Elliptic bindings for dynamically convex Reeb flows on the real projective three-space

Abstract

The first result of this paper is that every contact form on $\mathbb {R}P^3$ sufficiently $C^\infty$-close to a dynamically convex contact form admits an elliptic–parabolic closed Reeb orbit which is 2-unknotted, has self-linking number $-1/2$ and transverse rotation number in (1 / 2, 1]. Our second result implies that any p-unknotted periodic orbit with self-linking number $-1/p$ of a dynamically convex Reeb flow on a lens space of order p is the binding of a rational open book decomposition, whose pages are global surfaces of section. As an application we show that in the planar circular restricted three-body problem for energies below the first Lagrange value and large mass ratio, there is a special link consisting of two periodic trajectories for the massless satellite near the smaller primary—lunar problem—with the same contact-topological and dynamical properties of the orbits found by Conley (Commun Pure Appl Math 16:449–467, 1963) for large negative energies. Both periodic trajectories bind rational open book decompositions with disk-like pages which are global surfaces of section. In particular, one of the components is an elliptic–parabolic periodic orbit.

Appendices

Appendix 1: Proof of Proposition 4.3

The proof, which consists of showing that $\mathcal {J}_\mathrm{fast}(\alpha ,y)$ is open and closed in $\mathcal {J}$, is a generalization of the arguments contained in [35, Appendix]. We only include this proof here since we now make more general assumptions than in [35] and need to keep track of the point $y \in M{\setminus } x(\mathbb {R})$.

1.1 $\mathcal {J}_\mathrm{fast}(\alpha ,y)$ is closed

Let $J\in \mathcal {J}$ and $J_n\in \mathcal {J}_\mathrm{fast}(\alpha ,y)$ satisfy $J_n\rightarrow J$ in $C^\infty$. The data $(\alpha ,J)$ and $(\alpha ,J_n)$ induce almost complex structures ${\tilde{J}}$ and ${\tilde{J}}_n$ on $\mathbb {R}\times M$, respectively, as explained in Sect. 2.3. Consider ${\tilde{u}}_n=(a_n,u_n)$ embedded fast finite-energy ${\tilde{J}}_n$-holomorphic planes asymptotic to P satisfying $y\in u_n(\mathbb {C})$. Arguing as in Lemma 4.4 we obtain

$$\begin{aligned} u_n(\mathbb {C}) \cap x(\mathbb {R}) = \emptyset \quad \ \ \forall n. \end{aligned}$$

(84)

Let $\sigma >0$ be a number smaller than all positive periods of closed $\alpha$-Reeb orbits. After reparametrization and translation in the $\mathbb {R}$-direction, we may assume that

$$\begin{aligned} \int _\mathbb {D}u_n^*d\alpha = T-\sigma , u_n(0)=y\quad \text {and} \quad a_n(2)=0. \end{aligned}$$

(85)

Up to a subsequence, we find a finite set $\Gamma \subset \mathbb {D}$ and a finite-energy ${\tilde{J}}$-holomorphic map ${\tilde{u}}=(a,u) : \mathbb {C}{\setminus } \Gamma \rightarrow \mathbb {R}\times M$ such that ${\tilde{u}}_n \rightarrow {\tilde{u}}$ in $C^\infty _\mathrm{loc}(\mathbb {C}{\setminus } \Gamma )$. There is no loss of generality to assume that $\Gamma$ consists of negative punctures of ${\tilde{u}}$. Thus $\infty$ is a positive puncture. Using results on cylinders with small contact area it can be proved that ${\tilde{u}}$ is non-constant and its asymptotic limit at $\infty$ is P. Using “soft-rescalling” at the negative punctures, it can be shown that the asymptotic limits of ${\tilde{u}}$ at the punctures in $\Gamma$ are contractible closed $\alpha$-Reeb orbits.

We claim that

$$\begin{aligned} \int _{\mathbb {C}{\setminus }\Gamma } u^*d\alpha > 0. \end{aligned}$$

(86)

We prove this indirectly. If (86) is not true then we find a non-constant complex polynomial Q such that $Q^{-1}(0)=\Gamma$ and ${\tilde{u}}=F\circ Q$, where $F:\mathbb {C}{\setminus }\{0\}\rightarrow \mathbb {R}\times M$ is the map $F(e^{2\pi (s+it)})=(T_\mathrm{min}s,x(T_\mathrm{min}t))$. This implies, in particular, that $\Gamma \ne \emptyset$ and that the degree of Q is precisely p. If $\#\Gamma >1$ then the asymptotic limit at some point of $\Gamma$ is $(x,mT_\mathrm{min})$ for some $1\le m<p$ and, as observed above, this orbit must be contractible. But this is impossible: since $x(\mathbb {R})$ is p-unknotted the loop $t\in \mathbb {R}/\mathbb {Z}\mapsto x(T_\mathrm{min}t)$ induces an element of $\pi _1(M)$ of order p and, consequently, the loop $t\in \mathbb {R}/\mathbb {Z}\mapsto x(mT_\mathrm{min})$ can not be contractible because $m<p$. This proves that $\#\Gamma =1$. If $0\not \in \Gamma$ then $y=u(0)\in x(\mathbb {R})$, absurd. Thus $\Gamma =\{0\}$ and we can estimate $T = \int _{\partial \mathbb {D}} u^*\lambda = \lim _n \int _{\mathbb {D}} u_n^*d\lambda = T-\sigma$, a contradiction.

Let us enumerate the negative punctures in $\Gamma$ as $z_1,\dots ,z_N$, and let $P_i=(x_i,T_i)$ be the asymptotic limit of ${\tilde{u}}$ at $z_i$. Choose a $d\alpha$-symplectic trivialization B of $u^*\xi$ with the following property: it extends at every puncture $z_i$ (or $\infty$) to a $d\alpha$-symplectic trivialization of $({x_i}_{T_i})^*\xi$ (or of $(x_T)^*\xi$) coming from a capping disk. Here we used the assumption that $c_1(\xi )$ vanishes on $\pi _2(M)$.

It follows from [36, Lemma 4.9] that $\mu _{CZ}(P_i)\ge 2$ for every i. We sketch the argument below. The reason for this is that there is a holomorphic building that arises as the limit of the planes ${\tilde{u}}_n$, up to choice of a further subsequence. In our particular situation, the building has the simpler structure of a bubbling-off tree of finite-energy punctured spheres, see [36, Section 4] for instance.

The fundamental mechanism is the following: if ${\tilde{v}}$ is a punctured finite-energy sphere in the tree with precisely one positive puncture and if it is asymptotic to a closed Reeb orbit $P_+$ at its positive puncture satisfying $\mu _{CZ}(P_+)\le 1$, then there exists at least one negative puncture where the corresponding asymptotic limit satisfies $\mu _{CZ}\le 1$. The proof of this last claim is as follows. If, by contradiction, $\mu _{CZ}\ge 2$ for every asymptotic limit at a negative puncture of ${\tilde{v}}$ then we analyze two cases. In case the contact area of ${\tilde{v}}$ does not vanish we get $\mathrm{wind}_\pi ({\tilde{v}})<0$, absurd. In case the contact area of ${\tilde{v}}$ vanishes, there are negative punctures and all asymptotic limits (including $P_+$) are iterates of a common prime orbit $P_0$. Let $m\ge 1$ be such that $P_+=P_0^m$, fix a negative puncture $z_i$ and let j be such that the asymptotic limit at $z_i$ is $P_i=P_0^j$. Note that $m\ge j$, $P_0^m$ and $P_0^j$ are contractible, and if N is the least common multiple of m and j then trivializations of $\xi$ along $P_0^m,P_0^j$ coming from capping disks iterate (by N / m, N / j respectively) to trivializations of $\xi$ along $P_0^N$ which are homotopic since $c_1(\xi )$ vanishes on $\pi _2(M)$. Hence

$$\begin{aligned}&2\frac{N}{j} \le \mu _{CZ}((P_0^j)^{N/j}) = \mu _{CZ}(P_0^N) = \mu _{CZ}((P_0^m)^{N/m}) \le 2\frac{N}{m}-1 \le 2\frac{N}{j} - 1 \end{aligned}$$

This contradiction concludes the proof of the fundamental mechanism.

