Closed geodesics on bumpy spheres

Question

Main question:

• Does every bumpy Riemannian metric on a sphere have at least three short and prime closed geodesics, for some reasonable definition of short?

E.g., a geodesic $\gamma$ could be called short if every diffeomorphism from the standard sphere maps some great circle to a loop at least as long as $\gamma$.

I vaguely remember reading this result, but I've recently searched for it, and I haven't found it. It might not actually exist.

Side questions on similar results:

• This holds for all two-dimensional Riemannian spheres by the work of Lyusternik-Schnirelmann (1929) and Ballmann (1978), but Wikipedia now credits Grayson (1989) with the first correct proof. Are there issues with Ballmann's argument? I admit to not reading German; my only time through Ballmann’s paper was when Charlie Frohman informally, and very kindly, translated it for me one afternoon.

• Rademacher has recently showed that, on an open and $C^2$-dense set of metrics, the number of prime closed geodesics of length at most $t$ grows exponentially (see https://arxiv.org/abs/2208.03044). Can the methods of that paper be used to prove the existence of multiple short closed geodesics?

Answer 1

I can only comment on the first side question. From an outsider's perspective, the (twentieth-century) literature on simple closed geodesics is, quoting Jost, 'unusually rich in errors and controversies about the validity of techniques and results' [Jos89]. It's a pity, because the arguments are beautiful and would deserve a more unambiguous account. (Ironically, this quotation is from a paper that would later be claimed to be incorrect.)

Here is an (incomplete) account of the history of the Lyusternik–Schnirelmann theorem—every metric on $S^2$ has at least three simple closed geodesics—, as I've come to understand it. (I would appreciate corrections from people working in the field.)

• It seems universally accepted that the original proof of Lyusternik–Schnirelmann was incomplete [Hin84,Tai92]. Lyusternik later wrote extended and revised versions of the theory, which however still came short of a proof of the Lyusternik–Schnirelmann theorem [Tai92].
• Ballmann [Bal78] completed their proof; the validity of the argument seems accepted in the literature. For example, Hingston [Hin84] writes that the proof 'was only recently completed by Ballmann'. Likewise, Taimanov [Tai92] concedes that a 'proof of the Lyusternik–Shnirel'man theorem in the spirit of [Lyu47] is given there', but complains that 'the paper is written very concisely and certain details are not clarified'.
• Moreover, Taimanov seems not entirely unjustified in pointing out that Ballmann published his result 'in a journal not easily available' [Tai92]. (I haven't been able to find an online version of Ballmann's paper.) Being unaware of Ballmann's work, Taimanov published his own version of a 'complete proof of the Lyusternik-Shnirel'man theorem' in 1992. (I don't know the experts' opinion on this paper.) By then, there had been two other attempts at filling the gaps in the original argument of Lyusternik and Schnirelmann, which both appeared in 1989.
• One was by Jost [Jos89]. After Hingston pointed out an error in the paper, Jost published a correction, which Taimanov [Tai92] however claims is 'unsatisfactory'. Taimanov also refers to a 'number of important gaps' not addressed in the correction. (I don't know what Jost's opinion is on these alleged gaps.)
• The second was by Grayson [Gra89], using curve-shortening flow. I haven't found any references to gaps in the argument; I think this proof is largely (universally?) accepted as correct.

There are other references to incorrect results in the literature, notably concerning a topological lemma by Alber (sometimes spelled Al'ber). However, my impression is that these are mostly not immediately related to questions about simple closed geodesics, so I'll conclude the discussion here.

[Bal78] W. Ballmann, Der Satz von Lusternik und Schnirelmann, Bonner Math. Schriften 102 (1978), pp. 1-25.

[Gra89] M. A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), pp. 71-111.

[Hin84] N. Hingston, Equivariant Morse theory and closed geodesics, J. Differential Geometry 19 (1984) pp. 85-116.

[Jos89] J. Jost, A nonparametric proof of the theorem of Lusternik and Schnirelmann, Arch. Math. (Basel) 53 (1989) pp. 497-509.

[Lyu47] L. A. Lyusternik, Topology of functional spaces and the calculus of variations in the large, Trudy Mat. Inst. Steklov 19 (1947).

[Tai92] I. A. Taimanov, On the existence of three nonselfintersecting closed geodesics on manifolds homeomorphic to the two-sphere, Russian Acad. Sci. Izv. Math. 40 (1993) pp. 565-590.

Answer 2

For any Riemannian metric on the 2-sphere there exist 3 simple closed geodesics of length at most 20d, where d is the diameter of the 2-sphere. This result is proved here: https://arxiv.org/pdf/1410.8456.pdf

Answer 3

Grayson's proof of the existence of solutions of the curve shortening flow definitely filled the gap in Lusternik-Schnirelmann's original work (the gap was indeed about their construction by-hand of a flow on the loop space that shrinks loops without creating self-intersections).

Grayson's result is based on non-trivial, but solid, PDE techniques. Alternative shrinking flows constructed by hands in the literature always turned out to be highly complicated (I know several of them: Ballmann, Jost, Klingenberg, Hass-Scott, and there are probably more), to the point that it is hard to assess their correctness.

From my viewpoint, what matters is that Lusternik-Schnirelmann's initial idea for the theorem of the 3 simple closed geodesics was correct, and the technical difficulty in the construction of a shrinking flow on the space of simple loops has now been overcome at least by means of Grayson's work.