Motivation

A global surface of section reduces a flow on a closed 3-manifold to an area-preserving map of a surface. It is a compact embedded surface whose boundary consists of periodic orbits, whose interior is transverse to the flow, and which every other trajectory hits infinitely often in forward and backward time. Poincaré introduced the idea for the restricted three-body problem. Once a section is a disk, results on area-preserving disk maps (Brouwer, Franks) give periodic orbits and other structure for the whole flow.

For Hamiltonian flows on star-shaped energy surfaces in $\mathbb{R}^4$, equivalently Reeb flows on the tight 3-sphere, it is natural to ask which periodic orbits bound such a disk. Hryniewicz's criterion answers this for dynamically convex flows, with no genericity assumption. The answer is purely topological: a periodic orbit bounds a disk-like global section exactly when it is unknotted with self-linking number $-1$. The criterion is used in celestial mechanics: Joung and van Koert apply it to validated periodic orbits of the restricted three-body problem (arXiv:2407.19159).

Timeline.

• 1998. Hofer, Wysocki and Zehnder prove that every dynamically convex contact form on $S^3$ has some periodic orbit $P_0$, with Conley–Zehnder index $3$, that bounds a disk-like global section. That disk is a page of an open book adapted to the flow. Strictly convex energy surfaces in $\mathbb{R}^4$ are dynamically convex (Ann. of Math. 148).
• 2008/2012. Hryniewicz proves the "unknotted, self-linking $-1$" characterization in the non-degenerate case (arXiv:0812.4076).
• 2010/2011. Hryniewicz and Salomão treat non-degenerate tight contact forms on $S^3$. Two extra conditions appear there: $\mu_{CZ}\ge 3$, and linking with every orbit of index $2$ (arXiv:1006.0049).
• 2011/2014. Hryniewicz removes non-degeneracy for dynamically convex forms (arXiv:1105.2077, Theorem 1.7). In the same paper, Theorem 1.8 shows that any orbit coming from a fixed point of the first-return map of a disk-like section is again such a binding.
• 2012. Albers, Fish, Frauenfelder, Hofer and van Koert use this circle of ideas to get disk-like sections in the planar circular restricted three-body problem (arXiv:1103.3881).

Setting

Use coordinates $x=(q_1,p_1,q_2,p_2)$ on $\mathbb{R}^4$, the Liouville form $\lambda_0=\tfrac12\sum_j (q_j\,dp_j-p_j\,dq_j)$ and the symplectic form $\omega_0=d\lambda_0=\sum_j dq_j\wedge dp_j$.

Let $H:\mathbb{R}^4\to\mathbb{R}$ be smooth and set $S=H^{-1}(1)$. Assume that every ray from the origin meets $S$ exactly once, and that it crosses $S$ transversally:

$$dH(x)\,x>0\qquad(x\in S).$$

Then $S$ is a strictly star-shaped hypersurface diffeomorphic to $S^3$. Every contact form on $S^3$ that matters below arises this way, up to diffeomorphism (see Formalization scope).

The Hamiltonian vector field $X_H$ is defined by $\iota_{X_H}\omega_0=-dH$. With this sign, $\lambda_0(X_H)=\tfrac12\,dH(x)\,x>0$ on $S$. So $X_H|_S$ is a positive multiple of the Reeb vector field of the contact form $\lambda_0|_S$, and its orbits are the Reeb orbits reparametrized. A periodic orbit $P=(x,T)$ is a solution with $x(T)=x(0)$ and $T>0$. It is prime when $T$ is its least positive period. The contact structure is $\xi=\ker\lambda_0|_S$.

• Conley–Zehnder index. Fix the global frame of $\xi$ given by the quaternionic rotations of $\nabla H$. Along $P$, the linearized flow restricted to $\xi$ is a path $\varphi:[0,1]\to Sp(1)$ with $\varphi(0)=I$. The winding interval $I(\varphi)$ collects the total rotations of all nonzero vectors, measured in turns. Then $\mu_{CZ}(P)$ is the lower semicontinuous index of Hryniewicz's §2.1.1. In particular,

$$\mu_{CZ}(P)\ge 3 \iff \min I(\varphi)>1,$$

that is, every nonzero transverse vector turns by more than one full turn.

