hence}\hskip 14.22636pt\gamma_{1}(t)=a\cos(\omega t)+b\sin(\omega t).$

Combining $\gamma_{1}(0)=\gamma_{1}(1)=0$ and $\gamma_{1}(t)>0$ for
$t\in(0,1)$ we deduce $a=0$, $b>0$ and $\omega=\pi$. Using
$\dot{\gamma}_{2}=\omega\gamma_{1}$ we get
$\gamma_{2}(t)=\gamma_{2}(0)+b(1-\cos(\pi t))$. Finally, since
$|\dot{\gamma}|=L$, we conclude $b=\frac{L}{\pi}$. ∎

###  A.2 First Variation of the Willmore Energy

Let $I=[s_{0},s_{1}]$ be an interval in $\mathbb{R}$ and
$\gamma=(\gamma_{1},\gamma_{2})\in C^{\infty}(I,\mathcal{H}^{2})$. We assume
that the axisymmetric surface defined by

$f:I\times[0,2\pi)\rightarrow\mathbb{R}^{3},\
f(s,\theta):=\begin{bmatrix}\gamma_{1}(s)\cos\theta\\\
\gamma_{1}(s)\sin\theta\\\ \gamma_{2}(s)\end{bmatrix}$

is smooth. Given a variation
$\phi:(-\epsilon_{0},\epsilon_{0})\rightarrow\mathcal{H}^{2}$ of $\gamma$, we
can define a variation of $f$ by putting

$\Phi:(-\epsilon_{0},\epsilon_{0})\times
I\times[0,2\pi)\rightarrow\mathbb{R}^{3},\
\Phi(\epsilon,s,\theta):=\begin{bmatrix}\phi_{1}(\epsilon,s)\cos\theta\\\
\phi_{1}(\epsilon,s)\sin\theta\\\ \phi_{2}(\epsilon,s)\end{bmatrix}.$

Considering Equations (2.12) for the metric, (2.15) for the principal
curvatures of $f$ and the formula for $W[f]$ from Equation (2.4), we observe
that $W$ only depends on $s$ and not on $\theta$. This justifies the notation
$W[\gamma]$ for the very same function. Using the formulas for the normals of
$\Sigma_{\gamma}$ and $\gamma$ from Equation (2.13) we further have
$\langle\Phi^{\prime}(0),n\rangle=\langle\phi^{\prime}(0),\nu\rangle$. Using
the surface element from Equation (2.12), we can rewrite the surface integral
in Equation (2.3) as

$\int_{I\times\mathbb{S}^{1}}W[f]\langle\Phi^{\prime}(0),n\rangle
d\mu_{f}=2\pi\int_{s_{0}}^{s_{1}}W[\gamma]\langle\phi^{\prime}(0),\nu\rangle\gamma_{1}|\dot{\gamma}|ds.$

Let us now consider the boundary term. We assume that $\gamma$ is not closed –
that is $\gamma(s_{0})\neq\gamma(s_{1})$. Additionally, we assume that
$\gamma_{1}(s_{0})\neq 0\neq\gamma_{1}(s_{1})$. In this case
$\partial\Sigma_{\gamma}\neq\emptyset$ and

$\partial\Sigma_{\gamma}=\set{f(s,\theta)\ }{\ s\in\set{s_{0},s_{1}},\
\theta\in[0,2\pi)}.$

We focus on the boundary component at $s=s_{0}$. The interior conormal is
given by

$\eta=\frac{1}{|\dot{\gamma}(s)|}\frac{\partial}{\partial
s}\bigg{|}_{s=s_{0}}\hskip 14.22636pt\textrm{and}\hskip
14.22636ptDf\eta=\frac{1}{|\dot{\gamma}|}\frac{\partial f}{\partial
s}\bigg{|}_{s=s_{0}}.$ (A.7)

The tangential part of $\Phi^{\prime}(0)$ is
$\Phi^{\prime}(0)-\langle\Phi^{\prime}(0),n\rangle n$ and

$\langle\Phi^{\prime}(0)-\langle\Phi^{\prime}(0),n\rangle
n,Df\eta\rangle=\frac{1}{|\dot{\gamma}(s_{0})|}\langle\Phi^{\prime}(0),\frac{\partial
f}{\partial
s}\bigg{|}_{s=s_{0}}\rangle=\frac{1}{|\dot{\gamma}(s_{0})|}\langle\phi^{\prime}(0),\dot{\gamma}(s_{0})\rangle.$
(A.8)

Combining Equations (A.7) and (A.8) and using the formula fom Equation (2.5),
we get

$\displaystyle\int_{s_{0}\times\mathbb{S}^{1}}\omega(\eta)dS_{f}=$
$\displaystyle\frac{1}{2}\left[\frac{\varphi(s_{0})H^{\prime}(s_{0})-\varphi^{\prime}(s_{0})H(s_{0})}{|\dot{\gamma}(s_{0})|}-\frac{1}{2}\frac{H(s_{0})^{2}}{|\dot{\gamma}(s_{0})|}\langle\dot{\gamma}(s_{0}),\phi^{\prime}(0)\rangle\right]\int_{0}^{2\pi}\gamma_{1}(s_{0})d\theta$
$\displaystyle=$
$\displaystyle\pi\left[\frac{\varphi(s_{0})H^{\prime}(s_{0})-\varphi^{\prime}(s_{0})H(s_{0})-\frac{1}{2}H(s_{0})^{2}\langle\dot{\gamma}(s_{0}),\phi^{\prime}(0)\rangle}{|\dot{\gamma}(s_{0})|}\right]\gamma_{1}(s_{0}).$

Computing the boundary term at $s=s_{1}$ is achieved by a similar computation.
In total, we get the formula

$\displaystyle\delta\mathcal{W}[\gamma]\phi^{\prime}(0)=$ $\displaystyle
2\pi\int_{s_{0}}^{s_{1}}W[\gamma]\langle\phi,\nu\rangle\gamma_{1}|\dot{\gamma}|ds$
(A.9)
$\displaystyle-\pi\left[\frac{\varphi(s)H^{\prime}(s)-\varphi^{\prime}(s)H(s)-\frac{1}{2}H^{2}(s)\langle\dot{\gamma}(s),\phi^{\prime}(0)\rangle}{|\dot{\gamma}(s)|}\right]\gamma_{1}(s)\bigg{|}_{s=s_{0}}^{s=s_{1}}.$

###  A.3 Existence of Variations

######  Lemma A.1.

Let $\sigma\in(0,1)$, $\gamma\in\mathcal{F}_{\sigma}$ and $\eta\in
C^{\infty}_{0}((0,1))$ such that $\delta\mathcal{I}[\gamma]\eta=0$. Then there
exists
$\Phi:(-\epsilon_{0},\epsilon_{0})\rightarrow\mathcal{F}^{\prime}_{\sigma}$
such that $\Phi^{\prime}(0)=\eta$.

###### Proof.

Since $\sigma\in(0,1)$, Lemma 4.3 provides $\psi_{0}\in C^{\infty}_{0}((0,1))$
such that $\delta\mathcal{I}[\gamma]\psi_{0}=1$. For small
$\epsilon,\rho\in\mathbb{R}$ we have
$\gamma+\epsilon\eta+\rho\psi_{0}\in\mathcal{P}^{\prime}$ as
$\operatorname{supp}(\eta),\ \operatorname{supp}(\psi_{0})\subset(0,1)$. Now,
we define the operator

$Q(\epsilon,\rho):=\mathcal{I}[\gamma+\epsilon\eta+\rho\psi_{0}].$

Clearly $Q(0,0)=\sigma$ and

$\frac{d}{d\rho}\bigg{|}_{\rho=0}Q(0,\rho)=\delta\mathcal{I}[\gamma]\psi_{0}=1\neq
0.$

So, by the implicit function theorem, there exist $\epsilon_{0},\rho_{0}>0$
and a smooth map
$\kappa:(-\epsilon_{0},\epsilon_{0})\rightarrow(-\rho_{0},\rho_{0})$ such that
for $\epsilon\in(-\epsilon_{0},\epsilon_{0})$ and
$\rho\in(-\rho_{0},\rho_{0})$ we have

$Q(\epsilon,\rho)=\sigma\hskip 14.22636pt\Leftrightarrow\hskip
14.22636pt\rho=\kappa(\epsilon).$

Moreover $\kappa^{\prime}(0)=0$ as

$0=\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}Q(\epsilon,\kappa(\epsilon))=\delta\mathcal{I}[\gamma]\eta+\kappa^{\prime}(0)\delta\mathcal{I}[\gamma]\psi_{0}=\kappa^{\prime}(0).$

Therefore $\Phi(\epsilon):=\gamma+\epsilon\eta+\kappa(\epsilon)\psi_{0}$
provides a variation as claimed. ∎

Proof of Lemma 4.9.
Using $\varphi^{\prime}(0)=0$ and $\operatorname{supp}(\psi_{0})\subset(0,1)$,
we have
$\gamma+\epsilon\chi\cdot(\varphi\circ\gamma_{1})e_{2}+\rho\psi_{0}\in\mathcal{P}^{\prime}$.
This allows the definition of the operator

$Q(\epsilon,\rho):=\mathcal{I}[\gamma+\epsilon\chi\cdot(\varphi\circ\gamma_{1})e_{2}+\rho\psi_{0}].$

Note $Q(0,0)=\sigma$ and

$\frac{d}{d\rho}\bigg{|}_{\rho=0}Q(0,\rho)=\delta\mathcal{I}[\gamma]\psi_{0}=1\neq
0.$

So, by the implicit function theorem, there exist $\epsilon_{0},\rho_{0}>0$
and a diffeomorphism
$\kappa:(-\epsilon_{0},\epsilon_{0})\rightarrow(-\rho_{0},\rho_{0})$ such that
for $\epsilon\in(-\epsilon_{0},\epsilon_{0})$ and
$\rho\in(-\rho_{0},\rho_{0})$ we have

$Q(\epsilon,\rho)=\sigma\hskip 14.22636pt\Leftrightarrow\hskip
14.22636pt\rho=\kappa(\epsilon).$

Moreover we have
$\kappa^{\prime}(0)=-\delta\mathcal{I}[\gamma]\left(\chi\cdot(\varphi\circ\gamma_{1})e_{2}\right)$
as

$0=\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}Q(\epsilon,\kappa(\epsilon))=\delta\mathcal{I}[\gamma]\left(\chi\cdot(\varphi\circ\gamma_{1})e_{2}\right)+\kappa^{\prime}(0)\delta\mathcal{I}[\gamma]\psi_{0}.$

Therefore
$\Phi(\epsilon):=\gamma+\epsilon\chi\cdot(\varphi\circ\gamma_{1})e_{2}+\kappa(\epsilon)\psi_{0}$
provides a variation as claimed. ∎

###  A.4 Proof of Theorem 3.12

Schygulla’s proof in [18] is based on studying the scaled catenoid. For $a>0$
he considers

$g_{a}:\mathbb{R}\times[0,2\pi)\rightarrow\mathbb{R}^{3},\
g_{a}(s,\theta):=\left(a\cosh\left(\frac{s}{a}\right)\cos(\theta),a\cosh\left(\frac{s}{a}\right)\sin(\theta),s\right).$

Next, he introduces the inversion $I(x):=e_{3}+\frac{x-e_{3}}{|x-e_{3}|^{2}}$,
defines $f_{a}:=I\circ g_{a}$ and puts

$\Sigma_{a}:=f_{a}(\mathbb{R}\times[0,2\pi))\cup\set{e_{3}}.$

$\Sigma_{a}$ is smooth away from $e_{3}$. Additionally, there exist $R>0$,
functions $u^{\pm}:B_{R}(0)\rightarrow\mathbb{R}$ and a neighbourhood of
$e_{3}$ in which
$\Sigma_{a}=\operatorname{graph}(u^{+})\cup\operatorname{graph}(u^{-})$.
Direct computation shows $u^{\pm}\in C^{1,\alpha}(B_{R}(0))\cap
W^{2,p}(B_{R}(0))$ for all $\alpha\in(0,1)$ and $p\in[1,\infty)$. Another
direct computation shows $\mathcal{W}[\Sigma_{a}]=8\pi$.

We now sketch the proof of Theorem 3.12 by following Schygulla’s arguments and
commenting on the necessary alterations. For details, we refer to Lemma 1 in
Schygulla’s article [18].

