The unit codisc bundle is a Liouville domain

Let M be a compact smooth Riemannian manifold without boundary, with cotangent bundle T*M = TauCeti.CotangentBundle I M, Liouville form λ = p dq and canonical symplectic form ω = -dλ, which is nondegenerate (TauCeti.CotangentBundle.isSymplectic_symplecticForm). The radial field Z = p ∂_p, the Euler vector field of the bundle, is the Liouville field of the primitive -λ of ω (TauCeti.CotangentBundle.symplecticForm_interior_vectorBundleEulerField).

The unit codisc bundle D*M = {‖p‖ ≤ 1} is compact, and it is the sublevel set of the norm of covectors, which is continuous on T*M and smooth off the zero section. Along the fibre ray t ↦ t • z the norm is t ↦ |t| ‖z.2‖, so the differential of the norm sends Z to ‖z.2‖ (TauCeti.CotangentBundle.mfderiv_norm_snd_vectorBundleEulerField), which is 1 > 0 on the unit cosphere bundle S*M = {‖p‖ = 1}. Hence (D*M, -λ) is a Liouville domain in (T*M, ω) with Liouville field p ∂_p (TauCeti.CotangentBundle.isLiouvilleDomain_codiscBundle). It is the Liouville domain on which the symplectic homology and the capacities of cotangent bundles are computed; its boundary is S*M, of restricted contact type (TauCeti.CotangentBundle.isRestrictedContactType_cosphereBundle).

The flow of p ∂_p is the fibre dilation p ↦ e^t p, defined for all time, and it shrinks every covector into D*M. So T*M is the completion of D*M (TauCeti.CotangentBundle.isLiouvilleCompletion_codiscBundle): every covector outside D*M is e^t p for exactly one unit covector p and one time t > 0.

Main results

• TauCeti.CotangentBundle.isLiouvilleDomain_codiscBundle: D*M is a Liouville domain for the canonical symplectic form, with primitive -λ.
• TauCeti.CotangentBundle.isLiouvilleCompletion_codiscBundle: T*M is its completion.

References

• K. Cieliebak, Y. Eliashberg, From Stein to Weinstein and Back, AMS Colloquium Publications 59, 2012, Section 11.1 (the unit codisc bundle as a Liouville domain).
• J.-F. Barraud, M. Damian, V. Humilière, A. Oancea, Floer homology with DG coefficients. Applications to cotangent bundles, arXiv:2404.07953, §1.2.

theorem TauCeti.CotangentBundle.isLiouvilleDomain_codiscBundle {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I (↑⊤) E fun (x : M) => TangentSpace I x] [CompactSpace M] : IsLiouvilleDomain (symplecticForm I M) (-liouvilleForm I M) (codiscBundle I M)

The unit codisc bundle is a Liouville domain. For a compact smooth Riemannian manifold M without boundary, the unit codisc bundle D*M is a Liouville domain in T*M for the canonical symplectic form ω = -dλ, with primitive -λ: its Liouville field, the radial field p ∂_p, points strictly outwards along the unit cosphere bundle, the level ‖p‖ = 1 of the norm of covectors.

theorem TauCeti.CotangentBundle.isLiouvilleCompletion_codiscBundle {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type u_2} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] {M : Type u_3} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [Bundle.RiemannianBundle fun (x : M) => TangentSpace I x] [IsContMDiffRiemannianBundle I (↑⊤) E fun (x : M) => TangentSpace I x] [CompactSpace M] [T2Space M] : IsLiouvilleCompletion (symplecticForm I M) (-liouvilleForm I M) (codiscBundle I M)

The cotangent bundle is the completion of the unit codisc bundle. For a compact smooth Riemannian manifold M without boundary, T*M is a completion of the Liouville domain D*M: the flow p ↦ e^t p of its Liouville field p ∂_p is defined for all time, and carries every covector into D*M.