}(\epsilon)].$

###### Proof.

The proof is by contradiction. Assuming that the lemma is false, there exists
$\epsilon>0$ such that after passing to a subsequence, we have
$\dot{\gamma}_{1}^{(k)}(t_{k})<L_{k}(1-\epsilon)$ for a sequence
$t_{k}\rightarrow 0^{+}$ and where $L_{k}:=L[\gamma^{(k)}]$. By Lemma 5.4 we
have $\gamma^{(k)}\rightarrow\kappa$ in $W^{1,2}((0,1))$ and in
$C^{\infty}_{\operatorname{loc}}((0,1)\backslash\\{\frac{1}{2}\\})$ and
therefore $L_{k}\rightarrow L_{*}:=L[\kappa]$. For small $\delta>0$, we recall
$I_{\delta}:=(\delta,\frac{1}{2}-\delta)\cup(\frac{1}{2}+\delta,1-\delta)$.
Denoting the principal curvatures of $\kappa$ by $k_{i}^{*}$ we have

$\lim_{\delta\rightarrow
0^{+}}\frac{2\pi}{4}\int_{I_{\delta}}(k_{1}^{*}+k_{2}^{*})^{2}\kappa_{1}|\dot{\kappa}|dt=\mathcal{W}[\kappa]=8\pi.$
(5.7)

For each $\delta>0$ there exists $k_{0}(\delta)>0$ such that $t_{k}<\delta$
for all $k\geq k_{0}(\delta)$. Using that
$\mathcal{W}[\gamma^{(k)}]=\beta(\sigma_{k})\leq 8\pi$ we estimate

$\displaystyle 8\pi\geq\mathcal{W}[\gamma^{(k)}]=$
$\displaystyle\frac{2\pi}{4}\int_{0}^{1}(k_{1}^{(k)}+k_{2}^{(k)})^{2}\gamma_{1}^{(k)}|\dot{\gamma}^{(k)}|dt$
$\displaystyle\geq$
$\displaystyle\frac{2\pi}{4}\int_{I_{\delta}}(k_{1}^{(k)}+k_{2}^{(k)})^{2}\gamma_{1}^{(k)}|\dot{\gamma}^{(k)}|dt+\frac{2\pi}{4}\int_{0}^{t_{k}}(k_{1}^{(k)}+k_{2}^{(k)})^{2}\gamma_{1}^{(k)}|\dot{\gamma}^{(k)}|dt.$
(5.8)

We derive a suitable estimate for the second integral. By Corollary 4.12 we
know $\gamma^{(k)}\in C^{\infty}([0,t_{k}])$ and using Equation (2.21), we get

$k_{1}^{(k)}k_{2}^{(k)}\gamma_{1}^{(k)}|\dot{\gamma}^{(k)}|=-\frac{1}{L_{k}}\ddot{\gamma}_{1}^{(k)}\hskip
14.22636pt\textrm{for $t\in[0,t_{k}]$}.$

Using this identity, $\dot{\gamma}_{1}^{(k)}(0)=L_{k}$ and
$\dot{\gamma}_{1}^{(k)}(t_{k})\leq L_{k}(1-\epsilon)$ we may compute

$\displaystyle\int_{0}^{t_{k}}(k_{1}^{(k)}+k_{2}^{(k)})^{2}\gamma_{1}^{(k)}|\dot{\gamma}^{(k)}|dt\geq
2\int_{0}^{t_{k}}k_{1}^{(k)}k_{2}^{(k)}|\dot{\gamma}^{(k)}|\gamma_{1}^{(k)}dt=-\frac{2}{L_{k}}\int_{0}^{t_{k}}\ddot{\gamma}_{1}^{(k)}(t)dt$
$\displaystyle=$
$\displaystyle-\frac{2}{L_{k}}\left(\dot{\gamma}_{1}^{(k)}(t_{k})-\dot{\gamma}_{1}^{(k)}(0)\right)\geq
2L_{k}-2(L_{k}-\epsilon)=2\epsilon.$

Inserting into Estimate (5.8), we deduce that for all $\delta>0$ and $k\geq
k_{0}(\delta)$ we have

$8\pi\geq\frac{2\pi}{4}\int_{I_{\delta}}(k_{1}^{(k)}+k_{2}^{(k)})^{2}\gamma_{1}^{(k)}|\dot{\gamma}^{(k)}|dt+\pi\epsilon.$

Since $\gamma^{(k)}\rightarrow\kappa$ in $C^{r}(I_{\delta})$ for all
$r\in\mathbb{N}$, we can send $k\rightarrow\infty$ and deduce that for all
$\delta>0$

$8\pi\geq\frac{2\pi}{4}\int_{I_{\delta}}(k_{1}^{*}+k_{2}^{*})^{2}\kappa_{1}|\dot{\kappa}|dt+\pi\epsilon.$

In view of Equation (5.7), we arrive at a contradiction by letting
$\delta\rightarrow 0^{+}$. ∎

Putting $L_{*}:=L[\kappa]$ we have $L_{k}\rightarrow L_{*}$. So, by Lemma 5.6,
there exists $k_{0}>0$ and $t_{0}>0$ such that for all $k\geq k_{0}$ and
$t\in[0,t_{0}]$ we have $\dot{\gamma}_{1}^{(k)}(t)\geq\frac{1}{4}L_{*}>0$.
Consequently, each $\gamma_{1}^{(k)}|_{[0,t_{0}]}$ is a smooth diffeomorphism
onto its image. We also note that

$\gamma_{1}^{(k)}(t_{0})=\int_{0}^{t_{0}}\dot{\gamma}_{1}^{(k)}(t)dt\geq\frac{t_{0}L_{*}}{4}=:r_{0}.$

This implies that for all $k\geq k_{0}$

$u_{k}(r):[0,r_{0}]\rightarrow\mathbb{R},\
u_{k}(r):=\gamma_{2}^{(k)}\circ\left(\gamma_{1}^{(k)}\right)^{-1}(r)$ (5.9)

is well-defined and smooth. Since $\gamma^{(k)}\rightarrow\kappa$ in
$C^{\infty}_{\operatorname{loc}}((0,1)\backslash\\{0\\})$, we deduce that

$u_{k}(r)\rightarrow u^{*}(r):=R-\sqrt{R^{2}-r^{2}}\hskip 14.22636pt\textrm{
in $C^{\infty}_{\operatorname{loc}}((0,r_{0}]))$}\hskip 5.69046pt\textrm{
where $R=\frac{1}{\sqrt{8\pi}}$.}$ (5.10)

We wish to prove that the convergence is even in $C^{\infty}([0,r_{0}]))$.

######  Corollary 5.7.

For all $\epsilon>0$ there exist $k_{0}(\epsilon)\in\mathbb{N}$ and
$r_{1}(\epsilon)\in(0,r_{0}]$ such that $|u_{k}^{\prime}(r)|\leq\epsilon$ for
all $k\geq k_{0}(\epsilon)$ and $r\in[0,r_{1}(\epsilon)]$.