###### Proof.

Let $\varphi_{\pm}\in C^{\infty}_{0}(B_{R})$ be axially symmetric and smooth
functions – that is $\varphi(x)=\varphi(Tx)$ for all
$T\in\operatorname{SO}(2)$ – that satisfy $\varphi_{\pm}(0)=\pm 1$. For small
$\epsilon>0$, we define variations

$\Phi_{\pm}^{\epsilon}:B_{R}(0)\rightarrow\mathbb{R}^{3},\
\Phi^{\epsilon}_{\pm}(x):=(x,u_{\pm}(x)+\epsilon\varphi_{\pm}(x)).$

Following Schygulla’s computations, it follows that there exists $c_{0}>0$
such that

$\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0}\mathcal{W}[\Phi_{\pm}^{\epsilon}]=\mp
c_{0}\varphi_{\pm}(0).$

We modify $\Sigma_{a}$ and construct a surface $\Sigma_{a}^{\epsilon}$ by
replacing the graphs of $u^{\pm}$ with the graphs of
$u_{\pm}+\epsilon\varphi_{\pm}$ with small $\epsilon>0$. Consequently
$\mathcal{W}[\Sigma_{a}^{\epsilon}]<8\pi-\epsilon c_{0}$ for small
$\epsilon>0$. Additionally, $\Sigma_{a}^{\epsilon}$ can be parameterized over
$\mathbb{S}^{2}$ and is axially symmetric. Finally, we note
$\mathcal{I}[\Sigma_{a}]\rightarrow 0$, as $a\rightarrow 0^{+}$. Given
$\sigma\in(0,1)$ we can therefore choose $a$ and $\epsilon$ so small such that
simultaneously

$\displaystyle\mathcal{W}[\Sigma_{a}^{\epsilon}]$
$\displaystyle\leq\mathcal{W}[\Sigma_{a}]-\epsilon c_{0}=8\pi-\epsilon c_{0},$
$\displaystyle\mathcal{I}[\Sigma_{a}^{\epsilon}]$ $\displaystyle<\sigma.$

Approximating $\Sigma_{a}^{\epsilon}$ with smooth surfaces, we deduce that
there exists a smooth, axially symmetric embedding
$F_{0}:\mathbb{S}^{2}\rightarrow\mathbb{R}^{3}$ with $\mathcal{W}[F_{0}]<8\pi$
and $\mathcal{I}[F_{0}]<\sigma$. Theorem 5.2 in [12] allows us to deduce that
the Willmore flow $F(t)$ with initial datum $F(0)=F_{0}$ exists for all times
and converges to a sphere. As $F_{0}$ is axially symmetric and the solution to
the Willmore flow is unique (Proposition 1.1 in [11]), it follows that $F(t)$
is axially symmetric for all $t\geq 0$. Since $\mathcal{I}[F(0)]<\sigma$ and
$\mathcal{I}[F(t)]\rightarrow 1$ as $t\rightarrow\infty$, there exists a time
$t_{0}>0$ such that $\mathcal{I}[F(t_{0})]=\sigma$. Finally, by definitions of
$\beta(\sigma)$ and the dissipation of Willmore energy along the Willmore
flow, we get

$\beta(\sigma)\leq\mathcal{W}[F(t_{0})]\leq\mathcal{W}[F(0)]<8\pi.$

∎

###  A.5 The Monotonicity Formula

Let $\Sigma$ be a smooth $2$-dimensional manifold without boundary, $f\in
C^{1}(\Sigma,\mathbb{R}^{3})$ be an immersion and $\mu_{\Sigma}$ denote the
surface measure on $\Sigma$ that is induced by $f$. For $p\in\mathbb{R}^{3}$,
the _muliplicity_ of $p$ is defined as

$\theta^{2}_{f}(p):=\lim_{r\rightarrow 0^{+}}\frac{\mu_{\Sigma}(B_{r}(p))}{\pi
r^{2}}.$ (A.10)

If $\Sigma$ is compact, this expression is well-defined and

$\theta^{2}_{f}(p)=|\set{x\in\Sigma\ }{\ f(x)=p}|.$ (A.11)

The following result is due Simon [20]. Its applicability in the present
setting has essentially been established by Kuwert and Schätzle [12], Appendix
A.

######  Lemma A.2 (Monotonicity Formula).

Let $\gamma\in\mathcal{P}\cap C^{\infty}((0,1),\mathcal{H}^{2})$. The surface
$f_{\gamma}:[0,1]\times[0,2\pi)\rightarrow\mathbb{R}^{3}$ as defined in
Equation (2.11) is $C^{1}$-immersed. Putting
$\theta^{2}(p):=\theta^{2}_{f_{\gamma}}(p)$, we have

$\frac{1}{\pi}\int_{\mathbb{S}^{2}}\left|\frac{1}{4}\vec{H}+\frac{(f_{\gamma}(x)-p)^{\perp}}{|f_{\gamma}(x)-p|^{2}}\right|d\mu_{\Sigma_{\gamma}}(x)\leq\frac{1}{4\pi}\mathcal{W}[f_{\gamma}]-\theta^{2}(p).$

Here $\vec{H}$ is the almost everywhere defined mean curvature vector of
$f_{\gamma}$ and $(f_{\gamma}(x)-p)^{\perp}$ is the component of
$f_{\gamma}(x)-p$ that is orthogonal to $Df_{\gamma}(x)(T_{x}\mathbb{S}^{2})$.

###### Proof.

We put $f:=f_{\gamma}$ and $\mu:=\mu_{\Sigma_{\gamma}}$. We use the
monotonicity formula proved in Appendix A of [12]. We may apply their results
as the mean curvature $H$ is in $L^{2}(\mu)$. Indeed, by the definition of the
class $\mathcal{P}$ (see Definition 3.1)

$\int_{\mathbb{S}^{2}}H^{2}d\mu\leq
2\int_{\mathbb{S}^{2}}k_{1}^{2}+k_{2}^{2}d\mu<\infty.$

Using Equation (A3) from [12], we get

$\displaystyle\sigma^{-2}\mu(B_{\sigma}(x))+\int_{f^{-1}(B_{\rho}(p)\backslash
B_{\sigma}(p))}\left|\frac{1}{4}\vec{H}+\frac{(f(x)-p)^{\perp}}{|f(x)-p|^{2}}\right|^{2}d\mu$
$\displaystyle=$
$\displaystyle\rho^{-2}\mu(B_{\rho}(x))+\frac{1}{16}\int_{f^{-1}(B_{\rho}(p)\backslash
B_{\sigma}(p))}\vec{H}^{2}d\mu+R_{p,\rho}-R_{p,\sigma}$ (A.12)

where for $\alpha=\sigma,\rho$

$R_{p,\alpha}:=\frac{1}{2\alpha^{2}}\int_{f^{-1}(B_{\alpha}(p))}\langle(x-p),\vec{H}\rangle
d\mu.$

The Equation after (A6) in [12] implies $R_{p,\sigma}\rightarrow 0$ as
$\sigma\rightarrow 0^{+}$. We note that $\mu(B_{\rho}(p))\leq A[f]<\infty$ by
definition of the class $\mathcal{P}$. Hence
$\rho^{-2}\mu_{\Sigma}(B_{\rho}(p))\rightarrow 0$ for $\rho\rightarrow\infty$.
So Equation (A13) in [12] implies $\lim_{\rho\rightarrow\infty}R_{p,\rho}=0$.
Letting $\rho\rightarrow\infty$ and $\sigma\rightarrow 0^{+}$ in Equation
(A.12) then gives

$\liminf_{\sigma\rightarrow
0^{+}}\left[\sigma^{-2}\mu_{\Sigma}(B_{\sigma}(x))+\int_{\mathbb{S}^{2}\backslash
f^{-1}(B_{\sigma}(p))}\left|\frac{1}{4}\vec{H}+\frac{(f(x)-p)^{\perp}}{|f(x)-p|^{2}}\right|^{2}d\mu\right]\leq\frac{1}{4}\mathcal{W}[f].$

The lemma follows by using Fatou’s lemma and Equation (A.10). ∎

######  Lemma A.3.

Let $\gamma\in\mathcal{P}\cap C^{\infty}((0,1))$ and denote by $\vec{H}$ the
almost everywhere defined mean curvature vector of $f_{\gamma}$. Suppose that
there exists $p\in\mathbb{R}^{3}$ such that

$\frac{1}{4}\vec{H}(s)=-\frac{(f_{\gamma}(s,\theta)-p)^{\perp}}{|f_{\gamma}(s,\theta)-p|^{2}}.$
(A.13)

Then $\Sigma_{\gamma}$ is either a sphere or the inversion of a scaled and
vertically translated catenoid. In particular, only the second case is
possible if $\mathcal{W}[\Sigma_{\gamma}]=8\pi$.

###### Proof.

By vertical translation of $\gamma$ and rotational invariance, we can assume
$p=Re_{1}$ for some $R\geq 0$. Let $L:=L[\gamma]$. Using Equation (2.13), we
compute

$\displaystyle(f_{\gamma}(s,\theta)-Re_{1})\cdot n(s,\theta)$
$\displaystyle=\begin{bmatrix}\gamma_{1}\cos\theta-R\\\
\gamma_{1}\sin\theta\\\
\gamma_{2}\end{bmatrix}\cdot\frac{1}{L}\begin{bmatrix}-\dot{\gamma}_{2}\cos\theta\\\
-\dot{\gamma}_{2}\sin\theta\\\
\dot{\gamma}_{1}\end{bmatrix}=\frac{-\gamma_{1}\dot{\gamma}_{2}+\dot{\gamma}_{1}\gamma_{2}+R\dot{\gamma}_{2}\cos\theta}{L}.$

A quick computation shows
$|f_{\gamma}(s,\theta)-Re_{1}|^{2}=\gamma_{1}^{2}+\gamma_{2}^{2}+R^{2}-2R\gamma_{1}\cos\theta$.
Inserting into Equation (A.13), we get

$\frac{1}{4}H(s)\left(\gamma_{1}^{2}+\gamma_{2}^{2}+R^{2}-2R\gamma_{1}\cos\theta\right)=-\frac{-\gamma_{1}\dot{\gamma}_{2}+\dot{\gamma}_{1}\gamma_{2}+R\dot{\gamma}_{2}\cos\theta}{L}.$
(A.14)

#### The case $R>0$

Comparing the coefficients of $\cos\theta$ in Equation (A.14)and using $R\neq
0$ we derive
$\frac{\dot{\gamma}_{2}}{L}=\frac{1}{2}H(s)\gamma_{1}(s)$. By using Equation
(2.20), we get

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

We use $\dot{\gamma}_{2}^{2}=L^{2}-\dot{\gamma}_{1}^{2}$ and get
$-\ddot{\gamma}_{1}\gamma_{1}=\dot{\gamma}_{2}^{2}=L^{2}-\dot{\gamma}_{1}^{2}$.
Next, we put $x(t):=\gamma_{1}(\frac{t}{L})$. $x$ satisfies the following
problem

$\left\\{\begin{aligned} \ddot{x}x&=\dot{x}^{2}-1,\\\ x(0)&=x(L)=0,\\\
\dot{x}(0)&=1,\\\ x(t)&>0\hskip 14.22636pt\textrm{for
}t\in(0,L).\end{aligned}\right.$ (A.15)

Since $|\dot{\gamma}(t)|\leq L$ we have $|\dot{x}|\leq 1$. Suppose that there
exists $t_{0}\in(0,L)$ such that $\dot{x}(t_{0})^{2}=1$. Then by the Picard-
Lindelöff theorem $x(t)=x(t_{0})\pm(t-t_{0})$. This, however, contradicts
$x(0)=x(L)=0$. So $x^{2}(t)<1$ for all $t\in(0,L)$ and hence the following
computation is admissible for $t\in(0,L)$:

$-\frac{1}{2}\frac{d}{dt}\ln(1-\dot{x}^{2})=\frac{\dot{x}\ddot{x}}{1-\dot{x}^{2}}=-\frac{\dot{x}}{x}=-\frac{d}{dt}\ln(x)$

Integrating and subsequent exponentiation proves that there exists a positive
constant $K$ such that $1-\dot{x}^{2}=Kx^{2}$. Differentiating this identity
gives $\dot{x}(\ddot{x}+Kx)=0$ Since $\dot{x}(0)=1$ we have $\dot{x}\neq 0$
for small times $t\in[0,t_{0}]$ and get

$x(t)=a\sin(\sqrt{K}t)+b\cos(\sqrt{K}t)\hskip 14.22636pt\textrm{for
}t\in[0,t_{0}].$

As $x(t)>0$, Equation (A.15) and the Picard-Lindelöff theorem imply that this
formula is valid not only on $[0,t_{0}]$ but for all $t\in[0,L]$. Since
$x(0)=x(L)=0$ we get $b=0$ and $\sqrt{K}=L^{-1}n\pi$ for some
$n\in\mathbb{N}$. Moreover, as $x>0$ on $(0,L)$ and $\dot{x}(0)=1$, we get

$x(t)=\frac{L}{\pi}\sin\left(\frac{\pi}{L}t\right)\hskip 14.22636pt\textrm{for
all $t\in[0,L]$}.$

Let $y(t):=\gamma_{2}(\frac{t}{L})$. As
$\dot{y}^{2}=1-\dot{x}^{2}=\sin^{2}(\frac{\pi}{L}t)$, we have
$\dot{y}(t)=\pm\sin(\frac{\pi}{L}t)$ and hence

$y(t)=y(0)\mp\frac{L}{\pi}\cos\left(\frac{\pi}{L}t\right).$

The formulas for $x(t)$ and $y(t)$ imply that $\gamma$ parameterises a circle
of radius $\frac{L}{\pi}$ with center at $(0,y(0))$. In particular
$\mathcal{W}[\gamma]=4\pi$.

