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 C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 1 / 37

Eva Miranda and Daniel Peralta–Salas C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 2 / 37

Motivating examples from celestial mechanics C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 3 / 37

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. C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 4 / 37

Restricted planar circular 3-body problem II 1 − µ µ Time-dependent potential: U ( q , t ) = ∣ q − q E ( t )∣ + ∣ q − q M ( t )∣ Time-dependent Hamiltonian: H ( q , p , t ) = ∣ p ∣ 2 ( q , p ) ∈ R 2 ∖ { q E , q M } × R 2 2 − U ( q , t ) , Rotating coordinates: Time independent Hamiltonian H ( q , p ) = ∣ p ∣ 2 1 − µ µ 2 − ∣ q − q E ∣ + ∣ q − q M ∣ + p 1 q 2 − p 2 q 1 H has 5 critical points: L i Lagrange points ( H ( L 1 ) ≤ ⋅⋅⋅ ≤ H ( L 5 ) ) Periodic orbits of X H ? Perturbative methods (dynamical systems) or.... contact topology! C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 5 / 37

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 L X ω = ω . Proposition Let X be a Liouville vector field transverse to Σ . Then ( Σ ,α = ι X ω ) is a contact manifold. If Σ = H − 1 ( c ) , then R α ≅ X H ∣ H = c . Conjecture (Weinstein conjecture) Let ( M ,α ) closed contact manifold. Then R α admits periodic orbits. C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 6 / 37

Contact Geometry of the RPC3BP For c < H ( L 1 ) , Σ c = H − 1 ( c ) has 3 connected components: Σ E c (the satellite stays close to the earth), Σ M c (to the moon), or it is far away. Proposition (Albers–Frauenfelder–Koert–Paternain) For c < H ( L 1 ) , X = ( q − q E ) ∂ ∂ q is transverse to Σ E c . Hence ( Σ E c ,ι X ω ) is contact. But Weinstein conjecture does not apply because of non-compactness (collision!) ✴ C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 7 / 37

Moser regularization of the restricted 3-body problem c ≅ R P ( 3 ) . E Via Moser’s regularization Σ E c can be compactified to Σ Theorem (Albers–Frauenfelder–Koert–Paternain) For any value c < H ( L 1 ) , the regularized RPC3BP has a closed orbit with energy c. ✱ C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 8 / 37

But... Where are those periodic orbits? Maybe on the collision set? Keep track of the singularities in the geometric structure? ... b m -symplectic and b m -contact geometry! C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 9 / 37

Or manifold at infinity? Consider the canonical change of coordinates to polar coordinates: ( q , p ) ↦ ( r ,α, P r , P α ) McGehee change of coordinates: r = 2 x 2 , where x ∈ R + Non-canonical, the symplectic form becomes singular: ω = − 4 x 3 dx ∧ d α + dP r ∧ dP α This is a b 3 -symplectic form. Dynamics of X H ? C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 10 / 37

b m -symplectic and b m -contact geometry C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 11 / 37

Introducing b -symplectic b -symplectic structures can be seen as symplectic structures modeled over a Lie algebroid (the b -cotangent bundle). A vector field v is a b -vector field if v p ∈ T p Z for all p ∈ Z . The b -tangent bundle b TM is defined by Γ ( U , b TM ) = { b-vector fields on ( U , U ∩ Z ) } C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 12 / 37

b -cotangent bundle Consider a hypersurface Z = f − 1 ( 0 ) of M , the critical set b X ( M ) = { v.f. tangent to Z } = ⟨ f ∂ ∂ ∂ ∂ x n − 1 ⟩ ∂ f , ∂ x 1 ,..., Serre–Swan: There exists a bundle b TM such that Γ ( b TM ) = b X ( M ) . The dual: b T ∗ M and forms: b Ω k ( M ) = Γ ( Λ k ( b T ∗ M )) . C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 13 / 37

Extending differential calculus ω ∈ b Ω k ( M ) can be decomposed ω = α ∧ df f + β where α ∈ Ω k − 1 ( M ) ,β ∈ Ω k ( M ) . C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 14 / 37

Extending differential calculus ω ∈ b Ω k ( M ) can be decomposed ω = α ∧ df f + β where α ∈ Ω k − 1 ( M ) ,β ∈ Ω k ( M ) . Extension of the exterior derivative by defining d ( α ∧ df f + β ) ∶ = d α ∧ df f + d β. C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 14 / 37

Extending differential calculus ω ∈ b Ω k ( M ) can be decomposed ω = α ∧ df f + β where α ∈ Ω k − 1 ( M ) ,β ∈ Ω k ( M ) . Extension of the exterior derivative by defining d ( α ∧ df f + β ) ∶ = d α ∧ df f + d β. C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 14 / 37

b -symplectic and b -contact manifolds Definition ([GMP]) A b -symplectic form on W 2 n is ω ∈ b Ω 2 ( W ) such that d ω = 0, ω is non-degenerate. Definition A manifold ( M 2 n + 1 ,α ) where α ∈ b Ω 1 ( M ) is b -contact if α ∧ ( d α ) n ≠ 0. C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 15 / 37

b -symplectic and b -contact manifolds Definition ([GMP]) A b m -symplectic form on W 2 n is ω ∈ b m Ω 2 ( W ) such that d ω = 0, ω is non-degenerate. Definition A manifold ( M 2 n + 1 ,α ) where α ∈ b m Ω 1 ( M ) is b m -contact if α ∧ ( d α ) n ≠ 0. C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 15 / 37

b-Symplectic b-Contact Symplectic Contact C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 16 / 37

Poisson b-Symplectic b-Contact Symplectic Contact C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 16 / 37

Jacobi Poisson b-Symplectic b-Contact Symplectic Contact C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 16 / 37

Local study of b m -contact manifolds I Example z + xdy ) , R α = z ∂ ( R 3 , dz ∂ z ( R 3 , dx + y dz z ) , R α = ∂ ∂ x The Reeb vector field R α is defined by the equations ⎧ ⎪ ⎪ ι R α d α = 0 ⎨ ⎪ ι R α α = 1 ⎪ ⎩ C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 17 / 37

Local study of b m -contact and b m -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 b m -symplectic manifold and X ∈ b m X ( W ) such that L X ω = ω and X ⋔ Σ . Then ( Σ ,ι X ω ) is b m -contact with critical set Z = Z ∩ Σ . ̃ C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 18 / 37

Dynamics on b m -contact manifolds C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 19 / 37

The Reeb vector field R α is defined by the equations ⎧ ⎪ ι R α d α = 0 ⎪ ⎨ ⎪ ι R α α = 1 . ⎪ ⎩ The Reeb vector field can vanish! Do there exists plugs? C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 20 / 37

A trap is a smooth vector field on the manifold D n − 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 . C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 21 / 37

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 b m -contact? C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 22 / 37

Theorem There exists traps for the b m -Reeb flow. Z C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 23 / 37

Theorem There exists traps for the b m -Reeb flow. Z Question: Existence/Non-existence of periodic Reeb orbits away and on Z ? C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 23 / 37

Proposition z + β ) be a b m -contact manifold of dimension 3 . Then the Let ( M ,α = u dz restriction on Z of the 2 -form Θ = ud β + β ∧ du is symplectic and the Reeb vector field is Hamiltonian with respect to Θ with Hamiltonian function u, i.e. ι R Θ = du. This is highly 3-dimensional! C´ edric Oms (UPC) The singular Weinstein conjecture 7 August 2020 24 / 37