Now, using this fundamental mechanism explained above, we will argue to conclude that $\mu _{CZ}(P_i)\ge 2$ for every i. If not, there is $i_0$ such that $\mu _{CZ}(P_{i_0})\le 1$. The orbit $P_{i_0}$ is the asymptotic limit at the unique positive puncture of a finite-energy punctured sphere in the building. By the fundamental mechanism, this punctured sphere must have a negative puncture where it is asymptotic to an orbit satisfying $\mu _{CZ}\le 1$. Repeating this procedure, we go down the tree one level at a time until we reach a leaf, that is, a finte-energy plane asymptotic to an orbit satisfying $\mu _{CZ}\le 1$; this absurd concludes the argument.

Fix $R_0>1$ such that $\pi _\alpha \circ du$ does not vanish on $\mathbb {C}{\setminus } B_{R_0}$. Since ${\tilde{u}}_n \rightarrow {\tilde{u}}$ in $C^\infty _\mathrm{loc}(\mathbb {C}{\setminus } \Gamma )$, if n is large enough we can find

• (i) a smooth homotopy $h_n:[0,1]\times \mathbb {R}/\mathbb {Z}\rightarrow M$ satisfying $h_n(0,t)=u_n(R_0e^{i2\pi t})$ and $h_n(1,t)=u(R_0e^{i2\pi t})$,

• (ii) a non-vanishing section $\kappa _n$ of $(h_n)^*\xi$ satisfying $\kappa _n(0,t)=\pi _\alpha \cdot \partial _ru_n(R_0e^{i2\pi t})$ and $\kappa (1,t)=\pi _\alpha \cdot \partial _ru(R_0e^{i2\pi t})$, and

• (iii) a non-vanishing section $Z_n$ of $(h_n)^*\xi$ satisfying $Z_n(1,t)=B(R_0e^{i2\pi t})$.

Above $\partial _r$ denotes radial partial derivative with respect to usual polar coordinates on $\mathbb {C}{\setminus }0$. The vector field $Z_n(0,t)$ along $u_n(R_0e^{i2\pi t})$ lies on $\xi$ and extends as a non-vanishing section of $(u_n|_{B_{R_0}})^*\xi$. This is so by iii) and by the fact that $c_1(\xi )$ vanishes on $\pi _2(M)$. Hence, since $\mathrm{wind}_\pi ({\tilde{u}}_n)=0$ for all n we get $\mathrm{wind}(\kappa _n(0,t),Z_n(0,t))=1$. By the homotopy invariance of winding numbers we get

$$\begin{aligned} 1=\mathrm{wind}(\kappa _n(1,t),Z_n(1,t))=\mathrm{wind}(\pi _\alpha \cdot \partial _ru(R_0e^{i2\pi t}),B(R_0e^{i2\pi t})). \end{aligned}$$

Since $R_0$ can be chosen arbitrarily large we conclude that

$$\begin{aligned} \mathrm{wind}_\infty ({\tilde{u}},\infty ,B)=1. \end{aligned}$$

(87)

This follows from the asymptotic formula from Theorem 2.7.

The next step is to use (87) to conclude that $\mu _{CZ}(P_i)=2 \ \forall i$. Suppose by contradiction that $\mu _{CZ}(P_{i_0})\ge 3$ for some $i_0$. Then $\mathrm{wind}_\infty ({\tilde{u}},z_{i_0},B) \ge 2$. Since $\mu _{CZ}(P_i)\ge 2 \ \forall i$ we have $\mathrm{wind}_\infty ({\tilde{u}},z_i,B)\ge 1 \ \forall i$. This leads to the following absurdity:

$$\begin{aligned} \begin{array}{ll} &{} \mathrm{wind}_\pi ({\tilde{u}}) \\ &{}\quad = \mathrm{wind}_\infty ({\tilde{u}}) + N - 1 \\ &{}\quad = \mathrm{wind}_\infty ({\tilde{u}},\infty ,B) - \mathrm{wind}_\infty ({\tilde{u}},z_{i_0},B) - \sum _{i\ne i_0} \mathrm{wind}_\infty ({\tilde{u}},z_i,B) + N - 1 \\ &{}\quad \le 1 - 2 - (N-1) + N - 1 = -1. \end{array} \end{aligned}$$

Now we can propagate the above arguments down the bubbling-off tree in order to prove that every leaf is a finite-energy plane with asymptotic limit satisfying $\mu _{CZ}=2$. In fact, let ${\tilde{v}}=(b,v)$ be a finite-energy sphere in the second level of the tree. Then ${\tilde{v}}$ has exactly one positive puncture $\infty$ where it is asymptotic to the same asymptotic limit ${\bar{P}}$ as ${\tilde{u}}$ is at one of its negative punctures. We proved above that $\mu _{CZ}({\bar{P}})=2$. Every asymptotic limit of ${\tilde{v}}$ at a negative puncture must satisfy $\mu _{CZ}\ge 2$ since, otherwise, we can argue as before to find a leaf of the tree that is a finite-energy plane asymptotic to an orbit with $\mu _{CZ}\le 1$, absurd. If the contact area of ${\tilde{v}}$ does not vanish then $\mathrm{wind}_\infty ({\tilde{v}},\infty ,B') = 1$ where $B'$ is a symplectic trivialization² of $v^*\xi$ that extends at the punctures to trivializations coming from capping disks for its asymptotic limits. This is so because $\mu _{CZ}({\bar{P}})=2$. A calculation as above, using $\mathrm{wind}_\infty ({\tilde{v}},\infty ,B')=1$, will provide a contradiction if one the asymptotic limits of ${\tilde{v}}$ at a negative puncture satisfies $\mu _{CZ}>2$. If the contact area of ${\tilde{v}}$ vanishes then there is a prime closed Reeb orbit $P_0$ such that ${\bar{P}} = P_0^m$ and every asymptotic limit of ${\tilde{v}}$ at a negative puncture is of the form $P_0^j$ with $j\le m$. Suppose, again by contradiction, that an asymptotic limit $P'$ of ${\tilde{v}}$ at one of its negative punctures satisfies $\mu _{CZ}(P')>2$. Define j by $P' = P_0^j$. We know that $j\le m$. Denote by N the least common multiple of j and m. Of course, every asymptotic limit of ${\tilde{v}}$ is contractible and, consequently, so is $P_0^N$. We get a contradiction as follows

$$\begin{aligned}&2\frac{N}{m} = \mu _{CZ}((P_0^m)^{N/m}) =\mu _{CZ}(P_0^N) = \mu _{CZ}((P_0^j)^{N/j}) \ge 2\frac{N}{j} + 1 \ge 2\frac{N}{m} + 1. \end{aligned}$$

We concluded that, in all cases, every asymptotic limit of ${\tilde{v}}$ at a negative puncture satisfies $\mu _{CZ}=2$. This must be so for all curves in the second level of the tree. The same arguments apply to curves in the lower levels when we go down one level at a time. Thus, all leafs are planes asymptotic to orbits with $\mu _{CZ}=2$, as desired.

We are finally ready to conclude our compactness argument, this last step will make use of the linking hypotheses made in Proposition 4.3. The important remark is that for every plane ${\tilde{v}}=(b,v):\mathbb {C}\rightarrow \mathbb {R}\times M$ associated to a leaf of the tree there are sequences $A_n,B_n \in \mathbb {C}$, $c_n\in \mathbb {R}$, such that ${\tilde{u}}_n(A_nz+B_n) +c_n \rightarrow {\tilde{v}}$ in $C^\infty _\mathrm{loc}$. Let us assume by contradiction that the tree has at least two levels. Then the plane ${\tilde{v}}$ satisfies $\mathrm{wind}_\infty ({\tilde{v}})=1$ because its asymptotic limit $P_*$ satisfies $\mu _{CZ}(P_*)=2$. Consequently ${\tilde{v}}$ is an immersion transverse to the Reeb flow. Suppose that $P_*$ is not geometrically distinct from P. Setting $P_\mathrm{min}=(x,T_\mathrm{min})$ we have $P=P_\mathrm{min}^p$ and find j such that $P_* = P_\mathrm{min}^j$. If N is the least common multiple of j and p, then using $\mu _{CZ}(P)\ge 3$ and $\mu _{CZ}(P_*)=2$ we compute as above

$$\begin{aligned}&2\frac{N}{j} = \mu _{CZ}(P_*^{N/j}) = \mu _{CZ}(P_\mathrm{min}^N) = \mu _{CZ}(P^{N/p}) \ge 2\frac{N}{p} + 1 \ \Rightarrow \ j<p. \end{aligned}$$

But $P_\mathrm{min}$ induces an element of $\pi _1$ with order precisely p, contradicting contractibility of $P_*$. Hence $P_*$ is geometrically distinct from P. The period of $P_*$ is the Hofer energy of ${\tilde{v}}$ and, consequently, is not larger than T. By hypothesis $P_*$ is not contractible in $M{\setminus } x(\mathbb {R})$. This forces intersections of ${\tilde{v}}$ with the ${\tilde{J}}$-complex surface $\mathbb {R}\times x(\mathbb {R})$. By positivity and stability of intersections we find intersections of the maps ${\tilde{u}}_n$ with $\mathbb {R}\times x(\mathbb {R})$ for $n\gg 1$, contradicting (84). We have proved that the tree has exactly one level, consisting of its root. In other words, $\Gamma =\emptyset$ and ${\tilde{u}}$ is a plane. It is asymptotic to P and (87) tells us that it is fast. Clearly $u(0)=y$.