#### The Case $R=0$

Let
$I:\mathbb{R}^{3}\backslash\set{0}\rightarrow\mathbb{R}^{3}\backslash\set{0}$,
$I(x):=\frac{x}{|x|^{2}}$. Then $I^{*}\delta=\frac{1}{|x|^{4}}\delta$ and

$H[I\circ
f_{\gamma}]=H[f_{\gamma},I^{*}\delta]=|f_{\gamma}|^{2}H[f_{\gamma}]+4\langle
f_{\gamma},n\rangle=0.$

This shows that the surface defined by the profile curve
$\Gamma(s):=\frac{\gamma(s)}{|\gamma(s)|^{2}}$ is a minimal surface. Let
$\alpha$ denote the maximal solution to the problem

$\left\\{\begin{aligned}
\dot{\alpha}(t)&=\frac{|\gamma(\alpha(t))|^{2}}{L},\\\
\alpha(0)&=\frac{1}{2}.\end{aligned}\right.$

We define $\Gamma^{*}(t):=\Gamma(\alpha(t))$ and compute

$|\dot{\Gamma}^{*}(s)|=\left|\left(\frac{\dot{\gamma}}{|\gamma|^{2}}-2\frac{\langle\gamma,\dot{\gamma}\rangle}{|\gamma|^{4}}\gamma\right)\dot{\alpha}\right|=1.$

Consequently, $\Gamma^{*}$ is parameterized by arc length and satisfies
$H[\Gamma^{*}]=0$. Following the arguments after Equation (6.6) we get
$\Gamma_{1}^{*}(t)=\sqrt{a+bt+t^{2}}$ and
$\dot{\Gamma}_{2}^{*}(t)\Gamma_{1}^{*}(t)=c$ for constants $a>0$ and
$b,c\in\mathbb{R}$. By potentially redefining
$\Gamma^{*}(t)\rightarrow\Gamma^{*}(-t)$ we may further assume $b\geq 0$.
Since $|\dot{\Gamma}^{*}|=1$ we have

$1=(\dot{\Gamma}_{1}^{*})^{2}+(\dot{\Gamma}_{2}^{*})^{2}=\frac{c^{2}+(t+\frac{b}{2})^{2}}{(\Gamma_{1}^{*})^{2}}\hskip
14.22636pt\textrm{and hence}\hskip
14.22636ptt^{2}+bt+a=(t+\frac{b}{2})^{2}+c^{2}.$

If $c=0$ we deduce $\Gamma^{*}(t)=(t+\frac{b}{2},\Gamma_{2}^{*}(0))$ for
$t\geq-\frac{b}{2}$. Hence $\Sigma_{\Gamma^{*}}$ is part of a a plane $P$
orthogonal to the $x_{3}$-axis and consequently $\Sigma_{\gamma}$ is part of a
sphere or, if $0\in P$, part of $P$. The second case is impossible. Indeed, if
$\gamma(s)=(\gamma_{1}(s),c_{0})$ for some $c_{0}\in\mathbb{R}$, we get
$L=|\dot{\gamma}|=|\dot{\gamma}_{1}|$. Since $\gamma_{1}(0)=0$ and
$\gamma_{1}(t)\geq 0$ for all $t\in[0,1]$ we get $\gamma_{1}(t)=Lt$, which
contradicts $\gamma_{1}(1)=0$. So, if $c=0$, $\Sigma_{\gamma}$ is part of a
sphere and by the definition of the class $\mathcal{P}$, it then follows that
$\Sigma_{\gamma}$ is a sphere and hence $\mathcal{W}[\Sigma_{\gamma}]=4\pi$.
If $c\neq 0$, we have

$\displaystyle\Gamma_{1}^{*}(t)$
$\displaystyle=\sqrt{\left(t+\frac{b}{2}\right)^{2}+c^{2}},$
$\displaystyle\Gamma_{2}^{*}(t)$
$\displaystyle=\Gamma_{2}^{*}(0)+\int_{0}^{t}\frac{cdt}{\sqrt{(t+\frac{b}{2})^{2}+c^{2}}}=\Gamma_{2}^{*}(0)-|c|\operatorname{asinh}(\frac{b}{2c})+|c|\operatorname{asinh}(\frac{t}{c}+\frac{b}{2c}).$

This is the profile curve of a scaled and vertically translated catenoid.
Hence, $\Sigma_{\gamma}$ is the inversion of a scaled and vertically
translated catenoid and $\mathcal{W}[\Sigma_{\gamma}]=8\pi$. ∎

##  Appendix B Technicalities regarding the Direct Method

###  B.1 Lower Semi Continuity

The proof of Equations (3.8) and (3.9) is the consequence of two lemmas.
First, we establish the following:

######  Lemma B.1.

Let $\rho_{n},\rho\in C^{0}([0,1],\mathbb{R}^{+}_{0})$ and
$f_{n},f:[0,1]\rightarrow\mathbb{R}$ be measurable such that
$\rho_{n}\rightarrow\rho$ in $C^{0}([0,1])$ and $f_{n}\rho_{n}\rightarrow
f\rho$ weakly in $L^{1}((0,1))$. Further assume $f^{2}\rho\in L^{1}((0,1))$
and $\|f_{n}^{2}\rho_{n}\|_{L^{1}((0,1))}\leq C<\infty$ for all
$n\in\mathbb{N}$. Then

$\int_{0}^{1}f^{2}(t)\rho(t)dt\leq\liminf_{n\rightarrow\infty}\int_{0}^{1}f_{n}^{2}(t)\rho_{n}(t)dt.$

All integrals are well-defined since $f^{2}\rho\geq 0$ and
$f_{n}^{2}\rho_{n}\geq 0$.

###### Proof.

For $R>0$ we put $\phi_{R}:=f1_{|f|\leq R}\in L^{\infty}((0,1))$. Using
$f_{n}\rho_{n}\rightarrow f\rho$ weakly in $L^{1}((0,1))$ and
$\rho_{n}\rightarrow\rho$ in $C^{0}([0,1])$, we estimate

$\displaystyle\int_{0}^{1}f(t)\phi_{R}(t)\rho(t)dt=$
$\displaystyle\lim_{n\rightarrow\infty}\int_{0}^{1}f_{n}(t)\phi_{R}(t)\rho_{n}(t)dt$
$\displaystyle\leq$
$\displaystyle\liminf_{n\rightarrow\infty}\left[\left(\int_{0}^{1}f_{n}(t)^{2}\rho_{n}(t)dt\right)^{\frac{1}{2}}\left(\int_{0}^{1}\phi_{R}(t)^{2}\rho_{n}(t)dt\right)^{\frac{1}{2}}\right]$
$\displaystyle\leq$
$\displaystyle\liminf_{n\rightarrow\infty}\left(\int_{0}^{1}f_{n}(t)^{2}\rho_{n}(t)dt\right)^{\frac{1}{2}}\left(\int_{0}^{1}\phi_{R}(t)^{2}\rho(t)dt\right)^{\frac{1}{2}}.$

Note that $\phi_{R}\rightarrow f$ pointwise and $|\phi_{R}|\leq|f|$. As
$|f|^{2}\rho\in L^{1}((0,1))$ we can let $R\rightarrow\infty$ and get the
claimed estimate. ∎

######  Lemma B.2.

Let $\gamma^{(n)},\gamma^{*}\in\mathcal{P}^{w}$ such that
$\gamma^{(n)}\rightarrow\gamma^{*}$ in $\mathcal{P}^{w}$ and

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

Then $k_{i}^{(n)}\gamma_{1}^{(n)}\rightarrow k_{i}^{*}\gamma_{1}^{*}$ weakly
in $L^{1}((0,1))$ for $i=1,2$.

###### Proof.

Let $\phi\in L^{\infty}((0,1))$, $L_{n}:=L[\gamma^{(n)}]$ and
$L_{*}:=L[\gamma^{*}]$. Then $L_{n}\rightarrow L_{*}$ and, as
$\gamma^{*}\in\mathcal{P}^{w}$, we have $L_{*}>0$. We put
$R=\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}$ and estimate

$\left|\int_{0}^{1}\langle\ddot{\gamma}^{(n)},R\dot{\gamma}^{(n)}-R\dot{\gamma}^{*}\rangle\gamma_{1}^{(n)}\phi
dt\right|\leq\|\phi\|_{L^{\infty}}\left[\int_{0}^{1}|\ddot{\gamma}^{(n)}|^{2}\gamma_{1}^{(n)}dt\right]^{\frac{1}{2}}\left(\sup_{n}\gamma_{1}^{(n)}\right)^{\frac{1}{2}}\|\dot{\gamma}^{(n)}-\dot{\gamma}^{*}\|_{L^{2}}.$

$\||\ddot{\gamma}^{(n)}|^{2}\gamma_{1}^{(n)}\|_{L^{1}((0,1))}$ is bounded by
combining the assumptions of the lemma with Lemma 3.9. Using
$\gamma^{(n)}\rightarrow\gamma^{*}$ in $C^{0}([0,1])$ and $W^{1,2}((0,1))$, we
deduce that

$\lim_{n\rightarrow\infty}\left|\int_{0}^{1}\langle\ddot{\gamma}^{(n)},R\dot{\gamma}^{(n)}-R\dot{\gamma}^{*}\rangle\gamma_{1}^{(n)}\phi
dt\right|=0.$ (B.1)

Note that $R\dot{\gamma}^{*}\phi\in L^{\infty}$. So, by weak convergence in
$L^{1}$, we get

$\lim_{n\rightarrow\infty}\int_{0}^{1}\langle\ddot{\gamma}^{(n)},R\dot{\gamma}^{*}\rangle\gamma_{1}^{(n)}\phi
dt=\lim_{n\rightarrow\infty}\int_{0}^{1}\langle\ddot{\gamma}^{*},R\dot{\gamma}^{*}\rangle\gamma_{1}^{*}\phi
dt.$

In view of Equation (B.1) as well as Equation (2.15), we get
$k_{1}^{(n)}\gamma_{1}^{(n)}\rightarrow k_{1}^{*}\gamma_{1}^{*}$ weakly in
$L^{1}((0,1))$.

Next, we take care of $k_{2}^{(n)}$. As
$\dot{\gamma}^{(n)}\rightarrow\dot{\gamma}^{*}$ in $L^{2}((0,1))$ we get

$\int_{0}^{1}k_{2}^{(n)}\gamma_{1}^{(n)}(t)\phi(t)dt=\frac{2\pi}{L_{n}}\int_{0}^{1}\dot{\gamma}_{2}^{(n)}(t)\phi(t)dt\rightarrow\int_{0}^{1}k_{2}^{*}\gamma_{1}^{*}(t)\phi(t)dt.$

∎

Proof of Estimates (3.8) and (3.9).
The proof for both estimates is essentially the same. To prove Estimate (3.8),
we pass to a subsequence that realizes the $\liminf$ and assume without loss
of generality that the $\liminf$ is finite. Consequently

$\sup_{n\in\mathbb{N}}\int_{0}^{1}\left(k_{1}^{(n)}+k_{2}^{(n)}\right)^{2}2\pi|\dot{\gamma}^{(n)}|\gamma_{1}^{(n)}dt<\infty.\\\
$ (B.2)

We wish to apply Lemma B.1 with the functions $f_{n}=k_{1}^{(n)}+k_{2}^{(n)}$,
$f=k_{1}^{*}+k_{2}^{*}$, $\rho_{n}=\gamma_{1}^{(n)}$ and
$\rho=\gamma_{1}^{*}$. To justify this choice we first use Lemma B.2 to get
$(k_{1}^{(n)}+k_{2}^{(n)})\gamma_{1}^{(n)}\rightarrow(k_{1}^{*}+k_{2}^{*})\gamma_{1}^{*}$
weakly in $L^{1}((0,1))$. Additionally we have Estimate (B.2) and
$(k_{1}^{*}+k_{2}^{*})^{2}\gamma_{1}^{*}\in L^{1}((0,1))$ by definition of the
class $\mathcal{P}^{w}$. Estimate (3.8) now follows from Lemma B.1. To get
Estimate (3.9), the only change is to take $f_{n}=k_{i}^{(n)}$ and
$f=k_{i}^{*}$ for $i=1,2$. ∎

###  B.2 Compactness

The following lemma is due to Choski and Veneroni [4] (see Lemma 5). We
present a modified proof.

