The singular Weinstein conjecture C edric Oms Universidad Polit - - PowerPoint PPT Presentation

The singular Weinstein conjecture

C´ edric Oms

Universidad Polit´ ecnica de Catalunya Friday Fish

7 August 2020

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Eva Miranda and Daniel Peralta–Salas

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Motivating examples from celestial mechanics

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Restricted planar circular 3-body problem I

Simplified version of the general 3-body problem: One of the bodies has negligible mass. The other two bodies move in circles following Kepler’s laws for the 2-body problem. The motion of the small body is in the same plane.

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Restricted planar circular 3-body problem II

Time-dependent potential: U(q,t) =

1−µ ∣q−qE(t)∣ + µ ∣q−qM(t)∣

Time-dependent Hamiltonian: H(q,p,t) = ∣p∣2

2 − U(q,t),

(q,p) ∈ R2 ∖ {qE,qM} × R2 Rotating coordinates: Time independent Hamiltonian H(q,p) = ∣p∣2

2 − 1−µ ∣q−qE∣ + µ ∣q−qM∣ + p1q2 − p2q1

H has 5 critical points: Li Lagrange points (H(L1) ≤ ⋅⋅⋅ ≤ H(L5)) Periodic orbits of XH? Perturbative methods (dynamical systems) or.... contact topology!

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Level-sets of Hamiltonians

Let (W ,ω) be a symplectic manifold and Σ ⊂ W hypersurface.

Definition

A Liouville vector field is a v.f. X ∈ X(W ) such that LXω = ω.

Proposition

Let X be a Liouville vector field transverse to Σ. Then (Σ,α = ιXω) is a contact manifold. If Σ = H−1(c), then Rα ≅ XH∣H=c.

Conjecture (Weinstein conjecture)

Let (M,α) closed contact manifold. Then Rα admits periodic orbits.

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Contact Geometry of the RPC3BP

For c < H(L1), Σc = H−1(c) has 3 connected components: ΣE

c (the

satellite stays close to the earth), ΣM

c (to the moon), or it is far away.

Proposition (Albers–Frauenfelder–Koert–Paternain)

For c < H(L1), X = (q − qE) ∂

∂q is transverse to ΣE c .

Hence (ΣE

c ,ιXω) is contact.

But Weinstein conjecture does not apply because of non-compactness (collision!) ✴

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Moser regularization of the restricted 3-body problem

Via Moser’s regularization ΣE

c can be compactified to Σ E c ≅ RP(3).

Theorem (Albers–Frauenfelder–Koert–Paternain)

For any value c < H(L1), the regularized RPC3BP has a closed orbit with energy c. ✱

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But...

Where are those periodic orbits? Maybe on the collision set? Keep track of the singularities in the geometric structure? ...bm-symplectic and bm-contact geometry!

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Or manifold at infinity?

Consider the canonical change of coordinates to polar coordinates: (q,p) ↦ (r,α,Pr,Pα) McGehee change of coordinates: r = 2

x2 , where x ∈ R+

Non-canonical, the symplectic form becomes singular: ω = − 4 x3 dx ∧ dα + dPr ∧ dPα This is a b3-symplectic form. Dynamics of XH?

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bm-symplectic and bm-contact geometry

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Introducing b-symplectic

b-symplectic structures can be seen as symplectic structures modeled

• ver a Lie algebroid (the b-cotangent bundle).

A vector field v is a b-vector field if vp ∈ TpZ for all p ∈ Z. The b-tangent bundle bTM is defined by Γ(U, bTM) = { b-vector fields

• n (U,U ∩ Z) }

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b-cotangent bundle

Consider a hypersurface Z = f −1(0) of M, the critical set

bX(M) = {v.f. tangent to Z} = ⟨f ∂ ∂f , ∂ ∂x1 ,..., ∂ ∂xn−1 ⟩

Serre–Swan: There exists a bundle bTM such that Γ(bTM) = bX(M). The dual: bT ∗M and forms: bΩk(M) = Γ(Λk(bT ∗M)).

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Extending differential calculus

ω ∈ bΩk(M) can be decomposed ω = α ∧ df f + β where α ∈ Ωk−1(M),β ∈ Ωk(M).

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Extending differential calculus

ω ∈ bΩk(M) can be decomposed ω = α ∧ df f + β where α ∈ Ωk−1(M),β ∈ Ωk(M). Extension of the exterior derivative by defining d(α ∧ df f + β) ∶= dα ∧ df f + dβ.

