Teichmüller harmonic map flow from cylinders

Abstract

We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that also in this new setting and for arbitrary initial data, solutions of the Teichmüller harmonic map flow exist for all times. Furthermore, for solutions for which a three-point-condition does not degenerate as $t\rightarrow \infty$, we show convergence along a sequence $t_i\rightarrow \infty$ to a critical point of the area given either by a minimal cylinder or by two minimal discs spanning the given boundary curves.

Appendix

1.1 A.1 Courant–Lebesgue Lemma and properties of $H_{\Gamma ,*}^1(C_0)$

Throughout the paper we made use of the Courant–Lebesgue Lemma of which we use the following version, see e.g. [9, Lemma 3.1.1] or [17, Lemma 4.4]

Lemma A.1

Let $D_r(0)^+=\{x\in {\mathbb R}^2: {\vert x\vert } \le r, x_1\ge 0\}$ and let $u\in H^1(D_r(0)^+,{\mathbb R}^n)$ be any map that has energy $E(u,g_{\text {eucl}})\le E_0$, $E_0$ any fixed number. Then for any $\delta \in (0,\min (r,{\frac{1}{2}}))$ there exists $\rho \in (\delta ,\sqrt{\delta })$ so that $u|_{\partial D_\rho (0)^+}$ is absolutely continuous and so that the estimate

$$\begin{aligned} {\vert u(x)-u(y)\vert } \le C\cdot {\vert \log (\delta )\vert } ^{-1/2},\quad \text { for all } x,y \in \partial D_\rho ^+:= \{ y: {\vert y\vert } =\rho ,\quad y_1\ge 0\} \end{aligned}$$

holds true with a constant C that depends only on $E_0$.

We use in particular the following consequence for maps satisfying the three-point-condition

Corollary A.2

Let $u_i\in H^1_{\Gamma ,*}(C_0)$ be a sequence of maps that have uniformly bounded energy $E(u_i,g_i)\le E_0<\infty$ with respect to metrics $g_i=h_{b_i,\phi _i}^*G_{\ell _i}$ for which $\sup {\vert b_i^\pm \vert } <1$. Then the traces $u_i \vert _{{\partial C_0}}$ are equicontinuous.

Proof of Corollary A.2

Pulling back the maps and metrics by the diffeomorphism $h_{b_i,\phi _i}^{-1}$ one can reduce this corollary to the corresponding claim for the metrics $G_{\ell _i}$ and for maps $\tilde{u}_i$ so that the functions $\varphi _i^\pm$ that describe the traces $\tilde{u}_i\vert _{{\partial C_0}^\pm }=\alpha ^\pm \circ \varphi _i^\pm$ are such that there are points $\tilde{\theta }_{i,k}$ with

$$\begin{aligned} \varphi _i(\tilde{\theta }_{i,k})=\frac{2\pi }{3}k \quad \text {and}\quad {\vert \tilde{\theta }_{i,k+1}-\tilde{\theta }_{i,k}\vert } \ge c, \quad k=0,1,2 \end{aligned}$$

(A.1)

where $c>0$ depends only on $1-\sup {\vert b^\pm \vert }$ and where $\tilde{\theta }_{i,3}:=\tilde{\theta }_{i,0}+2\pi$.

We then remark that the upper bound on $\ell$ given by (3.10) implies that the induced metrics on the boundary of $(C_0,G_\ell )$ are all equivalent and that the numbers $Y(\ell )$ are bounded away from zero. Given any point $p=(\pm 1, \bar{\theta })$ we can thus apply Lemma A.1 on the set described by $\{(\pm 1,\bar{\theta })\}\mp D_r^+(0)$ in collar coordinates, with $r>0$ depending only on the upper bounds on $\ell$ and ${\vert b^\pm \vert }$, namely chosen so that $r< \min (c/2,Y(\ell ))$. The proof then follows by a standard argument: Given that the parametrisations are weakly monotone and that (A.1) does not permit that more than one of the three points $\alpha _\pm (\theta _k)$ is contained in the image of the small arc $(s,\theta )\in \{\pm 1\}\times [\bar{\theta }-r,\bar{\theta }+r]$, we obtain the desired bound on the modulus of continuity from the Courant–Lebesgue Lemma. $\square$

The above lemma implies in particular that any map u that is obtained as weak $H^1$ limit of a sequence of maps $u_i\in H_{\Gamma ,*}^1(C_0)$ is again an element of $H_{\Gamma ,*}^1(C_0)$.

