Proof. (i) Let us recall that $R(p,q)$ generates the cogeodesic flow of speed $1$ on $\{1\}\times M$, and is extended to $T^{*}L\backslash L$ by $\mathbb {R}^{+}_*$-invariance. So $R(q,p)$ is horizontal in $\{1\}\times M$, and since the horizontal distribution is also $\mathbb {R}^{+}_*$-invariant, $R(q,p)\in H_{(q,p)}$ for all $(q,p)\in T^{*}L$. Moreover, the $\mathbb {R}^{+}_*$-action is trivial in $q$ (meaning that $\pi (\tau \cdot (q,p))=q$), so

\[ \pi(R(q,p))=\pi \bigg(R\bigg(q,\frac p{\|p\|}\bigg)\bigg)=\bigg(\frac {p}{\|p\|}\bigg)^{\sharp}. \]

It follows that $\pi (R)^{\sharp }= p/{\|p\|}= \partial /{\partial r}$.

(ii) $H\subset T(\{r\}\times M)$ because parallel transport preserves the metric. It is also Lagrangian since the Levi-Civita connection has no torsion [Reference Grigor'yantsGri98].

(iii) The fact that $\ker \alpha \cap H_{(q,p)} = \{V\in H_{(q,p)}\, |\, p(\pi (V))=0\}$ and $\ker \alpha \cap F_{(q,p)} = F_{(q,p)} \cap$ $T_{(q,p)}M=\{V\in F_{(q,p)}\,|\, g_q(V,p) =0\}$ are immediate by definition. Now let $V \in \ker \alpha \cap H_{(q,p)}$. Since $\sharp$ is an isometry by definition of $g_q$ on $T_q^{*}L$ and $p(\pi (V))=0$, we have the chain of equalities

\[ g_q(\pi(V)^{\sharp},p) = \Vert p\Vert_g \,g_q(\pi(V)^{\sharp},\pi(R)^{\sharp}) = \Vert p\Vert_g \,g_q(\pi(V),\pi(R)) = p(\pi(V)) = 0, \]

which means that $\pi (V)^{\sharp }\in \ker \alpha \cap F_{(q,p)}$. By dimension considerations we see that $\sharp \circ \pi$ is indeed an isomorphism.

(iv) This is obvious by dimension considerations, since by (3), we have $\dim \ker \alpha \cap F_{(q,p)}=n-1=\dim \ker \alpha \cap H_{(q,p)}$.

Proof. By definition, $J_gR(q,p)=-r (\partial /{\partial r})$ when $\Vert p\Vert _g\gg 1$ and $J_g$ preserves $\ker \alpha$. Since $H_{(q,p)}$ is Lagrangian, showing that $J_{g|\ker \alpha }(q,p)$ is compatible with $d\lambda (q,p)$ amounts to proving that $d\lambda (V,J_g V) > 0$ for $V \in \ker \alpha \cap H_{(q,p)}$. Moreover, since $d\lambda$ is invariant by the lifts of diffeomorphisms of $L$ to $T^{*}L$, a straightforward computation shows that we can assume $g_{q}=\sum dq_i^{2}$. Then, for $V\in \ker \alpha \cap H_{(q,p)}$,

\begin{align*} d\lambda(V,J_gV) &= \sum dp_i\wedge dq_i (V,J_gV)\\ &={-}\sum dp_i(J_gV)dq_i(\pi(V))& \text{(because $J_gV \in F_{(q,p)}$, so $dq_i(J_gV)=0$)}\\ &=\sum dp_i(g(\pi(V),\cdot))dq_i(\pi(V))\\ &=\sum dq_i(\pi(V))^{2}>0 & \text{(because $g_q=\sum dq_i^{2}$).} \end{align*}

Finally, to show that $J_g$ is dilation invariant at infinity, it is enough to work on $\ker \alpha \cap H$ (because $H=(\ker \alpha \cap H)\oplus \langle R \rangle$). If $V\in \ker \alpha \cap H$, then $J_gV\in F\cap TM$, so the dilations on $(0,\infty )\times M$ act by the identity on both $V$ and $J_gV$.

Proof. We consider canonical coordinates $(q,p) \in T^{*}L$ determined by a local choice of coordinates $q = (q_1, \ldots , q_n)$ on $L$. One can show that the horizontal distribution $H_{(q,p)}$ of the connection $\nabla ^{*}$ is spanned by the vectors

(5)\begin{equation} \frac{\partial}{\partial q_i}^{H} = \frac{\partial}{\partial q_i} + \sum_{k,l} \Gamma^{k}_{il}(q) p_k \frac{\partial}{\partial p_l}, \quad i = 1, \ldots, n, \end{equation}

where $\Gamma ^{k}_{il}$ are the Christoffel symbols of the metric $g$. Since $d\sigma ({\partial }/{\partial p_l}) = -({\partial }/{\partial p_l})$, we see that $\sigma$ preserves the lifts $ ({\partial }/{\partial q_i})^{H}$ and hence the horizontal distribution. A brief calculation then reveals that $\sigma$ is $J_g$-antiholomorphic.

In $T^{*}L \backslash L$ one easily sees that $J_g$ preserves the contact distribution $\ker \alpha$ and is compatible with $d\alpha$, hence the sets $\{\|p\|_g\leq r\}$ are $J_g$-pseudoconvex for $r > 0$.

Proof. Point (i) is a general and classical fact that relies on the ellipticity of the $\bar \partial _{J_g}$-operator on the space of curves with boundary on totally real submanifolds (see also [Reference Ivashkovich and SukhovIS10]). Points (ii) and (iii) are specific to our situation. They rely on the special form of $J_g$ that makes the involution $\sigma (q,p):=(q,-p)$ anti-holomorphic; see Proposition 3.6.

