Spectral characterizations of Besse and Zoll Reeb flows Marco - - PowerPoint PPT Presentation

Spectral characterizations of Besse and Zoll Reeb flows

Marco Mazzucchelli (CNRS and ´ Ecole normale sup´ erieure de Lyon)

Joint work with:

• Stefan Suhr
• Daniel Cristofaro-Gardiner
• Viktor Ginzburg, Basak Gurel

The closed geodesics conjectures

(M, g)

The closed geodesics conjectures

(M, g)

◮ Every closed Riemannian manifold (M, g) of dim(M) ≥ 2 has infinitely many closed geodesics. ◮ Every closed Finsler manifold (M, F) has at least dim(M) many closed geodesics. Widely open for M = Sn (except S2)

The closed geodesics conjectures

(M, g)

◮ Every closed Riemannian manifold (M, g) of dim(M) ≥ 2 has infinitely many closed geodesics. ◮ Every closed Finsler manifold (M, F) has at least dim(M) many closed geodesics. Widely open for M = Sn (except S2) Subconjecture: Every closed (M, g) or (M, F) with dim(M) > 2 has at least two closed geodesics. Open for M = Sn (except 1 ≤ n ≤ 4).

Zoll Riemannian manifolds

◮ A closed Riemannian manifold (M, g) is Zoll if all its geodesics are closed and have the same length ℓ.

Zoll Riemannian manifolds

◮ A closed Riemannian manifold (M, g) is Zoll if all its geodesics are closed and have the same length ℓ. ◮ Prime length spectrum of (M, g): σp(M, g) =

• length(γ)
• γ prime closed geodesic of (M, g)

Zoll Riemannian manifolds

◮ A closed Riemannian manifold (M, g) is Zoll if all its geodesics are closed and have the same length ℓ. ◮ Prime length spectrum of (M, g): σp(M, g) =

• length(γ)
• γ prime closed geodesic of (M, g)
• Example:

(S2, ground)

σp(S2, ground) = {2π}.

Zoll Riemannian manifolds

◮ A closed Riemannian manifold (M, g) is Zoll if all its geodesics are closed and have the same length ℓ. ◮ Prime length spectrum of (M, g): σp(M, g) =

• length(γ)
• γ closed geodesic of (M, g)
• Conjecture: If σp(M, g) = {ℓ}, then (M, g) is Zoll.

Remark: The conjecture implies that every (M, g) admits at least two closed geodesics.

Zoll Riemannian manifolds

◮ A closed Riemannian manifold (M, g) is Zoll if all its geodesics are closed and have the same length ℓ. ◮ Prime length spectrum of (M, g): σp(M, g) =

• length(γ)
• γ closed geodesic of (M, g)
• Conjecture: If σp(M, g) = {ℓ}, then (M, g) is Zoll.

Remark: The conjecture implies that every (M, g) admits at least two closed geodesics. Theorem (Mazzucchelli, Suhr, 2017; claimed by Lusternik, 1960s) The conjecture is true for (S2, g). Indeed, slightly more is true: if every simply closed geodesic of (S2, g) has length ℓ, then every geodesic of (S2, g) is simply closed and has length ℓ.

Reeb flows on contact manifolds

◮ (Y 2n+1, λ) closed contact manifold, φt : Y → Y Reeb flow

Reeb flows on contact manifolds

◮ (Y 2n+1, λ) closed contact manifold, φt : Y → Y Reeb flow

λ 1-form on Y , λ ∧ dλn volume form R Reeb vector field on Y , λ(R) ≡ 1, dλ(R, ·) ≡ 0 φt flow of R

Reeb flows on contact manifolds

◮ (Y 2n+1, λ) closed contact manifold, φt : Y → Y Reeb flow

λ 1-form on Y , λ ∧ dλn volume form R Reeb vector field on Y , λ(R) ≡ 1, dλ(R, ·) ≡ 0 φt flow of R

◮ Closed Reeb orbit: γ(t) = φt(z) such that γ(t) = γ(t + τ) τγ := minimal period of γ

z = φτ(z) φt(z)

Reeb flows on contact manifolds

◮ (Y 2n+1, λ) closed contact manifold, φt : Y → Y Reeb flow

λ 1-form on Y , λ ∧ dλn volume form R Reeb vector field on Y , λ(R) ≡ 1, dλ(R, ·) ≡ 0 φt flow of R

