Dense existence of periodic Reeb orbits and ECH spectral invariants

In this paper, we prove (1): for any closed contact three-manifold with a $C^\infty$-generic contact form, the union of periodic Reeb orbits is dense, (2): for any closed surface with a $C^\infty$-generic Riemannian metric, the union of closed geodesics is dense. The key observation is $C^\infty$-closing lemma for 3D Reeb flows, which follows from the fact that the embedded contact homology (ECH) spectral invariants recover the volume.


Introduction
For any contact manifold (Y, λ), where λ is the contact form, the Reeb vector field R is defined by equations dλ(R, · ) = 0 and λ(R) = 1.
The first result of this paper is that, for any closed contact three-manifold with a C ∞generic contact form, the union of periodic Reeb orbits is dense in the manifold. To state the result more formally, recall that a set F in a topological space X is called residual, if F contains a countable intersection of open and dense sets in X. is residual in C ∞ (Y, R >0 ) with respect to the C ∞ -topology.
Remark 1.2. For any closed contact manifold (not necessarily three-dimensional) with a C 2 -generic contact from, the union of periodic Reeb orbits is dense; this is an easy consequence of the Hamiltonian C 1 -closing lemma by Pugh-Robinson (see [8] Corollary 11.4). On the other hand, there exists an example of a Hamiltonian system whose C ∞small perturbations have no nonconstant periodic orbits (see [4]).

Remark 1.3.
There is a series of results establishing the existence of infinitely many periodic orbits for C ∞ -generic Hamiltonian/Reeb/geodesic flows. See [3] and the references therein.
Remark 1.4. In the preliminary version of this paper, we required additional assumptions in Theorem 1.1, to directly apply Theorem 1.2 in [2]. It was pointed out by Michael Hutchings [7] that we can drop these assumptions using Theorem 1.3 in [2] and the arguments in [1] Section 2.6.
We also prove a similar result for closed geodesics (throughout this paper, all closed geodesics are assumed to be nonconstant). For any Riemannian metric g on a manifold Σ and f ∈ C ∞ (Σ, R >0 ), we define the metric f g by v f g := f (p) v g (p ∈ Σ, v ∈ T p Σ). Theorem 1.5. For any closed Riemannian surface (Σ, g), {f ∈ C ∞ (Σ, R >0 ) | the union of closed geodesics of (Σ, f g) is dense in Σ} is residual in C ∞ (Σ, R >0 ) with respect to the C ∞ -topology. Remark 1.6. As is clear from the proof of Theorem 1.5 (see Section 4), one can also prove the following variant of Theorem 1.5: for any closed surface Σ, let G (Σ) denote the space of all C ∞ -Riemannian metrics on Σ. Then, {g ∈ G (Σ) | the union of closed geodesics of (Σ, g) is dense in Σ} is residual in G (Σ) with respect to the C ∞ -topology.
To prove these theorems, we use spectral invariants in the theory of embedded contact homology (ECH). After some preliminaries in Section 2, we prove Theorem 1.1 in Section 3, and Theorem 1.5 in Section 4. The key observation is C ∞ -closing lemma for 3D Reeb flows (Lemma 3.1), which follows from the fact that the ECH spectral invariants recover the volume [2].