###### Proof.

For small $\epsilon>0$ we take $k_{0}(\epsilon)$ and $t_{0}(\epsilon)$ as in
Lemma 5.6 and restrict ourselves to $k\geq k_{0}(\epsilon)$. We put
$\rho_{k}(\epsilon):=\gamma_{1}^{(k)}(t_{0}(\epsilon))$. Note that for
$r\in[0,\rho_{k}(\epsilon)]$ we have
$(\gamma^{(k)}_{1})^{-1}(r)\in[0,t_{0}(\epsilon)]$. Hence
$\dot{\gamma}_{1}^{(k)}((\gamma^{(k)}_{1})^{-1}(r))\geq L_{k}(1-\epsilon)$ and
therefore

$\left|\dot{\gamma}_{2}^{(k)}\left(\left(\gamma^{(k)}_{1}\right)^{-1}(r)\right)\right|=\left(L_{k}^{2}-\left(\dot{\gamma}_{1}^{(k)}\left(\left(\gamma^{(k)}_{1}\right)^{-1}(r)\right)\right)^{2}\right)^{\frac{1}{2}}\leq
L_{k}\sqrt{1-(1-\epsilon)^{2}}.$

So, for all $k\geq k_{0}(\epsilon)$ and $r\leq\rho_{k}(\epsilon)$ we get

$|u_{k}^{\prime}(r)|=\left|\dot{\gamma}_{2}^{(k)}\left(\left(\gamma_{1}^{(k)}\right)^{-1}(r)\right)\cdot\frac{1}{\dot{\gamma}_{1}^{(k)}\left(\left(\gamma_{1}^{(k)}\right)^{-1}(r)\right)}\right|\leq\frac{L_{k}\sqrt{2\epsilon-\epsilon^{2}}}{L_{k}(1-\epsilon)}=\frac{\sqrt{2-\epsilon}}{1-\epsilon}\sqrt{\epsilon}.$

This implies that the lemma is proven as soon as a uniform lower bound for
$\rho_{k}(\epsilon)$ is established. To do so, we use the defining property of
$t_{0}(\epsilon)$ from Lemma 5.6 and estimate

$\rho_{k}(\epsilon)=\gamma_{1}^{(k)}(t_{0}(\epsilon))=\int_{0}^{t_{0}(\epsilon)}\dot{\gamma}_{1}^{(k)}(t)dt\geq
t_{0}(\epsilon)L_{k}(1-\epsilon).$

Since $L_{k}\rightarrow L_{*}>0$ we have $\inf_{k}L_{k}>0$ and so
$\rho_{k}(\epsilon)\geq r_{1}(\epsilon)>0$ independent of $k$. ∎

Corollary 5.7 allows us to essentially repeat the proof of Lemma 4.7 to
establish a uniform bound for $u_{k}^{\prime\prime}$.

######  Lemma 5.8.

There exists $C>0$ and $r_{0}>0$ such that $|u_{k}^{\prime\prime}(r)|\leq C$
for all $k\in\mathbb{N}$ and $r\in[0,r_{0}]$.

###### Proof.

As each individual $u_{k}\in C^{\infty}([0,r_{0}))$, we only need to prove a
uniform bound $|u_{k}^{\prime\prime}|\leq C$ for all $k\geq k_{1}$ with some
arbitrary $k_{1}$. To do so, we follow the proof of Lemma 4.7 and put
$w_{k}:=u_{k}^{\prime}$ and $v_{k}:=\sqrt{1+w_{k}^{2}}$. In view of Equation
(4.10), we have

$w_{k}^{\prime\prime}+\left(\frac{w_{k}}{r}\right)^{\prime}-\varphi(w_{k})=\frac{1}{2}v_{k}^{5}a_{k}r+b_{k}v_{k}^{4}w_{k}.$
(5.11)

Here we used that $\lambda_{k}=0$ by Lemma 4.10. Also we note that
$a_{k}=\pm_{\gamma^{(k)}}4\sqrt{\pi}\Lambda_{k}$ and
$b_{k}=\sigma_{k}\Lambda_{k}$. Corollary 5.5 gives $\Lambda_{k}\rightarrow 0$
as $k\rightarrow\infty$ so that $a_{k}$ and $b_{k}$ are bounded. Additionally,
Corollary 5.7 and $u_{k}\rightarrow u^{*}$ in $C^{\infty}((0,r_{0}])$ together
imply that $w_{k}=u_{k}^{\prime}$ is bounded.

Step 1
Note $w_{k}\varphi(w_{k})\geq 0$ by the definition of $\varphi(w)$ in Equation
(4.9). We multiply Equation (5.11) by $w_{k}$, choose any $\rho\in(0,r_{0})$
and integrate to get

$\int_{r}^{\rho}w_{k}^{\prime\prime}w_{k}+\left(\frac{w_{k}}{t}\right)^{\prime}w_{k}dt\geq\frac{a_{k}}{2}\int_{r}^{\rho}v_{k}^{5}tw_{k}dt+b_{k}\int_{r}^{\rho}v_{k}^{4}w_{k}^{2}dt\geq-C.$

$C$ is uniform over $k$ as $a_{k}$, $b_{k}$ and $w_{k}$ are bounded.
Performing the same calculations that lead to Estimate (4.11), we derive

$\displaystyle\int_{r}^{\rho}(w_{k}^{\prime}(t))^{2}+\frac{w_{k}(t)^{2}}{2t^{2}}dt$
$\displaystyle\leq
C(\rho)+\frac{w_{k}(r)^{2}}{2r}-w_{k}(r)\frac{w_{k}(r)+rw_{k}^{\prime}(r)}{r}$
$\displaystyle=C(\rho)+\frac{w_{k}(r)^{2}}{2r}-w_{k}(r)\frac{(rw_{k}(r))^{\prime}}{r}.$
(5.12)

The constant $C(\rho)$ includes the values $w_{k}(\rho)$ and
$w_{k}^{\prime}(\rho)$ but is uniform over $k$. Indeed, as $u_{k}\rightarrow
u^{*}$ in $C^{\infty}_{\operatorname{loc}}((0,r_{0}])$, we have
$w_{k}(\rho)\rightarrow(u^{*})^{\prime}(\rho)$ and
$w_{k}^{\prime}(\rho)\rightarrow(u^{*})^{\prime\prime}(\rho)$.

Step 2
Let again $\rho\in(0,r_{0})$. This time we integrate Equation (5.11) directly
and use $w_{k}^{\prime}+\frac{w_{k}}{r}=\frac{1}{r}(w_{k}r)^{\prime}$ to get

$\frac{(w_{k}t)^{\prime}}{t}\bigg{|}_{t=\rho}-\frac{(w_{k}r)^{\prime}}{r}=\int_{r}^{\rho}\varphi(w_{k})(t)+\frac{1}{2}a_{k}v_{k}^{5}t+b_{k}v_{k}^{4}w_{k}dt.$
(5.13)

Given $\epsilon>0$, we use Corollary 5.7 and choose
$k_{0}(\epsilon)\in\mathbb{N}$ and
$\rho(\epsilon):=r_{1}(\epsilon)\in(0,r_{0}]$ such that
$|w_{k}(r)|\leq\epsilon$ for all $k\geq k_{0}$ and $r\in(0,\rho(\epsilon))$.
This allows us to perform the same manipulations that lead to Estimate (4.13)
and get

$\displaystyle\left|\frac{(rw_{k}(r))^{\prime}}{r}\right|$ $\displaystyle\leq
C(\epsilon)+\epsilon\int_{r}^{\rho}w_{k}^{\prime}(t)^{2}+\frac{w_{k}(t)^{2}}{2t^{2}}dt\hskip
14.22636pt\textrm{for all }r\in(0,\rho(\epsilon)).$ (5.14)

Again $C(\epsilon)$ contains the boundary values $w_{k}(\rho(\epsilon))$ and
$w_{k}^{\prime}(\rho(\epsilon))$ which are, however, controlled uniformly over
$k$.