It only remains to be checked that ${\tilde{u}}$ is an embedding. It must be somewhere injective since, otherwise, we would factor it through a somewhere injective plane via a complex polynomial of degree at least 2, forcing critical points of ${\tilde{u}}$, but this is impossible since ${\tilde{u}}$ is an immersion. Hence the set A of self-intersection points of ${\tilde{u}}$ is discrete. If $A\ne \emptyset$ then again by positivity and stability of intersections we would find self-intersections of ${\tilde{u}}_n$ for n large, but this is impossible because the ${\tilde{u}}_n$ are embeddings. We have finally proved that $\mathcal {J}_\mathrm{fast}(\alpha ,y)$ is $C^\infty _\mathrm{loc}$-closed.

1.2 $\mathcal {J}_\mathrm{fast}(\alpha ,y)$ is open

Most of the argument can be found in [35, Appendix A]. Let $J_0\in \mathcal {J}_\mathrm{fast}(\alpha ,y)$ and ${\tilde{u}}_0=(a_0,u_0):\mathbb {C}\rightarrow \mathbb {R}\times M$ be an embedded fast finite-energy ${\tilde{J}}_0$-holomorphic plane asymptotic to P and satisfying $y\in u_0(\mathbb {C})$. After rotating the domain we can assume that $u(Re^{i2\pi t}) \rightarrow x(Tt)$ as $R\rightarrow +\infty$.

Fix $l\ge 1$ and let $\mathcal {K}^l$ be the space of $C^l$-sections K of the vector bundle $\mathcal {L}(\xi )$ satisfying $J_0K+KJ_0=0$ and $d\alpha (\cdot ,K\cdot )+d\alpha (K\cdot ,\cdot )=0$ on $\xi |_p$, for every $p\in M$. With the $C^l$-norm this space becomes a Banach space. If $r>0$ is small enough then every K belonging to the ball of radius r centered at the origin in $\mathcal {K}^l$ induces a complex structure $J=J_0\exp (-J_0K)$ on $\xi$ of class $C^l$ which is $d\alpha$-compatible. The set $\mathcal {U}_r$ of J that arise in this way contains a neighborhood of $J_0$ in $\mathcal {J}$, and admits the structure of a (trivial) Banach manifold via the above explained identification with a ball in $\mathcal {K}^l$.

Fix any $\mathbb {R}$-invariant metric g on $\mathbb {R}\times M$ for which ${\tilde{J}}_0$ is a pointwise isometry, like for instance $g = da \otimes da + \lambda \otimes \lambda + d\lambda (\cdot ,J_0\cdot )$. Then the normal bundle N of ${\tilde{u}}_0(\mathbb {C})$ is a ${\tilde{J}}_0$-invariant vector bundle over ${\tilde{u}}_0(\mathbb {C})$. Denoting by $\pi _M:\mathbb {R}\times M \rightarrow M$ the projection onto the second factor, clearly the bundle ${\tilde{\xi }} := \pi _M^*\xi$ is also ${\tilde{J}}_0$-invariant. In view of the asymptotic formula we can find $R_0$ large enough and a ${\tilde{J}}_0$-invariant subbundle $L\subset {\tilde{u}}_0^*T(\mathbb {R}\times M)$ which coincides with ${\tilde{u}}_0^*{\tilde{\xi }} = u_0^*\xi$ on $\mathbb {C}{\setminus } B_{R_0}$, and coincides with ${\tilde{u}}_0^*N$ on $B_{R_0-1}$. Here we denoted by $B_\rho$ the ball of radius $\rho$ in $\mathbb {C}$ centered at the origin.

Now let $(U,\Psi )$ be a Martinet tube at P. As explained in Definition 2.5 there are associated coordinates $(\theta ,x_1,x_2)\in \mathbb {R}/\mathbb {Z}\times B$ where $\alpha \simeq g(d\theta +x_1dx_2)$ for some smooth function g satisfying $g(\theta ,0,0)\equiv T_\mathrm{min}$, $dg(\theta ,0,0)=0$ for all $\theta$.

Consider the p-covering space $\mathbb {R}/p\mathbb {Z}\times B \rightarrow \mathbb {R}/\mathbb {Z}\times B$, and denote the $\mathbb {R}/p\mathbb {Z}$-coordinate by $\theta '$. The projection is $(\theta ',x_1,x_2) \mapsto (\theta ,x_1,x_2)$ and we denote by $\alpha '$ the lift of $\alpha$ to $\mathbb {R}/p\mathbb {Z}\times B$, namely $\alpha ' = g(\theta ,x_1,x_2)(d\theta '+x_1dx_2)$. The lift of $J_0|_U$ to a $d\alpha '$-compatible complex structure on $\xi ' := \ker \alpha '$ will be denoted by $J_0'$. Perhaps after making $R_0$ larger we can assume that ${\tilde{u}}_0(\mathbb {C}{\setminus } B_{R_0}) \subset \mathbb {R}\times U$ and, consequently, we can consider a lift $(\Psi \circ u_0)' : \mathbb {C}{\setminus } B_{R_0} \rightarrow \mathbb {R}/p\mathbb {Z}\times B$ of $\Psi \circ u_0$. There exists a $(d\alpha ',J_0')$-unitary frame $\{n_1',n_2'\}$ of $\xi '$ on $\mathbb {R}/p\mathbb {Z}\times B$ such that the $n_i' \circ (\Psi \circ u_0)'$ form a frame with the following important property: after identifying $(\Psi \circ u_0)'^*(\xi ') \simeq u_0^*\xi |_{\mathbb {C}{\setminus } B_{R_0}} = L|_{\mathbb {C}{\setminus } B_{R_0}}$ in the obvious manner, $\{n_1'\circ (\Psi \circ u_0)',n_2'\circ (\Psi \circ u_0)'\}$ extends to a complex frame of L. This extended frame of L will be denoted by $\{n_1,n_2\}$.

Note that for |z| large enough $L_z = \xi |_{u_0(z)}$, and taking the limit as $|z|\rightarrow \infty$ the frame $\{n_1,n_2\}$ induces a $(d\alpha ,J_0)$-unitary frame $\{{\bar{n}}_1,{\bar{n}}_2\}$ of $(x_T)^*\xi$. This follows from the construction of $\{n_1,n_2\}$ explained above. Let $\{e_1,e_2\}$ be a $d\alpha$-symplectic frame of $(x_T)^*\xi$ induced by some capping disk for P. It follows from [29, Theorem 1.8] that

$$\begin{aligned} \mathrm{wind}(t\mapsto e_1(t),t\mapsto n_1(t)) = -1. \end{aligned}$$

(88)

Let $\exp$ be the exponential map associated to the metric g. In view of the asymptotic formula and of the $\mathbb {R}$-invariance of g we can find a small open ball $B'\subset \mathbb {C}$ centered at the origin such that the map

$$\begin{aligned} \Phi : \mathbb {C}\times B' \rightarrow \mathbb {R}\times M&\Phi (z,w) = \exp _{{\tilde{u}}_0(z)}(\mathfrak {R}[w]n_1(z) + \mathfrak {I}[w]n_2(z)) \end{aligned}$$

(89)

is an immersion onto a neighborhood of ${\tilde{u}}_0(\mathbb {C})$. For every $J \in \mathcal U_r$ we can consider the $\mathbb {R}$-invariant almost complex structure ${\tilde{J}}$ on $\mathbb {R}\times M$ determined by $(\alpha ,J)$. Denoting ${\bar{J}} = \Phi ^*{\tilde{J}}$, then ${\bar{J}}$ can be seen as a smooth $\mathbb {R}^{4\times 4}$-valued function on $\mathbb {C}\times B'$ which can be written in $2\times 2$ blocks as

$$\begin{aligned} {\bar{J}}(z,w) = \begin{pmatrix} j_1(z,w) &{}\quad \Delta _1(z,w) \\ \Delta _2(z,w) &{}\quad j_2(z,w) \end{pmatrix}. \end{aligned}$$

