Abstract
Pseudo-holomorphic curves in symplectizations, as introduced by Hofer in 1993, and then developed by Hofer, Wysocki, and Zehnder, have brought new insights to Hamiltonian dynamics, providing new approaches to some classical questions in Celestial Mechanics. This short survey presents some recent developments in Reeb dynamics based on the theory of pseudo-holomorphic curves in symplectizations, focusing on transverse foliations near critical energy surfaces.
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Acknowledgements
P. Salomão acknowledges the support of NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai. P. Salomão is partially supported by FAPESP 2016/25053-8 and CNPq 306106/2016-7.
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Communicated by Claudio Gorodski.
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Appendices
Appendix A: The action functional
Let M be a smooth 3-manifold equipped with a contact form \(\lambda\). Denote by \(\varphi _t,t\in {\mathbb {R}},\) the Reeb flow of \(\lambda\).
The action functional associated with \(\lambda\) is defined on the space of smooth curves \(\gamma : {\mathbb {R}}/ {\mathbb {Z}}\rightarrow M\) as
Let \(\eta \in \Gamma (\gamma ^*TM)\) be a vector field along \(\gamma\) and let \(u:(-\epsilon ,\epsilon )\times {\mathbb {R}}/ {\mathbb {Z}}\rightarrow M\) be a variation of \(\gamma\) so that \(u(0,t)=\gamma (t)\) for all \(t\in {\mathbb {R}}/ {\mathbb {Z}}\) and \(u_s(0,\cdot )=\eta\). We compute the first variation of \({\mathcal {A}}\) at \(\gamma\) in the direction of \(\eta\)
Since \(d\lambda |_{\xi =\ker \lambda }\) is nondegenerate, this implies that \(\gamma\) is a critical point of \({\mathcal {A}}\) if and only if \({{\dot{\gamma }}} \subset \ker d\lambda |_\gamma .\) In particular, if \(\dot{\gamma }\) never vanishes, then \(\gamma\) is a critical point of \({\mathcal {A}}\) if and only if \(\gamma\) is a reparametrization of a closed Reeb orbit.
Now let \(P=(x,T)\) be a closed Reeb orbit of \(\lambda\). Then \(x_T = x(T\cdot ):{\mathbb {R}}/ {\mathbb {Z}}\rightarrow M\) is a critical point of \({\mathcal {A}}\). Let \(\eta \in \Gamma (x_T^*\xi )\) and let \(u:(-\epsilon ,\epsilon ) \times {\mathbb {R}}/ {\mathbb {Z}}\rightarrow M\) be a variation of \(x_T\) so that \(u_s(0,\cdot )=\eta\). We can assume that \(u_s(s,\cdot )\subset \xi , \forall s\). Using a finite covering of a tubular neighborhood of \(x_T({\mathbb {R}})\), we may assume that \(x_T\) is an embedding. Let \(\Xi\) be a vector field extending \(\eta\) in a neighborhood of \(x({\mathbb {R}})\).
The second variation of \({\mathcal {A}}\) at \(x_T\) in the direction of \(\eta\) is
We have used the usual relation \(d\alpha (X,Y)=X \cdot \alpha (Y) - Y \cdot \alpha (X) - \alpha ([X,Y])\), where X, Y are vector fields and \(\alpha\) is a 1-form.
Observe that the last expression obtained in (7) depends only on the linearized flow along P
A similar analysis of the action functional can be found in [32].
Appendix B: The asymptotic operator
Let \(P=(x,T)\) be a closed orbit of the Reeb flow of \(\lambda\) and let \(J:\xi \rightarrow \xi\) be a \(d\lambda\)-compatible complex structure. The asymptotic operator associated with P and J is the linear operator
where \({\mathcal {L}}_{\dot{x}_T} \eta\) is defined in the previous section.
It is an exercise to check that \(A_{P,J}\) admits 0 as an eigenvalue if and only if P is degenerate.
Since J preserves \(d\lambda\) we see from (7) that
where \(J_t = J|_{x_T(t)}\).
If \(\nu \in {\mathbb {R}}\) is an eigenvalue of \(A_{P,J}\) and \(\eta _\nu\) is a non-trivial \(\nu\)-eigenfunction, then
where \(g_J(\cdot , \cdot ) = d\lambda (\cdot , J \cdot )\) is the positive-definite inner product on \(\xi\) induced by \(d\lambda\) and J. Hence \(d^2{\mathcal {A}}(x_T) \cdot (\eta _\nu ,\eta _\nu )\) vanishes if and only if \(\nu =0\) and, otherwise, its sign coincides with the sign of \(\nu\).