Let $u:(D\backslash \{0\},\partial D)\to (T^{*}L,L)$ be a $J_g$-holomorphic map asymptotic to $\tilde \gamma$ at $0$. The map $v:\mathbb {C}\backslash \mathrm {int}\,D \to T^{*}L$ defined by $v(z):=\sigma \circ u( 1/{\bar z})$ is $J_g$-holomorphic and coincides with $u$ on $\partial D$ (because $u(\partial D)\subset L=\text {Fix}\, (\sigma )$). Thus, the map

\[\begin{array}{rcccl} {w} & : & {\mathbb{C}\backslash\{0\}} & \longrightarrow & {T^{*}L} \\ & & {z} & \longmapsto & {\left\{\begin{array}{@{}l} u(z)\ \text{if}\ z\in D\backslash\{0\} \\ v(z)\ \text{if}\ z \in \mathbb{C}\backslash D \end{array}\right.}\end{array}\]

is holomorphic on $\mathbb {C}\backslash \partial D$, continuous on $\partial D$, and since $u$ and $v$ are smooth up to $\partial D$, $w$ is $J_g$-holomorphic on $\partial D$ as well. Thus, $w: \mathbb {C}\backslash \{0\}\to T^{*}L$ is $J_g$-holomorphic. Since $w$ is asymptotic to $\tilde \gamma$ at $0$, it has no critical points near $0$ and [Reference McDuff and SalamonMS12, Lemma 2.4.1] guarantees that $w$ has only finitely many critical points in $D \backslash \{0\}$, where it coincides with $u$. This proves (ii).

We prove (iii) by contradiction. First assume there exists a sequence $z_n\in \mathrm {int}\, D\backslash \{0\}$ that converges to a point $z \in \mathrm {int}\, D\backslash \{0\}$ which is not a critical point of $u$, and such that $u(z_n) \in L$ for all $n$ (hence $u(z) \in L$). We define

\[ \Omega:=\{\xi\in \mathrm{int}\,D\backslash\{0\} \,|\, \exists\,\xi'\in \mathbb{C} \backslash D : u(\xi)=v(\xi')\}. \]

In particular, $\Omega \supset u^{-1}(L)\cap \mathrm {int}\,D\backslash \{0\} \ni z_n$. By [Reference McDuff and SalamonMS12, Lemma 2.4.3], $\Omega$ contains a neighborhood of each of its accumulation points in $\mathrm {int}\,D\backslash \{0\}$ that are not critical points of $u$. It therefore contains a neighborhood of $z$ by assumption. Notice that the asymptotic behavior of $u$ at $0$ implies that $\Omega$ does not meet some neighborhood of the puncture $0$. Now let $c:[0,1]\to D$ be a continuous curve between $c(0)=z$ and $c(1)=0$ that avoids the (finitely many) critical points of $u$. Let ${t_*} :=\sup \{t\,|\, c([0,t])\subset \Omega \}$. Since $z\in {\mathrm {int}\,\Omega }$ and $c(\textrm {Op}\,(1))\cap \Omega = \emptyset$, we have $0<{t_*}<1$. Now let $t_n < {t_*}$ be any sequence that converges to $t_*$. By assumption $c(t_n)\in \Omega$, so there exists $\xi _n'\in {\mathbb {C} \backslash D}$ such that $v(\xi _n')=u(c(t_n))$. Obviously $\xi _n'$ cannot accumulate at $\infty$ because $v(\xi _n') = u(c(t_n))\to u(c(t_*))\in T^{*}L$. Nor can it accumulate at a point $\xi '\in {\mathbb {C} \backslash D}$ because then, by [Reference McDuff and SalamonMS12, Lemma 2.4.3], $c(t_*)$ would be an accumulation point of $\Omega$, so would be contained in ${\mathrm {int}\,\Omega }$. Thus $c(t)$ would belong to $\Omega$ for $t<{t_*}+\varepsilon$, $\varepsilon >0$, which contradicts our definition of ${t_*}$. We therefore conclude that, after extraction, $\xi _n'\to \xi '\in \partial {D}$, so $u(c(t_*))=\lim u(c(t_n)) =\lim v(\xi _n')=v(\xi ')\in v(\partial D)\subset L$. This shows that any curve that joins $z$ to $0$ in ${D} \backslash \textrm {Crit} (u)$ intersects $u^{-1}(L)$. Since $\textrm {Crit}(u)$ is finite, we conclude that the connected component $U$ of $z$ in ${D}\backslash u^{-1}(L)$ does not contain $0$. Thus $u_{|U}:(U,\partial U)\to (T^{*}L,L)$ is $J_g$-holomorphic. If $\partial U$ is smooth, we can conclude by Stokes’ theorem. In general, we can invoke the pseudoconvexity of the hypersurfaces $\{\|p\|_g=r\}$ for $r > 0$ (see Proposition 3.6) to conclude that $u_{|U}$ takes values in $L$. Since $U$ is open and $L$ is totally real, we conclude that $u_{|U}$ is constant, which implies that $u$ is constant. But this contradicts the fact that $u$ is asymptotic to $\tilde \gamma$ at $0$.

This contradiction shows that $u^{-1}(L)$ can only accumulate at $\textrm {Crit}(u)\cup \partial {D}$. On the other hand, since $L$ is totally real, any non-critical point of $u$ in $\partial {D}$ is a point where the intersection of $u$ with $L$ is clean. These points cannot therefore be accumulated by points of $u^{-1}(L)\cap {\mathrm {int}\, D\backslash \{0\}}$. This completes the proof of (iii).

