Existence results for closed Finsler geodesics via spherical complexities

We apply topological methods and a Lusternik-Schnirelmann-type approach to prove existence results for closed geodesics of Finsler metrics on spheres and projective spaces. The main tool in the proofs are spherical complexities, which have been introduced in earlier work of the author. Using them, we show how pinching conditions and inequalities between a Finsler metric and a globally symmetric metric yield the existence of multiple closed geodesics as well as upper bounds on their lengths.


INTRODUCTION
In [Mes19], the author has introduced integer-valued homotopy-invariants based on spaces of continuous maps from spheres Given a closed manifold M, these invariants can be used to estimate numbers of orbits of critical points of G-invariant differentiable functions on Hilbert submanifolds of C 0 (S n , M), where G is a subgroup of O(n + 1) and where we consider the O(n + 1)-action by reparametrization, i.e. the one induced by the standard O(n + 1)-action on S n . For n = 1, this leads to estimates of critical orbits of O(2)-invariant or SO(2)-invariant functions on Hilbert manifolds of free loops in M.
A typical situation in which this method applies is the study of energy functionals of Riemannian or Finsler metrics on H 1 (S 1 , M), the Hilbert manifold of free loops in M that is locally modelled on the Sobolev space H 1 (S 1 , R dim M ) = W 1,2 (S 1 , R dim M ). The critical points of such functionals are precisely the closed geodesics of the metric under consideration and methods from Morse theory and Lusternik-Schnirelmann theory have been very successfully implemented to derive existence results for closed geodesics in the previous decades, we refer e.g. to [Oan15] for an overview. Here, we only want to present certain recent results that are close to or intersect with the results shown in this note. In the following, we will always let λ denote the reversibility of the Finsler metric under consideration, see [Rad04b] or [Rad04a] for the definition of reversibility. For x ∈ R we further let ⌈x⌉ denote the smallest integer that is bigger than or equal to x.
• In [LD09] and [DL10], Y. Long and H. Duan have proven that every Finsler metric on S 3 and S 4 admits two distinct prime closed geodesics.
Date: March 17, 2020. 1 • Duan has shown in [Dua15] that any Finsler metric on S n whose flag curvature satisfies λ 2 (1+λ) 2 < K ≤ 1 admits three distinct prime closed geodesics. While Rademacher is using methods of comparison theory and the Fadell-Rabinowitz index to show his results, Duan and Long apply a classification method of closed geodesics by their linearized Poincaré maps and Morse-theoretic arguments for bumpy metrics. Note that while Duan and Long obtain very general results, their approach does not yield any upper bound on the length of the closed geodesics whose existence they have shown.
We want to derive similar results using more topological methods, namely using spherical complexities. The author has shown in [Mes19] that the invariants introduced therein satisfy a Lusternik-Schnirelmann-type theory and established a relationship between these invariants and critical points of differentiable functions, which we want to summarize for the case of free loop spaces. Given a closed manifold M let , it is shown that under mild additional assumptions, the number SC M ( f a ) − 1 provides a lower bound on the number of non-constant critical G-orbits in f a . Relating the invariants SC M to the topological notion of sectional categories of fibrations, one can establish lower bound on the numbers SC M ( f a ) in terms of cohomology rings of the free loop space of M. In this note, we apply this topological approach and results from [Mes19] to derive existence results for closed geodesics of Finsler metrics of positive flag curvature on spheres and projective spaces. A first result on the existence of closed Finsler geodesics using spherical complexities was shown by the author in [Mes19,Theorem 7.4]. Foundations on the notion of Finsler metrics and their geodesics can be found in [She01] or [Rad04a].