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Extending differential calculus

ω ∈ bΩk(M) can be decomposed ω = α ∧ df f + β where α ∈ Ωk−1(M),β ∈ Ωk(M). Extension of the exterior derivative by defining d(α ∧ df f + β) ∶= dα ∧ df f + dβ.

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b-symplectic and b-contact manifolds

Definition ([GMP])

A b-symplectic form on W 2n is ω ∈ bΩ2(W ) such that dω = 0, ω is non-degenerate.

Definition

A manifold (M2n+1,α) where α ∈ bΩ1(M) is b-contact if α ∧ (dα)n ≠ 0.

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b-symplectic and b-contact manifolds

Definition ([GMP])

A bm-symplectic form on W 2n is ω ∈ bmΩ2(W ) such that dω = 0, ω is non-degenerate.

Definition

A manifold (M2n+1,α) where α ∈ bmΩ1(M) is bm-contact if α ∧ (dα)n ≠ 0.

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b-Symplectic Symplectic b-Contact Contact

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Poisson b-Symplectic Symplectic b-Contact Contact

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Jacobi Poisson b-Symplectic Symplectic b-Contact Contact

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Local study of bm-contact manifolds I

Example

(R3, dz

z + xdy), Rα = z ∂ ∂z

(R3,dx + y dz

z ), Rα = ∂ ∂x

The Reeb vector field Rα is defined by the equations ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ιRαdα = 0 ιRαα = 1

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Local study of bm-contact and bm-symplectic manifolds II

One can prove Darboux theorem, analyze the induced structure on the critical set...see [MO1].

Proposition

Let (W ,Z,ω) be a bm-symplectic manifold and X ∈ bmX(W ) such that LXω = ω and X ⋔ Σ. Then (Σ,ιXω) is bm-contact with critical set ̃ Z = Z ∩ Σ.

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Dynamics on bm-contact manifolds

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The Reeb vector field Rα is defined by the equations ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ιRαdα = 0 ιRαα = 1. The Reeb vector field can vanish! Do there exists plugs?

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A trap is a smooth vector field on the manifold Dn−1 × [0,1] such that

1 the flow of the vector field is given by

∂ ∂t near the boundary of

∂D × [0,1], where t is the coordinate on [0,1];

2 there are no periodic orbits contained in D × [0,1]; 3 the orbit entering at the origin of the disk D × {0} does not leave

D × [0,1] again. If the vector field additionally satisfies entry-exit matching condition, that is that the orbit entering at (x,0) leaves at (x,1) for all x ∈ D ∖ {0}, then the trap is called a plug.

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Weinstein conjecture: There are no plugs. Eliashberg–Hofer: non-existence of traps for dim=3. Geiges–Roettgen–Zehmisch: existence in higher dimension. Traps and plugs for bm-contact?

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Theorem

There exists traps for the bm-Reeb flow. Z

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Theorem

There exists traps for the bm-Reeb flow. Z Question: Existence/Non-existence of periodic Reeb orbits away and on Z?

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Proposition

Let (M,α = u dz

z + β) be a bm-contact manifold of dimension 3. Then the

restriction on Z of the 2-form Θ = udβ + β ∧ du is symplectic and the Reeb vector field is Hamiltonian with respect to Θ with Hamiltonian function u, i.e. ιRΘ = du. This is highly 3-dimensional!

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Infinitely many periodic orbits on Z

Proposition

Let (M,α) be a 3-dimensional bm-contact manifold and assume the critical hypersurface Z to be closed. Then there exists infinitely many periodic Reeb orbits on Z.

Proof.

1 α = u dz

z + β

2 u is non-constant on Z 3 Rα is Hamiltonian on Z for −u, 4 u−1(p) where p regular is a circle, 5 Rα periodic on u−1(p). C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 25 / 37

No periodic orbits away from Z

There are compact bm-contact manifolds (M,Z) of any dimension for all m ∈ N without periodic Reeb orbits on M ∖ Z.

Example

S3 ⊂ (R4,ω = dx1

x1 ∧ dy1 + dx2 ∧ dy2)

X = 1

2x1 ∂ ∂x1 + y1 ∂ ∂y1 + 1 2(x2 ∂ ∂x2 + y2 ∂ ∂y2 ) Liouville v.f.

Rα = 2x2

1 ∂ ∂x1 − x1y1 ∂ ∂y1 + 2x2 ∂ ∂y2 − 2y2 ∂ ∂x2

On Z = S2: rotation, Away from Z, no periodic orbits.

