The multiplicity and elliptic-orbit conjectures for arbitrary Reeb flows on the standard contact sphere

Two long-standing conjectures in symplectic dynamics assert that every contact form on the standard contact sphere $(S^{2n+1},\xi_{\mathrm{std}})$ carries at least $n+1$ simple (geometrically distinct, non-iterated) closed Reeb orbits, and at least one elliptic closed Reeb orbit, i.e. an orbit whose linearized Poincaré return map has all eigenvalues on the unit circle. The multiplicity conjecture is a theorem in dimension three ($n=1$) but open for every $n\ge 2$ without auxiliary hypotheses such as convexity, dynamical convexity, or nondegeneracy; the elliptic-orbit conjecture is open in every dimension, including dimension three, without such hypotheses. As of the 2022–2026 literature, both conjectures remain open: the multiplicity conjecture is now proved within the dynamically convex class (Çineli–Ginzburg–Gürel, to appear in Annals of Mathematics), but for $n\ge 2$, for completely general possibly degenerate contact forms, not even the existence of a second closed orbit is known, and the elliptic-orbit conjecture is open in all dimensions, including three, except under symmetry, finiteness, or dynamical-convexity assumptions. The problem asks to settle both conjectures for all $n\ge 1$: prove each in full generality or refute it by an explicit contact form.

Background

A contact form on a $(2n+1)$-dimensional manifold is a $1$-form $\beta$ for which $\beta\wedge(d\beta)^{n}$ never vanishes; its kernel is a contact structure, and there is a unique Reeb vector field $R_{\beta}$ characterized by $\beta(R_{\beta})=1$ and $d\beta(R_{\beta},\cdot)=0$. Closed trajectories of $R_{\beta}$, called closed Reeb orbits, are the periodic solutions of this first-order system and the central dynamical objects of symplectic geometry: Rabinowitz's theorem guarantees at least one closed orbit on every star-shaped hypersurface, and the Weinstein conjecture — that every closed contact manifold carries at least one closed orbit — is a theorem in dimension three (Taubes) and in many other classes, but the number of orbits and their stability type are understood only under strong auxiliary hypotheses. On the standard contact sphere $(S^{2n+1},\xi_{\mathrm{std}})\subset\mathbb{R}^{2n+2}$, where $\xi_{\mathrm{std}}$ is the kernel of the restriction of the Liouville form $\lambda=\frac{1}{2}\sum_{i}(x_{i}\,dy_{i}-y_{i}\,dx_{i})$, every contact form supporting $\xi_{\mathrm{std}}$ is of the shape $f\lambda|_{S^{2n+1}}$ for a positive function $f$; the correspondence $\beta=f\lambda|_{S^{2n+1}}\leftrightarrow\Sigma_{\beta}=\{\sqrt{f(x)}\,x\}$ identifies arbitrary contact forms on the sphere with arbitrary smooth compact star-shaped hypersurfaces of $\mathbb{R}^{2n+2}$ (hypersurfaces enclosing the origin and meeting every ray from it exactly once), so the question is equally about Hamiltonian flows on such energy surfaces.

Two attributes of closed orbits matter here. An orbit is simple (or prime) if it is not an iterate of a shorter closed orbit, and orbits are counted up to geometric distinctness. An orbit is elliptic if every eigenvalue of its linearized Poincaré return map — the $2n\times 2n$ symplectic matrix obtained by linearizing the flow on a section transverse to the orbit and following it once around — has modulus one; elliptic orbits are the candidates for KAM stability, in contrast to hyperbolic orbits, for which all eigenvalues have modulus different from one.

Two conjectures, studied since the 1980s for convex energy surfaces (the multiplicity problem is recorded in Ekeland's 1990 monograph; pinching results go back to Ekeland–Lasry) and stated for arbitrary contact forms by Ginzburg–Macarini and by Abreu–Macarini, predict a universal minimum for Reeb dynamics on the standard contact sphere: every contact form on $(S^{2n+1},\xi_{\mathrm{std}})$ carries (M) at least $n+1$ simple closed Reeb orbits and (E) at least one elliptic closed orbit. Both bounds are sharp: an ellipsoid in $\mathbb{R}^{2n+2}$ with pairwise rationally independent squared semi-axes has exactly $n+1$ simple closed orbits, all elliptic.