######  Lemma B.3.

Let $\gamma\in\mathcal{P}^{w}$, $L:=L[\gamma]$ and $(a,b)\subset[0,1]$ such
that $\gamma_{1}(t)>0$ for all $t\in(a,b)$. Then

$\displaystyle|\dot{\gamma}_{1}(b^{-})-\dot{\gamma}_{1}(a^{+})|$
$\displaystyle\leq\frac{L}{2\pi}\int_{a}^{b}|k_{1}k_{2}|2\pi
L\gamma_{1}(t)dt,$
$\displaystyle|\dot{\gamma}_{2}(b^{-})-\dot{\gamma}_{2}(a^{+})|$
$\displaystyle\leq\sqrt{\frac{L}{2\pi}}\left(\int_{a}^{b}k_{1}^{2}2\pi
L\gamma_{1}dt\right)^{\frac{1}{2}}\left(\int_{a}^{b}\frac{\dot{\gamma}_{1}^{2}}{\gamma_{1}}dt\right)^{\frac{1}{2}}.$

In particular, if $\dot{\gamma}$ is continuous at $a$ and $b$, the one sided
limits can be replaced by $\dot{\gamma}(a)$ and $\dot{\gamma}(b)$
respectively.

Before presenting the proof, we have the following remark: In the proof, we
use Theorem 3.3 to deduce that $\dot{\gamma}$ has one-sided limits at $a$ and
$b$. The analog of Theorem 3.3 in [4] is Lemma 6 and the proof uses Lemma 5
(the analog of our Lemma B.3). However, the continuity statement we use here
is Corollary 2 in [4] and proven without using the analog of our Lemma B.3.

###### Proof.

Let $\epsilon>0$ be small. Then $\gamma_{1}\geq c_{0}(\epsilon)>0$ on
$[a+\epsilon,b-\epsilon]$ so that $\gamma\in W^{2,2}((a+\epsilon,b-\epsilon))$
by Definition 3.2. Using Equation (2.21), we compute

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

Now we let $\epsilon\rightarrow 0^{+}$. To do so, we use $|k_{1}k_{2}|2\pi
L\gamma_{1}\in L^{1}((0,1))$ by definition of the class $\mathcal{P}^{w}$ and
Theorem 3.3 to get the first estimate. For the second estimate, we argue
similarly and write

$\displaystyle\frac{|\dot{\gamma}_{2}(b-\epsilon)-\dot{\gamma}_{2}(a+\epsilon)|}{L}$
$\displaystyle=\frac{1}{L}\left|\int_{a+\epsilon}^{b-\epsilon}\ddot{\gamma}_{2}(t)dt\right|$
$\displaystyle=\left|\int_{a+\epsilon}^{b-\epsilon}k_{1}\dot{\gamma}_{1}dt\right|$
$\displaystyle=\frac{1}{\sqrt{2\pi
L}}\left|\int_{a+\epsilon}^{b-\epsilon}k_{1}\frac{\dot{\gamma}_{1}}{\sqrt{\gamma_{1}}}\sqrt{2\pi
L\gamma_{1}}dt\right|$ $\displaystyle\leq\frac{1}{\sqrt{2\pi
L}}\left[\left(\int_{a}^{b}k_{1}^{2}2\pi
L\gamma_{1}dt\right)^{\frac{1}{2}}\left(\int_{a}^{b}\frac{\dot{\gamma}_{1}^{2}}{\gamma_{1}}dt\right)^{\frac{1}{2}}\right].$

Note that while valid, we may have estimated by $+\infty$ in the last step.
Letting $\epsilon\rightarrow 0^{+}$, the second estimate claimed in the lemma
follows. ∎

For the proof of Theorem 3.11, we require the space of bounded variations.

######  Definition B.4.

Let $\alpha,\beta\in\mathbb{R}$. For a measurbale function
$f:[\alpha,\beta]\rightarrow\mathbb{R}$ we put

$V_{\alpha}^{\beta}(f):=\sup\left\\{\sum_{i=0}^{N-1}|f(t_{i+1})-f(t_{i})|\
\bigg{|}\ N\in\mathbb{N}\textrm{ and
}\alpha=t_{0}<t_{1}<...<t_{N-1}<t_{N}=\beta\right\\}.$

$f$ is said to have _bounded variation_ if $V_{\alpha}^{\beta}(f)<\infty$. If
additionally $f\in L^{1}((\alpha,\beta))$, we define
$\|f\|_{\operatorname{BV}((\alpha,\beta))}:=\|f\|_{L^{1}((\alpha,\beta))}+V_{\alpha}^{\beta}(f)$.
The space of all $f$ with $\|f\|_{\operatorname{BV}((\alpha,\beta))}<\infty$
is denoted by $\operatorname{BV}((\alpha,\beta))$.

We stress that to define $V_{\alpha}^{\beta}(f)$, we consider a function
$f:[\alpha,\beta]\rightarrow\mathbb{R}$ and not the equivalence class of all
functions $\tilde{f}$ such that $\tilde{f}=f$ almost everywhere. We require
the following theorem. For a proof, we refer to Theorem 2.35 and Proposition
2.38 in [13].

######  Theorem B.5 (Helly’s Selection Theorem).

Let $(f_{n})$ be a sequence of measurable functions
$f_{n}:[\alpha,\beta]\rightarrow\mathbb{R}$ and $c\in[\alpha,\beta]$ such that
$|f_{n}(c)|+V_{\alpha}^{\beta}(f_{n})\leq C<\infty$ for all $n\in\mathbb{N}$.
Then, there exists a subsequence $f_{n_{a}}$ and a measurable function
$f:[\alpha,\beta]\rightarrow\mathbb{R}$ such that $f_{n_{a}}\rightarrow f$
pointwise and

$V_{\alpha}^{\beta}(f)\leq\liminf_{a\rightarrow\infty}V_{\alpha}^{\beta}(f_{n_{a}}).$

We establish the following corollary:

######  Corollary B.6.

Let $(f_{n})\subset\operatorname{BV}((\alpha,\beta))$ be a sequence such that
$\|f_{n}\|_{\operatorname{BV}((\alpha,\beta))}\leq C<\infty$. Then, there
exist $f\in\operatorname{BV}((\alpha,\beta))$ and a subsequence $(f_{n_{a}})$
such that $f_{n_{a}}\rightarrow f$ pointwise and in $L^{1}((\alpha,\beta))$.

###### Proof.

We first claim that $f_{n}(\alpha)$ is bounded. Once this is shown, we can use
Helly’s selection theorem to get a subsequence $f_{n_{a}}$ such that
$f_{n_{a}}\rightarrow f$ pointwise. For $x\in(\alpha,\beta)$ we estimate

$\displaystyle|f_{n_{a}}(x)|$
$\displaystyle\leq|f_{n_{a}}(\alpha)|+|f_{n_{a}}(\alpha)-f_{n_{a}}(x)|$
$\displaystyle\leq|f_{n_{a}}(\alpha)|+|f_{n_{a}}(\alpha)-f_{n_{a}}(x)|+|f_{n_{a}}(x)-f_{n_{a}}(\beta)|$
$\displaystyle\leq\sup_{a\geq
0}\left(|f_{n_{a}}(\alpha)|+V_{\alpha}^{\beta}(f_{n_{a}})\right).$

Therefore $f_{n_{a}}$ is bounded in $L^{\infty}((\alpha,\beta))$ and hence
$f\in L^{\infty}((\alpha,\beta))$. Using Lebesgue’s theorem, we obtain

$0=\lim_{a\rightarrow\infty}\int_{\alpha}^{\beta}|f(t)-f_{n_{a}}(t)|dt.$

To prove that $f_{n}(\alpha)$ is bounded, let $x_{n}\in[\alpha,\beta]$ such
that $|f_{n}(x_{n})|<\inf_{[\alpha,\beta]}|f_{n}|+1$. We have the estimate

$C\geq\int_{\alpha}^{\beta}|f_{n}(x)|dx\geq\int_{\alpha}^{\beta}|f_{n}(x_{n})|-1dx=|f_{n}(x_{n})|-(\beta-\alpha).$

So, $f_{n}(x_{n})$ is bounded and we can estimate

$|f_{n}(\alpha)|\leq|f_{n}(x_{n})|+|f_{n}(x_{n})-f_{n}(\alpha)|+|f_{n}(\beta)-f_{n}(x_{n})|\leq|f_{n}(x_{n})|+V_{\alpha}^{\beta}(f_{n})\leq
C.$

∎

Let $U\subset\mathbb{R}$ be open. Then there exist at most countably many and
uniquely determined disjoint intervals $I_{j}=(\alpha_{j},\beta_{j})$ such
that $U=\bigcup_{j}I_{j}$. This is e.g. shown in [16], Chapter 2, Section 1,
Theorem 9. Using this fact, we extend the notion of the space of bounded
variations to general open sets of $\mathbb{R}$.

######  Definition B.7.

Let $U\subset\mathbb{R}$ be open, $m\in\mathbb{N}\cup\set{\infty}$ and
$U=\bigcup_{j=1}^{m}(\alpha_{j},\beta_{j})$ be the unique decomposition of $U$
into disjoint, open intervals. For a measurable function
$f:U\rightarrow\mathbb{R}$, we put

$V_{U}(f):=\sum_{j=1}^{m}V_{\alpha_{j}}^{\beta_{j}}(f).$

$f$ is said to have _bounded variation_ if $V_{U}(f)<\infty$. If additionally
$f\in L^{1}(U)$, we define
$\|f\|_{\operatorname{BV}(U)}:=\|f\|_{L^{1}(U)}+V_{U}(f)$. The space of all
$f$ with $\|f\|_{\operatorname{BV}(U)}<\infty$ is denoted by
$\operatorname{BV}(U).$

By a usual diagonal argument, we get the following corollary from Corollary
B.6

######  Corollary B.8.

Let $U\subset\mathbb{R}$ be open and $(f_{n})\subset\operatorname{BV}(U)$ such
that $\|f_{n}\|_{\operatorname{BV}(U)}\leq C<\infty$. Then there exists
$f\in\operatorname{BV}(U)$ and a subsequence $(f_{n_{a}})$ such that
$f_{n_{a}}\rightarrow f$ in $L^{1}(U)$.

Finally, we establish Theorem 3.11. The following proof is due to Choski and
Veneroni [4] (see Proposition 2) with only minor modifications.

###### Proof of Theorem 3.11..

We put $L_{n}:=L[\gamma^{(n)}]$. By the assumptions of Theorem 3.11,
$\gamma^{(n)}$ is bounded in $C^{0}([0,1])$ and combining the assumptions of
Theorem 3.11 with Theorem 3.5 shows that $\dot{\gamma}^{(n)}$ is bounded in
$L^{\infty}((0,1))$. In particular, we can assume without loss of generality
that $L_{n}\rightarrow L_{*}\in[0,\infty)$.

Step 1 - Weak Limit
Since $\gamma^{(n)}$ is bounded in $C^{0}$ and $|\dot{\gamma}^{(n)}|$ is
bounded in $L^{\infty}((0,1))$ and therefore also in $L^{2}((0,1))$, we can
use the Arzelà-Ascoli theorem to deduce that, after passing to a subsequence,
$\gamma^{(n)}\rightarrow\gamma^{*}$ in $C^{0}([0,1])$ and
$\dot{\gamma}^{(n)}\rightarrow\dot{\gamma}^{*}$ weakly in $L^{2}((0,1))$.
Next, we prove by contradiction that $L_{*}>0$. Suppose $L_{*}=0$, then, as
$\gamma^{(n)}\rightarrow\gamma^{*}$ in $C^{0}([0,1])$, we get

$\theta\leq\liminf_{n\rightarrow\infty}A[\gamma^{(n)}]=2\pi\liminf_{n\rightarrow\infty}\int_{0}^{1}\gamma_{1}^{(n)}(t)L_{n}dt=0.$

Step 2 - Convergence in $W^{1,2}$
By Lemma B.3, the sequence $\dot{\gamma}^{(n)}_{1}$ is bounded in
$\operatorname{BV}((0,1))$ since

$V_{0}^{1}(\dot{\gamma}_{1}^{(n)})\leq\frac{L_{n}}{2\pi}\int_{0}^{1}|k_{1}^{(n)}k_{2}^{(n)}|2\pi
L_{n}\gamma_{1}^{(n)}(t)dt\leq C.$

Using Corollary B.6, we deduce that there exists $\sigma\in L^{1}((0,1))$ such
that $\dot{\gamma}^{(n)}_{1}\rightarrow\sigma$ strongly in $L^{1}((0,1))$
along a subsequence and since $\dot{\gamma}^{(n)}\rightarrow\dot{\gamma}^{*}$
weakly in $L^{2}((0,1))$, we get $\sigma=\gamma_{1}^{*}$. As
$\dot{\gamma}^{(n)}_{1}$ is bounded in $L^{\infty}((0,1))$, we get,
$\dot{\gamma}_{1}^{(n)}\rightarrow\dot{\gamma}_{1}^{*}$ in $L^{p}((0,1))$ for
all $p\in[1,\infty)$. Indeed

$\int_{0}^{1}|\dot{\gamma}_{1}^{(n)}-\dot{\gamma}_{1}^{(m)}|^{p}dt\leq\|\dot{\gamma}_{1}^{(n)}-\dot{\gamma}_{1}^{(m)}\|_{L^{\infty}}^{p-1}\|\dot{\gamma}_{1}^{(n)}-\dot{\gamma}_{1}^{(m)}\|_{L^{1}((0,1))}\leq
C\|\dot{\gamma}_{1}^{(n)}-\dot{\gamma}_{1}^{(m)}\|_{L^{1}((0,1))}$

shows that $\dot{\gamma}_{1}^{(n)}$ is Cauchy in $L^{p}((0,1))$ and hence
converges to some limit. By uniqueness of the limit in $L^{1}((0,1))$ we get
$\dot{\gamma}_{1}^{(n)}\rightarrow\dot{\gamma}_{1}^{*}$ in $L^{p}((0,1))$ for
all $p\in[1,\infty)$.