◮ Closed Reeb orbit: γ(t) = φt(z) such that γ(t) = γ(t + τ) τγ := minimal period of γ

z = φτ(z) φt(z)

◮ Action spectra: σp(Y , λ) =

• τγ
• γ periodic Reeb orbit

Reeb flows on contact manifolds

◮ (Y 2n+1, λ) closed contact manifold, φt : Y → Y Reeb flow

λ 1-form on Y , λ ∧ dλn volume form R Reeb vector field on Y , λ(R) ≡ 1, dλ(R, ·) ≡ 0 φt flow of R

◮ Closed Reeb orbit: γ(t) = φt(z) such that γ(t) = γ(t + τ) τγ := minimal period of γ

z = φτ(z) φt(z)

◮ Action spectra: σp(Y , λ) =

• τγ
• γ periodic Reeb orbit
• σ(Y , λ) =
• n τγ
• n ∈ N, γ periodic Reeb orbit

Reeb flows on contact manifolds

◮ (Y 2n+1, λ) closed contact manifold, φt : Y → Y Reeb flow ◮ Closed Reeb orbit: γ(t) = φt(z) such that γ(t) = γ(t + τ) τγ := minimal period of γ ◮ Action spectra: σp(Y , λ) =

• τγ
• γ periodic Reeb orbit
• σ(Y , λ) =
• n τγ
• n ∈ N, γ periodic Reeb orbit
• Example: Y = S∗M unit cotangent bundle of (M, F) or (M, g),

λ Liouville form, φt geodesic flow

Besse and Zoll Reeb flows

(Y , λ) closed, X Reeb vector field, φt : Y → Y Reeb flow

Besse and Zoll Reeb flows

(Y , λ) closed, X Reeb vector field, φt : Y → Y Reeb flow ◮ (Y , λ) is Besse when every Reeb orbit is periodic.

Besse and Zoll Reeb flows

(Y , λ) closed, X Reeb vector field, φt : Y → Y Reeb flow ◮ (Y , λ) is Besse when every Reeb orbit is periodic. Wadsley’s thm: If (Y , λ) Besse, then φτ =id for some τ > 0.

x y z x y z

Besse and Zoll Reeb flows

(Y , λ) closed, X Reeb vector field, φt : Y → Y Reeb flow ◮ (Y , λ) is Besse when every Reeb orbit is periodic. Wadsley’s thm: If (Y , λ) Besse, then φτ =id for some τ > 0.

x y z x y z

◮ (Y , λ) is Zoll when every Reeb orbit is periodic with the same minimal period τ, i.e. φτ = id, fix(φt) = ∅ ∀t ∈ (0, τ).

x y z x y z

Besse and Zoll Reeb flows

(Y , λ), X Reeb vector field, φt : Y → Y Reeb flow ◮ (Y , λ) is Besse when every Reeb orbit is periodic. ◮ (Y , λ) is Zoll when every Reeb orbit is periodic with the same minimal period τ

Besse and Zoll Reeb flows

(Y , λ), X Reeb vector field, φt : Y → Y Reeb flow ◮ (Y , λ) is Besse when every Reeb orbit is periodic. ◮ (Y , λ) is Zoll when every Reeb orbit is periodic with the same minimal period τ Example: ellipsoid Y = E(a, b) =

• (z1, z2) ∈ C2
• |z1|2

a

+ |z1|2

b

= 1

π

• a, b > 0

λ = i 4

• j=1,2
• zj dzj − zj dzj
• φt(z1, z2) = (ei2πt/az1, ei2πt/bz2)

Besse and Zoll Reeb flows

(Y , λ), X Reeb vector field, φt : Y → Y Reeb flow ◮ (Y , λ) is Besse when every Reeb orbit is periodic. ◮ (Y , λ) is Zoll when every Reeb orbit is periodic with the same minimal period τ Example: ellipsoid Y = E(a, b) =

• (z1, z2) ∈ C2
• |z1|2

a

+ |z1|2

b

= 1

π

• a, b > 0

λ = i 4

• j=1,2
• zj dzj − zj dzj
• φt(z1, z2) = (ei2πt/az1, ei2πt/bz2)

◮ If b/a ∈ Q then (Y , λ) is Besse ◮ If a = b then (Y , λ) is Zoll

Besse and Zoll Reeb flows in dimension 3

(Y 3, λ) closed, X Reeb vector field, φt : Y → Y Reeb flow

Besse and Zoll Reeb flows in dimension 3

(Y 3, λ) closed, X Reeb vector field, φt : Y → Y Reeb flow Theorem (Cristofaro-Gardiner, Hutchings, 2016) Every (Y 3, λ) has at least two closed Reeb orbits.