Step 3
Let $\epsilon>0$ be arbitrary. By taking $\rho=\rho(\epsilon)$ small enough in
Estimate (5.12) we may insert Estimate (5.14) to get

$\displaystyle\int_{r}^{\rho}w_{k}^{\prime}(t)^{2}+\frac{w_{k}(t)^{2}}{2t^{2}}dt$
$\displaystyle\leq
C(\epsilon)+\frac{w_{k}(r)^{2}}{2r}+|w_{k}(r)|\left(C(\epsilon)+\epsilon\int_{r}^{\rho}w_{k}^{\prime}(t)^{2}+\frac{w_{k}(t)^{2}}{2t^{2}}dt\right).$

By potentially shrinking $\rho$ we can assume without loss of generality that
$|w_{k}|\leq\epsilon$ on $[0,\rho]$ and so, in particular
$|w_{k}(r)|\leq\epsilon$. Choosing $\epsilon$ small enough, we can absorb the
integral on the right to the left and obtain that for some small enough $\rho$
and $r\in(0,\rho)$

$\int_{r}^{\rho}w_{k}^{\prime}(t)^{2}+\frac{w_{k}(t)^{2}}{2t^{2}}dt\leq
C\left(\frac{w_{k}(r)^{2}}{2r}+|w_{k}(r)|+1\right).$ (5.15)

Step 4
Let $\rho$ be small enough so that Estimate (5.15) holds. Inserting Estimate
(5.15) into Estimate (5.14) with $\epsilon=1$ and using that $|w_{k}|$ is
bounded uniformly, we get

$\displaystyle\left|\frac{(rw_{k}(r))^{\prime}}{r}\right|\leq$ $\displaystyle
C+C\left(\frac{w_{k}(r)^{2}}{2r}+|w_{k}(r)|+1\right)\leq
C\left(\frac{w_{k}(r)^{2}}{2r}+1\right).$

We multiply by $r$ and choose any $\mu\in(0,1)$. By using Corollary 5.7 and
potentially shrinking $\rho$ we can assume that $|w_{k}|\leq C^{-1}\mu$ on
$[0,\rho]$. Hence

$|(rw_{k})^{\prime}|\leq C(w_{k}(r)^{2}+r)\leq\mu|w_{k}|+Cr.$

Using the same justification as in the proof of Lemma 4.7, we compute

$r|w_{k}|^{\prime}+|w_{k}|=(r|w_{k}|)^{\prime}=|rw_{k}|^{\prime}\leq|(rw_{k})^{\prime}|\leq\mu|w_{k}|+Cr\hskip
14.22636pt\textrm{for all $r\in(0,\rho]$},$

multiply with $r^{-\mu}$ and get

$(r^{1-\mu}|w_{k}|)^{\prime}=r^{1-\mu}|w_{k}|^{\prime}+(1-\mu)r^{-\mu}|w_{k}|\leq
Cr^{1-\mu}.$

We integrate this inequality from $0$ to $r$ and get

$r^{1-\mu}|w_{k}(r)|\leq C\int_{0}^{r}t^{1-\mu}dt\leq C(\mu)r^{2-\mu}\hskip
14.22636pt\textrm{and hence}\hskip 14.22636pt|w_{k}(r)|\leq Cr.$ (5.16)

We now deviate from the proof of Lemma 4.7.

Step 5
Inserting (5.16) into (5.15), we get

$\int_{r}^{\rho}w_{k}^{\prime}(t)^{2}+\frac{w_{k}(t)^{2}}{2t^{2}}dt\leq C.$

Inserting this into Estimate (5.14) with $\epsilon=1$, we obtain
$|(w_{k}(r)r)^{\prime}|\leq Cr$. Combining this with Estimate (5.16), we get

$r|w_{k}^{\prime}(r)|\leq|rw_{k}^{\prime}(r)+w_{k}(r)|+|w_{k}(r)|\leq\left|(w_{k}(r)r)^{\prime}\right|+|w_{k}(r)|\leq
Cr.$

Hence $w_{k}^{\prime}(r)$ is bounded on $[0,\rho]$. The lemma follows by
recalling that $u_{k}\rightarrow u^{*}$ in $C^{\infty}([\rho,r_{0}])$. ∎

Corollary 5.7 and Lemma 5.8 prove that $u_{k}^{\prime}$ and
$u_{k}^{\prime\prime}$ are bounded. To deduce $u_{k}\rightarrow u^{*}$ in
$C^{\infty}([0,r_{0}])$, we prove that all higher derivatives are bounded as
well.

######  Lemma 5.9.

$u_{k}\rightarrow u^{*}$ in $C^{\infty}([0,r_{0}])$.

###### Proof.

In view of Equation (5.10) it suffices to prove that $(u_{k})\subset
C^{m}((0,r_{0}))$ is bounded for all $m\geq 0$. We have
$\gamma^{(k)}\rightarrow\kappa$ in $C^{0}([0,1])$ and hence $\gamma_{2}^{(k)}$
is a bounded sequence of functions. By the definition of $u_{k}$ in Equation
(5.9) we deduce that $(u_{k})\subset C^{0}([0,r_{0}])$ is bounded. Combining
this with Corollary 5.7 and Lemma 5.8, we obtain that $(u_{k})\subset
C^{2}([0,r_{0}])$ is bounded.

We put $w_{k}:=u_{k}^{\prime}$ and $v_{k}:=\sqrt{1+w_{k}^{2}}$. By Equation
(5.11)

$w_{k}^{\prime\prime}+\left(\frac{w_{k}^{\prime}}{r}\right)^{\prime}=\frac{1}{2}v_{k}^{5}a_{k}r+b_{k}v_{k}^{4}w_{k}+\frac{5w_{k}}{2(1+w_{k}^{2})}(w_{k}^{\prime})^{2}+\frac{w_{k}^{3}}{2r^{2}}(3+w_{k}^{2})=:f_{k}(r).$

To prove that $(w_{k})\subset C^{m}([0,r_{0}])$ is bounded for all $m\geq 1$
we argue by induction. $m=1$ has already been established. Assuming that
$(w_{k})\subset C^{m}([0,r_{0}])$ is bounded for some $m\geq 1$, we first use
$w_{k}(0)=u_{k}^{\prime}(0)=0$ to rewrite

$f_{k}(r)=\frac{1}{2}v_{k}^{5}a_{k}r+b_{k}v_{k}^{4}w_{k}+\frac{5w_{k}}{2(1+w_{k}^{2})}(w_{k}^{\prime})^{2}+\frac{w_{k}}{2}(3+w_{k}^{2})\left(\int_{0}^{1}w_{k}^{\prime}(sr)ds\right)^{2}.$
(5.17)

As $(w_{k})\subset C^{m}([0,r_{0}])$ is bounded, Equation (5.17) implies that
$(f_{k})\subset C^{m-1}([0,r_{0}])$ is bounded. Also $f\in
C^{\infty}((0,r_{0}])$. So, using Lemma D.2 we deduce that
$(w_{k}^{(m+2)})\subset C^{0}([0,r_{0}])$ is bounded. Noting that
$w_{k}^{(m+1)}(r_{0})$ is bounded since $u_{k}\rightarrow u^{*}$ in
$C^{\infty}_{\operatorname{loc}}((0,r_{0}])$, we may use Lemma D.2 to get that
$w_{k}^{(m+1)}\subset C^{0}([0,r_{0}])$ is bounded and the inductive step
follows.

Since $u_{k}^{\prime}=w_{k}$ and $(u_{k})\subset C^{0}([0,r_{0}])$ is bounded,
we have shown that $(u_{k})\subset C^{m}([0,r_{0}])$ is bounded for all
$m\in\mathbb{N}_{0}$. ∎

Finally, we prove Theorem 5.1 In view of Lemma 5.4, we only need to prove that
$\gamma^{(k)}$ is bounded in $C^{m}([0,t_{0}]\cup[1-t_{0},1])$ for some small
$t_{0}>0$ and all $m\in\mathbb{N}_{0}$.