Proof. Since both sides of the equality (7) are additive with respect to a partition of $\textrm {Op}\,([-1,1],D^{+})$, we can freely assume that $u$ takes values in a small chart of $T^{*}L$ localized on $L$, which amounts to assuming that $L=\mathbb {R}^{n}$. Let $\gamma :=u_{|[-1,1]}$ and $\gamma _\varepsilon :=u_{|D^{+}\cap \{\textrm {Im}\, z=\varepsilon \}}$. Taking into account that for the coordinates $x + iy \in D^{+}$,

\[ \partial_y u(t,0) = J_g \partial_x u(t,0) = J_g \dot \gamma (t) ={-}g_{\gamma(t)}(\dot \gamma(t),\cdot)\in T^{*}_{\gamma(t)}\mathbb{R}^{n}, \]

we have

\begin{align*} & \dot \gamma_\varepsilon(t)=\dot \gamma (t)+O(\varepsilon),\\ & \gamma_\varepsilon(t)=(\gamma(t),-\varepsilon g_{\gamma(t)}(\dot \gamma(t),\cdot))+O(\varepsilon^{2}). \end{align*}

Thus $p(\gamma _\varepsilon (t)):= p_{\gamma _\varepsilon (t)} = -\varepsilon g_{\gamma (t)}(\dot \gamma (t),\cdot )+O(\varepsilon ^{2})$ and $\|p(\gamma _\varepsilon (t))\|_g=\varepsilon \|\dot \gamma (t)\|_g+O(\varepsilon ^{2})$. If $\dot \gamma (t)\neq 0$, this implies

\[ \alpha(\gamma_\varepsilon(t)) := \alpha_{\gamma_\varepsilon(t)} = \frac p{\|p\|_g}(\gamma_\varepsilon(t)) ={-}g_{\gamma(t)}\bigg(\frac {\dot \gamma(t)}{\|\dot \gamma(t)\|_g},\,\cdot\, \bigg)+O(\varepsilon). \]

Notice now that, using the reflection argument as in Lemma 3.7, one immediately sees that the critical set of the restriction of $u$ to any compact set of $(-1,1)$ is a finite set, so the Lebesgue measure $\mathcal {L}eb(\textrm {Crit}(\gamma ))=0$. Thus, one can cover $\textrm {Crit}(\gamma )$ by an open set $J_\delta \subset (-1,1)$ of total length $\delta$. We can also assume that $J_\delta$ contains some neighborhood of $\{-1,1\}$, so that for some $\varepsilon (\delta )>0$, $\gamma _\varepsilon (t)$ is well defined for $t\in (-1,1)\backslash J_\delta$ for all $\varepsilon < \varepsilon (\delta )$. Then, for $\varepsilon _n<\varepsilon (\delta )$, we have

\begin{align*} \displaystyle\int_{({-}1,1)\backslash J_\delta+i\varepsilon_n} u^{*}\alpha \displaystyle &= \int_{({-}1,1)\backslash J_\delta} {\alpha_{\gamma_{\varepsilon_n}(t)} (\dot \gamma_{\varepsilon_n}(t))} \,dt \\ &={-}\int_{({-}1,1)\backslash J_\delta} {g_{\gamma(t)}}\bigg(\frac{\dot\gamma(t)}{{\|\dot\gamma(t)\|_g}}, \dot\gamma(t)+O(\varepsilon_n) \bigg) + O(\varepsilon_n)\,dt\\ &={-}\int_{(1,-1)\backslash J_\delta}\|\dot\gamma(t)\|_g + O(\varepsilon_n)\,dt. \end{align*}

Notice also that our $O(\varepsilon _n)$ depends on $u$ and $\delta$, but is uniform in $t$, so

\[ \lim_{n\to \infty}\int_{({-}1,1)\backslash J_\delta+i\varepsilon_n} u^{*}\alpha ={-}\ell_g(\gamma(({-}1,1) \backslash J_\delta)). \]

Now letting $\delta$ go to $0$, the right-hand side of this equality obviously converges to $-\ell _g(\gamma )$ because $\gamma$ is smooth. On the other hand,

\[ \bigg|\int_{J_\delta+i\varepsilon_n}u^{*}\alpha\bigg| \leq \int_{J_\delta}\bigg| \frac p{{\|p\|_g}}{\bigg|_{\gamma_{\varepsilon_n}(t)}}(\dot\gamma_{\varepsilon_n}(t))\bigg|dt \leq \int_{J_\delta} {\| \dot \gamma_{\varepsilon_n}(t)\|_g}\,dt \leq C\delta, \]

where $C$ is an upper bound for ${\|du\|_g}$. This shows equality (7).

Proof. Let $u \in \widehat {\mathcal {M}}(J_g,\beta )$. By Lemma 3.7 we have $u^{-1}(L)=\mathcal {I}\cup \partial {D}$, where $\mathcal {I} \subset {\mathrm {int}\,D\backslash \{0\}}$ can only accumulate at critical points of $u$. On $\Omega := {D\backslash (u^{-1}(L) \cup \{0\})}$ we can write $u=(a,\tilde u)\in {(0, \infty )} \times M$. Calling $P:TM\to \ker \alpha$ the projection along the Reeb vector field and putting $V=\langle V,R\rangle R+P(V)$ for $V\in TM$, the equations for a $J_g$-holomorphic map can be written

\begin{gather*} d a = \chi(a)\langle d\tilde u\circ j, R\rangle, \\ J_g(u) P (d\tilde u) = P(d\tilde u\circ j). \end{gather*}

As a result, if ${s + it}$ are local holomorphic coordinates near some point in $\Omega$,

\begin{align*} u^{*}d\alpha\bigg(\frac{\partial}{\partial s},\frac{\partial}{\partial t}\bigg) &= \tilde u^{*}d\alpha\bigg(\frac{\partial}{\partial s},\frac{\partial}{\partial t}\bigg)= d\alpha\bigg(\frac{\partial \tilde u}{\partial s},\frac{\partial \tilde u}{\partial t}\bigg)\\ & = d\alpha\bigg(P\frac{\partial \tilde u}{\partial s},P \frac{\partial \tilde u}{\partial t}\bigg) = d\alpha\bigg(P\frac{\partial \tilde u}{\partial s}, J_g P\frac{\partial \tilde u}{\partial s}\bigg) \geqslant 0, \end{align*}

and equality holds if and only if $P ({\partial \tilde u}/{\partial s}) =P ({\partial \tilde u}/{ \partial t})=0$.