Besse and Zoll Reeb flows in dimension 3

(Y 3, λ) closed, X Reeb vector field, φt : Y → Y Reeb flow Theorem (Cristofaro-Gardiner, Hutchings, 2016) Every (Y 3, λ) has at least two closed Reeb orbits. Theorem (Cristofaro-Gardiner, Mazzucchelli, 2019) ◮ (Y 3, λ) is Besse if and only if σ(Y , λ) ⊂ rN for some r > 0

Besse and Zoll Reeb flows in dimension 3

(Y 3, λ) closed, X Reeb vector field, φt : Y → Y Reeb flow Theorem (Cristofaro-Gardiner, Hutchings, 2016) Every (Y 3, λ) has at least two closed Reeb orbits. Theorem (Cristofaro-Gardiner, Mazzucchelli, 2019) ◮ (Y 3, λ) is Besse if and only if σ(Y , λ) ⊂ rN for some r > 0 ◮ (Y 3, λ) is Zoll if and only if σp(Y , λ) = {τ}

Riemannian and Finsler surfaces

(M2, F) closed Finsler surface

0x Tx M {F = 1}

Riemannian and Finsler surfaces

(M2, F) closed Finsler surface

0x Tx M {F = 1}

• Corollary. σ(M2, F) ⊂ rZ for some r > 0 if and only if F is Besse

and M = S2 or RP2.

Riemannian and Finsler surfaces

(M2, F) closed Finsler surface

0x Tx M {F = 1}

• Corollary. σ(M2, F) ⊂ rZ for some r > 0 if and only if F is Besse

and M = S2 or RP2. (M, g) closed Riemannian surface.

Riemannian and Finsler surfaces

(M2, F) closed Finsler surface

0x Tx M {F = 1}

• Corollary. σ(M2, F) ⊂ rZ for some r > 0 if and only if F is Besse

and M = S2 or RP2. (M, g) closed Riemannian surface. Corollary. ◮ If M is orientable, then σ(M, g) ⊂ rZ for some r > 0 if and

• nly if M = S2 and g Zoll.

◮ If M is non-orientable, then σ(M, g) ⊂ rZ for some r > 0 if and only if M = RP2 and g has constant curvature.

(Hard) open questions

(Y 2n+1, λ) closed contact manifold of dimension 2n + 1 > 3 σp(Y , λ) = prime action spectrum σ(Y , λ) = action spectrum ◮ (Weinstein’s conjecture) Does (Y , λ) have closed Reeb orbits? ◮ If yes, does it have more than one?

(Hard) open questions

(Y 2n+1, λ) closed contact manifold of dimension 2n + 1 > 3 σp(Y , λ) = prime action spectrum σ(Y , λ) = action spectrum ◮ (Weinstein’s conjecture) Does (Y , λ) have closed Reeb orbits? ◮ If yes, does it have more than one? ◮ If yes, does σp(Y , λ) = {τ} implies that (Y , λ) is Zoll? ◮ If yes, does σ(Y , λ) ⊂ rN for some r > 0 implies that (Y , λ) is Besse?

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) convex contact sphere

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) convex contact sphere

Y ⊂ Cn+1 convex hypersurface enclosing 0 λ = i

4

n+1

j=1

• zj dzj + zj dzj
• contact form on Y

Y Cn+1

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) convex contact sphere

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) convex contact sphere Ekeland-Hofer action selectors ck = ck(Y ) ∈ σ(Y , λ)

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) convex contact sphere Ekeland-Hofer action selectors ck = ck(Y ) ∈ σ(Y , λ) min σ(Y , λ) = c1 ≤ c2 ≤ c3 ≤ ...

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) convex contact sphere Ekeland-Hofer action selectors ck = ck(Y ) ∈ σ(Y , λ) min σ(Y , λ) = c1 ≤ c2 ≤ c3 ≤ ... Theorem (Ginzburg, G¨ urel, Mazzucchelli, 2019) ◮ ck = ck+n for some k if and only if (Y , λ) is Besse.