Next, we claim $\gamma_{1}^{*}(t)>0$ almost everywhere. To see this, let
$E:=\set{t\in[0,1]\ }{\ \gamma_{1}^{*}(t)=0}$. Clearly $E$ is compact and, by
uniform convergence, $\gamma_{1}^{(n)}\leq\epsilon$ on $E$ for all $n\geq
n_{0}(\epsilon)$. Using the second assumption of Theorem 3.11, we get

$C\geq\int_{0}^{1}\left(k_{2}^{(n)}\right)^{2}2\pi
L_{n}\gamma_{1}^{(n)}dt=\frac{2\pi}{L_{n}}\int_{0}^{1}\frac{\left(\dot{\gamma}_{2}^{(n)}\right)^{2}}{\gamma_{1}^{(n)}}dt\geq\frac{2\pi}{L_{n}\epsilon}\int_{E}\left(\dot{\gamma}_{2}^{(n)}\right)^{2}dt.$

This shows $\dot{\gamma}^{(n)}_{2}\rightarrow 0$ in $L^{2}(E)$. Combining this
with $\dot{\gamma}_{1}^{(n)}\rightarrow\dot{\gamma}_{1}^{*}$ in
$L^{2}((0,1))$, we get

$|E|L_{*}^{2}=\lim_{n\rightarrow\infty}|E|L_{n}^{2}=\lim_{n\rightarrow\infty}\int_{E}|\dot{\gamma}^{(n)}|^{2}dt=\int_{E}(\dot{\gamma}_{1}^{*})^{2}dt.$

Now assume $|E|>0$. Then $|\dot{\gamma}_{1}^{*}(t)|=L_{*}$ for almost all
$t\in E$. Using Rademacher’s theorem, we deduce that $\gamma$ is
differentiable at almost all $t\in E$. Since the classical and the weak
derivative agree almost everywhere, we deduce that there exists $t_{0}\in E$
such that $\gamma$ is differentiable in $t_{0}$ and
$|\dot{\gamma}^{*}_{1}(t_{0})|=L_{*}>0$. But by definition of $E$ we also have
$\gamma_{1}^{*}(t_{0})=0$, which implies that there exists $t\in[0,1]$ close
to $t_{0}$ such that $\gamma_{1}^{*}(t)<0$, which is a contradiction.

Next, we prove that along a subsequence
$\dot{\gamma}_{2}^{(n)}\rightarrow\dot{\gamma}_{2}$ in $L^{1}((0,1))$. For
$k\in\mathbb{N}$ we put $A_{k}:=\set{t\in[0,1]\ }{\ \gamma_{1}(t)>2k^{-1}}$.
For $n\geq n_{0}(k)$ we have $\gamma_{1}^{(n)}\geq\frac{1}{k}$ on $A_{k}$.
Using Lemma B.3, we deduce that $\dot{\gamma}_{2}^{(n)}$ is bounded in
$\operatorname{BV}(A_{k})$ for all $k$ with

$V_{A_{k}}(\dot{\gamma}_{2}^{(n)})\leq\sqrt{\frac{L_{n}}{2\pi}}\left(\int_{0}^{1}(k_{1}^{(n)})^{2}2\pi
L_{n}\gamma_{1}^{(n)}dt\right)^{\frac{1}{2}}k^{\frac{1}{2}}\left(\int_{0}^{1}(\dot{\gamma}_{1}^{(n)})^{2}dt\right)^{\frac{1}{2}}\leq
C\sqrt{k}.$

After passing to a subsequence we deduce
$\dot{\gamma}_{2}^{(n)}\rightarrow\dot{\gamma}_{2}^{*}$ in $L^{1}(A_{k})$ for
all $k\in\mathbb{N}$. We estimate

$\int_{0}^{1}|\dot{\gamma}_{2}^{(n)}-\dot{\gamma}_{2}^{*}|dt\leq\int_{A_{k}}|\dot{\gamma}_{2}^{(n)}-\dot{\gamma}_{2}^{*}|dt+(L_{n}+L_{*})|[0,1]\backslash
A_{k}|.$

$|[0,1]\backslash A_{k}|\rightarrow 0$ as $k\rightarrow\infty$ since $|E|=0$.
Thus $\dot{\gamma}_{2}^{(n)}\rightarrow\dot{\gamma}_{2}^{*}$ in $L^{1}((0,1))$
and, using the $L^{\infty}$-bound, also in $L^{p}((0,1))$ for all $1\leq
p<\infty$.

In particular we have $\dot{\gamma}^{(n)}\rightarrow\dot{\gamma}^{*}$ in
$L^{1}((0,1))$, which implies that for all $0\leq a<b\leq 1$

$L_{*}(b-a)=\lim_{n\rightarrow\infty}L_{n}(b-a)=\lim_{n\rightarrow\infty}\int_{a}^{b}|\dot{\gamma}^{(n)}|dt=\int_{a}^{b}|\dot{\gamma}^{*}|dt.$

Taking $a=0$ and $b=1$ shows $L[\gamma^{*}]=L_{*}$ and taking $b=a+\delta$ and
letting $\delta\rightarrow 0$ shows $|\dot{\gamma}^{*}(a)|=L_{*}$ for almost
every $a\in[0,1]$.

Step 3 - Limit in $\mathcal{P}^{w}$
Let $U\subset(0,1)$ be open such that $\gamma_{1}^{*}>0$ on $\bar{U}$. We
prove that $\gamma^{*}\in W^{2,2}(U)$. First, as
$\gamma^{(n)}\rightarrow\gamma^{*}$ in $C^{0}([0,1])$ we know that there
exists $n_{0}(U)$ such that for all $n\geq n_{0}(U)$ we have
$\inf_{U}\gamma_{1}^{(n)}\geq c_{0}>0$. By Lemma 3.9,
$(\ddot{\gamma}^{(n)})\subset L^{2}(U)$ is a bounded sequence and hence there
exists $\xi\in L^{2}(U)$ such that $\ddot{\gamma}^{(n)}\rightarrow\xi$ weakly
in $L^{2}(U)$ along a subsequence. For $\varphi\in C^{\infty}_{0}(U)$ we get

$\int_{U}\xi\varphi
dt=\lim_{n\rightarrow\infty}\int_{U}\ddot{\gamma}^{(n)}\varphi
dt=-\lim_{n\rightarrow\infty}\int_{U}\dot{\gamma}^{(n)}\dot{\varphi}dt=-\int_{U}\dot{\gamma}^{*}\dot{\varphi}dt.$

We deduce $\gamma^{*}\in W^{2,2}_{\operatorname{loc}}([\gamma_{1}^{*}>0])$. It
remains to prove that the curvature of $\gamma^{*}$ is bounded in $L^{2}$. As
$\gamma^{(n)}\rightarrow\gamma^{*}$ in $W^{1,2}((0,1))$ we have
$\dot{\gamma}^{(n_{a})}\rightarrow\dot{\gamma}^{*}$ pointwise almost
everywhere along a suitable subsequence and clearly
$\gamma^{(n)}\rightarrow\gamma^{*}$ pointwise. Hence, by Fatou’s lemma and the
assumptions of Theorem 3.11

$\displaystyle\int_{0}^{1}(k_{2}^{*})^{2}2\pi L\gamma_{1}^{*}dt$
$\displaystyle=\int_{0}^{1}\frac{(\dot{\gamma}^{*}_{2})^{2}}{\gamma_{1}^{*}L_{*}}2\pi
dt$
$\displaystyle\leq\liminf_{a\rightarrow\infty}\int_{0}^{1}\frac{(\dot{\gamma}^{(n_{a})}_{2})^{2}}{\gamma_{1}^{(n_{a})}L_{n_{a}}}2\pi
dt$
$\displaystyle=\liminf_{a\rightarrow\infty}\int_{0}^{1}\left(k_{2}^{(n_{a})}\right)^{2}2\pi
L_{n_{a}}\gamma_{1}^{(n_{a})}dt$ $\displaystyle\leq C.$

Next, we consider $k_{1}^{*}$. Let $U\subset(0,1)$ such that $\bar{U}\cap
E=\emptyset$. Since $(\ddot{\gamma}^{(n)})\subset L^{2}(U)$ is bounded and
$\gamma_{1}^{(n)}\rightarrow\gamma_{1}^{*}$ in $C^{0}([0,1])$, we may estimate

$\left|\int_{U}|\ddot{\gamma}^{(n)}|^{2}(\gamma_{1}^{*}-\gamma_{1}^{(n)})dt\right|\leq\|\gamma^{(n)}_{1}-\gamma_{1}^{*}\|_{C^{0}}\sup_{n}\|\ddot{\gamma}^{(n)}\|_{L^{2}(U)}^{2}\rightarrow
0.$ (B.3)

Along a subsequence $n_{a}$ we have
$\ddot{\gamma}^{(n_{a})}\rightarrow\ddot{\gamma}^{*}$ weakly in $L^{2}(U)$.
Let $\varphi\in C^{\infty}_{0}(U)$. Using Estimate (B.3), we can compute

$\displaystyle\int_{U}|\ddot{\gamma}^{*}|^{2}|\gamma_{1}^{*}|dt$
$\displaystyle=\lim_{a\rightarrow\infty}\int_{U}|\ddot{\gamma}^{(n_{a})}|^{2}\gamma_{1}^{*}dt=\lim_{a\rightarrow\infty}\int_{U}|\ddot{\gamma}^{(n_{a})}|^{2}\gamma_{1}^{(n_{a})}dt\leq\sup_{n}\int_{0}^{1}|\ddot{\gamma}^{(n)}|^{2}\gamma_{1}^{(n)}dt.$

Consequently $|\ddot{\gamma}^{*}|^{2}\gamma_{1}^{*}\in L^{1}(U)$ with norm
bounded independent of $U$. Using $|E|=0$ and Equation (2.21), we get

$\int_{0}^{1}(k_{1}^{*})^{2}2\pi\gamma_{1}^{*}|\dot{\gamma}^{*}|dt=\sup\left\\{\frac{2\pi}{L^{3}_{*}}\int_{U}|\ddot{\gamma}^{*}|^{2}\gamma_{1}^{*}dt\
\bigg{|}\ U\subset[0,1],\ \inf_{U}\gamma_{1}^{*}>0\right\\}<\infty.$

In total we have shown $\gamma^{*}\in\mathcal{P}^{w}$.

Step 4 - Convergence in $\mathcal{P}^{w}$
Considering Definition 3.10 and that $\gamma^{(n)}\rightarrow\gamma^{*}$ in
$C^{0}([0,1])$ and $W^{1,2}((0,1))$ has already been established, we are left
with proving that
$\ddot{\gamma}^{(n)}\gamma_{1}^{(n)}\rightarrow\ddot{\gamma}^{*}\gamma_{1}^{*}$
weakly in $L^{1}((0,1))$. Let $\xi\in\mathcal{P}^{w}$. By Theorem 3.3, there
exist $0=t_{0}<t_{1}<...<t_{N}=1$ such that $\xi_{1}(t)=0$ precisely when
$t=t_{i}$ for some $i$. For small $\epsilon>0$, we have $\xi\in
W^{2,2}(t_{i}+\epsilon,t_{i+1}-\epsilon)$. So, for $\phi\in
C^{\infty}([0,1])$, we have

$\int_{t_{i}+\epsilon}^{t_{i+1}-\epsilon}\ddot{\xi}\xi_{1}\phi
dt=\dot{\xi}\xi_{1}\phi\bigg{|}_{t_{i}+\epsilon}^{t_{i+1}-\epsilon}-\int_{t_{i}+\epsilon}^{t_{i+1}-\epsilon}\dot{\xi}(\dot{\xi}_{1}\phi+\xi_{1}\dot{\phi})dt.$

We sum over $i$ and wish to take $\epsilon\rightarrow 0^{+}$. As
$\xi\in\mathcal{P}^{w}$ we have $\ddot{\xi}\xi_{1}\in L^{1}((0,1))$ by Lemma
3.9. Using $\phi,\dot{\xi}\in L^{\infty}((0,1))$ and $\xi_{1}(t_{i})=0$, the
boundary term vanishes for $\epsilon\rightarrow 0^{+}$. So, we deduce

$\int_{0}^{1}\ddot{\xi}\xi_{1}\phi=-\int_{0}^{1}\dot{\xi}_{1}(\dot{\xi}_{1}\phi+\xi_{1}\dot{\phi})dt.$