In order to pass to the limit in the differential inequality (3.3) we use at several points in the paper that the tangent cones $T_u^+H_{\Gamma ,*}^1(C_0)$ depend continuously on u namely that

Lemma A.3

Let $u\in H_{\Gamma ,*}^1(C_0)$ and let $u^i\in H_{\Gamma ,*}^1(C_0)$ be any sequence that converges strongly in $H^1(C_0)$ to u. Then any element $v\in T^+_uH_{\Gamma ,*}^1(C_0)$ can be approximated by elements in $v^i\in T^+_{u^i}H_{\Gamma ,*}^1(C_0)$ in the sense that

$$\begin{aligned} v^i\rightarrow v \quad \text {strongly in } H^1(C_0). \end{aligned}$$

Indeed, writing $u\vert _{\partial C_\pm }=\alpha _\pm \circ \varphi _\pm$ respectively $u^i\vert _{\partial C_\pm }=\alpha _\pm \circ \varphi ^i_\pm$ and $v\vert _{\partial C_\pm }=\lambda _{\pm }\cdot \alpha _\pm '(\varphi _\pm )\cdot (\psi _\pm -\varphi _\pm )$ we can use that the traces converge both strongly in $H^{1/2}$ and uniformly and thus that also $v^i\vert _{\partial C_\pm }:=\lambda _{\pm }\cdot \alpha _\pm '(\varphi _\pm ^i)\cdot (\psi _\pm -\varphi ^i_\pm )$ converge to $v\vert _{\partial C_\pm }$ in $H^{1/2}$. The desired elements of $T^+_{u_i}H_{\Gamma ,*}^1(C_0)$ are then obtained as harmonic extensions of these traces similarly to the proof of Lemma 2.1 in [6].

As a consequence we obtain

Corollary A.4

Let $(u_i,g_i,f_i)\in H_{\Gamma ,*}^1(C_0)\times \widetilde{\mathcal {M}}\times L^2(C_0)$ be such that (3.3) is satisfied and assume that $g_i\rightarrow g$, $u_i\rightarrow u_\infty$ strongly in $H^1(C_0)$ and $f_i\rightharpoonup f$ weakly in $L^2$. Then (3.3) is satisfied also for the limit (u, g, f).

We finally outline how Proposition 3.6 can be derived from the corresponding estimates for maps from the disc proven by Duzaar and Scheven in [6, Theorem 8.3]

Sketch of the proof of Proposition 3.6

Because of the interior estimates of Lemma 3.4 it is sufficient to consider points p that are contained in $V^{\pm }=\{s: \pm s\in [\frac{1}{2},1]\}\times S^1$. We pull-back the maps and metrics first by $h_{b,\phi }^{-1}$, which we note maps $V^{\pm }$ onto itself, then by a fixed conformal diffeomorphism that maps a neighbourhood of $\partial D_1\subset (\overline{D_1}(0),g_{eucl})$ onto $V^{\pm }\subset (C_0,G_\ell )=(C_0,(h_{b,\phi }^{-1})^*g)$ and finally, to restore the three-point-condition, by the Möbius transform $M_{b^\pm ,\phi ^\pm }:\overline{D_1}\rightarrow \overline{D_1}$. The resulting triple $(\tilde{u},\tilde{g},\tilde{f})=\psi ^*(u,g,f)$ then satisfies equation (3.3) on a neighbourhood U of $\partial D_1$. By construction the metric $\tilde{g}$ is conformal to the Euclidian metric and we note that the corresponding conformal factor in $\tilde{g}=\lambda g_{eucl}$ is bounded uniformly both from above and away from zero since the bound on $1-{\vert b^{ \pm }\vert }$ allows us to control $M_{b^\pm ,\phi ^\pm }$ and since $\rho _\ell$ is controlled uniformly on $V^{\pm }$ even if $\ell \rightarrow 0$. Given that (3.3) holds true also for $(\tilde{u}, g_{eucl},\tilde{f}\lambda ^2)$ we can then apply Theorem 8.3 of [6] to obtain the claimed estimates on corresponding balls $D_r(p)$ in the Euclidean disc.