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) convex contact sphere Ekeland-Hofer action selectors ck = ck(Y ) ∈ σ(Y , λ) min σ(Y , λ) = c1 ≤ c2 ≤ c3 ≤ ... Theorem (Ginzburg, G¨ urel, Mazzucchelli, 2019) ◮ ck = ck+n for some k if and only if (Y , λ) is Besse. ◮ c1 = cn+1 if and only if (Y , λ) is Zoll.

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) convex contact sphere Ekeland-Hofer action selectors ck = ck(Y ) ∈ σ(Y , λ) min σ(Y , λ) = c1 ≤ c2 ≤ c3 ≤ ... Theorem (Ginzburg, G¨ urel, Mazzucchelli, 2019) ◮ ck = ck+n for some k if and only if (Y , λ) is Besse. ◮ c1 = cn+1 if and only if (Y , λ) is Zoll. ◮ Assume Y is δ-pinched for some δ ∈ (1, √ 2]. Then σ(Y , λ) ∩ (c1, δ2c1) = ∅ if and only if (Y , λ) is Zoll.

Y r R

R r < δ

Proof that ck = ck+n implies Besse

◮ a ∈ (1, 2) H : Cn+1 → R such that H|Y ≡ 1 and H(λ · ) = λaH.

Proof that ck = ck+n implies Besse

◮ a ∈ (1, 2) H : Cn+1 → R such that H|Y ≡ 1 and H(λ · ) = λaH.

Y H−1(h)

γ τ-periodic Reeb orbit on Y Γ(t) = h1/aγ(τt) Hamiltonian 1-periodic orbit on H−1(h), for some unique h = h(τ)

Proof that ck = ck+n implies Besse

◮ a ∈ (1, 2) H : Cn+1 → R such that H|Y ≡ 1 and H(λ · ) = λaH.

Y H−1(h)

γ τ-periodic Reeb orbit on Y Γ(t) = h1/aγ(τt) Hamiltonian 1-periodic orbit on H−1(h), for some unique h = h(τ)

◮ H∗ : Cn+1 → R dual function to H

Proof that ck = ck+n implies Besse

◮ a ∈ (1, 2) H : Cn+1 → R such that H|Y ≡ 1 and H(λ · ) = λaH.

Y H−1(h)

γ τ-periodic Reeb orbit on Y Γ(t) = h1/aγ(τt) Hamiltonian 1-periodic orbit on H−1(h), for some unique h = h(τ)

◮ H∗ : Cn+1 → R dual function to H

H(w) = max

z

• w, z − H(z)

Proof that ck = ck+n implies Besse

◮ a ∈ (1, 2) H : Cn+1 → R such that H|Y ≡ 1 and H(λ · ) = λaH.

Y H−1(h)

γ τ-periodic Reeb orbit on Y Γ(t) = h1/aγ(τt) Hamiltonian 1-periodic orbit on H−1(h), for some unique h = h(τ)

◮ H∗ : Cn+1 → R dual function to H

H(w) = max

z

• w, z − H(z)
• ◮ Clarke action functional

Proof that ck = ck+n implies Besse

◮ a ∈ (1, 2) H : Cn+1 → R such that H|Y ≡ 1 and H(λ · ) = λaH.

Y H−1(h)

γ τ-periodic Reeb orbit on Y Γ(t) = h1/aγ(τt) Hamiltonian 1-periodic orbit on H−1(h), for some unique h = h(τ)

◮ H∗ : Cn+1 → R dual function to H

H(w) = max

z

• w, z − H(z)
• ◮ Clarke action functional

Ψ : Lb

0(S1, Cn+1) → R,

Ψ(˙ Γ) =

• S1
• i ˙

Γ, Γ − H∗(−i ˙ Γ)

• dt,

b =

a a−1

Proof that ck = ck+n implies Besse

◮ a ∈ (1, 2) H : Cn+1 → R such that H|Y ≡ 1 and H(λ · ) = λaH.