The graph of a differentiable function $z\mapsto v(z)$ has ${\bar{J}}$-invariant tangent space if, and only if, $H(v,J)=0$ where

$$\begin{aligned} H(v,J) = \Delta _2(z,v) + j_2(z,v) \ dv - dv \ j_1(z,v) - dv \ \Delta _1(z,v) \ dv. \end{aligned}$$

(90)

Given $\gamma \in (0,1)$ and $\delta <0$ we can consider the space $C^{l,\gamma ,\delta }_0(\mathbb {C},\mathbb {C})$ defined as the set of maps $v:\mathbb {C}\rightarrow \mathbb {C}$ of class $C^{l,\gamma }$ such that $(s,t) \mapsto v(e^{2\pi (s+it)})$ is of class $C^{l,\gamma ,\delta }_0$ on $[0,+\infty )\times \mathbb {R}/\mathbb {Z}$; see Remark 4.14. Consider $Y = \mathbb {C}\times \mathcal {L}_\mathbb {R}(\mathbb {C})$ as a trivial vector bundle over $\mathbb {C}$. The space $C^{l,\gamma ,\delta }_0(Y)$ is defined as the space of $C^{l,\gamma }$-sections $z\mapsto A(z)$ such that $(s,t) \mapsto A(e^{2\pi (s+it)})e^{2\pi (s+it)}$ is of class $C^{l,\gamma ,\delta }_0$ on $[0,+\infty )\times \mathbb {R}/\mathbb {Z}$. The splitting $\mathcal {L}_\mathbb {R}(\mathbb {C}) = \mathcal {L}^{1,0}(\mathbb {C}) \oplus \mathcal {L}^{0,1}(\mathbb {C})$ as $\mathbb {C}$-linear and $\mathbb {C}$-antilinear maps induces splittings $Y = Y^{1,0} \oplus Y^{0,1}$ and $C^{l,\gamma ,\delta }_0(Y) = C^{l,\gamma ,\delta }_0(Y^{1,0}) \oplus C^{l,\gamma ,\delta }_0(Y^{0,1})$. The set $\mathcal {V}\subset C^{l,\gamma ,\delta }_0(\mathbb {C},\mathbb {C})$ of maps with image in $B'$ is clearly open. It is a standard procedure to check that

$$\begin{aligned} H:(v,J) \in \mathcal {V} \times \mathcal U_r \mapsto H(v,J) \in C^{l-1,\gamma ,\delta }(Y) \end{aligned}$$

defines a smooth map.

Let us write

$$\begin{aligned} {\bar{J}}_0 = \begin{pmatrix} j^0_1 &{}\quad \Delta _1^0 \\ \Delta _2^0 &{}\quad j_2^0 \end{pmatrix} \end{aligned}$$

so that $j^0_1(z,0)=j^0_2(z,0)=i$ and $\Delta ^0_1(z,0)=\Delta ^0_2(z,0)=0$. It follows that

$$\begin{aligned} D_1H(0,J_0)\zeta = i \ d\zeta - d\zeta \ i + D_2\Delta _2^0(z,0)\zeta . \end{aligned}$$

Differentiating the identity ${\bar{J}}_0^2=-I$ we get that $D := D_1H(0,J_0)$ takes values on $C^{l-1,\gamma ,\delta }_0(Y^{0,1})$.

Theorem 5.1

(Hofer et al. [29]) If $\delta <0$ is not in the spectrum of the asymptotic operator at P associated to $(\alpha ,J_0)$ then

$$\begin{aligned} D : C^{l,\gamma ,\delta }_0(\mathbb {C},\mathbb {C}) \rightarrow C^{l-1,\gamma ,\delta }_0(Y^{0,1}) \end{aligned}$$

is a Fredholm operator with index $\mu _{CZ}^\delta (P)-1$.

Here $\mu _{CZ}^\delta (P)$ denotes the Conley–Zehnder index associated to the linearized $\alpha$-Reeb flow along P represented by a $d\alpha$-symplectic frame induced by a capping disk, which takes the exponential weight $\delta$ into account in the following manner: $\mu _{CZ}^\delta (P) = 2k$ if $\delta$ lies between two eigenvalues of A with winding number k with respect to this frame, or $\mu _{CZ}^\delta (P)=2k+1$ if $\delta$ lies between eigenvalues with winding numbers k and $k+1$ with respect to this frame.

From now on we fix our choice of $\delta$: since $\rho (P)>1 \Leftrightarrow \mu _{CZ}(P)\ge 3$ we can place $\delta$ between eigenvalues with winding numbers 1 and 2, so that $\mu _{CZ}^\delta (P)=3$ and the index of D is 2.

The analysis of [29] shows that there exists a (trivial) smooth Banach bundle $\mathcal {E} \rightarrow \mathcal {V} \times \mathcal U_r$ with fibers modeled on $C^{l-1,\gamma ,\delta }(Y^{0,1})$ and a smooth section $\eta$ of $\mathcal {E}$ such that $\eta (v,J)=0 \Leftrightarrow H(v,J)=0$. Moreover, the partial vertical derivative $D_1\eta (0,J_0)$ at $(0,J_0)$ coincides with D.

Lemma 5.2

The operator D is surjective.

Proof

Consider the map $\sigma (s,t) = e^{2\pi (s+it)}$, fix $\zeta \in C^{l,\gamma ,\delta }(\mathbb {C},\mathbb {C})$ and denote $a(s,t)=\zeta \circ \sigma (s,t)$. If we evaluate $id\zeta - d\zeta i + D_2\Delta _2^0(z,0)\zeta$ at $z=\sigma (s,t)$ and apply it to $\partial _t\sigma$ we obtain $a_s+ia_t+C(s,t)a$ where $C(s,t)u=[D_2\Delta _2(\sigma (s,t),0)u]\partial _t\sigma (s,t)$. It is shown in [29] that $C(s,t) \rightarrow S(t)$ as $s\rightarrow +\infty$, where S(t) is a smooth 1-periodic path of symmetric $2\times 2$ matrices such that the linearized Reeb flow along $(x_T)^*\xi$ represented in the frame $\{{\bar{n}}_1,{\bar{n}}_2\}$ yields a path $\varphi (t) \in Sp(2)$ satisfying $-i\dot{\varphi }-S\varphi =0$. In view of (88) we get $\mu _{CZ}^\delta (\varphi )=1$ and conclude that all eigenvectors of the operator $A \simeq -i\partial _t-S$ associated to eigenvalues smaller than $\delta$ have winding number smaller than or equal to 0. Here we used the monotonicity of winding numbers along the spectrum of A; see [28].

If $\zeta \in \ker D$ is non-zero then it follows from asymptotic analysis as in [27, 34, 49] that for s large enough a(s, t) does not vanish and and looks like $e^{\lambda s}(u(t)+\epsilon (s,t))$ for some eigenvector u(t) associated to an eigenvalue $\lambda <0$ of A. Here $\epsilon (s,t)\rightarrow 0$ as $s\rightarrow +\infty$. In particular, the winding number of $t\mapsto a(s,t)$ is equal to the winding number m of u when s is large enough, and $\lambda <\delta$ since $\zeta \in C^{l-1,\gamma ,\delta }$. Thus $m\le 0$ and we can use Carleman’s similarity principle together with standard degree theory to conclude that $m=0$ and that any non-trivial $\zeta \in \ker D$ is nowhere vanishing.

We are finally in a position to conclude the argument: if there are three linearly independent vectors in $\ker D$ then a suitable combination of them will vanish at some point since $\dim _\mathbb {R}\mathbb {C}=2$. By the above analysis this linear combination is identically zero, absurd. Hence $\dim \ker D \le 2$. Since D has index 2 we conclude that $\dim \ker D=2$ and $\dim \mathrm{coker}\ D=0$, as desired. $\square$

The above lemma is an automatic transversality statement, it is found in [34], and also proved in [51]. As a consequence we find an open neighborhood $\mathcal {O}$ of $J_0$ in $\mathcal U_r$ such that, perhaps after shrinking $\mathcal {V}$, the universal moduli space

$$\begin{aligned} \mathcal {M}^\mathrm{univ} = \{ (v,J) \in \mathcal {V} \times \mathcal {O} \mid \eta (v,J)=0 \} \end{aligned}$$

is a smooth submanifold of $\mathcal {V} \times \mathcal {O}$. However, there is an important additional piece of information: perhaps after shrinking $\mathcal {V}$ and $\mathcal {O}$ even more we may assume that the projection $\Pi :(v,J) \in \mathcal {V} \times \mathcal {O} \mapsto J \in \mathcal {O}$ onto the second factor induces a submersion

$$\begin{aligned} \Pi |_{\mathcal {M}^\mathrm{univ}} : \mathcal {M}^\mathrm{univ} \rightarrow \mathcal {O} \end{aligned}$$

such that each fiber $\mathcal {M}(J) := \Pi ^{-1}(J) \cap \mathcal {M}^\mathrm{univ}$ is non-empty. Then each $\mathcal {M}(J)$ is a smooth (non-empty) manifold, and by [45, Lemma A.3.6] and Theorem 5.1 we have $\dim \mathcal {M}(J) = \mu _{CZ}^\delta (P)-1 = 2$ for all $J\in \mathcal {O}$.