Now fix $\varepsilon \ll 1$ such that $\partial D(0,1-\varepsilon )$ avoids $\mathcal {I}$. Since $\mathcal {I}$ can only accumulate at critical points of $u$ and $\textrm {Crit}(u)$ is finite, it is possible to find a finite number of disjoint disks $D(\zeta _j, \varepsilon _j) \Subset D(0,1-\varepsilon )$ such that $\mathcal {I} \cap D(0,1-\varepsilon ) \subset \bigcup _j D(\zeta _j, \varepsilon _j)$. Here the $\varepsilon _j$ are chosen small enough and the number of disjoint disks $k$ depends on these choices. Let also $\gamma _\varepsilon ^{j} := \partial D(\zeta _{j},\varepsilon _{j})$, $\gamma _\varepsilon := \partial D(0,\varepsilon )$, $\gamma _{1-\varepsilon }:=\partial D(0,1-\varepsilon )$, all those circles being oriented as the boundary of $\partial \Omega _\varepsilon := \partial (D(0, 1-\varepsilon ) \backslash (D(0,\varepsilon ) \cup \bigcup _j D(\zeta _j, \varepsilon _j)))$. Then

\[ {0 \leq \int_{\Omega_\varepsilon}u^{*}d\alpha}= \int_{\partial \Omega_\varepsilon} u^{*}\alpha = \sum_{j=1}^{k} \int_{\gamma_\varepsilon^{j}} u^{*}\alpha+\int_{\gamma_\varepsilon} u^{*}\alpha +\int_{\gamma_{1-\varepsilon}}u^{*}\alpha. \]

Since $u$ is asymptotic to ${\tilde \gamma }$ at $0$, a straightforward computation shows that

\[ \lim_{\varepsilon\to 0}\int_{\gamma_\varepsilon}u^{*}\alpha=\ell_g(\gamma). \]

On the other hand, since $\alpha = p/{{\| p \|_g}}$ is a bounded $1$-form on $T^{*}L\backslash L$, its integral over the small loops $u\circ \gamma _\varepsilon ^{j}$ tends to $0$ when $\varepsilon _j\to 0$ (recall that $u$ is smooth near the $\zeta _j$). Since $u$ is smooth on int $D\backslash \{0\}$ and $\mathcal {I}$ does not approach $0$, we can assume that

\[ \bigg|\sum_{j = 1}^{k} \int_{\gamma_\varepsilon^{j}} { u^{*}\alpha }\bigg| < \varepsilon \]

by decreasing $\varepsilon _j$ and increasing $k$ if necessary. Finally, taking orientation into account, Lemma 3.8 shows that

\[ \lim_{\varepsilon\to 0} \int_{\gamma_{1-\varepsilon}}u^{*}\alpha={-}\ell_g(u(\partial {D})). \]

Putting all these estimates together, we get

\[ 0\leq \int_\Omega u^{*}d\alpha=\ell_g(\gamma) - {\ell_g}(u(\partial {D})). \]

However, the reverse inequality holds as well, because the projection of $u$ to $L$ provides a singular chain between $\gamma$ and $u(\partial {D})$, and $\gamma$ is the geodesic of minimal length in its homology class. We therefore conclude that $u^{*}d\alpha =\tilde u^{*}d\alpha =0$ on $\Omega = {D}\backslash (\mathcal {I} \cup \{0\})$, so $\pi \circ d\tilde u=0$ on $\Omega$, which means that $\textrm {Im}\, du(z)\in \langle R, \partial/ {\partial r}\rangle$ for all $z\in \Omega$. Since $u$ is asymptotic to ${\tilde \gamma }$ at $0$, we see that $\textrm {Im}\, u=\textrm {Im}\, u_{\gamma ,g}$. But then $u=u_{\gamma ,g}$ modulo source reparameterization.

Proof. Until now, we have not made a distinction between the metric $g$ on the manifold and the induced metric on the cotangent bundle. In this proof alone, we need to make a clear distinction, so the metric in the cotangent bundle will be denoted $g^{\sharp }$. By definition of the Fermi coordinates the vector fields $( \partial /{\partial x_i})$ are parallel along the geodesic $\gamma$, so the vectors $( \partial /{\partial x_i})$ belong to the horizontal space at every point of the lift of $\gamma$, which are precisely the points $(x_n,y_n,0,0)$. Since $\alpha = \textrm {sign}(y_n)dx_n$ at these points, we thus see that $ \partial /{\partial x_i} \in \ker \alpha \cap H$ for $i\neq n$, so

\[ J_g\frac\partial{\partial x_i}(x_n,y_n,0,0)={-}\frac\partial{\partial y_i} \quad \forall\ i\neq n. \]

Since $R$ flows along the cogeodesic flow in such a way that its projection on $L$ has speed $1$, it is also clear that $R(x_n,y_n,0,0)= \partial /{ \partial x_n}$, so

\[ J_g\frac\partial{\partial x_n}(x_n,y_n,0,0)={-}\chi(y_n)\frac\partial{\partial y_n}. \]

This proves the first assertion of the lemma and shows as well that

\[ \frac\partial{\partial x_n}\bigg(J_g\frac\partial{\partial x_n}\bigg)(x_n,y_n,0,0)=0,\quad \frac\partial{\partial y_n}\bigg(J_g\frac\partial{\partial x_n}\bigg)(x_n,y_n,0,0) ={-}\chi'(y_n)\frac\partial{\partial y_n}. \]

In order to compute the other derivatives, we need to compute $J_g (\partial /{\partial x_n}) (x_n,y_n,x',y')$ at first order, which requires us to decompose $ \partial /{\partial x_n}$ along $\langle R, \partial /{\partial r}\rangle \oplus \ker \alpha \cap H \oplus \ker \alpha \cap F$.

