Completions of Liouville domains #
The completion of a Liouville domain (W, α) attaches to W the positive half of the
symplectization of its boundary: it is W ∪ (∂W × [1, ∞)), glued along ∂W = ∂W × {1}, with
the primitive r α|∂W on the cylindrical end. Symplectic homology, Hamiltonians linear at infinity
and the Viterbo transfer maps are defined on completions.
A Liouville domain here is a compact sublevel set W = {f ≤ c} inside an ambient manifold M
(TauCeti.IsLiouvilleDomain), and the completion is likewise realized inside M
(TauCeti.IsLiouvilleCompletion): M is a completion of W when the Liouville field Z of α,
the vector field with ι_Z ω = α, is complete, and its flow carries every point of M into W.
For instance, the cotangent bundle T*M of a compact Riemannian manifold without boundary is the
completion of its unit codisc bundle
(TauCeti.CotangentBundle.isLiouvilleCompletion_codiscBundle).
The Liouville field is unique, because the form is nondegenerate, and it points strictly out of
W along the boundary. So a trajectory of Z crosses ∂W only outwards: a point that is not in
the interior of W never returns to W under the forward flow
(TauCeti.IsLiouvilleDomain.flowOf_notMem), and when Z is complete the trajectories through
distinct boundary points are disjoint, each meeting the boundary only once
(TauCeti.IsLiouvilleDomain.injOn_flowOf). For a completion this
gives the cylindrical end in coordinates: every point x outside W is ψ_t(y) for exactly one
point y of the boundary ∂W and one time t > 0, where ψ is the Liouville flow
(TauCeti.IsLiouvilleCompletion.existsUnique_flowOf_eq). So M is the disjoint union of W and
the flow-out (y, t) ↦ ψ_t(y) of ∂W × (0, ∞), which is the end ∂W × (1, ∞) of the completion
in the coordinate r = e^t: the map (t, y) ↦ ψ_t(y) is a bijection from (0, ∞) × ∂W onto
M \ W (TauCeti.IsLiouvilleCompletion.bijOn_flowOf_Ioi_prod_frontier). This file proves this
point-set description of the end; the identification of the primitive on it with r α|∂W is in
TauCeti.Geometry.Symplectic.Liouville.Symplectization.
Main definitions #
TauCeti.IsLiouvilleCompletion:Mis a completion of the Liouville domainW.
Main results #
TauCeti.IsLiouvilleDomain.flowOf_notMem: the forward Liouville flow of a point outside the interior of a Liouville domain stays outside the domain.TauCeti.IsLiouvilleDomain.injOn_flowOf: for a complete Liouville field, the trajectories through the boundary are disjoint.TauCeti.IsLiouvilleDomain.preimage_flowOf: for a complete Liouville field, the images of a Liouville domain under the Liouville flow are Liouville domains.TauCeti.IsLiouvilleCompletion.existsUnique_flowOf_eq: the cylindrical end of a completion, each point outsideWlies on exactly one outgoing Liouville trajectory from∂W.TauCeti.IsLiouvilleCompletion.bijOn_flowOf_Ioi_prod_frontier,TauCeti.IsLiouvilleCompletion.bijOn_flowOf_Ici_prod_frontier: the Liouville flow map of a completion is a bijection from(0, ∞) × ∂WontoM \ W, and from[0, ∞) × ∂WontoM \ int W.TauCeti.IsLiouvilleCompletion.flowOf_flowOf: the Liouville flow of a completion is a global flow, satisfying the group law.TauCeti.IsLiouvilleCompletion.flowOf_eq_globalFlowDiffeomorph: its time-tmap is a diffeomorphism.TauCeti.IsLiouvilleCompletion.mpullback_globalFlowDiffeomorph_eq_exp_smul: this diffeomorphismψₜis conformally symplectic,ψₜ^* ω = e^t ω.TauCeti.IsLiouvilleCompletion.flowOf_neg_mem_interior: the backward Liouville flow carries the Liouville domain into its interior.
References #
- P. Seidel, A biased view of symplectic cohomology, Current Developments in Mathematics 2006, Section 3 (Liouville domains and their completions).
