| import Mathlib |
| import OAI.Analysis.BiholderTransport.Convexity.ActualCenterSemibound |
| import OAI.Analysis.BiholderTransport.Regularity.CenterRayLimit |
| import OAI.Analysis.BiholderTransport.Regularity.CenterSequenceCompact |
| import OAI.Analysis.BiholderTransport.Regularity.CenterPoleStationary |
| import OAI.Analysis.BiholderTransport.Calculus.TrueCenterGradient |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff NNReal |
|
|
| namespace WeakMTWTransport |
| section CenterTrueRayCompact |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty M] |
| [MeasurableSpace M] [BorelSpace M] |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] |
|
|
| omit [MeasurableSpace M] [BorelSpace M] in |
| lemma CenterSequenceData.true_ray_compact {a c:M} {u v:M → ℝ} {Φ:ℝ×ℝ → ℝ} {γ l:ℝ} |
| (hv:Continuous v) (hΦ:ContDiffAt ℝ ∞ Φ (γ,v c)) |
| (hder:deriv (fun s=>Φ (γ,s)) (v c)=l) (hl:0 < l) (hl1:l < 1) |
| {γj lj τj tj:ℕ → ℝ} {xj bj qj:ℕ → Model n} {zj:ℕ → M} |
| {F:ℕ → Model n → ℝ} |
| (D:CenterSequenceData a c u v Φ γj lj τj tj xj bj qj zj F) |
| {q:Model n} |
| (hq:(show TangentSpace 𝓘(ℝ,Model n) a from q)∈minimizingVectors a) |
| (he:riemannianExp a q=c) |
| (hγ:Tendsto γj atTop (𝓝 γ)) (hlm:Tendsto lj atTop (𝓝 l)) |
| (hτ:Tendsto τj atTop (𝓝 0)) (ht:Tendsto tj atTop (𝓝 1)) |
| (hx:Tendsto xj atTop (𝓝 (extChartAt 𝓘(ℝ,Model n) c c))) |
| (hb:Tendsto bj atTop (𝓝 (extChartAt 𝓘(ℝ,Model n) a a))) |
| (hτpos:∀ᶠ i in atTop,0 < τj i) (L:ℝ≥0) |
| (hLip:∀ᶠ i in atTop,LipschitzWith L (fun y=>Φ (γj i,v y))) : |
| ∃σ:ℕ → ℕ,StrictMono σ ∧ ∃r:Model n, |
| (show TangentSpace 𝓘(ℝ,Model n) c from r)∈minimizingVectors c ∧ |
| Tendsto (fun i=>(D.row (σ i)).r) atTop (𝓝 r) ∧ |
| Tendsto (fun i=>reverseRay (tangentScale (lj (σ i)) |
| (chartRay c (xj (σ i)) (D.row (σ i)).r))) atTop |
| (𝓝 (⟨a,q⟩:TangentBundle 𝓘(ℝ,Model n) M)) := by |
| obtain ⟨σ,hσ,r,hr,hrr⟩:=center_support_family_subseq D.row hx |
| have HP: a=riemannianExp c (l • r):= |
| (D.comp σ hσ.tendsto_atTop).pole hv hΦ hder hl hl1 hr |
| (hγ.comp hσ.tendsto_atTop) (hlm.comp hσ.tendsto_atTop) |
| (hτ.comp hσ.tendsto_atTop) (ht.comp hσ.tendsto_atTop) |
| (hx.comp hσ.tendsto_atTop) (hb.comp hσ.tendsto_atTop) hrr |
| (hσ.tendsto_atTop.eventually hτpos) L (hσ.tendsto_atTop.eventually hLip) |
| have HE:=reverse_minimizer_eq (contracted_minimizer_mem_injectivityDomain hr hl hl1) |
| hq HP.symm he |
| refine ⟨σ,hσ,r,hr,hrr,?_⟩ |
| have HT:=modified_reverseRay_tendsto (hx.comp hσ.tendsto_atTop) hrr |
| (hlm.comp hσ.tendsto_atTop) |
| have HE' : reverseRay (⟨c,l • r⟩ : TangentBundle 𝓘(ℝ,Model n) M) = |
| (⟨a,q⟩:TangentBundle 𝓘(ℝ,Model n) M) := by exact HE |
| rw [HE'] at HT |
| exact HT |
|
|
| end CenterTrueRayCompact |
| end WeakMTWTransport |
|
|
| end |
| end |
|
|
| end OAI |
|
|