Preliminary step: computation of $R$. $R$ is colinear to the Hamiltonian vector field associated to the function $H(x,y) := \|y\|^{2}_{g^{\sharp }_x} = \sum g_{ij}^{\sharp }(x)y_iy_j$, because both generate the cogeodesic flow on the level sets of $H$, which coincide with the hypersurfaces $\{r\}\times M$. This shows that for $r > 0$, $R$ is colinear to

\[ \sum_l\Bigg(\bigg(\sum_{i,j}\frac{\partial g_{ij}^{\sharp}}{\partial x_l}y_iy_j\bigg) \frac\partial{ \partial y_l}-2\bigg(\sum_jg_{lj}^{\sharp} y_j\bigg)\frac{\partial}{\partial x_l}\Bigg). \]

We specialize to two situations. When $x'=0$, we have $ {\partial g_{ij}^{\sharp }}/{\partial x_l}=0$ and $g_{lj}^{\sharp } =\delta _{lj}$, so for $y_n > 0$,

\[ R(x_n,y_n,0,y')\propto R':=\frac\partial{\partial x_n}+\sum_{{l\neq n}} \frac{y_l}{y_n}\frac\partial{\partial x_l}. \]

Taking into account again that $g_{ij}(x_n,0)=\delta _{ij}$ and that $\pi (R)^{\sharp }= \partial /{\partial r}$, we thus get

\[ \frac{\partial}{\partial r}(x_n,y_n,0,y')\propto \pi(R')^{\sharp}=g(\pi(R'),\cdot)= \frac\partial{ \partial y_n}+\sum_{l\neq n}\frac{y_l}{y_n}\frac\partial{\partial y_l}. \]

When $y'=0$, we get

\[ R(x_n,y_n,x',0)\propto \sum_l \frac{\partial g_{nn}^{\sharp} }{\partial x_l}y_n^{2} \frac\partial{ \partial y_l} - {2}\sum_l g_{ln}^{\sharp} y_n\frac\partial{\partial x_l}. \]

Taking into account that $g_{ln}^{\sharp } (x_n,x')=\delta _{ln}+O(\|x'\|^{2})$, we therefore get

\[ R(x_n,y_n,x',0)\propto R'=\frac\partial{\partial x_n}-{\frac{y_n}2}\sum_l \frac{\partial g_{nn}^{\sharp} }{ \partial x_l}\frac\partial{\partial y_l}+O(\|x'\|^{2}) \]

and

\[ \frac{\partial}{\partial r}(x_n,y_n,x',0)\propto \pi(R')^{\sharp}=g \bigg(\frac{\partial}{ \partial x_n},\cdot\bigg)+O(\|x'\|^{2})=\frac{\partial}{\partial y_n}+O(\|x'\|^{2}). \]

Computation of $ ({\partial }/{\partial y_i})(J_g ({\partial }/{\partial x_n}))$ for $i < n$. Using the preliminary step, we see that

\[ J_g\frac{\partial}{\partial x_n}(x_n,y_n,0,y')=J_g \bigg(R'-\sum_{l\neq n} \frac{y_l}{y_n}\frac{\partial}{ \partial x_l}\bigg). \]

Taking into account that $y_l\in O(\|y'\|)$ and $J_g ({\partial }/{\partial x_l}) = - ({\partial }/{ \partial y_l})$ for $l\neq n$ (using (5) and the fact that $x' = 0$), we thus get

\[ J_g\frac{\partial}{\partial x_n}(x_n,y_n,0,y') ={-}\chi(\|y\|_{g^{\sharp}})\pi(R')^{\sharp}+ \sum_{l\neq n}\frac{y_l}{ y_n}\frac{\partial}{\partial y_l}. \]

Since $y_n>0$, $\chi (\|y\|_{g^{\sharp }})=\chi (y_n)+O(\|y'\|^{2})$, so

\[ J_g\frac{\partial}{\partial x_n}(x_n,y_n,0,y')={-}\chi(y_n) \frac{\partial}{\partial y_n}+\sum_{l \neq n}\frac{y_l}{y_n}(1-\chi(y_n)) \frac{\partial}{\partial y_l}+O(\Vert y'\Vert^{2}). \]

Differentiating with respect to $y_l$, we get the announced formula for $ ({\partial }/{\partial y_l})(J_g ({\partial }/{\partial x_n}))$.

Computation of $ ({\partial }/{\partial x_i})(J_g ({\partial }/{\partial x_n}))$ for $i < n$. Using the preliminary step, we can write

\[ \frac{\partial}{\partial x_n}(x_n,y_n,x',0)=R' + {\frac{y_n}2}\sum_l \frac{\partial g_{nn}^{\sharp} }{ \partial x_l}\frac{\partial}{\partial y_l}+O(\|x'\|^{2}). \]

Noticing that $ {\partial g_{nn}^{\sharp } }/{\partial x_l} \in O(\|x'\|)$ for all $l$, $J_g ({\partial }/{\partial y_l})= ({\partial }/{\partial x_l})+O(\|x'\|)$ for all $l\neq n$, and $J_g ({\partial }/{\partial y_n})= (1/{\chi (y_n))} ({\partial }/{\partial x_n})+O(\|x'\|)$, we conclude that

\begin{align*} J_g\frac{\partial}{\partial x_n}(x_n,y_n,x',0)&=\displaystyle-\chi(y_n)\pi(R')^{\sharp}+ \frac{y_n}{{2} \chi(y_n)}\frac{\partial g_{nn}^{\sharp} }{\partial x_n}\frac{\partial}{\partial x_n}+ \frac{y_n}{2}\sum_l \frac{\partial g_{nn}^{\sharp} }{\partial x_l}\frac{\partial}{\partial x_l}+O(\|x'\|^{2}), \\ &\displaystyle={-}\chi(y_n)\frac{\partial}{\partial y_n}+\frac{y_n}{{2}\chi(y_n)} \frac{\partial g_{nn}^{\sharp} }{ \partial x_n}\frac{\partial}{\partial x_n}+ \frac{y_n}{2}\sum_{l\neq n} \frac{\partial g_{nn}^{\sharp} }{ \partial x_l} \frac{\partial}{\partial x_l}+O(\|x'\|^{2}). \end{align*}