We can then pull-back these estimates to give the claim of Proposition 3.6 since the bounds on $1-{\vert b^{\pm }\vert }$ and on $\rho$ (on $V^{\pm }$) give sufficient control on the involved metrics and diffeomorphisms. $\square$

1.2 A.2 Properties of hyperbolic collars and the horizontal family of metrics $G_\ell$

In this part of the appendix we collect some well known properties of hyperbolic collars, where we refer to the appendix of [14] and the references therein for more information, as well as properties of the hyperbolic cylinders $(C_0,G_\ell )$ that are used throughout the paper. We furthermore give the proof that the family of metrics described in Lemma 2.4 is horizontal, i.e. that $\frac{d}{d\ell } G_\ell \in Re(\mathcal{H}(G_\ell )).$

We first recall that the $\delta \text {-thin}$ part of a hyperbolic cylinder is described in collar coordinates $(s,\theta )\in (-Y(\ell ),Y(\ell ))$ by

$$\begin{aligned} (-\min (X_\delta (\ell ),Y(\ell )),\min (X_\delta (\ell ),Y(\ell ))) \times S^1 , \end{aligned}$$

where

$$\begin{aligned} X_\delta (\ell )= \frac{2\pi }{\ell }\left( \frac{\pi }{2}-\arcsin \left( \frac{\sinh \left( \frac{\ell }{2}\right) }{\sinh \delta }\right) \right) \end{aligned}$$

for $\delta \ge \ell /2$, respectively zero for smaller values of $\delta$.

Remark A.5

For the metrics $G_\ell =f_\ell ^*(\rho _\ell ^2(ds^2+d\theta ^2))$ this means that for each $\delta >0$ there exists a number $c_0(\delta )>0$ with $c_0(\delta )\rightarrow 0$ for $\delta \rightarrow 0$ so that $\delta \text {-thin}(C_0,G_\ell )$ is contained in $(-c_0(\delta ),c_0(\delta ))\times S^1$ (with respect to the fixed coordinates $(x,\theta )$ of $C_0$) for every $\ell >0$; or, said differently, for every $c_1>0$ there exists a number $\delta (c_1)>0$ so that

$$\begin{aligned} {{\mathrm{inj}}}_{G_\ell }(x,\theta )\ge \delta (c_1)\quad \text {for all}\quad {\vert x\vert } \ge c_1 \quad \text {and all}\quad \ell >0. \end{aligned}$$

As ${{\mathrm{inj}}}_G(x)\le \pi \rho (s_\ell (x))$ this also yields a uniform lower bound on the conformal factor on this set.

We also note that

$$\begin{aligned} {\vert dz^2\vert } _{g}=2\rho ^{-2}=\sqrt{2} {\vert Re(dz^2)\vert } \end{aligned}$$

(A.2)

and hence that the norms of $dz^2$ on $([-Y(\ell ),Y(\ell )]\times S^1, \rho ^2_\ell (ds^2+d\theta ^2))$ can be computed as

$$\begin{aligned} \Vert dz^2\Vert _{L^\infty }=\frac{8\pi ^2}{\ell ^{2}} \end{aligned}$$

(A.3)

and

$$\begin{aligned} \Vert dz^2\Vert _{L^2}^2=\frac{64\pi ^4}{\ell ^3}\cdot \left[ \sin \left( \text {atan}(\eta \ell ) \right) \cdot \cos \left( \text {atan}(\eta \ell ) \right) + \left( \frac{\pi }{2}-\text {atan}(\eta \ell )\right) \right] . \end{aligned}$$