We want to give an overview over the results of this article. Let M be a closed manifold and F : TM → [0, +∞) be a Finsler metric on M of reversibility λ whose flag curvature satisfies 0 < K ≤ 1. Denote the length of its shortest non-trivial closed geodesic by ℓ F . Then: • for M = S 2n , n ≥ 2, there will be two distinct closed geodesics of length less than 2ℓ F if one of the following holds: · F ≤ 1+λ λ √ g 1 , where g 1 is the round metric of curvature 1 (Theorem 3.1), · K > 9λ 2 4(1+λ) 2 (Theorem 3.2), • for M = S 2n+1 , n ∈ N, and given k, m ∈ N there will be 2m k distinct closed geodesics of length less than (k + 1)ℓ F if one of the following holds: • for M = CP n or M = HP n , n ≥ 3, there will be two distinct closed geodesics of length less than 2ℓ F if F < 1+λ λ √ g 1 , where g 1 is the globally symmetric Riemannian metric whose prime closed geodesics have length 2π (Theorem 5.1). All of these results are summarized in the following table: In section 2 we recall the constructions and main results from [Mes19] for the case of free loop spaces. These results will be applied to Finsler energy functionals on even-and odddimensional spheres, resp., in sections 3 and 4. Section 5 considers Finsler metrics on complex and quaternionic projective spaces.

SPHERICAL COMPLEXITIES ON LOOP SPACES
We recall some definitions and results about spherical complexities from [Mes19]. We will restrict to the setting of free loop spaces and slightly simplify the notation, e.g. the numbers SC X (A) appearing below are denoted by SC 1,X (A) in [Mes19]. Given a topological space X we let ΛX := C 0 (S 1 , X) denote its free loop space and let denote the set of contractible loops in X. We further put BX := C 0 (B 2 , X) where B 2 denotes the closed unit disk in R 2 . (2) Given A ⊂ Λ 0 X we put Remark 2.2. The numbers SC X (A) are related to the notion of sectional category (or Schwarz genus) of a fibration, see e.g. [CLOT03, Section 9.3]. Let r :  In [Mes19], the author has carried out a Lusternik-Schnirelmann-type theory for the numbers SC X and their higher-dimensional generalizations. We are going to give a brief account of the results that are relevant for the study of closed geodesics. We consider the O(2)-action on Λ 0 X given by reparametrization of loops and its restrictions to subgroups of O(2). Given a topological space X, we further let i.e. c 1 is the inclusion of constant loops. Given any function f : Y → R on a topological space Y we denote its closed and open sublevel sets, resp., by A result on the numbers of closed geodesics is given as a special case of this theorem. Given a closed manifold M we let Λ 1 M : , it inherits a Hilbert manifold structure. Moreover, it is well-known that Λ 1 M and Λ 0 M are homotopy-equivalent.
Definition 2.4. Let M be a closed manifold and F : TM → [0, +∞) be a Finsler metric. We let denote its energy functional. Given a ∈ R we further let N S (F, a) denote the number of SO(2)orbits of non-constant closed geodesics in E a F . If F is reversible, then we let N(F, a) denote the number of O(2)-orbits of non-constant closed geodesics in E a F .
It is well-known, see e.g. [Mer77], that E F is of class C 1,1 , satisfies the Palais-Smale condition and that its critical points are precisely the closed geodesics of F. From these observations one derives that Theorem 2.3 is applicable to f = E F and ϕ being the time-1 map of a negative gradient flow of E F . One obtains the following from Theorem 2.3: Thus, to ensure the existence of a certain number of closed geodesics in a sublevel set of E F it suffices to find corresponding lower bounds on the number SC M (E a F ) which we will do using the cohomology rings of Λ 0 M. Throughout this note, we will always let H * denote singular cohomology.
Definition 2.6 ( [FG08]). Let p : E → B be a fibration, A be an abelian group and let u ∈ H * (B; A) with u = 0. The sectional category weight of u, denoted by wgt p (u), is the largest k ∈ N 0 , such that f * u = 0 for all continuous maps f : The key observation about the sectional category weight of a fibration p is the following, see [FG08] for a proof: We want to apply this to the fibration r : BX → Λ 0 X and denote its weight function by Let r 0 be the zero map, r 1 := r and let r n : B n X → Λ 0 X be the n-fold fiberwise join of r with itself for each n ≥ 2. Then wgt 1 (u) = sup{n ∈ N | r * n u = 0}. Part b) of Proposition 2.7 shows that non-vanishing cup products can produce classes of weight two or bigger. In addition, the following lemma provides another criterion, that does not involve the cup product, for cohomology classes to have weight two or bigger. In the following, let Λ 2 X := C 0 (S 2 , X).