Y H−1(h)

γ τ-periodic Reeb orbit on Y Γ(t) = h1/aγ(τt) Hamiltonian 1-periodic orbit on H−1(h), for some unique h = h(τ)

◮ H∗ : Cn+1 → R dual function to H

H(w) = max

z

• w, z − H(z)
• ◮ Clarke action functional

Ψ : Lb

0(S1, Cn+1) → R,

Ψ(˙ Γ) =

• S1
• i ˙

Γ, Γ − H∗(−i ˙ Γ)

• dt,

b =

a a−1

◮ Crit(Ψ) \ {0} = ˙ Γ

• Γ 1-periodic Hamiltonian orbits
• Ψ(˙

Γ) = f (τ) := a

2

a−2

2 τ

(2−a)/a

Proof that ck = ck+n implies Besse

◮ Clarke action functional Ψ : Lb

0(S1, Cn+1) → R

Crit(Ψ) \ {0} = 1-periodic Hamiltonian orbits Ψ(˙ Γ) = f (τ) := a

2

a−2

2 τ

(2−a)/a

Proof that ck = ck+n implies Besse

◮ Clarke action functional Ψ : Lb

0(S1, Cn+1) → R

Crit(Ψ) \ {0} = 1-periodic Hamiltonian orbits Ψ(˙ Γ) = f (τ) := a

2

a−2

2 τ

(2−a)/a

◮ Ψ is S1-invariant

s · ˙ Γ = ˙ Γ(s + ·), ∀s ∈ S1, ˙ Γ ∈ Lb

0(S1, Cn+1)

Proof that ck = ck+n implies Besse

◮ Clarke action functional Ψ : Lb

0(S1, Cn+1) → R

Crit(Ψ) \ {0} = 1-periodic Hamiltonian orbits Ψ(˙ Γ) = f (τ) := a

2

a−2

2 τ

(2−a)/a

◮ Ψ is S1-invariant

s · ˙ Γ = ˙ Γ(s + ·), ∀s ∈ S1, ˙ Γ ∈ Lb

0(S1, Cn+1)

◮ H∗

S1(Lb 0(S1, Cn+1)) = H∗(CP∞) = 1, e, e2, e3, ...

Proof that ck = ck+n implies Besse

◮ Clarke action functional Ψ : Lb

0(S1, Cn+1) → R

Crit(Ψ) \ {0} = 1-periodic Hamiltonian orbits Ψ(˙ Γ) = f (τ) := a

2

a−2

2 τ

(2−a)/a

◮ Ψ is S1-invariant

s · ˙ Γ = ˙ Γ(s + ·), ∀s ∈ S1, ˙ Γ ∈ Lb

0(S1, Cn+1)

◮ H∗

S1(Lb 0(S1, Cn+1)) = H∗(CP∞) = 1, e, e2, e3, ...

◮ f (ck) := inf

• b ∈ R
• ek−1 = 0 in H∗

S1({Ψ < b})

Proof that ck = ck+n implies Besse

◮ Clarke action functional Ψ : Lb

0(S1, Cn+1) → R

Crit(Ψ) \ {0} = 1-periodic Hamiltonian orbits Ψ(˙ Γ) = f (τ) := a

2

a−2

2 τ

(2−a)/a

◮ Ψ is S1-invariant

s · ˙ Γ = ˙ Γ(s + ·), ∀s ∈ S1, ˙ Γ ∈ Lb

0(S1, Cn+1)

◮ H∗

S1(Lb 0(S1, Cn+1)) = H∗(CP∞) = 1, e, e2, e3, ...

◮ f (ck) := inf

• b ∈ R
• ek−1 = 0 in H∗

S1({Ψ < b})

• ◮ Apply Lusternik-Schnirelmann theory:

If ck = ck+n = c then en|U = 0 for all U ⊂ W 1,b(R/cZ, Y ) S1-invariant neighborhood of the space of c-periodic Reeb

• rbits

Proof that ck = ck+n implies Besse

(⋆) en|U = 0 for all S1-invariant neighborhood U ⊂ W 1,b(R/cZ, Y ) of the space of c-periodic Reeb orbits

Proof that ck = ck+n implies Besse

(⋆) en|U = 0 for all S1-invariant neighborhood U ⊂ W 1,b(R/cZ, Y ) of the space of c-periodic Reeb orbits ◮ With a bit of algebraic topology, (⋆) implies: Every sufficiently small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space of c-periodic Reeb orbits has non-zero cohomology H2n+1(W ).

