Duan–Long–Wang conjecture on the dichotomy of the number of prime closed geodesics on compact simply-connected Finsler manifolds with rational cohomology $T_{d,n+1}(x)$

On a compact simply-connected Finsler manifold $(M,F)$ whose rational cohomology is a truncated polynomial ring $H^*(M;\mathbb{Q})\cong T_{d,n+1}(x)=\mathbb{Q}[x]/(x^{n+1})$ with generator degree $d\ge 2$ and height $n+1\ge 2$ , Duan, Long and Wang (2016) conjectured that there are either infinitely many distinct prime closed geodesics, or exactly $\frac{dn(n+1)}{2}$ of them when $d$ is even and exactly $d+1$ when $d$ is odd; and in the finite case every prime closed geodesic must be elliptic with no eigenvalue of its linearized Poincaré map equal to $\pm 1$ . The predicted finite counts are realized by Katok–Ziller irreversible Finsler metrics on rank-one symmetric spaces, so the conjecture asserts these examples are extremal. The conjecture is open in general: it subsumes the Katok–Anosov conjecture on Finsler spheres (open for $d\ge 3$ in full generality) and, in its infinite branch for reversible metrics, the classical infinitely-many-closed-geodesics problem on the exceptional class of manifolds with bounded free-loop-space Betti numbers. Only restricted cases (bumpy metrics, curvature pinching, or the all-elliptic hypothesis) are known.

Background

A Finsler manifold is a smooth manifold with a norm on each tangent space that varies smoothly and comes from an inner product only in the special (Riemannian) case; a Finsler metric is called irreversible if $F(X)\ne F(-X)$ can occur, so that traversing a geodesic backwards need not give a geodesic, and its reversibility is the number $\lambda=\max\{F(-X)\mid F(X)=1\}\ge 1$ , equal to $1$ exactly for reversible metrics. A closed geodesic is prime if it is not an iterate of a shorter one, and two prime closed geodesics are distinct if they differ by more than a shift of parametrization. For a closed geodesic $c$ on an $n$ -dimensional manifold, the linearized Poincaré map $P_c$ is a symplectic matrix in $\mathrm{Sp}(2n-2)$ obtained by linearizing the geodesic flow around $c$ ; $c$ is elliptic if all eigenvalues of $P_c$ lie on the unit circle, irrationally elliptic if it is elliptic and no eigenvalue is a root of unity — equivalently, all rotation angles are irrational multiples of $\pi$ (the source paper [1] states its Remark 1.9(ii) with the weaker wording "all eigenvalues lying on the unit circle with none equal to $\pm 1$ ", which excludes only $\pm 1$ ; the source's wording is implied by, and strictly weaker than, the root-of-unity reading adopted here, the two differing exactly when $c$ has an eigenvalue that is a primitive $m$ -th root of unity of order $m\ge 3$ ) — and hyperbolic if none lie on the unit circle. A metric is bumpy if 1 is never an eigenvalue of the Poincaré map of any iterate of any closed geodesic — a generic condition. The Morse index $i(c)$ is the number of negative directions of the second variation (index form) of the energy functional at $c$ , and the flag curvature generalizes Riemannian sectional curvature to Finsler geometry. By the classical work of Gromoll–Meyer and Vigué-Poirrier–Sullivan (as recounted in the source paper [1]), the free loop space of a compact simply-connected manifold has unbounded Betti numbers (ranks of the rational homology groups) exactly unless $H^*(M;\mathbb{Q})\cong T_{d,n+1}(x)=\mathbb{Q}[x]/(x^{n+1})$ with $|x|=d\ge 2$ , height $n+1\ge 2$ , $\dim M=dn$ ; this class contains the spheres $S^d$ ( $n=1$ ), the complex projective spaces $\mathbb{C}P^n$ ( $d=2$ ), the quaternionic projective spaces $\mathbb{H}P^n$ ( $d=4$ ), and the Cayley plane $\mathbb{O}P^2$ ( $d=8$ , $n=2$ ), and is precisely the class where Morse-theoretic infinitude arguments for closed geodesics break down; note also that when $d$ is odd, graded commutativity forces $x^2=0$ , so $n=1$ and $M$ is rationally a sphere. For Riemannian metrics it is famously conjectured that every compact manifold carries infinitely many closed geodesics; on the exceptional class above this remains open except for Riemannian $S^2$ . For Finsler metrics the situation changes qualitatively: Katok [5] constructed irreversible Finsler metrics on rank-one symmetric spaces with only finitely many closed geodesics — on spheres exactly $2\lfloor (d+1)/2\rfloor$ of them — and Ziller [6] developed the geometry of these metrics on $\mathbb{C}P^n$ , $\mathbb{H}P^n$ and the Cayley plane, where the counts are $n(n+1)$ , $2n(n+1)$ and $24$ respectively, i.e. $\frac{dn(n+1)}{2}$ throughout, with all closed geodesics irrationally elliptic. For spheres this motivated the Katok–Anosov conjecture — every Finsler $S^d$ has at least $2\lfloor (d+1)/2\rfloor$ closed geodesics — proved by Bangert and Long only for $d=2$ . The main tool in higher dimensions is the Maslov-type index iteration theory for symplectic paths — the Maslov-type index is an integer-valued homotopy invariant of a path of symplectic matrices that counts signed crossings of the eigenvalue- $1$ variety — together with the common index jump theorem (CIJT) of Long–Zhu, which produces, for finitely many closed geodesics, common iterates whose Morse indices jump by prescribed common amounts; Duan–Long–Wang [1] enhanced it (the enhanced CIJT, or ECIJT) to give precise indices of carefully selected iterates of all closed geodesics simultaneously. Using the ECIJT, the source paper [1] proves only restricted precursors of its conjecture: for bumpy irreversible metrics with $d$ even, finitely many prime closed geodesics of positive Morse index, and $K\ge 0$ (which forces positive index), there are at least $\frac{dn(n+1)}{2}$ non-hyperbolic prime closed geodesics (Theorems 1.1 and 1.3); for bumpy odd spheres under an index or curvature-pinning condition there are at least $d+1$ (Theorems 1.5–1.6); and for bumpy metrics with $d$ even there is at least one non-hyperbolic closed geodesic or infinitely many (Theorem 1.8). Whether the exact dichotomy and the irrational ellipticity hold without these genericity and curvature hypotheses is the open content of the conjecture.