Lemma 2.8. Let X be a Hausdorff space, A be an abelian group, k ≥ 2 and u ∈ H k (Λ 0 X; A) with u = 0. If u ∈ ker c * 1 and H k−1 (Λ 2 X; A) = 0, then wgt 1 (u) ≥ 2. Proof. By part d) of Proposition 2.7 it suffices to show that r * 2 u = 0 under the given assumptions, where r 2 : B 2 X → Λ 0 X denotes the fiberwise join of r with itself. We have seen in [Mes19, Section 4.4] for a general fibration that this fiberwise join admits a Mayer-Vietoris sequence, which in our setting takes the form Since u ∈ ker c * 1 , and since, as shown in the proof of [Mes19, Theorem 1.13], there is a homotopy r ≃ c 1 • e 1 , where e 1 : BX → X, γ → γ(1), it holds that u ∈ ker r * . Hence, exactness of the top row yields r * 2 u ∈ im δ. But if H k−1 (Λ 2 X; A) = 0, then δ = 0 and the claim immediately follows.
Throughout the following, we let ι a : E <a F ֒→ Λ 0 M denote the inclusion of the open sublevel set for each a ∈ R.

admits k distinct SO(2)-orbits of non-constant contractible closed geodesics of energy less than a. If F is reversible, then it will admit k distinct O(2)-orbits of non-constant contractible closed geodesics of energy less than a.
Proof. If the critical values of E F are non-isolated in (−∞, a), then the statement is obvious. If they are isolated, then pick ε > 0 such that (a − ε, a) does not contain critical values of E F . The statement then follows from Theorem 2.9.
The main significance of Corollary 2.10 is that one might employ it to detect distinct closed geodesics. The following definition makes this notion precise.
(2) We call γ 1 and γ 2 positively distinct if they are either geometrically distinct or they lie in the same O(2)-orbit of ΛM, but not in the same SO(2)-orbit.
We will further let ℓ F > 0 denote the length of the shortest non-constant closed geodesic of F.
We want to show that under certain conditions on M and F, the Finsler metric admits multiple positively distinct closed geodesics and even geometrically distinct ones if F is reversible. We will do so using the following line of argument.

Strategy.
Let M be a closed manifold and F : TM → [0, +∞) be a Finsler metric that admits a contractible closed geodesic, e.g. for M simply connected.
1. Find an abelian group A, k ≥ 2 and u ∈ H * (Λ 0 M; A) with wgt 1 (u) ≥ k. 2. Find r ∈ N and a ∈ R so small that E <a F does not contain any m-fold iterated closed geodesic for m > r and show that ι * a u = 0. Then there will be k r positively distinct closed geodesics in E <a F . If F is reversible, then there will be k geometrically distinct closed geodesics in E <a F .
Comment on the strategy. Every closed geodesic γ of F satisfies E F (γ) ≥ ℓ 2 F . Thus, for each m ≥ 2, the m-fold iterate γ m of each closed geodesic satisfies Thus, if a ≤ (r + 1) 2 ℓ 2 F , then the sublevel set E <a F can contain at most r-fold iterated closed geodesics. Consequently, if we can ensure that E <F A contains k distinct SO(2)-orbits, the existence of k r distinct closed geodesics immediately follows since each closed geodesics can contribute at most r times to the count of the SO(2)-orbits.
Before we start to consider concrete examples, we want to conclude a section by presenting a result of Rademacher that we will use in all of the following examples. In the above line of argument, we were searching for energy levels a that satisfy inequalities of the form a ≤ r 2 ℓ 2 F for suitable r ∈ N. To find such numbers, we will need lower bounds for the lengths ℓ F , which have been established by Rademacher for Finsler metrics of positive curvature.