Proof of Theorem 5.1
Let $L_{*}:=L[\kappa]$ and $t_{0}>0$ such that $\gamma^{(k)}_{1}|_{[0,t_{0}]}$
are smooth diffeomorphisms for all $k\in\mathbb{N}$. By Lemma 5.6, we may
assume $\dot{\gamma}_{1}^{(k)}(t)\geq\frac{L_{*}}{2}>0$ for all $k\geq k_{0}$
and $t\in[0,t_{0}]$ after potentially shrinking $t_{0}$. We establish that
$(\gamma^{(k)})\subset C^{m}([0,t_{0}])$ is bounded for all $m\in\mathbb{N}$
by an inductive argument. For $m=0$ this follows from the fact that
$\gamma^{(k)}\rightarrow\kappa$ in $C^{0}([0,1])$. Now for the inductive step.
Suppose that for some $m\in\mathbb{N}_{0}$ we already know that
$\gamma^{(k)}\subset C^{m}([0,t_{0}])$ is bounded. Recalling
$\gamma_{2}^{(k)}(t)=u_{k}(\gamma_{1}^{(k)}(t))$ and
$L_{k}^{2}=|\dot{\gamma}^{(k)}(t)|^{2}$, we deduce

$L_{k}^{2}-\left(\dot{\gamma}_{1}^{(k)}(t)\right)^{2}=(u_{k}^{\prime}(\gamma_{1}^{(k)}(t))^{2}\left(\dot{\gamma}_{1}^{(k)}(t)\right)^{2}\hskip
14.22636pt\textrm{and so}\hskip
14.22636pt\left(\dot{\gamma}_{1}^{(k)}(t)\right)^{2}=\frac{L_{k}^{2}}{1+u_{k}^{\prime}(\gamma_{1}^{(k)}(t))^{2}}.$

Using that $\dot{\gamma}_{1}^{(k)}(t)\geq 0$ for all $t\in[0,t_{0}]$ we can
take the square root and obtain

$\dot{\gamma}_{1}^{(k)}(t)=\frac{L_{k}}{\sqrt{1+u_{k}^{\prime}(\gamma_{1}^{(k)}(t))^{2}}}\hskip
14.22636pt\textrm{when $t\in[0,t_{0}]$}.$ (5.18)

By Lemma 5.9 $(u_{k})\subset C^{l}([0,r_{0}])$ is bounded for all
$l\in\mathbb{N}$. Using Equation (5.18) it then follows that
$\gamma_{1}^{(k)}\subset C^{m+1}([0,t_{0}])$ is bounded. Since
$\gamma_{2}^{(k)}=u_{k}\circ\gamma_{1}^{(k)}$, we deduce that $\gamma^{(k)}$
is bounded in $C^{m+1}([0,t_{0}])$, which completes the inductive step.

Finally, we note that a similar discussion proves that $(\gamma^{(k)})$ is
bounded $C^{m}([1-t_{1},1])$ for some (potentially small) $t_{1}>0$ and all
$m\in\mathbb{N}$. Theorem 5.1 follows. ∎

##  6 Formation of the Catenoidal Neck

Throughout this section let $\sigma_{k}\rightarrow 0^{+}$ and
$\gamma^{(k)}\in\mathcal{F}_{\sigma_{k}}^{0+}$ such that
$\mathcal{W}[\gamma^{(k)}]=\beta(\sigma_{k})$. Additionally, let
$L_{k}:=L[\gamma^{(k)}]$ and

$\epsilon_{k}:=\inf_{t\in[\frac{1}{8},\frac{7}{8}]}\gamma_{1}^{(k)}(t).$

######  Lemma 6.1.

  1. (1)

If $(t_{k})\subset[\frac{1}{8},\frac{7}{8}]$ is a sequence of times such that
$\gamma_{1}^{(k)}$ has a local minimum at $t_{k}$, then
$t_{k}\rightarrow\frac{1}{2}$ and $\gamma_{1}^{(k)}(t_{k})\rightarrow 0$.
Moreover, there exists $k_{0}\in\mathbb{N}$ such that the minima are strict
for all $k\geq k_{0}$.

  2. (2)

$\epsilon_{k}\rightarrow 0^{+}$ as $k\rightarrow{\infty}$. Additionally,
$\tau_{k}\rightarrow\frac{1}{2}$ for every sequence
$(\tau_{k})\subset[\frac{1}{8},\frac{7}{8}]$ satisfying
$\gamma_{1}^{(k)}(\tau_{k})=\epsilon_{k}$.

###### Proof.

Clearly, the second part is a special case of the first part.

Now let $(t_{k})\subset[\frac{1}{8},\frac{7}{8}]$ be a sequence of local
minima for $\gamma_{1}^{(k)}$. After passing to a subsequence, we have
$t_{k}\rightarrow t^{*}\in[\frac{1}{8},\frac{7}{8}]$. We prove that
$t^{*}=\frac{1}{2}$ by contradiction. Clearly
$\dot{\gamma}_{1}^{(k)}(t_{k})=0$ and $\ddot{\gamma}_{1}^{(k)}(t_{k})\geq 0$.
If $t^{*}\neq\frac{1}{2}$, we can use $\gamma^{(k)}\rightarrow\kappa$ in
$C^{\infty}([0,1]\backslash\\{\frac{1}{2}\\})$ to obtain
$\dot{\kappa}_{1}(t_{*})=0$ and $\ddot{\kappa}_{1}(t^{*})\geq 0$. However,
since $\kappa_{1}(t)=|\sin(2\pi t)|$, this is impossible. So
$t^{*}=\frac{1}{2}$ and

$h_{k}:=\gamma_{1}^{(k)}(t_{k})\rightarrow\kappa_{1}(\frac{1}{2})=0.$

Finally, we prove that for large $k$, the minima at $t_{k}$ are strict. For
each $k$ we have $\dot{\gamma}_{1}^{(k)}(t_{k})=0$ and
$\ddot{\gamma}_{1}^{(k)}(t_{k})\geq 0$. If $\ddot{\gamma}_{1}^{(k)}(t_{k})>0$,
the local minimum at $t_{k}$ is strict. If $\ddot{\gamma}_{1}^{(k)}(t_{k})=0$
we also have $\dddot{\gamma}_{1}^{(k)}(t_{k})=0$ and
$\ddddot{\gamma}_{1}^{(k)}(t_{k})\geq 0$ as otherwise $t_{k}$ could not be a
local minimum of $\gamma_{1}^{(k)}$. If
$\dot{\gamma}_{1}^{(k)}(t_{k})=\ddot{\gamma}_{1}^{(k)}(t_{k})=\dddot{\gamma}_{1}^{(k)}(t_{k})=0$
and $\ddddot{\gamma}_{1}^{(k)}(t_{k})>0$ the local minimum at $t_{k}$ is
strict. We prove by contradiction that

$\dot{\gamma}_{1}^{(k)}(t_{k})=\ddot{\gamma}_{1}^{(k)}(t_{k})=\dddot{\gamma}_{1}^{(k)}(t_{k})=\ddddot{\gamma}_{1}^{(k)}(t_{k})=0$
(6.1)

along a subsequence is impossible. So let us assume that we had (6.1) along a
subsequence again denoted by $\gamma^{(k)}$. Since
$\dot{\gamma}_{1}^{(k)}(t_{k})=0$ we have
$|\dot{\gamma}_{2}^{(k)}(t_{k})|=L_{k}\neq 0$ so that differentiating
$|\dot{\gamma}^{(k)}(t)|^{2}=L_{k}^{2}$ implies
$\ddot{\gamma}_{2}^{(k)}(t_{k})=\dddot{\gamma}_{2}^{(k)}(t_{k})=\ddddot{\gamma}_{2}^{(k)}(t_{k})=0$.
Using Equation (2.15) we get

$k_{1}^{(k)}(t_{k})=\dot{k}_{1}^{(k)}(t_{k})=\ddot{k}_{1}^{(k)}(t_{k})=0=\dot{k}_{2}^{(k)}(t_{k})=\ddot{k}_{2}^{(k)}(t_{k})\hskip
14.22636pt\textrm{and}\hskip 14.22636pt|k_{2}^{(k)}(t_{k})|=\frac{1}{h_{k}}.$

This implies that $|H[\gamma^{(k)}](t_{k})|=\frac{1}{h_{k}}$,
$K[\gamma^{(k)}](t_{k})=0$ and $(\Delta_{g}H)[\gamma^{(k)}](t_{k})=0$.
Inserting into the Euler-Lagrange equation from Lemma 4.6, we get

$\frac{1}{2h_{k}^{3}}=\frac{1}{2}|H[\gamma^{(k)}](t_{k})|^{3}=2|W[\gamma^{(k)}](t_{k})|=\left|\Lambda_{k}\left(\pm_{k}4\sqrt{\pi}\pm_{k}\frac{\sigma_{k}}{h_{k}}\right)\right|.$
(6.2)

By Corollary 5.5 we have $\Lambda_{k}\rightarrow 0$. So, Equation (6.2)
implies
$\frac{1}{2}=|\Lambda_{k}(\pm_{k}4\sqrt{\pi}h_{k}^{3}\pm_{k}\sigma_{k}h_{k}^{2})|\rightarrow
0$ as $k\rightarrow\infty$, which is a contradiction. ∎

By Theorem 5.1, the curves $\gamma^{(k)}$ asymptotically look like two
circles. The fact that these are joined by a catenoidal shape is derived by
showing that near $t=\frac{1}{2}$, they solve approximately
$H[\gamma^{(k)}]=0$. The precise formulation of this observation is given in
the following lemma.