For $\ell$ small we thus have that

$$\begin{aligned} \Vert dz^2\Vert _{L^2}^2=\frac{32\pi ^5}{\ell ^3}+O(1), \end{aligned}$$

(A.4)

while for $\ell$ large

$$\begin{aligned} \Vert dz^2\Vert _{L^2}^2=\frac{128\pi ^4}{\eta \ell ^4}+O(\ell ^{-5}). \end{aligned}$$

(A.5)

We also recall the well known fact that if a metric g evolves by $\partial _tg=Re(\Psi )$ for a holomorphic quadratic differential $\Psi$ then the length of the central geodesic changes by

$$\begin{aligned} \frac{d\ell }{dt}=-\frac{2\pi ^2}{\ell } Re(c_0), \end{aligned}$$

(A.6)

where $c_0dz^2$ is the principal part in the Fourier expansion of $\Psi$, or in our case simply the coefficient in $\Psi =a_0dz^2, a_0\in {\mathbb R}$.

For large values of $\ell$ we can thus bound the evolution of $\ell$ along a horizontal curve by

$$\begin{aligned} \left| \frac{d\ell }{dt}\right| \le \frac{2\pi ^2}{\ell }\frac{\Vert \partial _t g\Vert _{L^2}}{\Vert dz^2\Vert _{L^2}}\le C\cdot \ell \Vert \partial _t g\Vert _{L^2} \end{aligned}$$

(A.7)

while for small values of $\ell$ we only obtain that

$$\begin{aligned} \left| \frac{d\ell }{dt}\right| \le C\cdot \ell ^{1/2} \Vert \partial _t g\Vert _{L^2} \end{aligned}$$

(A.8)

which allows for a degeneration of the metric along a curve of finite length.

Finally we explain how the formula for the horizontal families of metrics in ${\mathcal M}_{-1}$ claimed in Lemma 2.4 can be derived.

Proof of Lemma 2.4

Let $t \mapsto g(t)$ be a curve of metrics in ${\mathcal M}_{-1}$ which moves in horizontal direction, $\partial _tg(t)\in \text {Re}(\mathcal{H}(C_0,g(t)))$, and so that g is given as pull-back of a collar $([-Y(\ell ),Y(\ell )]\times S^1,\rho _{\ell }(s)^2(ds^2+d\theta ^2))$ by a suitable diffeomorphism $f_{\ell }:C_0\rightarrow [-Y(\ell ),Y(\ell )]\times S^1$ where both $Y(\ell )$ and $f_{\ell }$ need to be determined.

To begin with, we derive a differential equation for $Y(\ell )$ by computing the evolution of the width

$$\begin{aligned} w(\ell (t)):=\text {dist}_{g(t)}(\{-1\}\times S^1, \{1\}\times S^1) \end{aligned}$$

of the cylinder $(C_0,g(t))$.

Let t be any fixed time and let $(s,\theta )\in [-Y(\ell (t)),Y(\ell (t))]\times S^1$ be the corresponding collar coordinates. Then in these fixed coordinates, the evolution of g at time t is given by $a_0 (ds^2-d\theta ^2)$ where $a_0$ is related to the evolution of the length $\ell$ of the central geodesic by (A.6).

Thus the width of the collar, which at time t is simply given by the length of the geodesics $s\mapsto (s,\theta _0)$, evolves according to

$$\begin{aligned} \frac{d}{dt}w(\ell (t))= & {} \frac{d}{dt} \int _{-Y(\ell )}^{Y(\ell )}(g_{ss}(t))^{1/2} ds =\int _0^{Y(\ell )}(g_{ss}(t))^{-1/2}\cdot \partial _t g_{ss}(t) ds\\= & {} a_0\int _0^{Y(\ell )}\rho _{\ell }^{-1}(s) ds =a_0\frac{2\pi }{\ell }\int _0^{Y(\ell )}\cos \left( \frac{\ell s}{2\pi }\right) ds\\= & {} \left( \frac{2\pi }{\ell } \right) ^2 a_0\sin \left( \frac{\ell }{2\pi }Y(\ell )\right) =-\frac{2}{\ell }\sin \left( \frac{\ell }{2\pi }Y(\ell )\right) \frac{d\ell }{dt} \end{aligned}$$

where (A.6) is used in the last step. For $V(\ell )$ chosen so that $Y(\ell )=\frac{2\pi }{\ell }(\frac{\pi }{2}-V(\ell ))$, this formula reduces to

$$\begin{aligned} \frac{dw}{d\ell }=-\frac{2}{\ell }\cos (V(\ell )). \end{aligned}$$