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Periodic orbits away from Z?

Definition

(M3,ξ = ker α) is overtwisted if there exists D2 s.t. TD ∩ ξ defines a 1-dimensional foliation given by A contact manifold that is not overtwisted is called tight.

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Theorem (Hofer ’93)

Let (M3,α) a closed OT contact manifold. Then there exists a periodic

• rbit.

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Theorem (Hofer ’93)

Let (M3,α) a closed OT contact manifold. Then there exists a periodic

• rbit.

Not true for open OT manifolds!

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Theorem (Hofer ’93)

Let (M3,α) a closed OT contact manifold. Then there exists a periodic

• rbit.

Not true for open OT manifolds!

Definition

A bm-contact manifold is overtwisted if there exists an overtwisted disk away from the critical hypersurface Z.

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Theorem (Hofer ’93)

Let (M3,α) a closed OT contact manifold. Then there exists a periodic

• rbit.

Not true for open OT manifolds!

Definition

A bm-contact manifold is overtwisted if there exists an overtwisted disk away from the critical hypersurface Z.

Definition

A contact form α is R+-invariant around the critical set if there exists a contact vector field that α = u dz

zm + β, where u ∈ C∞(Z) and β ∈ Ω1(Z)

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Theorem

Let (M,α) be a closed bm-contact manifold with critical set Z. Assume there exists an overtwisted disk in M ∖ Z and assume that α is R+-invariant in a tubular neighbourhood around Z. Then there exists

1 a periodic Reeb orbit in M ∖ Z or 2 a family of periodic Reeb orbits approaching the critical set Z.

The proof is an adaptation of Hofer’s technique. Question: Other applications of this theorem?

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Z {0} × M

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Back to the motivating example

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Contact geometry of RPC3BP revisited

In rotating coordinates: H(q,p) = ∣p∣2

2 − 1−µ ∣q−qE∣ − µ ∣q−qM∣ + p1q2 − p2q1

Lemma

The vector field Y = p ∂

∂p is a Liouville vector field and is transverse to Σc

for c > 0. Symplectic polar coordinates: (r,α,Pr,Pα). McGehee change of coordinates: r = 2

x2 .

b3-symplectic form: −4dx

x3 ∧ dPr + dα ∧ dPα.

Is Σc b3-contact after McGehee? Can we apply the results on periodic

• rbits?

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b3-contact form in the RPC3BP

Theorem

After the McGehee change, the Liouville vector field Y = p ∂

∂p is a

b3-vector field that is everywhere transverse to Σc for c > 0 and the level-sets (Σc,ιY ω) for c > 0 are b3-contact manifolds. Topologically, the critical set is a cylinder and the Reeb vector field admits infinitely many non-trivial periodic orbits on the critical set.

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Proof.

Y transverse at the critical set? On critical set, Hamiltonian H = 1

2P2 r − Pα, so that

Y (H)∣H=c = P2

r − Pα = 1 2P2 r + c > 0;

b3-contact form α = (Pr dx

x3 + Pαdα)∣H=c with

Z = {(x,α,Pr,Pα)∣x = 0, 1

2P2 r − Pα = c};

Rα∣Z = XPr ; Cylinder is foliated by periodic orbits.

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Open questions and future work

Can those periodic orbits be continued away from the critical set?

Figure: A Singular periodic orbit a.k.a. heteroclinic

Conjecture (Singular Weinstein conjecture)

Let (M,α) be a compact bm-contact manifold. Then there exists always a singular periodic Reeb orbit. Recent work (joint with Miranda and Peralta-Salas: ”Generically”, the conjecture is satisfied.

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Thanks for listening

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References

Albers, Peter, Urs Frauenfelder, Otto Van Koert, and Gabriel P.

• Paternain. ”Contact geometry of the restricted three-body problem.”

Communications on pure and applied mathematics 65, no. 2 (2012): 229-263. Guillemin, Victor, Eva Miranda, and Ana Rita Pires. ”Symplectic and Poisson geometry on b-manifolds.” Advances in mathematics 264 (2014): 864-896. Miranda, Eva, and C´ edric Oms. ”The geometry and topology of contact structures with singularities.” arXiv preprint arXiv:1806.05638 (2018). Miranda, Eva, and C´ edric Oms. ”The singular Weinstein conjecture.” arXiv preprint arXiv:2005.09568 (2020).

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