######  Lemma 6.2.

Let $a_{k}\uparrow\frac{1}{2}$ and $b_{k}\downarrow\frac{1}{2}$. Then

$\lim_{k\rightarrow\infty}\mathcal{W}\left[\gamma^{(k)}\big{|}_{[a_{k},b_{k}]}\right]=\frac{\pi}{2}\lim_{k\rightarrow\infty}\int_{a_{k}}^{b_{k}}H[\gamma_{k}]^{2}\gamma_{1}^{(k)}|\dot{\gamma}^{(k)}|dt=0.$

###### Proof.

The proof is by contradiction. Assume that along a subsequence again denoted
by $\gamma^{(k)}$ we had

$\frac{\pi}{2}\int_{a_{k}}^{b_{k}}H[\gamma_{k}]^{2}\gamma_{1}^{(k)}|\dot{\gamma}^{(k)}|dt\geq\epsilon^{*}>0.$
(6.3)

For $\delta>0$ we put
$I_{\delta}:=[0,\frac{1}{2}-\delta)\cup(\frac{1}{2}+\delta,1]$. For each
$\delta>0$ there exists $k_{0}(\delta)$ such that
$I_{\delta}\cap(a_{k},b_{k})=\emptyset$ for all $k\geq k_{0}(\delta)$. Note
that $\mathcal{W}[\gamma^{(k)}]=\beta(\sigma_{k})\leq 8\pi$. Taking any
$\delta>0$ and $k\geq k_{0}(\delta)$, we can therefore estimate

$\displaystyle 8\pi\geq$ $\displaystyle\mathcal{W}[\gamma^{(k)}]$
$\displaystyle=$
$\displaystyle\frac{\pi}{2}\int_{0}^{1}H[\gamma^{(k)}]^{2}|\dot{\gamma}^{(k)}|\gamma_{1}^{(k)}dt$
$\displaystyle\geq$
$\displaystyle\epsilon^{*}+\frac{\pi}{2}\int_{I_{\delta}}H[\gamma^{(k)}]^{2}|\dot{\gamma}^{(k)}|\gamma_{1}^{(k)}dt.$

By Theorem 5.1, we have $\gamma^{(k)}\rightarrow\kappa$ in
$C^{\infty}(I_{\delta})$ as $k\rightarrow\infty$. So, letting
$k\rightarrow\infty$, we obtain

$8\pi\geq\epsilon^{*}+\frac{\pi}{2}\int_{I_{\delta}}H[\kappa]^{2}|\dot{\kappa}|\kappa_{1}dt=\epsilon^{*}+\mathcal{W}\left[\kappa\big{|}_{I_{\delta}}\right].$

We have $\mathcal{W}[\kappa]=8\pi$, so, letting $\delta\rightarrow 0^{+}$, we
arrive at a contradiction. ∎

The remaining arguments are essentially based on the following blow-up lemma.

######  Lemma 6.3 (Blow-Up Lemma).

Let $t_{k}\rightarrow\frac{1}{2}$ and put $h_{k}:=\gamma_{1}^{(k)}(t_{k})$.
Assume that there is $\rho_{0}>0$ independent of $k$ such that
$\gamma_{1}^{(k)}(t_{k}+h_{k}t)\geq\frac{1}{2}h_{k}$ for all
$|t|\leq\rho_{0}$. Then

$\Gamma^{(k)}(t):=\frac{1}{h_{k}}\left(\gamma^{(k)}(t_{k}+h_{k}t)-(0,\gamma_{2}^{(k)}(t_{k}))\right)$

is bounded in $C^{m}([-\rho,\rho])$ for all $\rho\in(0,\rho_{0})$ and
$m\in\mathbb{N}$. Additionally, if along some subsequence
$\Gamma^{(k_{l})}\rightarrow\Gamma^{*}$ in $C^{2}([-\rho,\rho])$, the limit
$\Gamma^{*}$ satisfies $H[\Gamma^{*}]=0$.

###### Proof.

By Lemma 4.6, the curves $\gamma^{(k)}$ satisfy an Euler Lagrange equation.
Using the well-known scaling of the Willmore operator and the mean curvature,
we get

$\displaystyle W[\Gamma^{(k)}](t)=$ $\displaystyle
h_{k}^{3}W[\gamma^{(k)}](t_{k}+h_{k}t)$ $\displaystyle=$ $\displaystyle
2\Lambda_{k}(\pm_{\gamma^{(k)}}4\sqrt{\pi}h_{k}^{3}-\sigma_{k}h_{k}^{3}H[\gamma^{(k)}](t_{k}+h_{k}t))$
$\displaystyle=$ $\displaystyle
2\Lambda_{k}(\pm_{\gamma^{(k)}}4\sqrt{\pi}h_{k}^{3}-\sigma_{k}h_{k}^{2}H[\Gamma^{(k)}](t)).$

Using Equation (A.9) and Equations (4.3), (4.4), (4.5), (4.6) we deduce that
for all $\phi\in C^{\infty}_{0}((-\rho,\rho))$

$\delta\mathcal{W}[\Gamma^{(k)}]\phi=2\Lambda_{k}\sigma_{k}h_{k}^{2}\delta
A[\Gamma^{(k)}]\phi-8\sqrt{\pi}\Lambda_{k}h_{k}^{3}\delta
V[\Gamma^{(k)}]\phi.$

$h_{k}\rightarrow 0^{+}$ since $t_{k}\rightarrow\frac{1}{2}$ and by Corollary
5.5 we also know $\Lambda_{k}\rightarrow 0$. We have
$|\dot{\Gamma}^{(k)}|=L_{k}\rightarrow L_{*}>0$ and hence $L_{k}+L_{k}^{-1}$
is bounded. By scaling invariance of the Willmore energy we have
$\mathcal{W}[\Gamma^{(k)}|_{(-\rho_{0},\rho_{0})}]\leq\mathcal{W}[\gamma^{(k)}]\leq
8\pi$. Next we exploit the assumption
$\gamma_{1}^{(k)}(t_{k}+h_{k}t)\geq\frac{1}{2}h_{k}$. First, by definition of
$\Gamma^{(k)}$, it implies $\Gamma_{1}^{(k)}(t)\geq\frac{1}{2}$ for all
$|t|<\rho_{0}$. Second, using Lemma 3.9, we get

$\int_{-\rho_{0}}^{\rho_{0}}|\ddot{\Gamma}^{(k)}(t)|^{2}dt=h_{k}\int_{t_{k}-h_{k}\rho_{0}}^{t_{k}+h_{k}\rho_{0}}|\ddot{\gamma}(s)|^{2}ds\leq
C.$

Finally, since by construction $\Gamma^{(k)}(0)=(1,0)$ and
$|\dot{\Gamma}^{(k)}|=L_{k}\leq C$, we obtain $|\Gamma^{(k)}|\leq C$. We have
proven that the assumptions of Theorem C.1 are satisfied with $M$ and
$\kappa(\Gamma^{(k)};(-\rho,\rho))\geq\kappa_{0}>0$ independent of $k$.
Therefore, the uniform $C^{m}$ bounds follow from Theorem C.1.