Since $\sum _{j = 1}^{n} g_{nj}^{\sharp } (x_n, x') g_{jn} (x_n, x') = 1$ and each factor for $j\neq n$ is $O(\|x'\|^{2})$, we see that $g_{nn}^{\sharp } = 1/{g_{nn}} + O(\Vert x'\Vert ^{4})$. Using $g_{nn}(x_n,x')=1+O(\|x'\|^{2})$, we obtain $ {\partial g_{nn}^{\sharp }}/{\partial x_l} = - (1/{g_{nn}^{2}}) ({\partial g_{nn}}/{\partial x_l}) + O(\|x'\|^{3}) = - ({\partial g_{nn}}/{\partial x_l})+O(\|x'\|^{2})$. Differentiating with respect to $x_j$, we get

\[ \frac\partial{\partial x_j}\bigg(J_g\frac\partial{\partial x_n}\bigg)(x_n,y_n,0,0) ={-}\frac{y_n}{2\chi(y_n)}\frac{\partial^{2} g_{nn}}{\partial x_n\partial x_j} \frac\partial{\partial x_n} -\frac{y_n}2\sum_{l\neq n}\frac{\partial^{2} g_{nn}}{\partial x_j\partial x_l} \frac\partial{\partial x_l}. \]

Notice, finally, that $ ({\partial g_{nn}}/{\partial x_j})(x_n,0)\equiv 0$, so $ {\partial ^{2} g_{nn}}/{\partial x_n\partial x_j}=0$. We thus get

\[ \frac\partial{\partial x_j}\bigg(J_g\frac\partial{\partial x_n}\bigg)(x_n,y_n,0,0) ={-}\frac{y_n}2\sum_{l\neq n}\frac{\partial^{2} g_{nn}}{\partial x_j \partial x_l} \frac\partial{\partial x_l}, \]

which is the third point of the lemma.

Proof. Notice that the coefficients of our partial differential equations depend only on the second derivatives of our metric, so they coincide for two metrics which are tangent of order $2$. We can therefore easily compute these coefficients under the assumptions of Proposition 3.14 and we find

\[ \frac{\partial^{2} g_{nn}}{\partial x_i\partial x_j}(t)=\left\{\begin{array}{@{}ll} {2}k & \text{ if } j=i\neq n,\\ 0 & \text{ otherwise.} \end{array}\right. \]

The solutions $z=(a,b)$ of (8) thus satisfy

(9)\begin{equation} \begin{aligned} \frac{\partial a_l}{\partial s} + \frac{\partial b_l}{\partial t} - k f(s)a_l & = 0 \quad \forall\ l\neq n, \\ \frac{\partial b_l}{\partial s}-\frac{\partial a_l}{\partial t} + \frac {1-\chi\circ f(s)}{f(s)} b_l & = 0 \quad \forall\ l \neq n, \\ \frac{\partial b_n}{\partial s}-\chi\circ f(s) \frac{\partial a_n}{\partial t} -\chi'\circ f(s)b_n & = 0, \\ \frac{\partial a_n}{\partial s}+\frac 1{\chi \circ f(s)}\frac{\partial b_n}{\partial t} & = 0. \end{aligned} \end{equation}

We need several changes of variables. First we define $\tilde a_n := a_n$, $\tilde b_n:= {b_n}/{\chi \circ f}$. Let $\rho (s):=\exp ({\int _0^{s}} ({1 - \chi \circ f(s'))}/{f(s')}ds'): [0, \infty ) \to \mathbb {R}$ be a solution of the differential equation

\[ \rho'(s)=\rho(s)\frac {1- \chi\circ f(s)}{f(s)}, \quad \rho(0) = 1, \]

and put $\tilde a_l:=\rho (s)a_l(s,t)$ and $\tilde b_l:=\rho (s)b_l(s,t)$ for $l = 1, \ldots , n-1$.

Since $f(s)$ tends to $\infty$ when $s$ goes to $\infty$ and $\chi (r)=r$ at infinity, we see that $\rho$ and $ 1/{\chi \circ f}$ tend to $0$ at $\infty$. As a result, the $\tilde a_l,\tilde b_l,\tilde a_n$ satisfy the same boundary conditions at $s=0,\infty$ as $a_l,b_l,a_n$ for $l\neq n$, but now $\tilde b_n(0,t)\equiv 0$ and $\tilde b_n( \infty ,t)=0$. Putting also $g(s) := - ({\rho '}/{\rho })(s) - kf(s)$, a straightforward computation shows that these functions now satisfy

(10)\begin{equation} \begin{array}{l} \displaystyle \dfrac{\partial \tilde a_l}{\partial s}+\dfrac{\partial \tilde b_l}{\partial t}+g(s) \tilde a_l=0,\\ \displaystyle \dfrac{\partial \tilde b_l}{\partial s}-\dfrac{\partial \tilde a_l}{\partial t}=0, \end{array}\quad \forall\ l\neq n,\quad \begin{array}{l} \displaystyle \dfrac{\partial \tilde a_n}{\partial s}+\dfrac{\partial \tilde b_n}{\partial t}=0, \\ \displaystyle \dfrac{\partial \tilde b_n}{\partial s}-\dfrac{\partial \tilde a_n}{\partial t}=0.\\ \end{array} \end{equation}

Let us first focus on our system for $l\neq n$. The functions $\tilde b_l$ satisfy

\[ \Delta \tilde b_l+g(s)\frac{\partial \tilde b_l}{\partial s} = 0, \]

together with the pseudoperiodicity and boundary conditions

\[ \tilde b_l(0)=0,\quad \tilde b_l(s)\underset{s \to \infty}\longrightarrow 0,\quad \tilde b'(s,t+\ell)=O\tilde b'(s,t). \]