As $\gamma^{(n)}\rightarrow\gamma^{*}$ in $C^{0}([0,1])$ and in
$W^{1,2}((0,1))$ we can compute

$\displaystyle\lim_{n\rightarrow\infty}\int_{0}^{1}\ddot{\gamma}^{(n)}\gamma_{1}^{(n)}\phi
dt=-\lim_{n\rightarrow\infty}\int_{0}^{1}\dot{\gamma}^{(n)}(\dot{\gamma}_{1}^{(n)}\phi+\gamma_{1}^{(n)}\dot{\phi})dt$
$\displaystyle=$
$\displaystyle-\int_{0}^{1}\dot{\gamma}^{*}(\dot{\gamma}_{1}^{*}\phi+\gamma_{1}^{*}\dot{\phi})dt=\int_{0}^{1}\ddot{\gamma}^{*}\gamma_{1}^{*}\phi
dt.$ (B.4)

Finally, we prove that Equation (B.4) is also valid for $\phi\in
L^{2}((0,1))$. Let $\phi\in L^{2}((0,1))$ and $\epsilon>0$. We choose
$\phi_{\epsilon}\in C^{\infty}([0,1])$ such that
$\|\phi_{\epsilon}-\phi\|_{L^{2}((0,1))}\leq\epsilon$ and estimate

$\left|\int_{0}^{1}\ddot{\gamma}^{*}\gamma_{1}^{*}(\phi-\phi_{\epsilon})dt\right|\leq\left(\int_{0}^{1}|\ddot{\gamma}^{*}|^{2}\gamma_{1}^{*}dt\right)^{\frac{1}{2}}\left(\sup\gamma_{1}^{*}\right)^{\frac{1}{2}}\|\phi-\phi_{\epsilon}\|_{L^{2}((0,1))}\leq
C\epsilon.$ (B.5)

The last step is justified by Lemma 3.9. Similarly, using Lemma 3.9 and the
assumptions of Theorem 3.11 we estimate

$\displaystyle\limsup_{n\rightarrow\infty}\left|\int_{0}^{1}\ddot{\gamma}^{(n)}\gamma_{1}^{(n)}(\phi-\phi_{\epsilon})dt\right|$
$\displaystyle\leq$
$\displaystyle\sup_{n\in\mathbb{N}}\left[\left(\int_{0}^{1}|\ddot{\gamma}^{(n)}|^{2}\gamma_{1}^{(n)}dt\right)^{\frac{1}{2}}\left(\sup\gamma_{1}^{(n)}\right)^{\frac{1}{2}}\right]\|\phi-\phi_{\epsilon}\|_{L^{2}((0,1))}\leq
C\epsilon.$ (B.6)

Combining Estimates (B.5) and (B.6) and letting $\epsilon\rightarrow 0^{+}$
shows that Equation (B.4) is valid for $\phi\in L^{2}((0,1))$ also. ∎

##  Appendix C General Regularity Theory

Let $U\subset\mathbb{R}$ be an open interval and $\gamma\in
W^{1,\infty}(U,\mathcal{H}^{2})\cap W^{2,2}(U,\mathcal{H}^{2})$. For
$\alpha,\beta\in\mathbb{R}$ we define the functional

$I(\phi):=\alpha\int_{0}^{1}\gamma_{1}^{2}(t)\dot{\phi}_{2}(t)+2\gamma_{1}(t)\dot{\gamma}_{2}(t)\phi_{1}(t)dt+\beta\int_{0}^{1}|\dot{\gamma}(t)|\phi_{1}(t)+\frac{\gamma_{1}(t)\langle\dot{\phi}(t),\dot{\gamma}(t)\rangle}{|\dot{\gamma}(t)|}dt.$
(C.1)

We assume that $\gamma$ satisfies the equation

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

The goal of this section is to prove the following regularity theorem:

######  Theorem C.1 (Regularity Theorem).

Let $\gamma\in W^{1,\infty}(U,\mathcal{H}^{2})\cap W^{2,2}(U,\mathcal{H}^{2})$
satisfy Equation (C.2). Further, we assume that $|\dot{\gamma}(t)|=L>0$ and

$\displaystyle|\alpha|+|\beta|+L+L^{-1}+\|\gamma\|_{C^{0}(U)}+\|\ddot{\gamma}\|_{L^{2}(U)}+\mathcal{W}[\gamma]\leq
M,$ (C.3) $\displaystyle\kappa(\gamma;U):=\inf_{t\in U}\gamma_{1}(t)>0.$ (C.4)

Then $\gamma\in C^{\infty}(U)$ and $\|\gamma\|_{C^{m}(U)}\leq
C(M,\kappa(\gamma;U),U,m)$. The constant is increasing in the first and
decreasing in the second slot.

The proof is split into several lemmas. We remark that Eichmann and Grunau
have carried out similar arguments in [7] for the Willmore equation without
the isoperimetric Lagrange multiplier.

######  Lemma C.2.

Under the assumptions of Theorem C.1 $\gamma\in W^{2,\infty}(U)$, $H\in
L^{\infty}(U)$ and

$\|\gamma\|_{W^{2,\infty}(U)}+\|H\|_{L^{\infty}(U)}\leq
C(M,\kappa(\gamma;U),U).$

###### Proof..

Let $U=(\tau_{1},\tau_{2})$. Throughout the proof $C$ will denote a constant
that is allowed to depend on $M$, $\kappa(\gamma;U)$ and $U$ and that may
change from line to line.

Step 1: Constructing an Appropriate Test Function
Given $\varphi=(\varphi_{1},\varphi_{2})\in C_{0}^{\infty}(U,\mathbb{R}^{2})$
define

$\zeta(t):=\int_{\tau_{1}}^{t}\int_{\tau_{1}}^{s}\varphi(r)drds.$

Let $\tau_{1}<r_{1}<r_{2}<\tau_{2}$ such that
$\operatorname{supp}(\varphi)\subset[r_{1},r_{2}]$. Then $\zeta(t)=0$ when
$t\in[\tau_{1},r_{1}]$ and $\zeta(t)=a+bt$ when $t\in[r_{2},\tau_{2}]$ where

$a=\int_{\tau_{1}}^{\tau_{2}}\int_{\tau_{1}}^{s}\varphi(r)drds-b\tau_{2}\hskip
14.22636pt\textrm{and}\hskip
14.22636ptb=\int_{\tau_{1}}^{\tau_{2}}\varphi(r)dr.$

We choose a smooth function $\chi\equiv 0$ on
$[\tau_{1},\tau_{1}+\frac{1}{3}(\tau_{2}-\tau_{1})]$ and $\chi\equiv 1$ on
$[\tau_{1}+\frac{2}{3}(\tau_{2}-\tau_{1}),\tau_{2}]$ and define

$\phi(t):=\zeta(t)-\chi(t)(a+bt).$ (C.5)

It is easy to see that $\phi\in C^{\infty}_{0}(U)$ and that
$\|\phi\|_{C^{1}(U)}\leq C\|\varphi\|_{L^{1}(U)}$.

Step 2: Weak Euler-Lagrange Equation
We compute

$I(\phi)=\delta\mathcal{W}[\gamma]\phi=\pi\int_{\tau_{1}}^{\tau_{2}}H\left(\delta
H\phi\right)\gamma_{1}|\dot{\gamma}|dt+\frac{\pi}{2}\int_{\tau_{1}}^{\tau_{2}}H^{2}\left(|\dot{\gamma}|\phi_{1}+\gamma_{1}\frac{\langle\dot{\gamma},\dot{\phi}\rangle}{|\dot{\gamma}|}\right)dt.$
(C.6)

Considering the definition of $I(\phi)$ it is easy to see that
$|I(\phi)|\leq\|\phi\|_{C^{1}}$. Using Estimates (C.3) and (C.4) we estimate

$\displaystyle\left|\int_{\tau_{1}}^{\tau_{2}}H^{2}\left(|\dot{\gamma}|\phi_{1}+\gamma_{1}\frac{\langle\dot{\gamma},\dot{\phi}\rangle}{|\dot{\gamma}|}\right)dt\right|\leq$
$\displaystyle\left(L+\|\gamma_{1}\|_{C^{0}}\right)\|\phi\|_{C^{1}}\int_{\tau_{1}}^{\tau_{2}}H^{2}dt$
$\displaystyle\leq$ $\displaystyle\frac{C\|\phi\|_{C^{1}}}{2\pi
L\kappa(\gamma;U)}\int_{\tau_{1}}^{\tau_{2}}H^{2}2\pi\gamma_{1}|\dot{\gamma}|dt$
$\displaystyle\leq$ $\displaystyle C\|\phi\|_{C^{1}}.$ (C.7)

We recall that $H=k_{1}+k_{2}$. Using the formulas for $k_{1}$ and $k_{2}$
from Equation (2.15) it is easy to derive the following equations:

$\displaystyle\delta k_{1}\phi$
$\displaystyle=\frac{\ddot{\phi}_{2}\dot{\gamma}_{1}+\ddot{\gamma}_{2}\dot{\phi}_{1}-\ddot{\phi}_{1}\dot{\gamma}_{2}-\ddot{\gamma}_{1}\dot{\phi}_{2}}{L^{3}}-3\frac{\ddot{\gamma}_{2}\dot{\gamma}_{1}-\ddot{\gamma}_{1}\dot{\gamma}_{2}}{L^{4}}\frac{\langle\dot{\gamma},\dot{\phi}\rangle}{L}$
(C.8) $\displaystyle\delta k_{2}\phi$
$\displaystyle=\frac{\dot{\phi}_{2}}{\gamma_{1}L}-\frac{\dot{\gamma}_{2}}{\gamma_{1}^{2}L}\phi_{1}-\frac{\dot{\gamma}_{2}}{\gamma_{1}L^{2}}\frac{\langle\dot{\gamma},\dot{\phi}\rangle}{L}$
(C.9)

Next, we estimate the terms in $\delta H\phi$ that do not contain
$\ddot{\phi}$. Using $|\dot{\gamma}|=L$ and Estimate (C.3) we get

$|\delta k_{2}\phi|\leq C\|\phi\|_{C^{1}}\hskip 14.22636pt\textrm{and}\hskip
14.22636pt\left|\frac{\ddot{\gamma}_{2}\dot{\phi}_{1}-\ddot{\gamma}_{1}\dot{\phi}_{2}}{L^{3}}-3\frac{\ddot{\gamma}_{2}\dot{\gamma}_{1}-\ddot{\gamma}_{1}\dot{\gamma}_{2}}{L^{4}}\frac{\langle\dot{\gamma},\dot{\phi}\rangle}{L}\right|\leq
C\|\phi\|_{C^{1}}|\ddot{\gamma}|.$ (C.10)

Using Equation (C.6) and inserting Estimates (C.7) and (C.10) yields

$\displaystyle\left|\int_{\tau_{1}}^{\tau_{2}}H(\ddot{\phi}_{2}\dot{\gamma}_{1}-\ddot{\phi}_{1}\dot{\gamma}_{2})\gamma_{1}|\dot{\gamma}|dt\right|\leq$
$\displaystyle
C\|\phi\|_{C^{1}}\left[1+\int_{\tau_{1}}^{\tau_{2}}|H|\gamma_{1}|\dot{\gamma}_{1}|dt+\int_{\tau_{1}}^{\tau_{2}}|H|\
|\ddot{\gamma}|\gamma_{1}|\dot{\gamma}_{1}|dt\right]$ $\displaystyle\leq$
$\displaystyle
C\|\phi\|_{C^{1}}(1+\|\ddot{\gamma}\|_{L^{2}{((\tau_{1},\tau_{2}))}})$
$\displaystyle\leq$ $\displaystyle C\|\phi\|_{C^{1}}.$ (C.11)

Here we have first used the Hölder inequality and the bound on the Willmore
energy from Estimate (C.3).