Proof that ck = ck+n implies Besse

(⋆) en|U = 0 for all S1-invariant neighborhood U ⊂ W 1,b(R/cZ, Y ) of the space of c-periodic Reeb orbits ◮ With a bit of algebraic topology, (⋆) implies: Every sufficiently small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space of c-periodic Reeb orbits has non-zero cohomology H2n+1(W ).

H2n+1(U)

• π∗
• H2n+1(W )

H2n+1(U′)

π∗

• H2n

S1(U)

H2n

S1(U′)

U ⊇ W ⊇ U′ neighborhoods of the space of c-periodic Reeb orbits; U, U′ are S1-invariant

Proof that ck = ck+n implies Besse

(⋆) en|U = 0 for all S1-invariant neighborhood U ⊂ W 1,b(R/cZ, Y ) of the space of c-periodic Reeb orbits ◮ With a bit of algebraic topology, (⋆) implies: Every sufficiently small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space of c-periodic Reeb orbits has non-zero cohomology H2n+1(W ). ◮ We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space of c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof ◮ The Reeb orbits are geodesics of a suitable Riemannian metric

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof ◮ The Reeb orbits are geodesics of a suitable Riemannian metric ◮ Z Y open neighborhood of fix(φc)

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof ◮ The Reeb orbits are geodesics of a suitable Riemannian metric ◮ Z Y open neighborhood of fix(φc) ◮ ι : Z ֒ → W 1,b(R/cZ, Y ), z → ιz

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof ◮ The Reeb orbits are geodesics of a suitable Riemannian metric ◮ Z Y open neighborhood of fix(φc) ◮ ι : Z ֒ → W 1,b(R/cZ, Y ), z → ιz

φc(z) = z ιz(t) = φt(z)

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof ◮ The Reeb orbits are geodesics of a suitable Riemannian metric ◮ Z Y open neighborhood of fix(φc) ◮ ι : Z ֒ → W 1,b(R/cZ, Y ), z → ιz

z′ φc(z) = z ιz(t)

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof ◮ The Reeb orbits are geodesics of a suitable Riemannian metric ◮ Z Y open neighborhood of fix(φc) ◮ ι : Z ֒ → W 1,b(R/cZ, Y ), z → ιz

z′ φt(z′) φc(z) = z ιz(t)

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof ◮ The Reeb orbits are geodesics of a suitable Riemannian metric ◮ Z Y open neighborhood of fix(φc) ◮ ι : Z ֒ → W 1,b(R/cZ, Y ), z → ιz

z′ φt(z′) φc(z) = z ιz(t)

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof ◮ The Reeb orbits are geodesics of a suitable Riemannian metric ◮ Z Y open neighborhood of fix(φc) ◮ ι : Z ֒ → W 1,b(R/cZ, Y ), z → ιz

z′ ιz′(t) φc(z) = z ιz(t)

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof ◮ The Reeb orbits are geodesics of a suitable Riemannian metric ◮ Z Y open neighborhood of fix(φc) ◮ ι : Z ֒ → W 1,b(R/cZ, Y ), z → ιz

z′ ιz′(t) φc(z) = z ιz(t)

◮ W ⊂ W 1,b(R/cZ, Y ) small tubular neighborhood of ι(Z)

Proof that ck = ck+n implies Besse

We are left to show: If some Reeb orbit of Y is not c-periodic, then there exists an arbitrarily small neighborhood W ⊂ W 1,b(R/cZ, Y ) of the space

• f c-periodic Reeb orbits with H2n+1(W ) = 0.

Proof ◮ The Reeb orbits are geodesics of a suitable Riemannian metric ◮ Z Y open neighborhood of fix(φc) ◮ ι : Z ֒ → W 1,b(R/cZ, Y ), z → ιz

z′ ιz′(t) φc(z) = z ιz(t)

◮ W ⊂ W 1,b(R/cZ, Y ) small tubular neighborhood of ι(Z) ◮ H2n+1(W ) ∼ = H2n+1(Z) = 0.

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) restricted contact type hypersurface of Cn+1

Y Cn+1

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) restricted contact type hypersurface of Cn+1

Y Cn+1

Ekeland-Hofer capacities ck = ck(Y ) = ck(fill(Y )) ∈ σ(Y , λ)

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) restricted contact type hypersurface of Cn+1

Y Cn+1

Ekeland-Hofer capacities ck = ck(Y ) = ck(fill(Y )) ∈ σ(Y , λ) c1 ≤ c2 ≤ c3 ≤ ...