We need to argue a little more in order to keep track of the point y. There is no loss of generality to assume that ${\tilde{u}}_0(0)=(0,y)$. We introduce the evaluation map

$$\begin{aligned} \mathrm{ev} : \mathbb {C}\times \mathcal {M}^\mathrm{univ} \rightarrow \mathbb {R}\times M&\mathrm{ev}(z,v,J) = \Phi (z,v(z)) \end{aligned}$$

where $\Phi$ is the map (89). This is easily proved to be a smooth map and $(0,0,J_0) \in \mathrm{ev}^{-1}(0,y)$.

Lemma 5.3

The map $\mathrm{ev}|_{\mathbb {C}\times \mathcal {M}(J_0)}:\mathbb {C}\times \mathcal {M}(J_0) \rightarrow \mathbb {R}\times M$ is non-singular at the point $(0,0,J_0)$.

Proof

Since ${\tilde{u}}_0$ is an embedding we only need to show that the partial derivative of $\mathrm{ev}|_{\mathbb {C}\times \mathcal {M}(J_0)}$ in the $\mathcal {M}(J_0)$-direction is transverse to ${\tilde{u}}_0(\mathbb {C}) = \mathrm{ev}(\mathbb {C}\times \{(0,J_0)\})$. This would not be the case precisely when there is a section in $\ker D$ which vanishes somewhere, but this possibility was ruled out by the argument used to prove Lemma 5.2. $\square$

It is an immediate consequence of Lemma 5.3 that the differential $d(\mathrm{ev})|_{(0,0,J_0)}$ of the map $\mathrm{ev}$ at the point $(0,0,J_0)$ is onto, so there is no loss of generality to assume that $\mathrm{ev}^{-1}(0,y)$ is a smooth submanifold. The codimension of $\mathrm{ev}^{-1}(0,y)$ is 4, and by Lemma 5.3 $\mathrm{ev}^{-1}(0,y)$ intersects $\mathbb {C}\times \mathcal {M}(J_0)$ transversally at the point $(0,0,J_0)$. In particular, since $\dim \mathbb {C}\times \mathcal {M}(J_0)=4$, it follows that the restriction to $T_{(0,0,J_0)}\mathrm{ev}^{-1}(0,y)$ of the linearization of the map $(z,v,J) \mapsto J$ at the point $(0,0,J_0)$ is surjective. This completes the proof that for every J close enough to $J_0$ in $\mathcal U_r$ there exists some $(v,J) \in \mathcal {M}(J)$ such that (0, y) belongs to the image of the map $z \mapsto \mathrm{ev}(z,v,J)$.

In order to finish the proof that $J_0$ is an interior point of $\mathcal {J}_\mathrm{fast}(\alpha ,y)$ it only remains to be shown that the map $z\mapsto \mathrm{ev}(z,v,J)$ can be reparametrized as an embedded fast finite-energy ${\tilde{J}}$-holomorphic plane, whenever J is close enough to $J_0$ in $\mathcal U_r$ and $(v,J) \in \mathcal {M}^\mathrm{univ}$. This argument uses the analysis of [29, Appendix] and has been spelled out in detail in [35, Appendix A]. We sketch it here: using the analysis of [29, Appendix] such a map $z\mapsto \mathrm{ev}(z,v,J)$ can be reparametrized as a finite-energy ${\tilde{J}}$-holomorphic plane ${\tilde{u}}$. It is clearly an embedded plane. Since $v\in C^{l,\gamma ,\delta }_0$ we can use asymptotic analysis as in [27, 34, 49] to show that the asymptotic eigenvalue of this plane is $\le \delta$, and hence $<\delta$ since $\delta$ does not belong to the spectrum of the corresponding asymptotic operator. In particular, $\mathrm{wind}_\infty ({\tilde{u}})\le 1$ and, by the similarity principle, we have $\mathrm{wind}_\infty ({\tilde{u}})=1 \Rightarrow \mathrm{wind}_\pi ({\tilde{u}})=0$, as desired.

Appendix 2: Proof of Proposition 4.15

The arguments here are almost identical to those of [35, Appendix B], we include them for the sake of completeness. We fix a sequence of smooth maps

$$\begin{aligned} K_n : [0,+\infty ) \times \mathbb {R}/\mathbb {Z}\rightarrow \mathbb {R}^{2m\times 2m} \quad \ \ (n\ge 1) \end{aligned}$$

(91)

and a map

$$\begin{aligned} K^\infty : \mathbb {R}/\mathbb {Z}\rightarrow \mathbb {R}^{2m\times 2m} \end{aligned}$$

(92)

such that $K^\infty (t)$ is symmetric for all t, and satisfying the following property: for every pair of sequences $n_l,s_l\rightarrow +\infty$ there exists $l_k\rightarrow +\infty$ and $c\in [0,1)$ such that

$$\begin{aligned} \lim _{k\rightarrow \infty } \Vert \partial ^{\beta _1}_s\partial ^{\beta _2}_t[K_{n_{l_k}}(s,t)-K^\infty (t+c)](s_{l_k},\cdot ) \Vert _{L^\infty (\mathbb {R}/\mathbb {Z})} = 0 \end{aligned}$$

(93)

for all $\beta _1,\beta _2\ge 0$.

From now on L will denote the unbounded self-adjoint operator

$$\begin{aligned} L : W^{1,2} \subset L^2 \rightarrow L^2&e(t) \mapsto -J_0\dot{e}(t)-K^\infty (t) e(t) \end{aligned}$$

(94)

where $W^{1,2} = W^{1,2}(\mathbb {R}/\mathbb {Z},\mathbb {R}^{2m})$, $L^2=L^2(\mathbb {R}/\mathbb {Z},\mathbb {R}^{2m})$ and $J_0$ is the standard complex structure

$$\begin{aligned} J_0 = \begin{pmatrix} 0 &{}\quad -I \\ I &{}\quad 0 \end{pmatrix} \end{aligned}$$

(95)

on $\mathbb {R}^{2m}$ (written in four $m\times m$ blocks). The first step towards the proof of the proposition is the following lemma.

Lemma 6.1

For each $s\ge 0$ and $n\ge 1$ consider the unbounded self-adjoint operator $L_n(s):W^{1,2}\subset L^2 \rightarrow L^2$ given by $L_n(s) = -J_0 \partial _t - S_n(s,t)$ where $S_n = \frac{1}{2}(K_n+K_n^T)$ is the symmetric part of $K_n$. If $\delta <0$ does not lie on the spectrum of the operator L (94) then we can find $s_0\ge 0$, $n_0\ge 1$ and $c>0$ such that

$$\begin{aligned} \Vert [L_n(s)-\delta ]e\Vert _{L^2} \ge c\Vert e\Vert _{L^2} \end{aligned}$$

for all $s\ge s_0$, $n\ge n_0$, $e\in W^{1,2}$.