Problem Statement

Prove or refute the following strengthening of the conjecture of Duan–Long–Wang (2016): on every compact simply-connected Finsler manifold $(M,F)$ with $H^*(M;\mathbb{Q})\cong T_{d,n+1}(x)=\mathbb{Q}[x]/(x^{n+1})$ for some $d\ge 2$ and $n\ge 1$ (so $\dim M=dn$ ; when $d$ is odd, graded commutativity forces $x^2=0$ , hence $n=1$ and $M$ is rationally a sphere), there exist either infinitely many distinct prime closed geodesics, or exactly $\frac{dn(n+1)}{2}$ distinct prime closed geodesics when $d\ge 2$ is even and exactly $d+1$ distinct prime closed geodesics when $d\ge 2$ is odd; and in the latter finite case, every prime closed geodesic must be irrationally elliptic (elliptic, with no eigenvalue of its linearized Poincaré map a root of unity, i.e. all rotation angles irrational multiples of $\pi$ ; it is strictly stronger than the wording by which [1] states its Remark 1.9(ii), namely "all eigenvalues on the unit circle with none equal to $\pm 1$ " — the latter is implied by, and does not imply, the former, the two differing exactly when the geodesic has an eigenvalue that is a primitive $m$ -th root of unity of order $m\ge 3$ — so a proof of the statement below gives the conjecture of [1] as a corollary). No genericity (bumpiness), curvature, reversibility, or Morse-index hypothesis may be assumed; a resolution must settle the dichotomy and the irrational-ellipticity conclusion in this generality. Note that the $n=1$ subcase is the Katok–Anosov conjecture for Finsler spheres, itself open for $d\ge 3$ without auxiliary hypotheses, and the infinite branch restricted to reversible (in particular Riemannian) metrics is the classical infinitely-many-closed-geodesics conjecture on this exceptional cohomology class.

References

Huagui Duan, Yiming Long, Wei Wang, The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds, Calculus of Variations and Partial Differential Equations 55 (2016), no. 6, Article 145. DOI: 10.1007/s00526-016-1075-7, https://doi.org/10.1007/s00526-016-1075-7 (preprint: arXiv:1510.02872, https://arxiv.org/abs/1510.02872)
Huagui Duan, Dong Xie, Multiplicity of closed geodesics on bumpy Finsler manifolds with elliptic closed geodesics, Journal of Functional Analysis 284 (2023), no. 8, Article 109861. DOI: 10.1016/j.jfa.2023.109861, https://doi.org/10.1016/j.jfa.2023.109861 (preprint: arXiv:2201.10975, https://arxiv.org/abs/2201.10975)
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