Estimate Auxiliary Terms
For the moment, we abbreviate $\xi:=\chi(a+bt)$. We use Estimate (C.3) to
estimate

$\displaystyle\left|\int_{\tau_{1}}^{\tau_{2}}H(\ddot{\xi}_{2}\dot{\gamma}_{1}-\ddot{\xi}_{1}\dot{\gamma}_{2})\gamma_{1}|\dot{\gamma}|dt\right|\leq
2\|\ddot{\xi}\|_{C^{0}}\int_{\tau_{1}}^{\tau_{2}}|H|\
|\dot{\gamma}|^{2}\gamma_{1}dt$ $\displaystyle\leq$ $\displaystyle
C\|\ddot{\xi}\|_{C^{0}}\sqrt{\mathcal{W}[\gamma]}\leq C\|\xi\|_{C^{2}}.$
(C.12)

We wish to use this estimate to replace $\phi$ in Estimate (C.11) with
$\zeta$. To do so, we recall $\phi=\zeta-\chi(a+bt)$. Using
$\ddot{\zeta}=\varphi$, Estimate (C.12) as well as the fact that $|a|+|b|\leq
C\|\varphi\|_{L^{1}}$ we get

$\left|\int_{\tau_{1}}^{\tau_{2}}H(\varphi_{2}\dot{\gamma}_{1}-\varphi_{1}\dot{\gamma}_{2})\gamma_{1}|\dot{\gamma}|dt\right|\leq
C\|\varphi\|_{L^{1}((0,1))}.$ (C.13)

In particular, choosing $\varphi=(\varphi_{1},0)$ we find that for all
$\varphi_{1}\in C^{\infty}_{0}((\tau_{1},\tau_{2}))$ we have

$\left|\int_{\tau_{1}}^{\tau_{2}}H\varphi_{1}\dot{\gamma}_{2}\gamma_{1}|\dot{\gamma}|dt\right|\leq
C\|\varphi_{1}\|_{L^{1}((\tau_{1},\tau_{2}))}.$ (C.14)

Regularity Improvement
Estimate (C.14) allows us to extend the integral expression
$\int_{\tau_{1}}^{\tau_{2}}H\dot{\gamma}_{2}\gamma_{1}|\dot{\gamma}|\varphi_{1}dt$
to a linear and continuous functional
$\Lambda:L^{1}((\tau_{1},\tau_{2}))\rightarrow\mathbb{R}$. As
$(L^{1}((\tau_{1},\tau_{2})))^{\prime}=L^{\infty}((\tau_{1},\tau_{2}))$ we may
deduce $H\dot{\gamma}_{2}\gamma_{1}|\dot{\gamma}|\in
L^{\infty}((\tau_{1},\tau_{2}))$ with

$\|H\dot{\gamma}_{2}\gamma_{1}|\dot{\gamma}|\
\|_{L^{\infty}((\tau_{1},\tau_{2}))}\leq C.$

As $\gamma_{1}\geq\kappa(\gamma;U)$ on $(\tau_{1},\tau_{2})$ and
$|\dot{\gamma}|=L$ we obtain
$\|H\dot{\gamma}_{2}\|_{L^{\infty}((\tau_{1},\tau_{2}))}\leq C$. Using
$\varphi=(0,\varphi_{2})$ one can derive a similar estimate for
$\|H\dot{\gamma}_{1}\|_{L^{\infty}((\tau_{1},\tau_{2}))}$. Combining both
estimates, we get

$\|H\|_{L^{\infty}((\tau_{1},\tau_{2}))}=\frac{\|H\dot{\gamma}\|_{L^{\infty}((\tau_{1},\tau_{2}))}}{L}\leq\frac{1}{L}(\|H\dot{\gamma}_{1}\|_{L^{\infty}((\tau_{1},\tau_{2}))}+\|H\dot{\gamma}_{2}\|_{L^{\infty}((\tau_{1},\tau_{2}))})\leq
C.$

Using the identities from Equation (2.20), we deduce $\gamma\in
W^{2,\infty}((\tau_{1},\tau_{2}))$ and the estimate
$\|\ddot{\gamma}\|_{L^{\infty}((\tau_{1},\tau_{2}))}\leq C$. ∎

######  Lemma C.3.

Under the assumptions of Theorem C.1 $\gamma\in W^{3,\infty}(U)$, $H\in
W^{1,\infty}(U)$ and

$\|\gamma\|_{W^{3,\infty}(U)}+\|H\|_{W^{1,\infty}(U)}\leq
C(M,\kappa(\gamma;U),U).$

###### Proof..

Let again $U=(\tau_{1},\tau_{2})$. As in the proof of Lemma C.2 $C$ always
refers to a constant depending on $M$, $\kappa(\gamma;U)$ and $U$. Also, the
rest of the proof is very similar to the proof of Lemma C.2 and we only sketch
the parts that need to be altered. First, the test function has to get
modified. Given $\varphi=(\varphi_{1},\varphi_{2})\in
C_{0}^{\infty}((\tau_{1},\tau_{2}),\mathbb{R}^{2})$ we put

$\zeta(t):=\int_{0}^{t}\varphi(r)dr.$

Let $[r_{1},r_{2}]=\operatorname{supp}\varphi$. Then $\zeta(t)=0$ for
$t\in[\tau_{1},r_{1}]$ and $\zeta(t)=a$ when $t\in[r_{2},\tau_{2}]$ where

$a=\int_{\tau_{1}}^{\tau_{2}}\varphi(r)dr.$

We choose the same smooth function $\chi$ as in the proof of Lemma C.2 and put
$\phi(t):=\zeta(t)-a\chi(t)$. It is easy to see that $\phi\in
C^{\infty}_{0}(U)$ and that $\|\phi\|_{C^{0}}\leq C\|\varphi\|_{L^{1}}$ and
$\|\dot{\phi}\|_{L^{1}}\leq C\|\varphi\|_{L^{1}}$. Testing the weak Euler-
Lagrange equation with $\phi$ as we did in Equation (C.6) and using the
estimates derived in Lemma C.2 we get

$\left|\int_{\tau_{1}}^{\tau_{2}}H\left(\delta
H\phi\right)\gamma_{1}|\dot{\gamma}|dt\right|\leq
C\left[\int_{\tau_{1}}^{\tau_{2}}H^{2}\left(|\dot{\gamma}|\
|\phi_{1}|+\gamma_{1}|\dot{\phi}|\right)dt+|I(\phi)|\right]\leq
C\|\phi\|_{W^{1,1}(U)}.$ (C.15)

Next, we take care of the terms in $\delta H$ that do not contain
$\ddot{\phi}$. Recalling Equations (C.8) and (C.9) and using the result of
Lemma C.2, we estimate

$\displaystyle\left|\delta
H\phi-\frac{\ddot{\phi}_{2}\dot{\gamma}_{1}-\ddot{\phi}_{1}\dot{\gamma}_{2}}{L^{3}}\right|$
$\displaystyle=|\delta
k_{2}\phi|+\left|\frac{\ddot{\gamma}_{2}\dot{\phi}_{1}-\ddot{\gamma}_{1}\dot{\phi}_{2}}{L^{3}}-3\frac{\ddot{\gamma}_{2}\dot{\gamma}_{1}-\ddot{\gamma}_{1}\dot{\gamma}_{2}}{L^{4}}\frac{\langle\dot{\gamma},\dot{\phi}\rangle}{L}\right|$
$\displaystyle\leq C\left(|\phi|+|\dot{\phi}|\right).$ (C.16)

Inserting into Estimate (C.15) and using the estimate for
$\|H\|_{L^{\infty}((\tau_{1},\tau_{2}))}$ from Lemma C.2, we get

$\left|\int_{\tau_{1}}^{\tau_{2}}H(\ddot{\phi}_{2}\dot{\gamma}_{1}-\ddot{\phi}_{1}\dot{\gamma}_{2})\gamma_{1}|\dot{\gamma}|dt\right|\leq
C\|\phi\|_{W^{1,1}(U)}.$

As in the proof of Lemma C.2, we now insert the formula for $\phi$ and
estimate the auxiliary terms to get

$\left|\int_{\tau_{1}}^{\tau_{2}}H(\dot{\varphi}_{2}\dot{\gamma}_{1}-\dot{\varphi}_{1}\dot{\gamma}_{2})\gamma_{1}|\dot{\gamma}|dt\right|\leq
C\|\phi\|_{W^{1,1,}}\leq C\|\varphi\|_{L^{1}}.$ (C.17)

Now we argue similarly to as we did in the final step of the proof of Lemma
C.2. First we take $\varphi=(\varphi_{1},0)$ and consider the densely defined
linear operator

$\Lambda:L^{1}((\tau_{1},\tau_{2}))\rightarrow\mathbb{R},\hskip
14.22636pt\Lambda(\varphi_{1}):=\int_{\tau_{1}}^{\tau_{2}}H\dot{\varphi}_{1}\dot{\gamma}_{2}\gamma_{1}|\dot{\gamma}|dt.$

Estimate (C.17) allows us to extend $\Lambda$ to a continuous linear map on
all of $L^{1}((\tau_{1},\tau_{2}))$. Now, by the Riesz representation theorem
(see e.g. [9], Theorem 19.2), there exists a function $h\in
L^{\infty}((\tau_{1},\tau_{2}))$ such that
$\Lambda(\varphi)=\int_{\tau_{1}}^{\tau_{2}}h\varphi dt$. Going back to
$\varphi\in C_{0}^{\infty}((\tau_{1},\tau_{2}))$ we get

$\int_{\tau_{1}}^{\tau_{2}}h\varphi_{1}dt=\int_{\tau_{1}}^{\tau_{2}}H\gamma_{1}\dot{\gamma}_{2}|\dot{\gamma}|\dot{\varphi}_{1}dt\hskip
14.22636pt\textrm{for all }\varphi_{1}\in
C^{\infty}_{0}((\tau_{1},\tau_{2})).$

Thus $H\gamma_{1}\dot{\gamma}_{2}|\dot{\gamma}|$ is weakly differentiable and
Estimate (C.17) implies

$\|(H\gamma_{1}\dot{\gamma}_{2}|\dot{\gamma}|)^{\prime}\|_{L^{\infty}((\tau_{1},\tau_{2}))}\leq
C.$

Using the results from Lemma C.2 as well as $\gamma_{1}\geq\kappa(\gamma;U)$
on $U=(\tau_{1},\tau_{2})$ we obtain $H\dot{\gamma}_{2}\in
W^{1,\infty}((\tau_{1},\tau_{2}))$ with
$\|H\dot{\gamma}_{2}\|_{W^{1,\infty}((\tau_{1},\tau_{2}))}\leq C$. Taking
$\varphi=(0,\varphi_{2})$ instead, we can derive a similar estimate for
$H\dot{\gamma}_{1}$. The lemma follows as in the proof of Lemma C.2 by using
the identities in Equation (2.20). ∎

######  Lemma C.4.

Under the assumptions of Theorem C.1 $\gamma\in W^{k+2,\infty}(U)$, $H\in
W^{k,\infty(U)}$ for all $k\in\mathbb{N}_{0}$ and

$\|\gamma\|_{W^{k+2,\infty}(U)}+\|H\|_{W^{k,\infty}(U)}\leq
C(M,\kappa(\gamma;U),U,k).$

###### Proof..

As before, $C$ will always refer to a constant depending on $M$,
$\kappa(\gamma;U)$ and $U$ and this time also $k$. The proof is inductive. The
case $k=1$ follows from Lemma C.3. Therefore it suffices to prove the
following implication for $k\geq 1$:

$\left\\{\begin{aligned} &H\in W^{k,\infty}(U)\\\ &\|H\|_{W^{k,\infty}(U)}\leq
C\\\ &\gamma\in W^{k+2,\infty}(U)\\\ &\|\gamma\|_{W^{k+2,\infty}(U)}\leq
C\end{aligned}\right\\}\hskip 14.22636pt\Rightarrow\hskip
14.22636pt\left\\{\begin{aligned} &H\in W^{k+1,\infty}(U)\\\
&\|H\|_{W^{k+1,\infty}(U)}\leq C\\\ &\gamma\in W^{k+3,\infty}(U)\\\
&\|\gamma\|_{W^{k+3,\infty}(U)}\leq C\end{aligned}\right\\}$ (C.18)

To establish (C.18) we take $\varphi=(\varphi_{1},\varphi_{2})\in
C_{0}^{\infty}(U,\mathbb{R}^{2})$ and choose the test function
$\phi:=\varphi^{(k-1)}$ Testing the weak version of the Euler-Lagrange
equation with $\phi$ as we did in Equation (C.6) gives

$\displaystyle\left|\int_{\tau_{1}}^{\tau_{2}}H\left(\delta
H\phi\right)\gamma_{1}|\dot{\gamma}|dt\right|\leq
C\left|\int_{\tau_{1}}^{\tau_{2}}H^{2}\left(|\dot{\gamma}|\phi_{1}+\gamma_{1}\frac{\langle\dot{\gamma},\dot{\phi}\rangle}{|\dot{\gamma}|}\right)dt\right|+C|I(\phi)|$
$\displaystyle\leq$ $\displaystyle
C\left|\int_{\tau_{1}}^{\tau_{2}}H^{2}\left(|\dot{\gamma}|\varphi^{(k-1)}_{1}(t)+\gamma_{1}\frac{\langle\dot{\gamma},\varphi^{(k)}(t)\rangle}{|\dot{\gamma}|}\right)dt\right|+C|I(\varphi^{(k-1)})|.$
(C.19)

First, we establish a suitable estimate for $I(\varphi^{(k-1)})$. By
assumption $\gamma\in W^{k+2,\infty}(U)$. Therefore we can integrate the
integrals in Equation (C.1) by parts to obtain $|I(\varphi^{(k-1)})|\leq
C\|\varphi\|_{L^{1}}$. Also using the assumption $H\in W^{k,\infty}(U)$ and
$\gamma\in W^{k+2,\infty}(U)$, we can rewrite the integral on the right in
Estimate (C.19) by integrating by parts several times and estimate it by
$C\|\varphi\|_{L^{1}}$. In total, we deduce