Since $\Delta h^{2}=2h\Delta h+2\Vert \nabla h\Vert ^{2}$, we get $\Delta \tilde b_l^{2}+g(s) ({\partial \tilde b_l^{2}}/{\partial s})=2\Vert \nabla \tilde b_l\Vert ^{2}\geq 0$. Summing these $l$ equations, we get

(11)\begin{equation} \Delta \Vert \tilde b'\Vert^{2}+g(s)\frac{\partial \Vert \tilde b'\Vert^{2}}{\partial s} \geq 0. \end{equation}

Now define $h_\varepsilon (s,t):=\Vert \tilde b' \Vert ^{2} + \varepsilon e^{- \alpha s- (\beta /2) t^{2}}$. An immediate computation gives

\[ \Delta h_\varepsilon+g(s)\frac{\partial h_\varepsilon}{\partial s} \geq \varepsilon(\alpha^{2} -\beta+\beta^{2}t^{2} - g(s)\alpha)e^{-\alpha s- (\beta/ 2)t^{2}}. \]

We recall that $g(s) = - ({\rho '}/{\rho })(s) - kf(s) = -(({1 - \chi \circ f})/{f})(s) - kf(s)$, that $f(s)\underset {s\to \infty }\longrightarrow +\infty$, and that $\chi (r)=r$ when $r\gg 1$. We therefore see that $-g$ is bounded from below on $[0,\infty )$, so the right-hand side of this last inequality is positive if $\alpha$ is large enough, which we assume henceforth. The function $h_\varepsilon$ is positive by definition, and does not have any maximum on $[0,\infty )\times \mathbb {R}$ (because its Laplacian is positive at each critical point). Let $(s_n,t_n)$ be a maximizing sequence for $h_\varepsilon$, that is, $h_\varepsilon (s_n,t_n)\to \sup h_\varepsilon$. Obviously, $s_n$ remains bounded because $h_\varepsilon$ converges uniformly to $0$ as $s$ goes to $\infty$. We also claim that thanks to the quasi-peridodicity ($\tilde b'(s,t+\ell ) = O\tilde b'(s,t$)), we can assume that $t_n\in [0,\ell ]$. Indeed, if Frac$( {t_n}/{\ell })$ denotes the fractional part of $ {t_n}/{\ell }$ (whose sign is the same as that of $t_n$), we have

\begin{align*} h_\varepsilon(s_n,t_n) = \Vert \tilde b'(s_n,t_n)\Vert^{2}+e^{-\alpha s_n^{2}- (\beta/ 2)t_n^{2}} & = \bigg\Vert \tilde b'\bigg(s_n,\ell\text{Frac}\bigg(\frac{t_n}\ell\bigg)\bigg)\bigg\Vert^{2} + e^{-\alpha s_n^{2}- (\beta / 2)t_n^{2}} \\ & \leq h_\varepsilon\bigg(s_n,\ell\text{Frac}\bigg(\frac{t_n}\ell\bigg)\bigg). \end{align*}

Since $(s_n,t_n)$ cannot accumulate at an interior point, we see that $s_n\to 0$, so $\sup h_\varepsilon =\varepsilon$. This implies that $h_\varepsilon \leq \varepsilon$, so $\Vert \tilde b'\Vert ^{2}\leq \varepsilon$. Since this holds for all $\varepsilon >0$, we conclude that $\tilde b'\equiv 0$, so $b'=0$. Thus, using the second equation of the system (9), we see that $a'=a'(s)$, and from the first equation, that

\[ \frac{d a'(s)}{ds} - k f(s)a'(s)=0. \]

This last equation gives

\[ a'(s) = a'(0)e^{k\int_0^{s}f(u)\,du}. \]

Letting $s\to \infty$, and taking into account that $f(s) > 0$ and $a'(\infty )=0$, we thus get $a'(0)=0$, and therefore $a'=0$.

The equations for $(\tilde a_n,\tilde b_n)$ are the standard Cauchy–Riemann equations, so $\tilde h(z):=\tilde a_n(z)+i\tilde b_n(z)$ is a holomorphic function. By the condition $\tilde b_n(0,t)=0$, Schwarz reflection provides an extension $h:\mathbb {C}\to \mathbb {C}$, and in view of the boundary conditions and periodicity, this extension is bounded. By Liouville's theorem, this function $h$ is constant, so $\tilde b_n\equiv 0$ and $\tilde a_n$ is a constant. We therefore get that $\ker D$ is one-dimensional, being parameterized by the sole value of the constant $\tilde a_n=a_n$. By Proposition 3.10, $D$ is surjective.

Proof. Since the Riemannian metrics with a unique minimizing geodesic in class $\beta$ are $\mathbb {\mathcal {C}}^{l}$-generic (for any $l$), we can assume that this uniqueness property holds for $g$. Then the minimizing geodesic $\gamma (\beta )$ is the $k$-cover of some primitive geodesic $\gamma '$ in a class $\beta '$ such that $k\beta '=\beta$ (we say that a geodesic is primitive if it is not a multiple-cover of another one). It is then immediate to check that $\gamma (\beta ')=\gamma '$. In dimension $2$, a primitive geodesic that minimizes the length in a homology class is always simple (i.e. injective). In dimension higher than $2$, the property of a primitive geodesic being simple is $\mathbb {\mathcal {C}}^{l}$-generic (in the metric), so a further perturbation of $g$ ensures that $\gamma '$ is simple. Summarizing this preparatory discussion, we can slightly perturb $g$ so as to ensure that $\gamma := \gamma (\beta ) = \gamma (k\beta ')$ for some $k\geq 1$ is the unique $\beta$-minimizing geodesic, while $\gamma ':=\gamma (\beta ')$ is a simple closed geodesic. We denote its length by $\ell '$ and let $\varphi :\mathbb {R}\times (-1,1)^{n-1} \looparrowright L$ be Fermi coordinates near $\gamma '$. Then

\[ \varphi^{*}g=\delta_{ij}+h_{ij}(x_n,x'),\quad |h_{ij}(x_n,x')|\leq C(x_n)\Vert x'\Vert^{2}. \]

The pseudoperiodicity of $\varphi$ has some immediate consequences.