On the other hand, we can directly compute $w(\ell (t))$ by working in collar coordinates of g(t) as

$$\begin{aligned} w(\ell )= & {} 2\int _0^{Y(\ell )}\rho _\ell (s)ds=2\int _0^{Y(\ell )}\frac{\ell }{2\pi \cos \left( \frac{\ell }{2\pi }s\right) } ds\\= & {} 2h\left( \frac{\ell }{2\pi }Y(\ell )\right) =2h\left( \frac{\pi }{2}-V(\ell )\right) \end{aligned}$$

where $h(x):=\log (\tan (\frac{x}{2}+\frac{\pi }{4}))$ is so that $h'(x)=\frac{1}{\cos (x)}$.

Thus

$$\begin{aligned} \frac{d w}{d\ell }=-2\frac{1}{\sin (V(\ell ))}\cdot \frac{d}{d\ell }(V(\ell )) \end{aligned}$$

meaning that V satisfies

$$\begin{aligned} \frac{1}{\ell }\cos (V(\ell ))=\frac{1}{\sin (V(\ell ))}\cdot \frac{d}{d\ell }(V(\ell )) \end{aligned}$$

or equivalently

$$\begin{aligned} \frac{2V'}{\sin (2V)}=\ell ^{-1}. \end{aligned}$$

Thus $V(\ell )=\text {atan}(c_0\cdot \ell )$ for some constant $c_0>0$ and therefore

$$\begin{aligned} Y(\ell )=\frac{2\pi }{\ell }\left( \frac{\pi }{2}-\text {atan}(c_0\ell )\right) . \end{aligned}$$

We can argue similarly to derive the formula for the diffeomorphism $f_\ell (x,\theta )=(s_\ell (x),\theta )$. Namely, we use that $\partial _tg=f_\ell ^*(a_0(ds^2-d\theta ^2))$ needs to agree with

$$\begin{aligned} \partial _tg= & {} \frac{d\ell }{dt}\cdot \frac{d}{d\ell } (f_\ell ^*(\rho _\ell ^2\cdot (ds^2+d\theta ^2)))\\= & {} -\frac{2\pi ^2}{\ell } a_0 \frac{d}{d\ell } \left[ \rho _\ell ^2(s_\ell (x))\cdot \left( \left( \frac{ \partial s_\ell }{\partial x} \right) ^2\cdot dx^2+d\theta ^2\right) \right] . \end{aligned}$$

Comparing the two expressions for $(\partial _t g)_{\theta \theta }$ immediately yields the condition that

$$\begin{aligned} -1=-\frac{2\pi ^2}{\ell }\frac{d}{d\ell }(\rho ^2_\ell \circ s_\ell ) \end{aligned}$$

and thus that $(\frac{\ell }{2\pi })^2-\rho _\ell ^2\circ s_\ell (x)$ is independent of $\ell$, or equivalently that there exists a constant c(x) with

$$\begin{aligned} \tan \left( \frac{\ell }{2\pi }s_\ell (x)\right) =\frac{c(x)}{\ell }. \end{aligned}$$

Finally one can determine c(x) so that $s_{\ell _0}(x)=x$ for the number $\ell _0>0$ for which $Y(\ell _0)=1$ and check that for the resulting map $f_\ell$ also the two expressions for $(\partial _t g)_{xx}$ agree. $\square$

We furthermore remark that the metrics $G_\ell$ converge locally smoothly away from the central geodesic $\{0\}\times S^1$ as $\ell \rightarrow 0$ with the limiting metric $G_0$ given by

$$\begin{aligned} G_0\vert _{C_\pm }=f_{\pm }^*(\rho _0(s)^2(ds^2+d\theta ^2)), \end{aligned}$$