In dimension three ($n=1$) the multiplicity statement (M) is a theorem: Cristofaro-Gardiner and Hutchings proved, via embedded contact homology (a Floer-type theory built from holomorphic curves in the symplectization of the contact manifold), that every contact form on any closed three-manifold has at least two closed Reeb orbits, with an independent proof for $S^{3}$ by Ginzburg–Hein–Hryniewicz–Macarini based on symplectically degenerate maxima (a degenerate critical-point mechanism for the action functional); the two-or-infinitely-many dichotomy on all closed contact three-manifolds with torsion first Chern class was recently completed by Cristofaro-Gardiner–Hryniewicz–Hutchings–Liu. The elliptic statement (E), in contrast, is open in every dimension, including dimension three: when a three-dimensional flow has only finitely many simple orbits, the dichotomy forces exactly two, both irrationally elliptic, but no unconditional result covers flows with infinitely many closed orbits, and the conjecture is completely open in any dimension bigger than one. For $n\ge 2$ every general result requires hypotheses. Under strict convexity plus antipodal symmetry, Liu–Long–Zhu obtained at least $n+1$ closed orbits and Dell'Antonio–D'Onofrio–Ekeland obtained an elliptic orbit; for general convex hypersurfaces Long–Zhu proved the multiplicity bound in the nondegenerate case. Convexity implies dynamical convexity (Hofer–Wysocki–Zehnder): every closed orbit has Conley–Zehnder index — the integer grading measuring the winding of the linearized flow, computable from the Poincaré return map — at least $n+2$, the value attained by ellipsoids. Within the dynamically convex class the multiplicity bound progressed from $\lceil(n+1)/2\rceil+1$ (Duan–Liu; Ginzburg–Gürel) to the full conjectured $n+1$ (Çineli–Ginzburg–Gürel 2024, using filtered symplectic homology — the Floer homology of a Liouville filling, whose positive equivariant part encodes closed Reeb orbits — and its persistence-module structure), so conjecture (M) is now proved under dynamical convexity; without any such hypothesis not even the existence of two closed orbits is known for possibly degenerate flows, and for nondegenerate flows the best general bound is two (Abreu–Gutt–Kang–Macarini). On the elliptic side, conditional results are: dynamically convex and antipodally symmetric forms have an elliptic orbit (Abreu–Macarini 2017); dynamically convex forms with exactly $n+1$ prime orbits have two irrationally elliptic ones (Li–Liu–Wang 2025); convex hypersurfaces with finitely many closed characteristics have at least two elliptic ones (Liu–Ou 2026). Abreu–Macarini (2022) proves sharp dynamical implications of convexity on $\mathbb{Z}_{p}$-symmetric spheres and lens spaces — forced elliptic and non-hyperbolic orbits under pinching conditions, and $C^{1}$-stable examples of dynamically convex symmetric contact forms not equivalent to convex ones by symmetry-preserving contactomorphisms — thereby showing that dynamical convexity alone does not reproduce the index thresholds that convexity provides, while leaving the two unconditional conjectures (M) and (E) untouched. These two conjectures form the natural strong form of the Weinstein conjecture on the one contact manifold whose topology is fixed and simplest, where the difficulty is purely dynamical.

Problem Statement

Let $n\ge 1$ and let $\beta$ be an arbitrary smooth contact form supporting the standard contact structure $\xi_{\mathrm{std}}$ on $S^{2n+1}$; equivalently, let $\Sigma\subset\mathbb{R}^{2n+2}$ be an arbitrary smooth compact star-shaped hypersurface carrying the Reeb flow of the restricted Liouville form. No nondegeneracy, convexity, dynamical-convexity, index, pinching, or symmetry assumption may be imposed. Settle, for each of the following two conjectures, whether it holds for all $n\ge 1$ and all such $\beta$ — by a complete proof, or by an explicit counterexample for some $n\ge 1$:

(M) Multiplicity: $\beta$ carries at least $n+1$ simple closed Reeb orbits (geometrically distinct, non-iterated periodic trajectories of the Reeb vector field $R_{\beta}$). The case $n=1$ is a theorem — every contact form on a closed three-manifold has at least two simple closed orbits — so the open content of (M) is the case $n\ge 2$, and a counterexample to (M) necessarily has $n\ge 2$.

(E) Ellipticity: $\beta$ carries at least one elliptic closed Reeb orbit, i.e. a closed orbit for which every eigenvalue of the linearized Poincaré return map has modulus one. This is open for every $n\ge 1$, including $n=1$: in dimension three, ellipticity is known only conditionally (for instance, a flow with only finitely many simple closed orbits has exactly two, both irrationally elliptic), and no known result excludes infinitely many closed orbits none of which is elliptic.

The problem is completely settled only when each of (M) and (E) has been either proved in this full generality or refuted; no implication between the two statements is known, and a resolution of one while the other stands is a partial result, not a completion. For calibration, the ellipsoid $\{z=(x_{1},\dots,x_{n+1},y_{1},\dots,y_{n+1})\in\mathbb{R}^{2n+2}:\frac{1}{2}\sum_{i=1}^{n+1}(x_{i}^{2}+y_{i}^{2})/r_{i}=1\}$ with $r_{i}/r_{j}\notin\mathbb{Q}$ for $i\ne j$ has exactly $n+1$ simple closed orbits, all elliptic, so neither bound can be improved.

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