The Katok–Anosov conjecture: sharp multiplicity of closed geodesics on Finsler spheres

Katok (1973) constructed non-reversible Finsler metrics on the $n$-sphere $S^n$ with exactly $2\lfloor(n+1)/2\rfloor$ distinct prime closed geodesics, and Anosov (1974) conjectured this number to be the sharp universal lower bound: every Finsler metric on $S^n$ carries at least $2\lfloor(n+1)/2\rfloor$ distinct prime closed geodesics. This Katok–Anosov conjecture — together with Ziller's weaker companion conjecture of at least $n$ — is proved for $S^2$, for bumpy (non-degenerate) metrics with positively pinched flag curvature, and for strongly pinched positively curved spheres in even dimensions, but it remains open in full generality for every $n\ge3$: without non-degeneracy or curvature hypotheses, not even two prime closed geodesics are currently known on every Finsler $n$-sphere.

Background

A Finsler metric on a smooth manifold $M$ is a continuous function $F:TM\to[0,\infty)$, smooth away from the zero section, positively homogeneous of degree one in the velocity ($F(x,\lambda y)=\lambda F(x,y)$ for $\lambda>0$), and such that the fiberwise second vertical derivatives of $F^2$ define a positive definite quadratic form; Riemannian metrics are the special reversible case $F(x,y)=\sqrt{g_x(y,y)}$. Unlike a Riemannian metric, a Finsler metric need not be reversible — $F(x,-y)$ may differ from $F(x,y)$ — so its geodesic flow has a preferred direction of travel. A closed geodesic is a periodic solution of the geodesic equation; it is prime if it is not an iterate $c^m(t)=c(mt)$, $m\ge2$, of another one, and two prime closed geodesics are distinct if they differ by more than a shift of the circle parameter. For a non-reversible metric the reversed curve of a closed geodesic need not be a geodesic, so direction of travel matters when counting.

A classical existence theorem guarantees at least one closed geodesic on every compact Riemannian manifold, and a long-standing conjecture asserts that every compact Riemannian manifold carries infinitely many prime closed geodesics; this has been proved except for the compact rank one symmetric spaces (CROSSs: spheres, real, complex and quaternionic projective spaces, and the Cayley projective plane), with the case of the Riemannian 2-sphere settled by Franks (1992) and Bangert (1993). In the Finsler category the analogous infinitude statement is false: Katok (1973) constructed non-reversible Finsler metrics on CROSSs — in particular on every sphere $S^n$ — with only finitely many prime closed geodesics, exactly $2\lfloor(n+1)/2\rfloor$ of them on $S^n$, all non-degenerate and elliptic; Ziller (1983) analyzed these examples in detail. Note that the count equals $n$ when $n$ is even and $n+1$ when $n$ is odd, i.e. $2k$ on both $S^{2k}$ and $S^{2k-1}$.

Motivated by these examples, Anosov conjectured in his 1974 ICM address that every Finsler metric on $S^n$ carries at least $2\lfloor(n+1)/2\rfloor$ distinct prime closed geodesics; the recent literature calls this the Katok–Anosov conjecture. Ziller conjectured the weaker bound $n$. Because Katok's examples attain the conjectured number, the Katok–Anosov bound would be optimal. The main attack tool is Morse theory of the energy functional on the free loop space $\Lambda S^n$ (where critical $S^1$-orbits of positive energy are precisely the closed geodesics), combined with iteration theory for the Morse indices of closed geodesics and index-jump theorems for symplectic matrix paths (the linearized Poincaré map of a closed geodesic is a symplectic matrix). Two structural parameters govern the known partial results: the reversibility $\lambda=\max\{F(-X)\mid F(X)=1\}\ge1$ introduced by Rademacher, which measures how far $F$ is from reversible, and the flag curvature $K$, the Finsler analogue of sectional curvature; a metric is called bumpy (non-degenerate) if 1 is not an eigenvalue of the linearized Poincaré map of any closed geodesic including iterates — a genericity condition.

The known partial results are: Bangert and Long (2010) proved the Katok–Anosov bound for $S^2$, i.e. at least two prime closed geodesics on every Finsler 2-sphere. For bumpy metrics with flag curvature pinched by $(\lambda/(\lambda+1))^2<K\le1$, Wang (2012) proved the full bound $2\lfloor(n+1)/2\rfloor$; Duan, Long and Wang (2016) obtained sharp optimal bounds on all bumpy, non-negatively curved CROSSs, and two closed geodesics on bumpy Finsler spheres of dimension greater than two were obtained by Duan and Long (2007) and Rademacher (2006). Without non-degeneracy, all results require positive flag-curvature pinching: at least three prime closed geodesics (Duan 2015), at least $\lfloor(n+1)/2\rfloor$ (Wang 2019), and most recently at least $n$ for $n\ge4$ under the stronger pinching $\left(\frac{2n-3}{n-1}\right)^2\left(\frac{\lambda}{\lambda+1}\right)^2<K\le1$ with $\lambda<\frac{n-1}{n-2}$ (Duan–Xie 2025) — which reaches the full Katok–Anosov bound precisely in even dimensions and falls one short of $n+1$ in odd dimensions. For completely arbitrary (possibly degenerate, unpinned) Finsler metrics on $S^n$ with $n\ge3$, not even the existence of a second prime closed geodesic is known in full generality: the two-geodesic theorems are either generic (bumpy, cf. Wang 2020) or curvature-pinned. Wang (Nonlinear Anal. 2012) proves the lower bound $n$ prime closed geodesics without self-intersections under metric-comparison hypotheses ($F^2<((\lambda+1)/\lambda)^2g_0$ with $g_0$ the round metric, plus a lower bound $l(S^n,F)\ge\pi(1+1/\lambda)$ on the shortest geodesic loop) or under flag-curvature pinching, and a stability bound of $2\lfloor n/2\rfloor$ non-hyperbolic prime closed geodesics under stronger pinching.

Problem Statement

Prove or refute the Katok–Anosov conjecture: every smooth Finsler metric $F$ on the $n$-sphere $S^n$ with $n\ge3$ (the case $n=2$ is already settled) carries at least

$$2\left\lfloor\frac{n+1}{2}\right\rfloor$$

distinct prime closed geodesics — equivalently, at least $2k$ on $S^{2k}$ and at least $2k$ on $S^{2k-1}$ for every $k\ge2$ — under no genericity, non-degeneracy, reversibility, or curvature-pinning hypothesis on $F$. A proof implies in particular the weaker conjecture of Ziller that every Finsler $n$-sphere carries at least $n$ prime closed geodesics (identical to the Katok–Anosov bound for even $n$, weaker by one for odd $n$). A refutation consists of a Finsler metric on some $S^n$, $n\ge3$, carrying fewer than $2\lfloor(n+1)/2\rfloor$ distinct prime closed geodesics; by Katok's examples, such a metric would have strictly fewer closed geodesics than the known extremal examples. Counting convention: closed geodesics are counted up to shift of the circle parameter and up to iteration (prime geodesics only), and for a non-reversible metric a geodesic and its reverse curve count as two distinct prime closed geodesics only when both are geodesics.

References

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