EVEN-DIMENSIONAL SPHERES
Let n ∈ N, let g 1 denote the round Riemannian metric on S n of constant curvature 1 and let E 1 : Λ 1 S n → R denote its energy functional. For every a ∈ R we let Λ ≤a S n := E −1 1 ((−∞, a]) denote the corresponding closed sublevel set of E 1 . The homology groups of free loop spaces of globally symmetric spaces can be computed using Morse-Bott theory of the energy functional, see [Zil77], which we want to discuss explicitly for (S n , g 1 ), see also [Oan15,Section 7] for another discussion of this case. Let T 1 S n denote the unit tangent bundle of S n with respect to g 1 . Using [Zil77, Theorem 5], the methods of its proof and the universal coefficient theorem one shows that for each field F there are isomorphisms of abelian groups H * (Λ ≤(2πm) 2 S n ; F) ∼ = H * (S n ; F) ⊕ Here, for a graded group G * = k∈Z G k and ℓ ∈ Z we let G * [ℓ] denote the shifted graded group given by (G * [ℓ]) k = G k+ℓ for all k ∈ Z. It further follows from the Morse-theoretic construction underlying these isomorphisms that the maps c * 1 : H * (ΛS n ; F) → H * (S n ; F) and j * (2πm) 2 : H * (ΛS n ; F) → H * (Λ ≤(2πm) 2 S n ; F) are the projections onto the corresponding summands under the isomorphisms in (3.1) and (3.2), where j (2πm) 2 : Λ ≤(2πm) 2 S n ֒→ ΛS n denotes the inclusion for each m ∈ N.
In the following, we consider even-dimensional spheres and odd-dimensional spheres separately, since the cohomology rings of ΛS n can be distinguished along these two cases.
Theorem 3.1. Let n ≥ 2 and let F : TS 2n → [0, +∞) be a Finsler metric on S 2n of reversibility λ. If the flag curvature of F satisfies 0 < K ≤ 1 and if then F will admit two positively distinct closed geodesics of length less than 2ℓ F . If F is reversible, then the geodesics can be chosen geometrically distinct.
Proof. As shown in [Hau14,Example 5.4.18], it holds that It follows from this observation and from (3.1) and (3.2) that H 4n−2 (ΛS 2n ; Z 2 ) = 0 and Since H 4n−2 (S 2n ; Z 2 ) = 0, it holds that x ∈ ker c * 1 , thus wgt 1 (x) ≥ 1. It further follows from the previous computation that j * 4π 2 u = 0. To apply Lemma 2.8 to the class x ∈ H 4n−2 (ΛS 2n ; Z 2 ), we note that the Z 2 -cohomology groups of Λ 2 S 2n = C 0 (S 2 , S 2n ) are well-known and it follows from [Sal04, Theorem 18.(1)] and [Nei10, Corollary 10.26.4(b)] that there is an isomorphism of abelian groups where the index of every class denotes its degree. This particularly shows that H 4n−3 (Λ 2 S 2n ; Z 2 ) = 0, so that Lemma 2.8 yields wgt 1 (x) ≥ 2. By assumption (3.3), it holds for each γ ∈ Λ 1 S 2n that By [Rad04b, Theorem 4], it follows from the flag curvature bounds of F that ℓ F ≥ 1+λ λ π, so the above chain of inequalities yields In particular, this shows that Λ ≤4π 2 S n ⊂ E <4ℓ 2 F F and it follows that j 4π 2 factors via the inclusion Since wgt 1 (x) ≥ 2, it follows from Corollary 2.10 that E <4ℓ 2 F F contains two distinct SO(2)-orbits of non-constant closed geodesics and even two distinct O(2)-orbits if F is reversible. Since the m-fold iterate, m ≥ 2, of a non-constant closed geodesic γ of F satisfies contains only prime non-constant closed geodesics. Thus, since it contains two SO(2)-orbits or O(2)-orbits, resp., of non-constant closed geodesics, it must contain two positively distinct closed geodesics of F and two geometrically distinct ones if F is reversible, which shows the claim.