(A.9)

for $f_\pm :C_\pm \rightarrow [0,\infty )\times S^1$ defined by

$$\begin{aligned} f_\pm (x,\theta )=\left( \lim _{\ell \rightarrow 0}Y(\ell )\mp s_\ell (x),\theta \right) = \left( \frac{2\pi }{\ell _0}\cdot \tan \left( \frac{\pi }{2}\mp \frac{\ell _0 x}{2\pi } \right) -2\pi \eta ,\theta \right) \end{aligned}$$

and $([0,\infty ),\rho _0^2(ds^2+d\theta ))$ the hyperbolic cusp described in Theorem 2.7, case II.

1.3 A.3 Properties of the diffeomorphisms $h_{b,\phi }$

Here we provide (a sketch of) the proof of the properties of the diffeomorphisms $h_{b,\phi }$ introduced in (4.2).

Proof of Lemma 4.3

We recall that for $x>0$ we can write $h_{b,\phi }(x,\theta )=(x,\lambda _1(x)f_{b^+}(\theta )+(1-\lambda _1(x))\theta +\lambda _2(x)\phi ^+)$. Thus $Y_{\phi ^+} =\lambda _2(x)\cdot \frac{\partial }{\partial \theta }$ is a Killing field on $\text {supp}(\lambda _1)\subset \{\lambda _2\equiv 1\}$ so in particular

$$\begin{aligned} L_{Y_{\text {Re}(b^+)}}G\perp L_{Y_{\phi ^+}}G \quad \text {as well as}\quad L_{Y_{\text {Im}(b^+)}}G\perp L_{Y_{\phi ^+}}G. \end{aligned}$$

To prove the other orthogonality relation we recall that a different choice of $\phi$ only results in a constant rotation on $\text {supp}(Y_{Arg (b)})=\text {supp}(Y_{{\vert b\vert } })$. It is thus enough to consider the case $\phi =0$ and we abbreviate in the following $h_b=h_{b,0}$. We also recall that all generating vectorfields have the form $Y=Y^\theta \cdot \frac{\partial }{\partial \theta }$ and we claim that for $x\ge 0$ and $\psi =Arg(b^+)$

$$\begin{aligned} Y_{Arg(b^+)}^\theta (h_b(x,\psi +\theta ))=Y_{Arg(b^+)}^\theta (h_b(x,\psi -\theta )) \end{aligned}$$

while

$$\begin{aligned} Y_{{\vert b^+\vert } }^\theta (h_b(x,\psi +\theta ))=-Y_{{\vert b^+\vert } }^\theta (h_b(x,\psi -\theta )). \end{aligned}$$

Given that the conformal factor of the collar metric is independent of $\theta$ this immediately results in the claimed orthogonality of $L_{Y_{Arg(b^+)}}G$ and $L_{Y_{\vert b^+\vert } }G$.

To prove the symmetry relation for $Y_{Arg(b^+)}$ or equivalently for $\frac{d}{d Arg(b^+)} f_{b^+}$ we first observe that for $b^+=a\cdot e^{i\psi }, a\in {\mathbb R}$, we have $M_{b^+}(e^{i(\psi +\theta )})=e^{i\psi }M_{a}(e^{i\theta })$ and thus

$$\begin{aligned} f_{b^+}(\theta )=\psi +f_{a}(\theta -\psi ). \end{aligned}$$

(A.10)

In particular

$$\begin{aligned} \frac{d}{d\text {Arg}(b^+)} f_{b^+}(\theta )=1-\partial _\theta f_a(\theta -\psi ). \end{aligned}$$

Differentiating the relation $M_a(e^{i\theta })=e^{if_a(\theta )}$ we get

$$\begin{aligned} \partial _\theta f_a(\theta )= & {} (iM_a(e^{i\theta }))^{-1}\cdot \left( \frac{d}{dz}M_a\right) (e^{i\theta })\cdot i\cdot e^{i\theta }\\= & {} \frac{1-a^2}{(1+a\cdot cos\theta )^2+a^2sin^2(\theta )}. \end{aligned}$$

Thus indeed

$$\begin{aligned} \frac{d}{d\text {Arg}(b^+)}f_{b^+}(\psi +\theta )=1-\partial _\theta f_a(\theta )=1-\partial _\theta f_a(-\theta )=\frac{d}{d\text {Arg}(b^+)}f_{b^+}(\psi -\theta ) \end{aligned}$$

which implies the claimed symmetry of $Y_{Arg(b)}$.