Besse and Zoll Reeb flows in higher dimension

(Y 2n+1, λ) restricted contact type hypersurface of Cn+1

Y Cn+1

Ekeland-Hofer capacities ck = ck(Y ) = ck(fill(Y )) ∈ σ(Y , λ) c1 ≤ c2 ≤ c3 ≤ ... Theorem (Ginzburg, G¨ urel, Mazzucchelli, 2019) If σ(Y , λ) is discrete and ck(Y ) = ck+n(Y ) =: c for some k ≥ 1, then (Y , λ) is Besse and c is a common period for its closed Reeb orbits.

Geodesic flows in higher dimension

◮ (M, g) closed Riemannian manifold

Geodesic flows in higher dimension

◮ (M, g) closed Riemannian manifold ◮ Energy functional E : ΛM = W 1,2(S1, M) → [0, ∞), E(γ) =

• S1 ˙

γ(t)2

gdt

Geodesic flows in higher dimension

◮ (M, g) closed Riemannian manifold ◮ Energy functional E : ΛM = W 1,2(S1, M) → [0, ∞), E(γ) =

• S1 ˙

γ(t)2

gdt

◮ Action selector associated to κ ∈ H∗

S1(ΛM, M), κ = 0

c(κ) := inf √ b

• κ = 0 in H∗

S1({E < b}, M)

• ∈ σ(M, g)

Geodesic flows in higher dimension

◮ (M, g) closed Riemannian manifold ◮ Energy functional E : ΛM = W 1,2(S1, M) → [0, ∞), E(γ) =

• S1 ˙

γ(t)2

gdt

◮ Action selector associated to κ ∈ H∗

S1(ΛM, M), κ = 0

c(κ) := inf √ b

• κ = 0 in H∗

S1({E < b}, M)

• ∈ σ(M, g)

◮ Assume M is a simply connected, spin, CROSS: M = Sn, CPn/2, HPn/4, or CaP2 (n = 16) except CPn/2 with n/2 even

Geodesic flows in higher dimension

◮ (M, g) closed Riemannian manifold ◮ Energy functional E : ΛM = W 1,2(S1, M) → [0, ∞), E(γ) =

• S1 ˙

γ(t)2

gdt

◮ Action selector associated to κ ∈ H∗

S1(ΛM, M), κ = 0

c(κ) := inf √ b

• κ = 0 in H∗

S1({E < b}, M)

• ∈ σ(M, g)

◮ Assume M is a simply connected, spin, CROSS: M = Sn, CPn/2, HPn/4, or CaP2 (n = 16) except CPn/2 with n/2 even ◮ αm, βm generators of Him

S1(ΛM, M) and Him+2n−2 S1

(ΛM, M) im = m i(M) + (m − 1)(n − 1)

Geodesic flows in higher dimension

M closed, simply connected, spin, CROSS αm, βm generators of Him

S1(ΛM, M) and Him+2n−2 S1

(ΛM, M) im = m i(M) + (m − 1)(n − 1)

Geodesic flows in higher dimension

M closed, simply connected, spin, CROSS αm, βm generators of Him

S1(ΛM, M) and Him+2n−2 S1

(ΛM, M) im = m i(M) + (m − 1)(n − 1) Theorem (Ginzburg, G¨ urel, Mazzucchelli, 2019) The following conditions are equivalent: (i) c(α1) = c(β1) (ii) c(αm) = c(βm) for all m ≥ 1 (iii) (M, g) is Zoll

Geodesic flows in higher dimension

M closed, simply connected, spin, CROSS αm, βm generators of Him

S1(ΛM, M) and Him+2n−2 S1

(ΛM, M) im = m i(M) + (m − 1)(n − 1) Theorem (Ginzburg, G¨ urel, Mazzucchelli, 2019) The following conditions are equivalent: (i) c(α1) = c(β1) (ii) c(αm) = c(βm) for all m ≥ 1 (iii) (M, g) is Zoll If M = Sn with n = 3, then (i) can be replaced by: (i’) c(αm) = c(βm) for some m ≥ 1

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