Proof

Let us argue by contradiction, assuming that there exist $n_l,s_l \rightarrow +\infty$ and vectors $e_l \in W^{1,2}$ such that $\Vert e_l\Vert _{L^2}=1 \ \forall l$, and

$$\begin{aligned} \Vert [L_{n_l}(s_l)-\delta ]e_l\Vert _{L^2} \rightarrow 0 \ \ \ \text {as} \ \ \ l\rightarrow \infty . \end{aligned}$$

In view of the assumptions, we find $l_k$ and $c\in [0,1)$ such that $S_{n_{l_k}}(s_{l_k},t) \rightarrow K^\infty (t+c)$ uniformly in t, as $k\rightarrow \infty$. Since

$$\begin{aligned} \partial _te_{l_k} = J_0([L_{n_{l_k}}(s_{l_k})-\delta ]e_{l_k}) + J_0S_{n_{l_k}}(s_{l_k},\cdot )e_{l_k} + \delta J_0e_{l_k} \end{aligned}$$

then $e_{l_k}$ is bounded in $W^{1,2}$. Thus, up to a further subsequence, we may assume that $e_{l_k} \rightarrow e$ in $L^2$ as $k\rightarrow \infty$, for some $e\in L^2$. Appealing again to the above equation we conclude that $e_{l_k}$ is a Cauchy sequence in $W^{1,2}$, in particular $e\in W^{1,2}$, $\Vert e\Vert _{L^2}=1$ and $\Vert e_{l_k}-e\Vert _{W^{1,2}} \rightarrow 0$. But all this implies that $[L_{n_{l_k}}(s_{l_k})-\delta ]e_{l_k} \rightarrow -J_0\partial _te-K^\infty (\cdot +c)e - \delta e$ as $k\rightarrow \infty$. In other words, $e_c(t)=e(t-c)$ satisfies $Le_c=\delta e_c$ and $\Vert e_c\Vert _{L^2}=1$. This is in contradiction with our hypothesis on $\delta$. $\square$

Lemma 6.2

Suppose that $\delta <0$ does not lie on the spectrum of the operator L (94), and let $\mu$ be largest number in $\sigma (L)$ below $\delta$. Then there exists $0<r<\delta -\mu$ and $n_1,s_1>0$ such that

$$\begin{aligned} \Vert X(s,\cdot )\Vert _{L^2} \le e^{(\delta -r)(s-s_1)} \Vert X(s_1,\cdot ) \Vert _{L^2} \end{aligned}$$

holds for all $s\ge s_1$ and for all solutions X of

$$\begin{aligned} \partial _sX+J_0\partial _tX+K_nX=0&\lim _{s\rightarrow +\infty } e^{-\delta s} \Vert X(s,\cdot )\Vert _{L^2} = 0 \end{aligned}$$

with $n\ge n_1$.

Proof

Let $S_n=\frac{1}{2}(K_n+K_n^T)$ and $A_n=\frac{1}{2}(K_n-K_n^T)$ be the symmetric and anti-symmetric parts of $K_n$. Then we have the following property: for all $\beta _1,\beta _2\ge 0$ and for all pairs of sequences $s_l,n_l\rightarrow +\infty$ we can find $l_k\rightarrow \infty$ and $c\in [0,1)$ such that

$$\begin{aligned} \begin{array}{ll} &{} \lim \limits _{k\rightarrow \infty } \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}[S_{n_{l_k}}(s,t)-S^\infty (t+c)](s_{l_k},\cdot )\Vert _{L^\infty (\mathbb {R}/\mathbb {Z})} = 0 \\ &{} \lim \limits _{k\rightarrow \infty } \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}A_{n_{l_k}}(s_{l_k},\cdot ) \Vert _{L^\infty (\mathbb {R}/\mathbb {Z})} = 0. \end{array} \end{aligned}$$

(96)

In particular, it follows that

$$\begin{aligned} \lim _{s,n\rightarrow +\infty } \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}S_n(s,\cdot ) \Vert _{L^\infty (\mathbb {R}/\mathbb {Z})} = 0, \quad \ \ \forall \beta _1\ge 1,\beta _2\ge 0, \end{aligned}$$

(97)

in the sense that for all $\beta _1\ge 1,\beta _2\ge 0$ and $\epsilon >0$ there are numbers $s(\epsilon ,\beta _1,\beta _2)$, $n(\epsilon ,\beta _1,\beta _2)$ such that

$$\begin{aligned} s\ge s(\epsilon ,\beta _1,\beta _2),n\ge n(\epsilon ,\beta _1,\beta _2) \Rightarrow \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}S_n(s,\cdot ) \Vert _{L^\infty (\mathbb {R}/\mathbb {Z})}\le \epsilon . \end{aligned}$$

Analogously,

$$\begin{aligned} \lim _{s,n\rightarrow +\infty } \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}A_n(s,\cdot ) \Vert _{L^\infty (\mathbb {R}/\mathbb {Z})} = 0, \quad \ \ \forall \beta _1\ge 0,\beta _2\ge 0. \end{aligned}$$

(98)

The function X solves a partial differential equation which depends on n: this is not explicit in the notation X(s, t) but the reader should not forget the vital role played by n. Consider $Y(s,t)=e^{-\delta s}X(s,t)$. Then

$$\begin{aligned} Y_s - (L_n(s)-\delta )Y + A_nY = 0, \end{aligned}$$

where $L_n(s)$ is the self-adjoint operator described in the statement of the previous lemma, and $\Vert Y(s,\cdot )\Vert _{L^2(\mathbb {R}/\mathbb {Z})}^2 \rightarrow 0$ as $s\rightarrow +\infty$. Setting $g(s)=\frac{1}{2}\Vert Y(s,\cdot )\Vert _{L^2(\mathbb {R}/\mathbb {Z})}^2$ then one quickly computes

$$\begin{aligned} g''(s) = 2\Vert (L_n(s)-\delta )Y\Vert _{L^2}^2 - 2\langle (L_n(s)-\delta )Y,A_nY \rangle _{L^2} - \langle (\partial _sS_n)Y,Y \rangle _{L^2}. \end{aligned}$$

(99)

For this one uses many times that $L_n(s)$ is self-adjoint.

The following notation will simplify the exposition below: given a function f of $(s,t) \in \mathbb {R}\times \mathbb {R}/\mathbb {Z}$ we write $\Vert f\Vert _{L^2,s}$ for the $L^2(\mathbb {R}/\mathbb {Z})$-norm of $f(s,\cdot )$.

We now follow [35, Lemma B.2] closely, giving more details. Let $n_0$, $s_0$ and $c>0$ be given by the previous lemma. In view of (98) we can find $n_1\ge n_0$, $s_1\ge s_0$ such that

$$\begin{aligned} n\ge n_1,s\ge s_1 \Rightarrow \Vert A_n(s,\cdot )\Vert _{L^\infty } < c. \end{aligned}$$

(100)

In particular

$$\begin{aligned} n\ge n_1,s\ge s_1 \Rightarrow c\Vert Y\Vert _{L^2,s} > \Vert A_n(s,\cdot )\Vert _{L^\infty }\Vert Y\Vert _{L^2,s} \ge \Vert A_nY\Vert _{L^2,s}. \end{aligned}$$

(101)

Using (99) and the previous lemma we estimate when $s\ge s_1,n\ge n_1$:

$$\begin{aligned} \begin{array}{ll} g''(s) &{}\ge 2c\Vert Y\Vert _{L^2,s}\Vert (L_n(s)-\delta )Y\Vert _{L^2,s} - 2\Vert A_nY\Vert _{L^2,s}\Vert (L_n(s)-\delta )Y\Vert _{L^2,s} \\ &{}\quad - \Vert \partial _sS_n(s,\cdot )\Vert _{L^\infty }\Vert Y\Vert _{L^2,s}^2 \\ &{}= 2\Vert (L_n(s)-\delta )Y\Vert _{L^2,s} \left( c\Vert Y\Vert _{L^2,s} - \Vert A_nY\Vert _{L^2,s} \right) \\ &{}\quad - \Vert \partial _sS_n(s,\cdot )\Vert _{L^\infty }\Vert Y\Vert _{L^2,s}^2 \\ &{}\ge 2c\Vert Y\Vert _{L^2,s} \left( c\Vert Y\Vert _{L^2,s} - \Vert A_n(s,\cdot )\Vert _{L^\infty }\Vert Y\Vert _{L^2,s} \right) \\ &{}\quad - \Vert \partial _sS_n(s,\cdot )\Vert _{L^\infty }\Vert Y\Vert _{L^2,s}^2 \\ &{}= 2 \left( c^2 - c\Vert A_n(s,\cdot )\Vert _{L^\infty } - \frac{1}{2}\Vert \partial _sS_n(s,\cdot )\Vert _{L^\infty } \right) \Vert Y\Vert _{L^2,s}^2 \\ &{}= 4g(s) \left( c^2 - c\Vert A_n(s,\cdot )\Vert _{L^\infty } - \frac{1}{2}\Vert \partial _sS_n(s,\cdot )\Vert _{L^\infty } \right) \end{array} \end{aligned}$$

(102)

Choose $0<\epsilon \ll 1$. By means of (97) and (98) we find $n_2\ge n_1,s_2\ge s_1$ such that

$$\begin{aligned} n\ge n_2,s\ge s_2 \Rightarrow c^2 - c\Vert A_n(s,\cdot )\Vert _{L^\infty } - \frac{1}{2}\Vert \partial _sS_n(s,\cdot )\Vert _{L^\infty } \ge (c-\epsilon )^2. \end{aligned}$$