On the other hand, again for $a\in {\mathbb R}$

$$\begin{aligned} \frac{d}{da}f_a(\theta ) =(iM_a(e^{i\theta }))^{-1}\cdot \left( \frac{d}{da}M_a(e^{i\theta })\right) =-\frac{2\cdot \sin (\theta )}{(1+a\cos \theta )^2+a^2\sin ^2(\theta )} \end{aligned}$$

so that for $b^+=a e^{i\psi }$

$$\begin{aligned} \frac{d}{d{\vert b^+\vert } }f_{b^+}(\psi +\theta )=\frac{d}{da}f_a(\theta )=-\frac{d}{da}f_a(-\theta ) =-\frac{d}{d{\vert b^+\vert } }f_{b^+}(\psi -\theta ) \end{aligned}$$

as claimed.

It remains to prove the estimates for $L_{Y_\phi ^+}G$ and $L_{Y_{{\vert b^+\vert } }}G$. As $Y_{\phi ^+}=\lambda _2(x)\frac{\partial }{\partial \theta }$, we have

$$\begin{aligned} L_{Y_{\phi ^+}}G_{\ell }=\lambda _2'(x)\rho ^2(s_\ell (x))\cdot (dx\otimes d\theta +d\theta \otimes dx). \end{aligned}$$

The claimed estimate thus follows from the uniform upper and lower bounds on $\rho _\ell \circ s_\ell$ on fixed cylinders $[\delta ,1]\times S^1\subset C_0$, $\delta >0$, that are valid for all $\ell \in (0,L_0]$, compare Remark A.5.

To analyse $L_{Y_{\vert b^+\vert } }G$, we first remark that

$$\begin{aligned} h_{b,\phi }^*G_\ell= & {} \rho _\ell ^2(s_\ell (x))\cdot \left[ \left[ \left( \frac{\partial s_\ell }{\partial x} \right) ^2+ \left( \frac{\partial h_{b,\phi }^\theta }{\partial x} \right) ^2 \right] dx^2\right. \nonumber \\&+ \left( \frac{\partial h_{b,\phi }^\theta }{\partial x} \right) \cdot \left( \frac{\partial h_{b,\phi }^\theta }{\partial \theta } \right) \cdot (dx\otimes d\theta +d\theta \otimes dx)\\&+\left. \left( \frac{\partial h_{b,\phi }^\theta }{\partial \theta } \right) ^2d\theta ^2 \right] \end{aligned}$$

and that in view of (A.10) we only need to consider the case that $b^+=a\in {\mathbb R}$.

As we only wish to prove a lower bound on $\Vert L_{Y_{{\vert b^+\vert } }}G\Vert _{L^2}$ it is enough to consider the subcylinder $[\frac{7}{8},1]\times S^1$ on which the above expression reduces to

$$\begin{aligned} h_{a}^*G_\ell =\rho ^2(s_\ell (x))\cdot \left[ \left( \frac{\partial s_\ell }{\partial x} \right) ^2dx^2+ \left( \frac{\partial f_{a}}{\partial \theta } \right) ^2 d\theta ^2\right] \end{aligned}$$

so that

$$\begin{aligned} \frac{d}{da}(h_a^*G)=2\rho ^2(s_\ell (x))\cdot \frac{\partial f_a}{\partial \theta }\cdot \frac{d}{da}\frac{\partial f_a}{\partial \theta }d\theta ^2. \end{aligned}$$