Preliminaries
In Section 2.1, we prove a preliminary result (Lemma 2.2) on action spectra of contact manifolds. In Sections 2.2 and 2.3, we recall the theory of ECH, in particular quantitative aspects, very briefly. For precise definitions and proofs, see [6] and the references therein.
2.1. Action spectra of contact manifolds. First we introduce the following notations for arbitrary set S ⊂ R.
We also define S + := m≥0 S m .
• S is of class CV, if there exists a C ∞ -manifold X and f ∈ C ∞ (X), such that S is contained in the set of all critical values of f . Lemma 2.1. If S is of class CV, then S + is a null (Lebesgue measure zero) set.
Proof. Suppose S is contained in the set of critical values of f ∈ C ∞ (X). Then, for each integer m ≥ 1, S m is contained in the set of critical values of X ×m → R; (x 1 , . . . , x m ) → f (x 1 ) + · · · + f (x m ), thus S m is a null set by Sard's theorem. Therefore S + is also a null set.
For any closed contact manifold (Y, λ), we set We define A : P(Y, λ) → R >0 by A (γ) := T γ . The image of this map Im A =: A (Y, λ) ⊂ R >0 is called the action spectrum of (Y, λ).
Proof. Let us abbreviate A (Y, λ) as A . It is easy to see that A is closed, and min A > 0. Thus A + is closed. On the other hand, we can show that A is of class CV. The proof is similar to the case of Hamiltonian periodic orbits (see [10] Lemma 3.8) and omitted. Then, A + is a null set by Lemma 2.1.
Let us assume that the contact form λ is nondegenerate, i.e. for any γ ∈ P(Y, λ), 1 is not an eigenvalue of the linearized Poincaré map of γ. Under this assumption, any periodic Reeb orbit of (Y, λ) is either elliptic or hyperbolic.
When Γ = 0, the empty set ∅ is an ECH generator. For any L ∈ R >0 , ECC L (Y, λ, Γ) denotes the subspace of ECC(Y, λ, Γ) which is generated by ECH generators Let ξ := ker λ be the contact distribution, and let d ∈ Z ≥0 denote the divisibility of In the following, we denote ECC by ECC * , to specify the relative grading.
To define a differential on ECC * (Y, λ, Γ), we fix an almost complex structure J on the symplectization Y × R, which satisfies several conditions (see [6] Section 1.3). The differential ∂ J , which decreases the grading by 1, is defined by counting the number of Jholomorphic currents with ECH indices 1 modulo R-translation. It is shown that ∂ 2 J = 0, and the homology group is denoted by ECH * (Y, λ, Γ, J).

(Continuity) For any sequence (f
The next lemma shows that spectral invariants are "action selectors". A special case of this lemma is proved in [1] Lemma 3.1 (a).

Lemma 2.4. For any
Proof. First we assume that λ is nondegenerate. Let us abbreviate c σ (Y, λ) by c, and This contradicts the definition of c. Next we consider the case that λ can be degenerate. Let us take a sequence (f j ) j≥1 in C ∞ (Y, R >0 ) such that lim j→∞ f j − 1 C 1 = 0, and f j λ is nondegenerate for any j. ( · C 1 is defined by fixing local charts on Y .) For each j, c σ (Y, f j λ) ∈ A (Y, f j λ) m(j) for some m(j). Now sup j m(j) < ∞, since inf j min A (Y, f j λ) > 0. Hence, up to subsequence, we obtain c σ (Y, f j λ) = a 1 j + · · · + a m j , where a 1 j , . . . , a m j ∈ A (Y, f j λ), and a l ∞ := lim j→∞ a l j exists for any 1 ≤ l ≤ m. The assumption lim j→∞ f j − 1 C 1 = 0 implies that a l ∞ ∈ A (Y, λ) for any l, thus c σ (Y, λ) = a 1 ∞ + · · · + a m ∞ ∈ A (Y, λ) m .

Proof of Theorem 1.1
The key observation is C ∞ -closing lemma for 3D Reeb flows (Lemma 3.1). We fix local charts on Y , and define f C l for any integer l ≥ 0 and f ∈ C l (Y ). For any f ∈ C ∞ (Y ), we set For any nonempty open set U in Y and ε > 0, there exists f ∈ C ∞ (Y ) such that f − 1 C ∞ < ε and there exists nondegenerate γ ∈ P(Y, f λ) which intersects U.
Once this claim is proved, we take g ∈ C ∞ (Y ) such that g C ∞ is sufficiently small, g| Im γ , dg| Im γ ≡ 0 (thus γ ∈ P(Y, e g (1 + th)λ)), and γ is nondegenerate as a periodic Reeb orbit with the contact form e g (1 + th)λ (this is possible since the linearized Poincaré map of γ is twisted by the Hessian of g). Then, f := e g (1 + th) satisfies the requirement in the lemma.
Let us prove Theorem 1.1. For any nonempty open set U ⊂ Y , let It is enough to prove the next lemma.
Remark 4.2. The closing problem for geodesic flows was already discussed in [8] Section 10 as an open problem (see also [9] Introduction). The C 1 -closing problem was solved in [9] Corollary 4 as a consequence of [9] Theorem 3, which claims that for any unit tangent vector (q, v) on a closed Riemannian manifold (of any dimension), one can create a periodic orbit of the geodesic flow which passes near (q, v) by a C 1 -small conformal perturbation of the metric. On the other hand, our Lemma 4.1 shows that for any point q on a closed Riemannian surface, one can create a closed geodesic which passes near q by a C ∞ -small conformal perturbation of the metric.