$\left|\int_{\tau_{1}}^{\tau_{2}}H\left(\delta
H\phi\right)\gamma_{1}|\dot{\gamma}|dt\right|\leq C\|\varphi\|_{L^{1}}.$
(C.20)

Next, we again estimate all terms in $\delta H\phi$ that do not contain
$\ddot{\phi}$. Using Equations (C.8) and (C.9), $\gamma\in W^{k+2,\infty}$ and
repeated integration by parts we get

$\displaystyle\left|\int_{\tau_{1}}^{\tau_{2}}H\left(\delta
H\phi-\frac{\dot{\gamma}_{1}\ddot{\phi}_{2}-\dot{\gamma}_{2}\ddot{\phi}_{1}}{L^{3}}\right)\gamma_{1}|\dot{\gamma}|dt\right|$
$\displaystyle\leq$
$\displaystyle\left|\int_{\tau_{1}}^{\tau_{2}}H\left(\delta
k_{2}\phi+\frac{\ddot{\gamma}_{2}\dot{\phi}_{1}-\ddot{\gamma}_{1}\dot{\phi}_{2}}{L^{3}}-3\frac{\ddot{\gamma}_{2}\dot{\gamma}_{1}-\ddot{\gamma}_{1}\dot{\gamma}_{2}}{L^{4}}\frac{\langle\dot{\gamma},\dot{\phi}\rangle}{L}\right)\gamma_{1}|\dot{\gamma}|dt\right|$
$\displaystyle\leq$ $\displaystyle C\|\varphi\|_{L^{1}}.$

Combining with Estimate (C.20), we find

$\left|\int_{\tau_{1}}^{\tau_{2}}H\frac{\dot{\gamma}_{1}\varphi^{(k+1)}_{2}-\dot{\gamma}_{2}\varphi^{(k+1)}_{1}}{L^{3}}\gamma_{1}|\dot{\gamma}|dt\right|\leq
C\|\varphi\|_{L^{1}((\tau_{1},\tau_{2}))}.$

As in the proof of Lemma C.2 this now implies $H\in
W^{k+1,\infty}((\tau_{1},\tau_{2}))$ and hence $\gamma\in
W^{k+3,\infty}((\tau_{1},\tau_{2}))$. This establishes Implication (C.18) and
thereby the lemma. ∎

##  Appendix D Calculus Lemmas

######  Lemma D.1.

Let $n\in\mathbb{N}_{0}$, $u\in C^{1}((0,1))\cap C^{0}([0,1))$, $h\in
C^{n}([0,1)$ and

$\frac{(xu(x))^{\prime}}{x}=h(x).$

Then $u\in C^{n+1}([0,1))$.

###### Proof..

First, we consider the case $n=0$. We have

$xu(x)=0\cdot u(0)+\int_{0}^{x}th(t)dt.$ (D.1)

So, for $x\in(0,1)$ we get

$u^{\prime}(x)=\frac{d}{dx}\left[\frac{1}{x}\int_{0}^{x}th(t)dt\right]=h(x)-\frac{1}{x^{2}}\int_{0}^{x}th(t)dt.$

We claim that $u^{\prime}(x)\rightarrow\frac{1}{2}h(0)$ as $x\rightarrow 0$.
Indeed

$\frac{1}{x^{2}}\int_{0}^{x}h(t)tdt=\int_{0}^{1}sh(sx)ds\rightarrow
h(0)\int_{0}^{1}sds=\frac{1}{2}h(0).$

Now we consider the case $n>0$. Rewriting Equation (D.1) we get

$u(x)=x\int_{0}^{1}h(tx)tdt$

and deduce $u\in C^{n}([0,1))$. Next, compute

$\displaystyle u^{\prime}(x)=$
$\displaystyle\int_{0}^{1}th(tx)dt+x\int_{0}^{1}t^{2}h^{\prime}(tx)dt$
$\displaystyle\int_{0}^{1}th(tx)dt+x\left[\frac{t^{2}h(tx)}{x}\bigg{|}_{t=0}^{t=1}-\int_{0}^{1}2t\frac{h(tx)}{x}dt\right]$
$\displaystyle=$ $\displaystyle h(x)-\int_{0}^{1}th(tx)dt.$

Clearly, this last formula is $C^{n}([0,1])$, which proves the lemma. ∎

######  Lemma D.2.

Let $r_{0}>0$, $f\in C^{0}([0,r_{0}])\cap C^{1}((0,r_{0}])$ and $w\in
C^{2}([0,r_{0}])$ such that

$w^{\prime\prime}(r)+\left(\frac{w}{r}\right)^{\prime}=f(r).$

Then $|w^{\prime\prime}(r)|\leq\frac{4}{3}\sup_{0\leq\rho\leq r}|f(\rho)|$ and
consequently
$|w^{\prime}(r)|\leq|w^{\prime}(r_{0})|+\frac{4}{3}r_{0}\sup_{0\leq\rho\leq
r_{0}}|f(\rho)|$.

###### Proof..

We compute

$\left(\frac{(wr)^{\prime}}{r}\right)^{\prime}=\left(w^{\prime}+\frac{w}{r}\right)^{\prime}=f.$

Integrating this equation from $r>0$ to $r_{0}$ shows

$-\frac{(w(r)r)^{\prime}}{r}+w^{\prime}(r_{0})+\frac{w(r_{0})}{r_{0}}=\int_{r}^{r_{0}}f(s)ds=\int_{0}^{r_{0}}f(s)ds-\int_{0}^{r}f(s)ds.$

We put
$c_{0}:=w^{\prime}(r_{0})+\frac{w(r_{0})}{r_{0}}-\int_{0}^{r_{0}}f(s)ds$. Then

$\frac{(rw(r))^{\prime}}{r}=c_{0}+\int_{0}^{r}f(s)ds.$

We multiply by $r$. Another integration from $r$ to $r_{0}$ shows

$\displaystyle r_{0}w(r_{0})-rw(r)$
$\displaystyle=\frac{1}{2}c_{0}(r_{0}^{2}-r^{2})+\int_{r}^{r_{0}}x\int_{0}^{x}f(s)dsdx$
$\displaystyle=-\frac{1}{2}c_{0}r^{2}+\frac{1}{2}c_{0}r_{0}^{2}+\int_{0}^{r_{0}}x\int_{0}^{x}f(s)dsdx-\int_{0}^{r}x\int_{0}^{x}f(s)dsdx.$

This shows that there exist constants $c_{1},c_{2}\in\mathbb{R}$ such that

$rw(r)=c_{1}r^{2}+c_{2}+\int_{0}^{r}\int_{0}^{x}xf(s)dsdx.$

Taking $r=0$ shows $c_{2}=0$. Next, we substitute $x=ry$ and obtain

$w(r)=c_{1}r+r\int_{0}^{1}\int_{0}^{ry}yf(s)dsdy.$

We compute two $r$-derivatives.

$\displaystyle w^{\prime\prime}(r)=$ $\displaystyle
2\int_{0}^{1}\frac{\partial}{\partial
r}\int_{0}^{yr}yf(s)dsdy+r\int_{0}^{1}\frac{\partial^{2}}{\partial
r^{2}}\int_{0}^{ry}yf(s)dsdy$ $\displaystyle=$ $\displaystyle
2\int_{0}^{1}y^{2}f(yr)dy+r\int_{0}^{1}y^{3}f^{\prime}(yr)dy$ $\displaystyle=$
$\displaystyle
2\int_{0}^{1}y^{2}f(yr)dy+\int_{0}^{1}y^{3}\frac{\partial}{\partial y}f(yr)dy$
$\displaystyle=$ $\displaystyle
2\int_{0}^{1}y^{2}f(yr)dy+y^{3}f(yr)\bigg{|}_{y=0}^{y=1}-\int_{0}^{1}3y^{2}f(yr)dy$
$\displaystyle=$ $\displaystyle-\int_{0}^{1}y^{2}f(yr)dy+f(r)$

Consequently

$|w^{\prime\prime}(r)|\leq\left(1+\int_{0}^{1}y^{2}dy\right)\sup_{0\leq\rho\leq
r}|f(\rho)|=\frac{4}{3}\sup_{0\leq\rho\leq r}|f(\rho)|.$

∎

######  Lemma D.3.

Let $f\in C^{1}((0,\rho))$ such that $f^{\prime}\in L^{1}((0,\rho))$. Then
$f\in C^{0}([0,\rho])$.

###### Proof..

We prove continuity at $x=0$. The case $x=\rho$ works similarly. Let
$x<y\in(0,\rho]$. Then

$|f(x)-f(y)|\leq\int_{x}^{y}|f^{\prime}(t)|dt\leq\int_{0}^{y}|f^{\prime}(t)|dt.$

This implies that $f(\epsilon_{n})$ is Cauchy for any $\epsilon_{n}\rightarrow
0^{+}$ and hence $f(0):=\lim_{x\rightarrow 0^{+}}f(x)$ is well-defined. To see
that $f$ is continuous at $x=0$, we use $f^{\prime}\in L^{1}((0,1))$ to
estimate

$|f(0)-f(y)|\leq\int_{0}^{y}|f^{\prime}(t)|dt\rightarrow 0,\hskip
14.22636pt\textrm{as $y\rightarrow 0^{+}$.}$

∎

## Acknowledgements

The author would like to thank Ernst Kuwert for the suggestion of this
interesting topic and the helpful guidance as well as Marius Müller for the
many helpful discussions.

## References

  * [1] Roberta Alessandroni and Ernst Kuwert.  Local solutions to a free boundary problem for the Willmore functional.  Calculus of Variations and Partial Differential Equations, 55(2):1–29, 2016.
  * [2] Peter B. Canham.  The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell.  Journal of theoretical biology, 26(1):61–81, 1970.
  * [3] Jingyi Chen and Yuxiang Li.  Radially symmetric solutions to the graphic willmore surface equation.  The Journal of Geometric Analysis, 27(1):671–688, 2017.
  * [4] Rustum Choksi and Marco Veneroni.  Global minimizers for the doubly-constrained helfrich energy: the axisymmetric case.  Calculus of Variations and Partial Differential Equations, 48(3):337–366, 2013.
  * [5] Heinz J. Deuling and Wolfgang Helfrich.  Red blood cell shapes as explained on the basis of curvature elasticity.  Biophysical journal, 16(8):861–868, 1976.
  * [6] Gerhard Dziuk and Klaus Deckelnick.  Error analysis of a finite element method for the willmore flow of graphs.  Interfaces and free boundaries, 8(1):21–46, 2006.
  * [7] Sascha Eichmann and Hans-Christoph Grunau.  Existence for willmore surfaces of revolution satisfying non-symmetric dirichlet boundary conditions.  Advances in Calculus of Variations, 12(4):333–361, 2019.
  * [8] Wolfgang Helfrich.  Elastic properties of lipid bilayers: theory and possible experiments.  Zeitschrift für Naturforschung, 28(11-12):693–703, 1973.
  * [9] Friedrich Hirzebruch and Winfried Scharlau.  Einführung in die Funktionalanalysis.  Spektrum Akademischer Verlag, 1991.
  * [10] Ernst Kuwert and Yuxiang Li.  Asymptotics of willmore minimizers with prescribed small isoperimetric ratio.  SIAM Journal on Mathematical Analysis, 50(4):4407–4425, 2018.
  * [11] Ernst Kuwert and Reiner Schätzle.  Gradient flow for the willmore functional.  Communications in Analysis and Geometry, 10(2):307–339, 2002.
  * [12] Ernst Kuwert and Reiner Schätzle.  Removability of point singularities of willmore surfaces.  Annals of Mathematics, 160(1):315–357, 2004.
  * [13] Giovanni Leoni.  A first course in Sobolev spaces.  American Mathematical Soc., 2017.
  * [14] Karl H. Mayer.  Algebraische Topologie.  Birkhäuser Verlag, Basel, 1989.
  * [15] Takeyuki Nagasawa and Izumi Takagi.  Bifurcating critical points of bending energy under constraints related to the shape of red blood cells.  Calculus of Variations and Partial Differential Equations, 16(1):63–111, 2003.
  * [16] Charles C. Pugh.  Real Mathematical Analysis.  Springer New York, 2003.
  * [17] Christian Scharrer.  Embedded delaunay tori and their willmore energy.  Nonlinear Analysis, 223:113010, 2022.
  * [18] Johannes Schygulla.  Willmore minimizers with prescribed isoperimetric ratio.  Archive for Rational Mechanics and Analysis, 203(3):901, 2012.
  * [19] Udo Seifert, Karin Berndl, and Reinhard Lipowsky.  Shape transformations of vesicles: Phase diagram for spontaneous-curvature and bilayer-coupling models.  Physical review A, 44(2):1182, 1991.
  * [20] Leon Simon.  Existence of surfaces minimizing the willmore functional.  Communications in Analysis and Geometry, 1(2):281–326, 1993.