Now assume that after passing to a subsequence
$\Gamma^{(k)}\rightarrow\Gamma^{*}$ in $C^{2}$. We use the invariance of the
Willmore energy under scaling and apply Lemma 6.2 to compute

$\displaystyle\frac{\pi}{2}\int_{-\rho}^{\rho}H[\Gamma^{*}]^{2}|\dot{\Gamma}^{*}|\Gamma^{*}_{1}dt=$
$\displaystyle\frac{\pi}{2}\lim_{k\rightarrow\infty}\int_{-\rho}^{\rho}H[\Gamma^{(k)}]^{2}|\dot{\Gamma}^{(k)}|\Gamma^{(k)}_{1}dt$
$\displaystyle=$
$\displaystyle\lim_{k\rightarrow\infty}\mathcal{W}[\Gamma^{(k)}|_{[-\rho,\rho]}]$
$\displaystyle=$
$\displaystyle\lim_{k\rightarrow\infty}\mathcal{W}[\gamma^{(k)}|_{[t_{k}-h_{k}\rho,t_{k}+h_{k}\rho]}]$
$\displaystyle=$ $\displaystyle 0.$

Since $\Gamma_{1}^{*}\geq\frac{1}{2}$ on $(-\rho,\rho)$ and
$|\dot{\Gamma}^{*}|=L_{*}>0$ we deduce $H[\Gamma^{*}]\equiv 0$. ∎

Next, we establish that for $k$ large enough, the minimal value $\epsilon_{k}$
is only attained once.

######  Lemma 6.4 (Proof of Theorem 1.3, parts (1) and (2)).

There exists $k_{0}\in\mathbb{N}$ such that for all $k\geq k_{0}$ there exists
a unique $\tau_{k}\in[\frac{1}{8},\frac{7}{8}]$ satisfying
$\gamma_{1}^{(k)}(\tau_{k})=\epsilon_{k}$. Moreover, there exists $\rho_{0}>0$
such that $\dot{\gamma}_{1}(t)<0$ for $t\in(\tau_{k}-\rho_{0},\tau_{k})$ and
$\dot{\gamma}_{1}(t)>0$ for $t\in(\tau_{k},\tau_{k}+\rho_{0})$.

Note that part (3) of Theorem 1.3 follows by combining Lemmas 6.1 and 6.4.

###### Proof.

We choose a sequence $(\tau_{k})\subset[\frac{1}{8},\frac{7}{8}]$ such that
$\gamma_{1}^{(k)}(\tau_{k})=\epsilon_{k}$ and prove by contradiction that
there exists $\rho_{0}>0$ such that $\dot{\gamma}_{1}^{(k)}(t)\neq 0$ for all
$t\in(\tau_{k}-\rho_{0},\tau_{k}+\rho_{0})\backslash\set{\tau_{k}}$. Once this
is shown, uniqueness follows since any other sequence $\tau_{k}^{\prime}$
would satisfy $|\tau_{k}-\tau_{k}^{\prime}|\rightarrow 0$ by Lemma 6.1.

Assuming the above claim was false and potentially passing to a subsequence
again denoted by $\gamma^{(k)}$, we obtain a sequence
$\tilde{\tau}_{k}\neq\tau_{k}$ satisfying
$|\tilde{\tau}_{k}-\tau_{k}|\rightarrow 0$ and
$\dot{\gamma}_{1}^{(k)}(\tilde{\tau}_{k})=0$. For fixed $k$ either
$\ddot{\gamma}_{1}^{(k)}(\tilde{\tau}_{k})\leq 0$ or
$\ddot{\gamma}_{1}^{(k)}(\tilde{\tau}_{k})>0$. Suppose that the second case is
true. Then $\gamma_{1}^{(k)}$ has a local minimum at $\tilde{\tau}_{k}$. By
Lemma 6.1, the minima are strict and hence we deduce that $\gamma_{1}^{(k)}$
must attain a local maximum at some $\hat{\tau}_{k}$ in between $\tau_{k}$ and
$\tilde{\tau}_{k}$. Then $\dot{\gamma}^{(k)}(\hat{\tau}_{k})=(0,\pm L_{k})$
and $\ddot{\gamma}_{1}^{(k)}(\hat{\tau}_{k})\leq 0$. In the first case,
$\tilde{\tau}_{k}$ has the same properties:
$\dot{\gamma}^{(k)}(\tilde{\tau}_{k})=(0,\pm L_{k})$ and
$\ddot{\gamma}_{1}^{(k)}(\tilde{\tau}_{k})\leq 0$.

So, in any case we have produced a sequence
$\tau_{k}\neq\hat{\tau}_{k}\rightarrow\frac{1}{2}$ such that after potentially
passing to another subsequence we have
$\ddot{\gamma}_{1}^{(k)}(\hat{\tau}_{k})\leq 0$ and
$\dot{\gamma}_{1}^{(k)}(\hat{\tau}_{k})=(0,\pm L_{k})$. We will assume that
$+$ is the correct sign. If ‘$-$’ is correct, the argument needs no
modification.

Since $\gamma^{(k)}\rightarrow\kappa$ in $C^{0}([0,1])$ we deduce
$h_{k}:=\gamma_{1}^{(k)}(\hat{\tau}_{k})\rightarrow\kappa_{1}(\frac{1}{2})=0$.
For a parameter $\rho_{0}>0$, which we will choose shortly, we define the
curve

$\Gamma^{(k)}:[-\rho_{0},\rho_{0}]\rightarrow\mathcal{H}^{2},\
\Gamma^{(k)}(t):=\frac{1}{h_{k}}\left(\gamma^{(k)}(\hat{\tau}_{k}+h_{k}t)-\left(0,\gamma^{(k)}_{2}(\hat{\tau}_{k})\right)\right).$

Using $L_{k}|=\dot{\gamma}^{(k)}|\rightarrow L[\kappa]=L_{*}$ we deduce that
$L_{k}$ is bounded and estimate

$\gamma_{1}^{(k)}(\hat{\tau}_{k}+h_{k}t)=h_{k}+h_{k}\int_{0}^{t}\dot{\gamma}_{1}^{(k)}(\hat{\tau}_{k}+h_{k}s)ds\geq
h_{k}(1-L_{k}|t|)\geq h_{k}(1-C\rho_{0}).$ (6.4)

Choosing $\rho_{0}$ small enough, we obtain
$\gamma^{(k)}_{1}(\hat{\tau}_{k}+h_{k}t)\geq\frac{1}{2}h_{k}$. We fix any
$\rho\in(0,\rho_{0})$. By Lemma 6.3 we deduce that $\Gamma^{(k)}$ is bounded
in $C^{m}([-\rho,\rho])$ for all $m\in\mathbb{N}$. Consequently there exists a
subsequence, again denoted by $\Gamma^{(k)}$, that converges in
$C^{2}([-\rho,\rho])$ to some limit $\Gamma^{*}$. Lemma 6.3 gives
$H[\Gamma^{*}]=0$. Since $\Gamma^{(k)}(0)=(1,0)$,
$\dot{\Gamma}^{(k)}(0)=(0,L_{k})$ and $\ddot{\Gamma}^{(k)}_{1}(0)\leq 0$ we
obtain $\Gamma^{*}(0)=(1,0)$, $\dot{\Gamma}^{*}(0)=(0,L_{*})$ and
$\ddot{\Gamma}^{*}_{1}(0)\leq 0$. Inserting into $H[\Gamma^{*}]=0$, we produce
the contradiction

$0=k_{1}[\Gamma^{*}](0)+k_{2}[\Gamma^{*}](0)=-\frac{\ddot{\Gamma}_{1}^{*}(0)\dot{\Gamma}^{*}_{2}(0)}{L_{*}^{3}}+\frac{\dot{\Gamma}_{2}^{*}(0)}{\Gamma_{1}^{*}(0)L_{*}}=-\frac{\ddot{\Gamma}_{1}^{*}(0)}{L_{*}^{2}}+1\geq
1.$

Consequently there exists $\rho_{0}>0$ such that
$\dot{\gamma}_{1}^{(k)}(t)\neq 0$ for all
$t\in(\tau_{k}-\rho_{0},\tau_{k}+\rho_{0})$. We argue for example that
$\dot{\gamma}_{1}^{(k)}>0$ when $t\in(\tau_{k},\tau_{k}+\rho)$. Indeed, if
this were not true, then $\dot{\gamma}_{1}(t)<0$ would have to be true. But
that would imply that for small $t>0$

$\epsilon_{k}=\inf_{[\frac{1}{8},\frac{7}{8}]}\gamma_{1}^{(k)}\leq\gamma_{1}^{(k)}(\tau_{k}+t)=\gamma_{1}^{(k)}(\tau_{k})+\int_{0}^{t}\dot{\gamma}_{1}^{(k)}(\tau_{k}+s)ds<\epsilon_{k}.$

∎

######  Lemma 6.5.

There exists $k_{1}\in\mathbb{N}$ such that for $k\geq k_{1}$, the curves
$\hat{\gamma}^{(k)}(s):=(\gamma_{1}^{(k)}(1-s),\gamma_{2}^{(k)}(1-s)-\gamma_{2}^{(k)}(1))$
satisfy $\hat{\gamma}^{(k)}\in\mathcal{F}_{\sigma}^{0+}$ and
$\mathcal{W}[\hat{\gamma}^{(k)}]=\beta(\sigma_{k})$.