• Dynamical convexity. The flow is dynamically convex if $\mu_{CZ}(P)\ge 3$ for every periodic orbit $P$ in $S$, prime or multiply covered.
• Disk-like global surface of section. A smoothly embedded closed disk $D\subset S$ such that $\partial D=x(\mathbb{R})$ for a periodic orbit $P$, $X_H$ is transverse to $D\setminus\partial D$, and every trajectory not contained in $\partial D$ meets $D$ at arbitrarily large positive and negative times. Then $P$ bounds $D$.
• Unknotted. $P$ is unknotted if $x(\mathbb{R})$ is the boundary of some smoothly embedded closed disk in $S$.
• Self-linking number. Push $x$ off itself along the global frame of $\xi$ to a disjoint loop $x'$. Then

$$\operatorname{sl}(P)=\operatorname{lk}(x,x')\in\mathbb{Z},$$

the linking number in $S\cong S^3$, with $S$ oriented as the boundary of the star-shaped domain it bounds. This agrees with Hryniewicz's Definition 1.5, which uses a section of $\xi$ over a spanning disk.

Formalization targets

Goal: Hryniewicz's criterion (Theorem 1.7, first sentence)

For every dynamically convex strictly star-shaped $S$ and every prime periodic orbit $\bar P$:

$$\bar P \text{ bounds a disk-like global surface of section} \iff \bar P \text{ is unknotted and } \operatorname{sl}(\bar P)=-1.$$

The statement fixes no constants and no non-degeneracy, and it covers every dynamically convex star-shaped surface.

Stronger: adapted open book (Theorem 1.7, second sentence)

If $\bar P$ is unknotted with $\operatorname{sl}(\bar P)=-1$, then $S\setminus \bar x(\mathbb{R})$ fibres smoothly over $\mathbb{R}/\mathbb{Z}$. Every fibre is the interior of a disk-like global surface of section whose oriented boundary is $\bar P$.

Further: new bindings from fixed points (Theorem 1.8)

Let $D_0$ be any disk-like global section. Every periodic orbit through a fixed point of the first-return map of $D_0\setminus\partial D_0$ is unknotted, has self-linking number $-1$, and so bounds the page of an adapted open book.

Significance

The result. The criterion turns a dynamical question into a topological check. It does not depend on whether the orbit is degenerate, and degenerate orbits are what one meets at bifurcations and on symmetric levels. Every periodic orbit in a strictly convex energy surface that is unknotted with $\operatorname{sl}=-1$ is a binding, so the flow is organised by many open books at once. Theorem 1.8 makes this concrete: the Hamiltonian flow twists around two different bindings, $\partial D_0$ and $\partial D_1$. In applications, numerically validated orbits become analytic global sections without a separate non-degeneracy check (Joung–van Koert, Theorems 1.2 and 1.5).

Formalizing it. The theorem is proved, in a 50-page paper that relies on Hofer–Wysocki–Zehnder's theory of pseudo-holomorphic curves in symplectizations. It has no machine-checked proof. Mathlib has no Conley–Zehnder index, no self-linking number, no linking number of curves in $S^3$, and no global surfaces of section. A Lean statement fixes every sign and orientation convention involved: the sign of $X_H$, the orientation of $S$, which push-off defines $\operatorname{sl}$, and the index of degenerate orbits. Each of these is easy to get wrong in prose. Formal proofs of the parts that use no holomorphic curves are valuable on their own: the necessity direction, the description of the index by winding intervals, and the explicit ellipsoid examples.

Difficulty

The obvious route is to approximate the contact form by non-degenerate forms $\lambda_k\to\lambda$ that keep $\bar P$ as an orbit, and then apply the non-degenerate theorems. This fails. The $\lambda_k$ need not be dynamically convex. They can have orbits of very high action with $\mu_{CZ}=2$ that are not linked with $\bar P$, so the Hryniewicz–Salomão criterion does not apply to $\lambda_k$ (Hryniewicz, p. 4). The families of planes that would give the pages for $\lambda_k$ have to be controlled directly as $k\to\infty$, and this is not a formal limit argument.

A second obstruction is genuinely global. A disk spanning $\bar P$ and transverse to the flow in its interior is easy to produce when $\operatorname{sl}(\bar P)=-1$. Showing that every trajectory returns to it is the whole content of the theorem, and no local or perturbative argument gives it.