In order to better describe the spectrum of the operator \(A_{P,J}\) we need some normalization. Choose a unitary trivialization \(\Psi : x_T^* \xi \rightarrow {\mathbb {R}}/ {\mathbb {Z}}\times {\mathbb {R}}^2\). This means that
Here (x, y) are coordinates in \({\mathbb {R}}^2\).
We claim that the operator
has the form
for some smooth loop \(t \mapsto S(t)\) of \(2\times 2\) symmetric matrices.
Indeed, recall that the Reeb flow of \(\lambda\) preserves \(d\lambda |_\xi\) and thus we find a smooth path of \(2\times 2\) symplectic matrices \(\Phi (t)\), satisfying \(\Phi (0)=I\) and
It represents the linearized flow of \(\frac{1}{T}\lambda\) restricted to \((\xi ,d\lambda |_\xi )\) along \(x_T\) in coordinates \(\Psi\). In particular, a solution \({\bar{\zeta }}(t)\in {\mathbb {R}}^2 \simeq \xi |_{x_T(t)}\) to the linearized flow with initial condition \(\bar{\zeta }(t+s)=\zeta (t+s) \in {\mathbb {R}}^2 \simeq \xi _{x_T(t+s)}\) satisfies
Note that \({\bar{\zeta }}(t)\) depends on s.
Denoting
let us compute \({\mathcal {L}}_{\dot{x}_T} \eta\) in coordinates \(\Psi\)
Now let
Since \(\Phi (t)\) is symplectic we have
Hence
Using the definition of S we compute
It follows from (12) that \(S(t)^T=S(t), \forall t\). The reader can easily check that (9) implies \(S(t+1)=S(t) \forall t.\) Finally, (8) follows from (10), (11) and the identity
We check the symmetry of \(A_{P,J}\) using coordinates \(\Psi\). The \(L^2\)-product on \(\Gamma (x_T^*\xi )\) induced by \(d\lambda\) and J takes the form
Integrating by parts, we obtain
The eigenvalues of \({\mathcal {L}}_S\) are real. A non-trivial \(\nu\)-eigenfunction \(\zeta _\nu\) of \({\mathcal {L}}_S\) satisfies the smooth linear ODE
and thus \(\zeta _\nu\) is smooth and never vanishes. In particular, \(\zeta _\nu\) has a well-defined winding number
where \(\zeta _\nu (t) = (r(t)\cos (\Theta (t)),r(t) \sin (\Theta (t))) \forall t,\) for continuous functions \(r>0,\Theta\). It does not depend on the \(\nu\)-eigenfunction.
If the linearized first return map \(D\varphi _T(x_T(0)):\xi |_{x_T(0)} \rightarrow \xi _{x_T(0)}\) is the identity map then it is always possible to choose \(J\in {\mathcal {J}}_+(\xi )\) and a trivialization \(\Psi\) so that \(S \equiv 0\). In this case, the eigenvalues of \({\mathcal {L}}_0=-J_0\frac{d}{dt}\) are
Each \(\nu _k\) admits a 2-dimensional eigenspace
whose eigenfunctions have winding number k.
The next theorem asserts that the general case is similar.
Theorem B.1
([19]) Let \(t\mapsto S(t)\) be a smooth loop of \(2\times 2\) symmetric matrices and let \({\mathcal {L}}_S\) be defined as in (8). Then
-
(i)
The spectrum \(\sigma ({\mathcal {L}}_S)\) consists of real eigenvalues which accumulate precisely at \(\pm \infty\).
-
(ii)
The winding number \(\mathrm{wind}(\nu )\in {\mathbb {Z}},\nu \in \sigma ({\mathcal {L}}_S),\) is independent of the \(\nu\)-eigenfunction.
-
(iii)
The map
$$\begin{aligned} \nu \mapsto \mathrm{wind}(\nu )\in {\mathbb {Z}}, \end{aligned}$$is a surjective increasing map. For every \(k\in {\mathbb {Z}}\), there exist precisely two eigenvalues \(\nu _k^1,\nu _k^2\), counting multiplicities, so that
$$\begin{aligned} \mathrm{wind}(\nu _k^1)=\mathrm{wind}(\nu _k^2)=k. \end{aligned}$$ -
(iv)
If \(\nu _1\ne \nu _2\) satisfy \(\mathrm{wind}(\nu _1)=\mathrm{wind}(\nu _2)\), then any two \(\nu _1,\nu _2\)-eigenfunctions are pointwise linearly independent.
Appendix C: The generalized Conley–Zehnder index
Let \(P=(x,T)\) be a closed Reeb orbit of \(\lambda\) and let \(\tau\) be a unitary trivialization of \(x_T^*\xi\). Let \({\mathcal {L}}_S\) be the operator defined in (8). Let
and let
These winding numbers satisfy \(0\le \mathrm{wind}^\tau _+(P)-\mathrm{wind}^\tau _-(P)\le 1\) and they do not depend on J.
