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110f4f3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 | 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
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