For the formalization, nothing in the proof of sufficiency avoids finite-energy pseudo-holomorphic planes: Fredholm theory, asymptotic analysis, bubbling-off and compactness all enter. None of this exists in Lean.

Formalization scope

• Ambient space. $\mathbb{R}^4$ is Fin 4 → ℝ with coordinates ordered $(q_1,p_1,q_2,p_2)$. $\lambda_0$, $\omega_0$ and $X_H$ are defined explicitly, with the sign conventions above.
• Energy surfaces. Star-shaped surfaces are smooth (ContDiff ℝ ⊤) functions $H$ with the ray condition and $dH(x)x>0$ on $H^{-1}(1)$. The Reeb flow of a general dynamically convex form on $S^3$ reduces to this case, up to diffeomorphism and positive time change: such a form is tight (Hofer–Wysocki–Zehnder), and every tight form on $S^3$ comes from a star-shaped hypersurface (Eliashberg 1992). That reduction is not part of the targets. Hryniewicz makes the same reduction (§3, first paragraph).
• Flow. The flow is the flow of $X_H$, not of the Reeb field. All notions in the targets are invariant under positive time change. Periodic orbits are solutions of $\dot x=X_H(x)$ on all of $\mathbb{R}$. "Prime" means the recorded period is least.
• Index. $\mu_{CZ}\ge 3$ is encoded by the winding-interval condition in the global quaternionic frame, with degenerate orbits included. Multiply covered orbits are included in dynamical convexity.
• Disks. Disks are smooth embeddings of the closed unit disk (injective, with injective differential up to the boundary). The return condition demands hits at arbitrarily large positive and negative times.
• Ruling out vacuous encodings. A version without the two-sided return condition, with a dynamical-convexity predicate that no surface satisfies, or with $\operatorname{sl}$ that is not a linking number of the push-off, proves a different theorem. The ellipsoid milestone below is a non-vacuity check on the definitions.
• Infrastructure. A complete development needs the following.
  • Reusable beyond this mission: the Conley–Zehnder index of paths in $Sp(1)$, the linking number of disjoint loops in $S^3$ (or in $\mathbb{R}^3$ after stereographic projection), and global flows of vector fields on compact level sets.
  • Specific to this proof: contact topology of spanning disks (characteristic foliations, elimination of singularities), and finite-energy planes in $\mathbb{R}\times S^3$ with their Fredholm, asymptotic and compactness theory.
• Contributions welcome. The definition layer, the index and linking-number libraries, the ellipsoid examples, Lemma 2.1, the necessity direction, Lemma 3.12, and any sub-step of the holomorphic-curve argument stated as an independent lemma.

Selected references

• H. Hofer, K. Wysocki, E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. 148 (1998), 197–289. https://doi.org/10.2307/120994
• U. L. Hryniewicz, Fast finite-energy planes in symplectizations and applications, Trans. Amer. Math. Soc. 364 (2012), 1859–1931. https://arxiv.org/abs/0812.4076
• U. L. Hryniewicz, P. A. S. Salomão, On the existence of disk-like global sections for Reeb flows on the tight 3-sphere, Duke Math. J. 160 (2011), 415–465. https://arxiv.org/abs/1006.0049
• U. L. Hryniewicz, Systems of global surfaces of section for dynamically convex Reeb flows on the 3-sphere, J. Symplectic Geom. 12 (2014), 791–862. https://arxiv.org/abs/1105.2077
• U. L. Hryniewicz, P. A. S. Salomão, Global surfaces of section for Reeb flows in dimension three and beyond, Proc. ICM 2018 (extended version). https://arxiv.org/abs/1712.01925
• P. Albers, J. W. Fish, U. Frauenfelder, H. Hofer, O. van Koert, Global surfaces of section in the planar restricted 3-body problem, Arch. Ration. Mech. Anal. 204 (2012), 273–284. https://arxiv.org/abs/1103.3881
• C. Joung, O. van Koert, Computational symplectic topology and symmetric orbits in the restricted three-body problem, Nonlinearity 38 (2025), 025015. https://arxiv.org/abs/2407.19159
• Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier 42 (1992), 165–192. https://doi.org/10.5802/aif.1288