###### Proof.

It is clear that $\hat{\gamma}^{(k)}\in\mathcal{P}$,
$\mathcal{W}[\gamma^{(k)}]=\mathcal{W}[\hat{\gamma}^{(k)}]$ and that
$\mathcal{I}[\gamma^{(k)}]=\mathcal{I}[\hat{\gamma}^{(k)}]$. Additionally
$\hat{\gamma}^{(k)}(0)=0$. Let $\epsilon>0$ satisfy

$0<\epsilon<\int_{0}^{1}\kappa_{2}(t)dt.$

Since $\gamma^{(k)}\rightarrow\kappa$ in $C^{0}([0,1])$ we can choose
$k_{1}(\epsilon)$ such that for all $k\geq k_{1}$

$\int_{0}^{1}\hat{\gamma}_{2}^{(k)}(t)dt=\int_{0}^{1}\gamma_{2}^{(k)}(t)dt-\gamma_{2}^{(k)}(1)\geq\int_{0}^{1}\kappa_{2}(s)ds-\kappa_{2}(1)-\epsilon=\int_{0}^{1}\kappa_{2}(s)ds-\epsilon>0.$

∎

By Lemma 6.4 we know that for $k$ large enough, $\gamma_{1}^{(k)}$ attains its
minimum on $[\frac{1}{8},\frac{7}{8}]$ only once, namely at $t=\tau_{k}$. At
$t=\tau_{k}$ we then have $\dot{\gamma}^{(k)}(\tau_{k})=(0,\pm_{k}L_{k})$.
Using Lemma 6.5, we can now modify the sequence $\gamma^{(k)}$ and ensure that
for all $k\geq k_{1}$

$\dot{\gamma}^{(k)}(\tau_{k})=(0,-L_{k}).$ (6.5)

######  Lemma 6.6 (Proof of Theorem 1.3, part (4)).

Let $\gamma^{(k)}$ now additionally satisfy Equation (6.5). For all $R>0$
there exists $k_{0}(R)\in\mathbb{N}$ such that for all $k\geq k_{0}(R)$ the
function

$\Gamma^{(k)}:[-R,R]\rightarrow\mathcal{H}^{2},\
\Gamma^{(k)}(t):=\frac{1}{\epsilon_{k}}\left(\gamma^{(k)}(\tau_{k}+\epsilon_{k}t)-(0,\gamma_{2}^{(k)}(\tau_{k}))\right)$

is well-defined. Additionally, putting

$\Gamma^{*}(t):=\left(\sqrt{1+L_{*}^{2}t^{2}},\
-\operatorname{asinh}(L_{*}t)\right)\hskip 14.22636pt\textrm{where
}L_{*}=\sqrt{\frac{\pi}{2}},$

we have $\Gamma^{(k)}\rightarrow\Gamma^{*}$ in $C^{m}([-R,R])$ for all
$m\in\mathbb{N}$.

###### Proof.

For $R>0$ we choose $k_{0}(R)$ so large that
$|\tau_{k}+\epsilon_{k}t|\in(\frac{1}{8},\frac{7}{8})$ for all $|t|\leq R+1$
and $k\geq k_{0}(R)$. By definition of $\epsilon_{k}$ we get
$\Gamma^{(k)}_{1}(t)\geq 1$ for all $t\in[-R,R]$. Therefore we can use Lemma
6.3 and deduce that $\Gamma^{(k)}$ is bounded in $C^{m}([-R,R])$ for all
$m\in\mathbb{N}$. We show that any subsequence contains another subsequence
that converges to the claimed limit. Let $\Gamma^{(k)}$ be some subsequence,
again denoted by $\Gamma^{(k)}$. By the uniform $C^{m}$ bound we can assume
that $\Gamma^{(k)}\rightarrow\Gamma^{*}$ in $C^{2}([-R,R])$ after potentially
passing to yet another subsequence and by Lemma 6.3, the limit satisfies
$H[\Gamma^{*}]=0$. By assumption we have
$\dot{\gamma}_{2}^{(k)}(\tau_{k})=-L_{k}$ and since $\tau_{k}$ is a minimum of
$\gamma_{1}^{(k)}$ we have $\dot{\gamma}_{1}^{(k)}(\tau_{k})=0$ and
$\ddot{\gamma}_{1}^{(k)}(\tau_{k})\geq 0$. By Theorem 5.1, we have
$L_{k}=L[\gamma^{(k)}]\rightarrow L[\kappa]=\sqrt{\frac{\pi}{2}}$. Exploiting
the $C^{2}$-convergence, we get

$\Gamma^{*}(0)=(1,0),\hskip 14.22636pt\dot{\Gamma}^{*}(0)=(0,-L_{*})\hskip
14.22636pt\textrm{and}\hskip 14.22636pt\ddot{\Gamma}^{*}_{1}(0)\geq 0.$

Since
$|\dot{\Gamma}^{*}(t)|=\lim_{k\rightarrow\infty}|\dot{\Gamma}^{(k)}(t)|=\lim_{k\rightarrow\infty}L_{k}=L_{*}$,
the limit $\Gamma^{*}$ is parameterized proportional to arc length. Using
Equation (2.20), we get

$\ddot{\Gamma}_{1}^{*}=\frac{(\dot{\Gamma}_{2}^{*})^{2}}{\Gamma_{1}^{*}}\hskip
14.22636pt\textrm{and}\hskip
14.22636pt\ddot{\Gamma}_{2}^{*}=-\frac{\dot{\Gamma}_{1}^{*}\dot{\Gamma}_{2}^{*}}{\Gamma_{1}^{*}}.$
(6.6)

Using the first identity and $|\dot{\Gamma}^{*}|^{2}=L_{*}^{2}$, we compute

$\frac{1}{2}\frac{d^{2}}{dt^{2}}\Gamma_{1}^{*}(t)^{2}=\ddot{\Gamma}_{1}^{*}\Gamma_{1}^{*}+(\dot{\Gamma}_{1}^{*})^{2}=(\dot{\Gamma}_{2}^{*})^{2}+(\dot{\Gamma}_{1}^{*})^{2}=L_{*}^{2}.$

So, using $\Gamma_{1}^{*}(0)=1$, $\dot{\Gamma}_{1}^{*}(0)=0$ and
$\Gamma_{1}^{*}(t)\geq 0$, we deduce
$\Gamma_{1}^{*}(t)=\sqrt{1+L_{*}^{2}t^{2}}$. Now, using the second identity in
Equation (6.6), we compute

$\frac{d}{dt}(\dot{\Gamma}_{2}^{*}\Gamma_{1}^{*})=\ddot{\Gamma}_{2}^{*}\Gamma_{1}^{*}+\dot{\Gamma}_{1}^{*}\dot{\Gamma}_{2}^{*}=0\hskip
14.22636pt\textrm{and hence}\hskip
14.22636pt\dot{\Gamma}_{2}^{*}(t)\Gamma_{1}^{*}(t)=\dot{\Gamma}_{2}^{*}(0)\Gamma_{1}^{*}(0).$

We have $\dot{\Gamma}_{2}^{*}(0)=-L_{*}$ and $\Gamma^{*}(0)=(1,0)$, which
implies