This and (102) together give

$$\begin{aligned} n\ge n_2,s\ge s_2 \Rightarrow g''(s) \ge 4(c-\epsilon )^2g(s). \end{aligned}$$

(103)

Now one uses the following fundamental fact about positive functions satisfying a differential inequality of the above type: if the non-negative $C^2$-function g defined on $[s_2,+\infty )$ satisfies (103) and $g(s) \rightarrow 0$ as $s\rightarrow +\infty$, then

$$\begin{aligned} g(s) \le g(s_2)e^{-2(c-\epsilon )(s-s_2)} \quad \ \ \forall s\ge s_2. \end{aligned}$$

The conclusion of the lemma follows since $2g(s)=e^{-2\delta s}\Vert X(s,\cdot )\Vert _{L^2}^2$. $\square$

Lemma 6.3

Under the hypotheses of Proposition 4.15

$$\begin{aligned} \limsup _{s,n\rightarrow +\infty } \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}K_n(s,\cdot )\Vert _{L^\infty (\mathbb {R}/\mathbb {Z})} < +\infty \end{aligned}$$

(104)

holds for all $\beta _1,\beta _2\ge 0$. In other words, for all $\beta _1,\beta _2\ge 0$ there exist numbers $s(\beta _1,\beta _2)$, $n(\beta _1,\beta _2)$ and $M(\beta _1,\beta _2)$ such that

$$\begin{aligned} s\ge s(\beta _1,\beta _2), n\ge n(\beta _1,\beta _2), t\in \mathbb {R}/\mathbb {Z}\ \ \Rightarrow \ \ |\partial _s^{\beta _1}\partial _t^{\beta _2}K_n(s,t)| \le M(\beta _1,\beta _2). \end{aligned}$$

Proof

If this lemma is not true then we find $\beta _1,\beta _2\ge 0$, sequences $n_l,s_l\rightarrow +\infty$ and $t_l\in \mathbb {R}/\mathbb {Z}$ such that $|\partial _s^{\beta _1}\partial _t^{\beta _2}K_{n_l}(s_l,t_l)| \rightarrow +\infty$. By the hypotheses of Proposition 4.15, $\exists$ $l_k$, $c\in [0,1)$ such that $|\partial _s^{\beta _1}\partial _t^{\beta _2}[K_{n_{l_k}}(s,t)-K^\infty (t+c)](s_{l_k},t_{l_k})| \rightarrow 0$, absurd. $\square$

Lemma 6.4

For every $k\ge 1$ there exists $n_k$ such that

$$\begin{aligned} n\ge n_k, \ \beta _1+\beta _2\le k \ \ \Rightarrow \ \ \lim _{s\rightarrow +\infty } e^{-\delta s} \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}X_n(s,\cdot )\Vert _{L^\infty (\mathbb {R}/\mathbb {Z})} = 0. \end{aligned}$$

(105)

Proof

Fix $p>1$. In this proof we shall need to make use of the standard elliptic estimate which holds for smooth functions $h:\mathbb {R}\times \mathbb {R}/\mathbb {Z}\rightarrow \mathbb {R}^{2m}$ with compact support

$$\begin{aligned} \Vert h\Vert _{W^{{\ell }+1,p}(\mathbb {R}\times \mathbb {R}/\mathbb {Z})} \le C_{\ell } ( \Vert {\bar{\partial }} h\Vert _{W^{{\ell },p}(\mathbb {R}\times \mathbb {R}/\mathbb {Z})} + \Vert h\Vert _{W^{{\ell },p}(\mathbb {R}\times \mathbb {R}/\mathbb {Z})} ) \end{aligned}$$

(106)

for the Cauchy–Riemann operator ${\bar{\partial }} = \partial _s+J_0\partial _t$; here the constant $C_\ell >0$ is independent of h.

Fix a smooth function $\phi :\mathbb {R}\rightarrow [0,1]$ satisfying $\phi |_{[-1,1]}\equiv 1$ and $\mathrm{supp}(\phi ) \subset (-2,2)$. For each $\tau \in \mathbb {R}$ set $\phi _\tau (s)=\phi (s-\tau )$ and $Q_\tau =[\tau -2,\tau +2]\times \mathbb {R}/\mathbb {Z}$. Now we use (104) to find for every $\ell \ge 1$ numbers $s_\ell ,n_\ell$ and $M_\ell$ such that

$$\begin{aligned} s\ge s_\ell ,n\ge n_\ell ,t\in \mathbb {R}/\mathbb {Z}\ \ \Rightarrow \ \ |\partial _s^{\beta _1}\partial _t^{\beta _2}K_n(s,t)| \le M_\ell \ \ \forall \beta _1,\beta _2 \quad \text {with} \ \beta _1+\beta _2\le \ell . \end{aligned}$$

(107)

Using this and (106) we can estimate for $\tau \ge s_{\ell }+2$ and $n\ge n_{\ell }$:

$$\begin{aligned} \begin{array}{ll} &{} \Vert X_n\Vert _{W^{{\ell }+1,p}([\tau -1,\tau +1]\times \mathbb {R}/\mathbb {Z})} \le \Vert \phi _\tau X_n\Vert _{W^{{\ell }+1,p}(\mathbb {R}\times \mathbb {R}/\mathbb {Z})} \\ &{}\quad \le C_{\ell }\left( \Vert {\bar{\partial }}(\phi _\tau X_n)\Vert _{W^{{\ell },p}(\mathbb {R}\times \mathbb {R}/\mathbb {Z})} + \Vert \phi _\tau X_n\Vert _{W^{{\ell },p}(\mathbb {R}\times \mathbb {R}/\mathbb {Z})} \right) \\ &{}\quad \le C'_{\ell }\left( \Vert {\bar{\partial }} X_n\Vert _{W^{{\ell },p}(Q_\tau )} + \Vert X_n\Vert _{W^{{\ell },p}(Q_\tau )} \right) \\ &{}\quad = C'_{\ell }\left( \Vert K_n X_n\Vert _{W^{{\ell },p}(Q_\tau )} + \Vert X_n\Vert _{W^{{\ell },p}(Q_\tau )} \right) \\ &{}\quad \le C''_{\ell } \Vert X_n\Vert _{W^{{\ell },p}(Q_\tau )} \end{array} \end{aligned}$$

(108)

where $C'_{\ell }$ depends on $C_{\ell }$ and the derivatives of $\phi$ up to order $\ell +1$, and $C''_{\ell }$ depends on $C'_{\ell }$ and $M_{\ell }$. Since $X_n\in E$ for all n, we find that $\Vert X_n\Vert _{W^{l,p}(Q_\tau )}$ decays like $e^{\delta \tau }$ as $\tau \rightarrow +\infty$, where l is the number used in the definition of E. An induction argument will tell us that for any $k\ge 1$, $\Vert X_n\Vert _{W^{k,p}(Q_\tau )}$ decays like $e^{\delta \tau }$ as $\tau \rightarrow +\infty$ if n is larger than some $n_k$. Then (105) follows from the Sobolev embedding theorem. $\square$

With the above lemmata at our disposal, we finally turn to the proof of the proposition. The number $\delta <0$ does not lie in $\sigma (L)$, and we take $\mu \in \sigma (L)$ satisfying $\mu <\delta$ and $(\mu ,\delta ] \cap \sigma (L) = \emptyset$.

Step 1 For every $m\ge 0$ we can find $s_m,n_m$ and $0<r_m<\delta -\mu$ such that

$$\begin{aligned} \left( \sum _{j=0}^m \Vert (\partial _s)^jX_n(s,\cdot ) \Vert _{L^2(\mathbb {R}/\mathbb {Z})}^2 \right) ^{\frac{1}{2}} \le e^{(\delta -r_m)(s-s_m)} \left( \sum _{j=0}^m \Vert (\partial _s)^jX_n(s_m,\cdot ) \Vert _{L^2(\mathbb {R}/\mathbb {Z})}^2\right) ^{\frac{1}{2}} \end{aligned}$$

holds if $s\ge s_m$ and $n\ge n_m$.