• • $\varphi _*\Vert x'\Vert$ is a well-defined function locally near $\gamma '$ on $L$. With slight abuse of notation, we therefore understand $\Vert x'\Vert$ as a function defined on both $\mathbb {R}\times (-1,1)^{n-1}$ and $L$ (near $\gamma '$).

• • Similarly, $\varphi _*\delta _{ij}$ is a well-defined tensor locally near $\gamma '$ on $L$, because the ambiguity in multi-valuedness of $\varphi$ is given by an orthogonal matrix $O$ in the fiber direction, whose derivative is again $O$, which precisely preserves the metric $\delta _{ij}$. By construction of the Fermi coordinates, $g=\varphi _*\delta _{ij}+O(\Vert x'\Vert ^{2})$.

• • $h_{ij}(x_n+{\ell '},x')=h_{ij}(x_n,Ox')$, so our bound on $h_{ij}$ is in fact uniform in $x_n$: $|h_{ij}(x_n,x')|$ $\leq C\Vert x'\Vert ^{2}$ for some constant $C$ large enough.

Now let $\rho :[0,\infty ) \to [0,1]$ be a smooth non-increasing function that equals $1$ near $0$, with $\text {Supp}\, \rho =[0,1]$, and define $\rho _\varepsilon (t):=\rho ( t/\varepsilon )$. The previous remarks show that

\[ g_\varepsilon:=\rho_\varepsilon(\Vert x'\Vert^{2}) (1+nC\Vert x'\Vert^{2})\varphi_*\delta_{ij}+(1- \rho_\varepsilon(\Vert x'\Vert^{2}))(1+\varepsilon)g \]

is a well-defined metric on $L$. We claim that it has the required properties provided $\varepsilon$ is chosen small enough. Indeed, since both $g$ and $(1+nC\Vert x'\Vert ^{2})\varphi _*\delta _{ij}$ are tangent to $\varphi _*\delta _{ij}$, the difference $\Vert g-g_\varepsilon \Vert _{\mathbb {\mathcal {C}}^{0}}$ is of order $\varepsilon$ for $\varepsilon$ small enough, so point (i) of the proposition holds. Next, we prove that $\gamma$ is the unique $\beta$-minimizing geodesic for $g_\varepsilon$. Notice that

\begin{align*} \varphi^{*}g_\varepsilon-\varphi^{*}g & = \rho_\varepsilon(\|x'\|^{2})((1+nC\|x'\|^{2})\delta_{ij}-(\delta_{ij}+h_{ij}(x_n,x'))) + (1-\rho_\varepsilon(\|x'\|^{2}))\varepsilon g\\ &\geq \rho_\varepsilon(\|x'\|^{2})(nC\|x'\|^{2}\delta_{ij}-h_{ij}(x_n,x')), \end{align*}

and the estimate $|h_{ij}(x_n,x')|\leq C\|x'\|^{2}$ holds. Thus, $\varphi ^{*}g_\varepsilon \geq \varphi ^{*}g$ for $\varepsilon$ small enough, with equality if and only if $x'=0$. Thus, $g_\varepsilon \geq g$ in the neighborhood of $\textrm {Im}\,\gamma '=\textrm {Im}\, \gamma$ endowed with our Fermi coordinates, with equality exactly on $\textrm {Im}\, \gamma$. On the other hand, if $\varepsilon$ is small enough, $g_\varepsilon =(1+\varepsilon )g$ outside this neighborhood. Thus we see that $g_\varepsilon \geq g$ on $L$, with equality exactly on $\textrm {Im}\, \gamma$. Then $\ell _{g_\varepsilon }(\gamma )=\ell _g(\gamma )=\ell$, while if $\gamma _2$ is any other closed connected curve in the class $\beta$, $\ell _{g_\varepsilon }({\gamma _2})>\ell _g({\gamma _2})\geq \ell$. We conclude that $\gamma$ is indeed the unique $g_\varepsilon$-minimizing geodesic in the class $\beta$. Moreover, in the multi-valued coordinates $\varphi$ near $\gamma$, we have $\varphi ^{*}g_\varepsilon =(1+nC\Vert x'\Vert ^{2})\delta _{ij}$. Finally, notice that although $\varphi$ does not provide Fermi coordinates for $g_\varepsilon$ near $\gamma '$, it does provide generalized Fermi coordinates because $\varphi (x_n+{\ell '},x')=\varphi (x_n,Ox')$ and $\varphi ^{*}g_\varepsilon$ is indeed tangent to $\delta _{ij}$ at order $2$ (see Definition 3.13). Point (ii) therefore holds for $g_\varepsilon$.

Proof. Let $u:(D\backslash \{0\},\partial D)\to (T^{*}L,L)$ be a $J$-holomorphic map asymptotic to the lift of the geodesic $\gamma (\beta )$, which is primitive by assumption. Notice that this lift is then automatically injective. Let $0<\varepsilon < r$ be a regular value of $\|u\|_g$, so that the subset $\{\,\|u\|_g = \varepsilon \,\} \subset D\backslash \{0\}$ is a finite union of closed circles. Let $\Omega$ be the connected component of a neighborhood of $0$ in $\{\,\|u\|_g > \varepsilon \,\}$ and $\Omega _1,\dots , \Omega _n$ the other components (there are finitely many). Let $v:=u_{|\Omega }$ and $v_i:=u_{|\Omega _i}$. Note that since $u(\partial D)\subset L$, none of these components touches $\partial D$.