$\dot{\Gamma}_{2}^{*}(t)=\frac{-L_{*}}{\Gamma_{1}^{*}(t)}=\frac{-L_{*}}{\sqrt{1+L_{*}^{2}t^{2}}}\hskip
14.22636pt\textrm{and hence}\hskip
14.22636pt\Gamma_{2}^{*}(t)=-\operatorname{asinh}(L_{*}t).$

∎

##  Appendix A Properties of Profile Curves

###  A.1 Proofs for Subsection 3.1

We begin by proving Lemma 3.6

###### Proof.

We put $L:=L[\gamma]$ and $d\mu_{\gamma}:=2\pi|\dot{\gamma}|\gamma_{1}dt$ so
that $k_{1}^{2}+k_{2}^{2}\in L^{1}(\mu_{\gamma})$ by definition of the class
$\mathcal{P}$. Hence $|k_{1}k_{2}|\in L^{1}(\mu_{\gamma})$ and we may estimate

$\displaystyle\mathcal{W}[\gamma]=$
$\displaystyle\frac{2\pi}{4}\int_{0}^{1}(k_{1}+k_{2})^{2}\gamma_{1}|\dot{\gamma}(t)|dt$
$\displaystyle\geq
2\pi\int_{0}^{1}k_{1}k_{2}|\dot{\gamma}|\gamma_{1}dt=\lim_{\epsilon\rightarrow
0^{+}}2\pi\int_{\epsilon}^{1-\epsilon}k_{1}k_{2}|\dot{\gamma}|\gamma_{1}dt.$
(A.1)

By definition of the class $\mathcal{P}$ we have $\gamma\in
W^{2,2}((\epsilon,1-\epsilon))$. So we may use Equation (2.21) and deduce

$2\pi\int_{\epsilon}^{1-\epsilon}k_{1}k_{2}|\dot{\gamma}|\gamma_{1}dt=-\frac{2\pi}{L}\int_{\epsilon}^{1-\epsilon}\ddot{\gamma}_{1}(t)dt=\frac{2\pi}{L}(\dot{\gamma}_{1}(\epsilon)-\dot{\gamma}_{1}(1-\epsilon)).$

We insert into Equation (A.1) and wish to let $\epsilon\rightarrow 0^{+}$. To
do so, note that $\dot{\gamma}$ is continuous by the definition of the class
$\mathcal{P}$. So, using Equation (3.2), we find

$\mathcal{W}[\Sigma_{\gamma}]\geq
2\pi\int_{0}^{1}k_{1}k_{2}|\dot{\gamma}|\gamma_{1}dt=\lim_{\epsilon\rightarrow
0^{+}}2\pi\int_{\epsilon}^{1-\epsilon}k_{1}k_{2}|\dot{\gamma}|\gamma_{1}dt=\frac{2\pi}{L}(L-(-L))=4\pi.$
(A.2)

Equality only occurs when $(k_{1}+k_{2})^{2}=4k_{1}k_{2}$ almost everywhere,
which is true only if $k_{1}=k_{2}$ almost everywhere. This gives

$\ddot{\gamma}_{2}\dot{\gamma}_{1}-\ddot{\gamma}_{1}\dot{\gamma}_{2}=L^{3}k_{1}=L^{3}k_{2}=\frac{\dot{\gamma}_{2}}{\gamma_{1}}L^{2}.$

Multiplying this equation by $\dot{\gamma}_{2}$ and using
$\langle\dot{\gamma},\ddot{\gamma}\rangle=0$ yields

$\ddot{\gamma}_{1}\gamma_{1}=\dot{\gamma}_{1}^{2}-L^{2}.$ (A.3)

As $\gamma_{1}>0$ on $(0,1)$ and $\gamma\in C^{1}([0,1])$, we deduce
$\gamma_{1}\in C^{2}((0,1))\cap C^{1}([0,1])$. Since $\dot{\gamma}_{1}\in
C^{0}([0,1])$, $\dot{\gamma}_{1}(0)=L$ and $\dot{\gamma}_{1}(1)=-L$ there
exists a non-empty, open interval $I\subset(0,1)$ such that
$\dot{\gamma}_{1}(t)\not\in\set{0,\pm L}$ for all $t\in I$. On $I$, the
following computation is valid:

$-\frac{1}{2}\frac{d}{dt}\ln(L^{2}-\dot{\gamma}_{1}^{2})=\frac{\dot{\gamma}_{1}\ddot{\gamma}_{1}}{L^{2}-\dot{\gamma}_{1}^{2}}\overset{\eqref{globalgamma1ode}}{=}-\frac{\dot{\gamma}_{1}}{\gamma_{1}}=-\frac{d}{dt}\ln(\gamma_{1}(t))$

From this, we deduce that there exists $K>0$ such that
$L^{2}-\dot{\gamma}_{1}^{2}=K\gamma_{1}^{2}(t)$ for all $t\in I$.
Differentiating gives $\dot{\gamma}_{1}(\ddot{\gamma}_{1}+K\gamma_{1})=0$. As
$\dot{\gamma}_{1}(t)\neq 0$ we deduce that for some $a,b\in\mathbb{R}$ we get

$\gamma_{1}(t)=a\cos(\sqrt{K}t)+b\sin(\sqrt{K}t)\hskip 14.22636pt\textrm{for
all $t\in I$}.$ (A.4)

Using $\gamma_{1}>0$ for all $t\in(0,1)$ and Equation (A.3), the Picard-
Lindeöff theorem implies that Equation (A.4) is valid for all $t\in(0,1)$.
Using $\dot{\gamma}_{2}^{2}=L^{2}-\dot{\gamma}_{1}^{2}$, it is then readily
seen that $\gamma$ traces out a semicircle and hence $\Sigma_{\gamma}$ is a
sphere. ∎

Note that in Equation (A.2), we have verified Lemma 3.7 as we have shown that

$\int_{\mathbb{S}^{2}}k_{1}k_{2}d\mu_{\Sigma_{\gamma}}=2\pi\int_{0}^{1}k_{1}k_{2}|\dot{\gamma}|\gamma_{1}dt=4\pi.$

Next, we prove Lemma 3.8

###### Proof.

First we note $H_{0}\neq 0$ as
$4\pi\leq\mathcal{W}[\gamma]=\frac{1}{4}A[\gamma]H_{0}^{2}$ by Lemma 3.6. Let
$L:=L[\gamma]$. Using the assumption $H=H_{0}$ and Equation (2.20), we get

$H_{0}\dot{\gamma}_{2}=-\frac{\ddot{\gamma}_{1}}{L}+\frac{\dot{\gamma}_{2}^{2}}{L\gamma_{1}}\hskip
14.22636pt\textrm{and}\hskip
14.22636ptH_{0}\dot{\gamma}_{1}=\frac{\ddot{\gamma}_{2}}{L}+\frac{\dot{\gamma}_{1}\dot{\gamma}_{2}}{L\gamma_{1}}.$
(A.5)

Using $\gamma\in C^{1}([0,1])$, $\gamma_{1}>0$ on $(0,1)$ and Equation (A.5),
we get $\gamma\in C^{\infty}((0,1))$. Using the second identity in Equation
(A.5), we compute

$\frac{d}{dt}(\dot{\gamma}_{2}\gamma_{1})=\ddot{\gamma}_{2}\gamma_{1}+\dot{\gamma}_{2}\dot{\gamma}_{1}=LH_{0}\gamma_{1}\dot{\gamma}_{1}\hskip
14.22636pt\textrm{and hence}\hskip
14.22636pt\dot{\gamma}_{2}\gamma_{1}=\frac{1}{2}LH_{0}\gamma_{1}^{2}.$ (A.6)

There is no integration constant since $\gamma_{1}(0)=0$. Using
$\gamma_{1}(t)>0$ for $t\in(0,1)$, we may deduce
$\dot{\gamma}_{2}=\frac{1}{2}LH_{0}\gamma_{1}$. Inserting into the first
identity in Equation (A.5) and putting $\omega:=\frac{LH_{0}}{2}$, we get

$\ddot{\gamma}_{1}+\omega^{2}\gamma_{1}=0\hskip 14.22636pt\textrm{and