Proof of Step 1

Differentiating (78) with respect to s yields

$$\begin{aligned} \begin{array}{ll} &{} \partial _sX_n+J_0\partial _tX_n+K_nX_n = 0 \\ &{} (\partial _s)^2X_n+J_0\partial _t\partial _sX_n+\partial _s(K_nX_n) = 0 \\ &{} \vdots \\ &{} (\partial _s)^{m+1}X_n+J_0\partial _t(\partial _s)^mX_n+(\partial _s)^m(K_nX_n) = 0 \end{array} \end{aligned}$$

This system of equations can be rewritten as a single equation in the form

$$\begin{aligned} \partial _sZ_n + \hat{J}_0\partial _tZ_n + \hat{K}_nZ_n = 0 \end{aligned}$$

where $Z_n = \begin{bmatrix} X_n,\partial _sX_n,\dots ,(\partial _s)^mX_n \end{bmatrix}^T$,

$$\begin{aligned} \hat{J}_0 = \begin{bmatrix} J_0&0&\dots&0 \\ 0&J_0&\dots&0 \\ \vdots&\vdots&\ddots&\vdots \\ 0&0&\dots&J_0 \end{bmatrix} \ \ \ \hat{K}_n = \begin{bmatrix} K_n&0&\dots&0 \\ *&K_n&\dots&0 \\ \vdots&\vdots&\ddots&\vdots \\ *&*&\dots&K_n \end{bmatrix} \end{aligned}$$

where every block-entry of the lower triangular block-matrix $\hat{K}_n$ below the diagonal is a term $\Delta _n(s,t)$ satisfying

$$\begin{aligned} \lim _{s,n\rightarrow +\infty } \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}\Delta _n(s,\cdot )\Vert _{L^\infty (\mathbb {R}/\mathbb {Z})} = 0 \quad \ \ \forall \beta _1,\beta _2\ge 0. \end{aligned}$$

We can now apply Lemma 6.2 to $Z_n$ in order to obtain get $s_m,n_m,r_m$ with the properties we desired. $\square$

Step 2 For every integer $q\ge 0$ we can find numbers $s_q$, $n_q$, and $0<r_q<\delta -\mu$, $c_q$ such that

$$\begin{aligned} \begin{array}{ll} \max _{\beta _1+\beta _2\le q} &{} \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}X_n(s,\cdot )\Vert _{L^\infty (\mathbb {R}/\mathbb {Z})}\\ &{} \le c_q e^{(\delta -r_q)(s-s_q)} \max _{\beta _1+\beta _2\le q+1} \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}X_n(s_q,\cdot ) \Vert _{L^\infty (\mathbb {R}/\mathbb {Z})} \end{array} \end{aligned}$$

(109)

holds for all $s\ge s_q$ and $n\ge n_q$.

Proof of Step 2

We will prove by induction that for every $m\ge 0$ there exist $k_m,s_m,n_m$ and $0<\rho _m<\delta -\mu$ such that

$$\begin{aligned} \begin{array}{ll} &{} \sqrt{\sum _{\beta _1+\beta _2\le m} \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}X_n(s,\cdot )\Vert _{L^2(\mathbb {R}/\mathbb {Z})}^2}\\ &{}\quad \le k_m e^{(\delta -\rho _m)(s-s_m)} \sqrt{\sum _{\beta _1+\beta _2\le m} \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}X_n(s_m,\cdot )\Vert _{L^2(\mathbb {R}/\mathbb {Z})}^2} \end{array} \end{aligned}$$

(110)

holds for when $s\ge s_m$ and $n\ge n_m$. If we succeed in proving (110) then the proof of Step 2 will be complete in view of the Sobolev embedding theorem, which tells us that $W^{1,2}(\mathbb {R}/\mathbb {Z}) \hookrightarrow L^\infty (\mathbb {R}/\mathbb {Z})$ continuously.

Now we proceed with the proof of (110). The case $m=0$ is a special case of Step 1. Assuming (110) holds for m we now show that it also holds for $m+1$. However, this induction step will be proved by a separate induction argument: we will show, using induction in $0\le j\le m+1$, that for every such j we can find $s',n'$, $c'$ and $0<\rho '<\delta -\mu$ such that

$$\begin{aligned} \Vert \partial _s^{m+1-j}\partial _t^jX_n(s,\cdot )\Vert _{L^2(\mathbb {R}/\mathbb {Z})} \le c'e^{(\delta -\rho ')(s-s')} \sqrt{\Sigma ' \Vert \partial _s^{\beta _1}\partial _t^{\beta _2}X_n(s',\cdot ) \Vert _{L^2(\mathbb {R}/\mathbb {Z})}^2} \end{aligned}$$

(111)

holds if $s\ge s'$ and $n\ge n'$, where the sum $\Sigma '$ indicates a sum over all numbers $\beta _1,\beta _2\ge 0$ satisfying either $\beta _1+\beta _2\le m$, or $\beta _1+\beta _2=m+1$ and $\beta _2\le j$. The case $j=0$ follows from Step 1 and our previous induction hypothesis. Fix $b\le m+1$ and assume that (111) holds for all $0\le j\le b-1$. By (78) we get

$$\begin{aligned} \begin{array}{ll} &{}\partial _s(\partial _s^{m+1-b}\partial _t^{b-1}X_n)+J_0\partial _s^{m+1-b}\partial _t^bX_n\\ &{}\quad = \partial _s^{m+1-b}\partial _t^{b-1}(\partial _sX_n+J_0\partial _tX_n) \\ &{}\quad = \partial _s^{m+1-b}\partial _t^{b-1}(-K_nX_n) \end{array} \end{aligned}$$

Hence

$$\begin{aligned} \partial _s^{m+1-b}\partial _t^bX_n = J_0(\partial _s^{m+2-b}\partial _t^{b-1}X_n + \partial _s^{m+1-b}\partial _t^{b-1}(K_nX_n)). \end{aligned}$$

This equation, the uniform asymptotic bounds (104) on derivatives of $K_n$ and the induction hypothesis together prove (111) for $0\le j\le b$. Hence (111) holds for all $0\le j\le m+1$, which together with (110) for m gives (110) for $m+1$. The proof of Step 2 is complete. $\square$

We are ready for the final

Step 3 Some subsequence of $X_n$ converges in $C^{l,\alpha ,\delta }_0$.

Proof of Step 3

Since $X_n$ is $C^\infty _\mathrm{loc}$-bounded we can find $X_\infty (s,t)$ smooth and assume, up to selection of a subsequence, that $X_n\rightarrow X_\infty$ in $C^\infty _\mathrm{loc}$. We will show now that

$$\begin{aligned} \lim _{n\rightarrow \infty } \ \ \left[ \sup _{(s,t)\in [0,+\infty )\times \mathbb {R}/\mathbb {Z}} e^{-\delta s}| \partial _s^{\beta _1}\partial _t^{\beta _2}[X_n-X_\infty ]| \right] = 0 \end{aligned}$$

(112)

holds for every $\beta _1,\beta _2\ge 0$. Let $\beta _1,\beta _2\ge 0$ and $\epsilon >0$ be fixed arbitrarily. By Step 2 we find $s',n'$ such that

$$\begin{aligned} \sup _{s\ge s', n\ge n'} e^{-\delta s}| \partial _s^{\beta _1}\partial _t^{\beta _2}X_n(s,t)| \le \epsilon /2. \end{aligned}$$

(113)

For this we used formula (109) and the fact that $X_n\rightarrow X_\infty$ in $C^\infty _\mathrm{loc}$. In particular, taking the limit as $n\rightarrow \infty$ one gets

$$\begin{aligned} \sup _{s\ge s'} e^{-\delta s}| \partial _s^{\beta _1}\partial _t^{\beta _2}X_\infty (s,t)| \le \epsilon /2. \end{aligned}$$

(114)

In view of the $C^\infty _\mathrm{loc}$-convergence we find $n''\ge n'$ such that

$$\begin{aligned} n\ge n'' \Rightarrow \sup _{(s,t) \in [0,s']\times \mathbb {R}/\mathbb {Z}} e^{-\delta s}| \partial _s^{\beta _1}\partial _t^{\beta _2}[X_n-X_\infty ]| \le \epsilon . \end{aligned}$$

(115)

Putting (113)–(115) together we obtain

$$\begin{aligned} n\ge n'' \Rightarrow \sup _{(s,t) \in [0,+\infty )\times \mathbb {R}/\mathbb {Z}} e^{-\delta s}| \partial _s^{\beta _1}\partial _t^{\beta _2}[X_n-X_\infty ]| \le \epsilon . \end{aligned}$$

(116)

Since $\epsilon >0$ was arbitrary, this proves (112). Since (112) holds for any $\beta _1,\beta _2\ge 0$ it follows that $X_\infty \in C^{l,\alpha ,\delta }_0$ and $X_n\rightarrow X_\infty$ in $C^{l,\alpha ,\delta }_0$. $\square$

The proof of Proposition 4.15 is now complete.