On this part of the cylinder we furthermore have that $g^{\theta \theta }=\rho ^{-2}\circ s_\ell \cdot (\frac{\partial f_a}{\partial \theta })^{-2}$ and that $\rho$ is bounded away from zero uniformly in $\ell$. Thus

$$\begin{aligned} \Vert L_{Y_{{\vert b\vert } }}G\Vert _{L^2(C_0,G)}^2=\Vert \frac{d}{da}(h_a^*G)\Vert _{L^2(C_0,h_a^*G)} \!\ge & {} \! 4\int _{\left[ \frac{7}{8},1\right] \times S^1} \left( \frac{\partial f_{a}^\theta }{\partial \theta } \right) ^{-2}\left| {\frac{d}{da}\frac{\partial f_a}{\partial \theta }}\right| ^2 dv_g\nonumber \\ \!\ge & {} \! c \int _{S^1} \left( \frac{\partial f_{a}^\theta }{\partial \theta } \right) ^{-1}\left| {\frac{d}{da}\frac{\partial f_a}{\partial \theta }}\right| ^2 d\theta \quad \end{aligned}$$

(A.11)

for some fixed constant $c>0$. As $\partial _\theta f_a=\frac{1-a^2}{(1+a\cos \theta )^2+a^2\sin ^2\theta }$, we compute

$$\begin{aligned} \partial _a\partial _\theta f_a= & {} -{\vert a e^{i\theta }+1\vert } ^{-4}\cdot \left[ 2a(1+a\cos \theta )^2+2a^3\sin ^2\theta \right. \\&\left. +\, (1-a^2)\left[ 2\cos (\theta )(1+a\cos \theta )+2a\sin ^2\theta \right] \right] \\= & {} -{\vert ae^{i\theta }+1\vert } ^{-4}\cdot \left[ 2a \sin ^2\theta +2(1+a\cos \theta )\cdot (a+\cos \theta ) \right] . \end{aligned}$$

We set $\varepsilon =1-a$ and remark that for $\varepsilon$ small and for $\theta$ given by $\theta =\pi +\lambda \cdot \varepsilon$

$$\begin{aligned} \left[ 2a \sin ^2\theta +2(1+a\cos \theta )\cdot (a+\cos \theta ) \right] \ge 2\cdot \left[ \lambda ^2\varepsilon ^2-\varepsilon ^2+O(\varepsilon ^3)\right] \end{aligned}$$

so that this expression is bounded away from 0 by $2\varepsilon ^2$ for angles $2\varepsilon \le {\vert \theta +\pi \vert } \le 3\varepsilon$.

Combined with (A.11) we thus find that

$$\begin{aligned} \Vert L_{Y_{{\vert b\vert } }}G\Vert _{L^2(C_0,G)}^2\ge & {} c\cdot \varepsilon ^4 \int _{\pi +2\varepsilon }^{\pi +3\varepsilon } \left( \frac{1-a^2}{{\vert ae^{i\theta }+1\vert } ^2} \right) ^{-1}\cdot {\vert ae^{i\theta }+1\vert } ^{-8} d\theta \ge c\varepsilon ^{-2}\\= & {} \frac{c}{(1-a)^2} \end{aligned}$$

for $1-a$ sufficiently small. This implies the claim of Lemma 4.3. $\square$

Sketch of Proof of Lemma 4.2

The property asked for in Lemma 4.2 is essentially a consequence of us choosing the diffeomorphisms as restrictions of Möbius transforms onto $S^1$ and the fact that given any two triples $(w_1,w_2,w_3)$ and $(z_1,z_2,z_3)$ of points on $S^1$ there is a unique Möbiustransform mapping $z_i$ to $w_i$. To be more precise, using the group property of the Möbius transforms one can reduce the claim of Lemma 4.2 to proving that for any distinct $\vartheta _{1,2,3}\in [0,2\pi )$ and any $a_0\in [0,1)$ the derivative of the map $(b,\psi )\mapsto (f_{b,\psi }(\vartheta _1), f_{b,\psi }(\vartheta _2), f_{b,\psi }(\vartheta _2))$ has full rank in the point $(b,\psi )=(a_0,0)\in {\mathbb C}\times {\mathbb R}$. A short calculation then verifies this claim.$\square$