Definition C.1
( [20]) The generalized Conley–Zehnder index of \(P=(x,T)\) with respect to \(\tau\) is defined as
This definition immediately implies that if \(\nu \in \sigma (A_{P,J})\), then
A more geometric definition of \(\mathrm{CZ}^\tau\) is as follows. Consider \(\Phi _t,t\in [0,1]\), the family of \(2\times 2\) symplectic matrices representing the linearized Reeb flow on \(\xi\) along P in coordinates induced by \(\tau\) as in the previous section.
For each initial condition \(0\ne {\bar{\zeta }}\) let \(\Theta _{\bar{\zeta }}(t)\) be a continuous argument of \(\Phi _t{\bar{\zeta }}\) with \(t\in [0,1]\). Let
and
be the interval containing the argument variations of all initial conditions.
The length of \(I^\tau (P)\) is less than \(\frac{1}{2}\) and thus, for each \(\epsilon >0\) small, either \(I^\tau (P)-\epsilon\) contains an integer k or is contained in between two consecutive integers k and \(k+1\). In the first case we define \({{\widetilde{\mu }}}^\tau (P) := 2k\) and in the second case, \({{\widetilde{\mu }}}^\tau (P)= 2k+1\).
It is a simple exercise to check that
Moreover, P is nondegenerate if and only if the boundary of \(I^\tau (P)\) does not contain an integer. For a proof to these facts, see [22].
Appendix D: The quaternionic trivialization
Let \(S \subset {\mathbb {R}}^4\) be a regular energy level of a Hamiltonian function H, that is \(S=H^{-1}(c), c \in {\mathbb {R}},\) and \(\nabla H|_S\) never vanishes. The quaternion group induces an orthonormal frame
spanned by the vector fields
Here, the \(4\times 4\) matrices \(A_i,i=1,2,3,\) are
where 0, I and J are the \(2 \times 2\) matrices
Observe that \(X_3\) is parallel to the Hamiltonian vector field \(X_H=A_3 \nabla H\).
Denote by \(\phi _t\) the Hamiltonian flow of \(X_H\) restricted to S. Since \(d\phi _t: TS \rightarrow TS\) preserves the line bundle \({\mathbb {R}}X_3\), one may restrict the study of the linearized flow to
As discussed in Sect. 1.1, if S has contact-type then \(X_H|_S\) is parallel to a Reeb vector field R of a contact form \(\lambda\) on S. In this case, the contact structure \(\xi =\ker \lambda\) is transverse to R and as a result we have \(\xi \simeq \mathrm{span} \{X_1,X_2\}\). This makes the quaternionic trivialization a useful tool for estimating Conley–Zehnder indices.
The orthonormal frame (13) induces a trivialization \(\Psi : TS \rightarrow S \times {\mathbb {R}}^3\)
which provides a simple form to analyze the transverse linearized flow along Hamiltonian trajectories.
Proposition D.1
In coordinates \((a_1,a_2)\in {\mathbb {R}}^2\) induced by the trivialization (16), a solution to the linearized flow along a non-constant trajectory of \(X_H\), projected to \(TS/{\mathbb {R}}X_3\), satisfies the equation
where J is given in (15), M is the symmetric matrix
and
The matrix \(\mathbf{H}=\mathbf{H}(x)\) is the Hessian of H at \(x\in S\), \(X_i, i=1,2,3\), is given by (14), and \(\langle \cdot ,\cdot \rangle\) is the standard inner product on \({\mathbb {R}}^4\).
Proof
Let \(x(t)\in S\) be a Hamiltonian trajectory, that is a solution to
A solution \(y(t)\in T_{x(t)}S\) to the linearized flow \(d\phi _t:TS\rightarrow TS\) along x(t) satisfies the linear differential equation
Substituting \(y=a_1X_1 + a_2X_2+a_3X_3\) in (19), we obtain
We may assume for simplicity that \(|\nabla H|=1\). In particular, it follows from (14) and (18) that
Taking the inner product of the expression (20) with \(X_1\), using (21) and the relations
we obtain
This is precisely the expression for \(\dot{a}_1\) in (17). Analogously one obtains
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de Paulo, N.V., Salomão, P.A.S. Reeb flows, pseudo-holomorphic curves and transverse foliations. São Paulo J. Math. Sci. 16, 314–339 (2022). https://doi.org/10.1007/s40863-022-00285-0
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DOI: https://doi.org/10.1007/s40863-022-00285-0