- K. Cieliebak, Y. Eliashberg, From Stein to Weinstein and Back, AMS Colloquium Publications 59, 2012, Section 11.1.
The Liouville flow never enters a Liouville domain. Let Z be the Liouville field,
ι_Z ω = α. If x is not an interior point of the Liouville domain W, then its trajectory under
the flow of Z lies outside W at every positive time t at which it is defined.
The Liouville trajectories through the boundary are disjoint. If the Liouville field Z
of a Liouville domain W is complete, then (y, t) ↦ ψ_t(y) is injective on ∂W × ℝ, where ψ
is the flow of Z.
The Liouville flow carries Liouville domains to Liouville domains. If the Liouville field
Z of a Liouville domain W is complete, then for every time t the preimage of W under the
time-t Liouville flow ψ_t, which is the image ψ_{-t}(W), is again a Liouville domain for the
same form and primitive: ψ_{-t}(W) = {f ∘ ψ_t ≤ c} when W = {f ≤ c}, and since ψ_t preserves
Z, the field Z crosses the level of f ∘ ψ_t outwards.
A manifold M is a completion of the Liouville domain W ⊆ M, for the form ω with
primitive α, when the Liouville field Z of α, the vector field with ι_Z ω = α, is complete
and its flow carries every point of M into W in some time. Then M is the union of W and of
the forward flow-out of its boundary, a cylindrical end
(TauCeti.IsLiouvilleCompletion.existsUnique_flowOf_eq).
- exists_liouville_outward : ∃ (Z : (x : M) → TangentSpace I x), (ContMDiff I I.tangent ↑⊤ fun (x : M) => ⟨x, Z x⟩) ∧ RoughForm.interior Z ↑ω = ↑α ∧ ∃ (f : M → ℝ) (c : ℝ), Continuous f ∧ W = {x : M | f x ≤ c} ∧ ∀ (x : M), f x = c → ContMDiffAt I (modelWithCornersSelf ℝ ℝ) (↑⊤) f x ∧ 0 < (d% f x) (Z x)
- exists_isCompleteVectorField : ∃ (Z : (x : M) → TangentSpace I x), RoughForm.interior Z ↑ω = ↑α ∧ IsCompleteVectorField Z ∧ ∀ (x : M), ∃ (t : ℝ), flowOf Z t x ∈ W
The Liouville field of a completion is complete, and its flow carries every point into the Liouville domain.
Instances For
The Liouville field of a completion is complete.
The Liouville flow of a completion carries every point into the Liouville domain.
The cylindrical end of a completion. Let M be a completion of the Liouville domain W,
with Liouville flow ψ. Every point x outside W is ψ_t(y) for exactly one point y of the
boundary of W and one time t > 0.
On a completion, the maximal Liouville flow is a global flow.
On a completion, the time-t map of the Liouville flow is the diffeomorphism
TauCeti.globalFlowDiffeomorph of the complete smooth Liouville field.
The Liouville flow of a completion is conformally symplectic: its time-t map ψₜ, a
diffeomorphism, satisfies ψₜ^* ω = e^t ω.
The group law for the Liouville flow of a completion: ψ_s (ψ_t x) = ψ_{s + t} x.
The backward Liouville flow of a completion carries the Liouville domain into its interior.
The backward Liouville flow of a completion preserves the Liouville domain.
On a completion, the Liouville trajectory of a boundary point lies in the Liouville domain exactly at nonpositive times.
Every point of a completion outside the interior of the Liouville domain is reached from the boundary by the forward Liouville flow, at a nonnegative time.
The cylindrical end of a completion. Let M be a completion of the Liouville domain W,
with Liouville flow ψ. The map (t, y) ↦ ψₜ(y) is a bijection from (0, ∞) × ∂W onto the
complement of W.
The closed cylindrical end of a completion. Let M be a completion of the Liouville
domain W, with Liouville flow ψ. The map (t, y) ↦ ψₜ(y) is a bijection from [0, ∞) × ∂W
onto the complement of the interior of W.