Next we will prove the existence of two closed geodesics on even-dimensional spheres dropping assumption (3.3) and imposing a much stronger pinching condition on the flag curvature of the Finsler metric instead.
Theorem 3.2. Let n ≥ 2 and let F : TS 2n → [0, +∞) be a Finsler metric on S 2n of reversibility λ. Let k, m ∈ N be given, such that the flag curvature K of F satisfies 9λ 2 4(1 + λ) 2 < K ≤ 1. Then F admits two positively distinct closed geodesics whose lengths are less than 2ℓ F . If F is reversible, then the geodesics can be chosen geometrically distinct.

ODD-DIMENSIONAL SPHERES
For odd-dimensional spheres it is possible to show the existence of more than two closed geodesics, since the rational cohomology of their free loop spaces has a rich ring structure that we want to make use of. As a disadvantage, we need stronger assumptions on the flag curvature than in the even-dimensional case to obtain a lower bound on ℓ F , see Theorem 2.12.
Theorem 4.1. Let n ∈ N, let F : TS 2n+1 → [0, +∞) be a Finsler metric on S 2n+1 , let λ denote its reversibility and let K denote its flag curvature. If for some k, m ∈ N, then F will admit 2m k positively distinct closed geodesics of length less than (k + 1)ℓ F . If F is reversible, then the geodesics can be chosen geometrically distinct.
Proof. It was shown in [VPS76], see also [ where deg x = 2n + 1 and deg y = 2n. Here, A(x) denotes the exterior Q-algebra generated by x. Since H 2n (S 2n+1 ; Q) = 0, it holds that y ∈ ker c * 1 , so parts a) and b) of Proposition 2.7 show that wgt 1 (y k ) ≥ k ∀k ∈ N.
We want to study the even powers y 2m ∈ H 4mn (ΛS 2n+1 ; Q), m ∈ N. We derive from the results of [Zil77] and the universal coefficient theorem that From this computation and the isomorphisms (3.2) and (3.1) we obtain which yields j * 4m 2 π 2 (y 2m ) = 0 for each m ∈ N. By the curvature condition, it follows from [Rad04b, Theorem 1] that ℓ F ≥ π(1 + 1 λ ). Using this estimate and the other assumption in (4.1), we compute for each γ ∈ Λ 1 S 2n+1 that This particularly implies that Λ ≤4m 2 π 2 S 2n+1 ⊂ E Since wgt 1 (y 2m ) ≥ 2m, it follows from Corollary 2.10 that E for some m ∈ N, then F will admit 2m positively distinct closed geodesics of length less than 2ℓ F . If F is reversible, then the geodesics can be chosen geometrically distinct.
We next want to derive an analogue of Theorem 3.2 for odd-dimensional spheres, in which we drop the assumed inequality between F and the round metric from Theorem 4.1 and replace it by a stronger pinching condition.
Proof. Consider the case K = C. In the proof of [Mes19, Theorem 7.4] applied to M = CP n , it was shown that there exists u ∈ H 3 (ΛCP n ; Q) with wgt 1 (u) ≥ 2. Using the results of [Zil77] and the universal coefficient theorem, one obtains In terms of the isomorphisms in (5.1) and (5.2) this shows that H 3 (ΛCP n ; Q) ∼ = H 2 (T 1 CP n ; Q)[−1] ∼ = H 3 (Λ ≤4π 2 CP n ; Q), which yields j * 4π 2 u = 0. The remainder of the proof is carried out in analogy with the proof of Theorem 3.1. In the case K = H, we can apply the same line of argument and obtain a class u ∈ H 7 (ΛHP n ; Q) with wgt 1 (u) ≥ 2. In analogy with the other case, one shows that j * 4π 2 u = 0 using the cohomology of T 1 HP n that was computed in [Zil77] and concludes the proof in the same way as for K = C.