| import Mathlib | |
| import OAI.Analysis.BiholderTransport.Regularity.StationaryPullback | |
| namespace OAI | |
| noncomputable section | |
| open Set Filter Manifold Bundle | |
| open scoped Topology ContDiff | |
| namespace WeakMTWTransport | |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace 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] | |
| section | |
| omit [CompactSpace M] [IsManifold 𝓘(ℝ,Model n) ∞ M] | |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) | |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] | |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] | |
| lemma fixedJoinMiddle_eq_slice {x : M} {h T : ℝ} | |
| {p : TangentSpace 𝓘(ℝ,Model n) x} | |
| (hB : ContDiffAt ℝ 2 (fixedJoinAction x h T p) (0,p)) | |
| (v w : TangentSpace 𝓘(ℝ,Model n) x) : | |
| fixedJoinMiddle x h T p v w=fderiv ℝ | |
| (fderiv ℝ (fun b => fixedJoinAction x h T p (0,b))) p v w := by | |
| let V := TangentSpace 𝓘(ℝ,Model n) x | |
| let I : V →L[ℝ] V×V := (0 : V →L[ℝ] V).prod (ContinuousLinearMap.id ℝ V) | |
| have H := second_fderiv_comp_affine (f := fixedJoinAction x h T p) I 0 p | |
| (by simpa [I] using hB) v w | |
| simpa [I,fixedJoinMiddle_apply] using H.symm | |
| lemma fixedJoinMiddle_symmetric {x : M} {h T : ℝ} | |
| {p : TangentSpace 𝓘(ℝ,Model n) x} | |
| (hB : ContDiffAt ℝ 2 (fixedJoinAction x h T p) (0,p)) | |
| (v w : TangentSpace 𝓘(ℝ,Model n) x) : | |
| fixedJoinMiddle x h T p v w=fixedJoinMiddle x h T p w v := | |
| (hB.isSymmSndFDerivAt (by norm_num)).eq (0,v) (0,w) | |
| end | |
| lemma fixedJoinMiddle_one_nonneg {x : M} {h : ℝ} | |
| {p : TangentSpace 𝓘(ℝ,Model n) x} (hh : 0<h) (hh1 : h<1) | |
| (hp : p∈minimizingVectors x) (hleft : h • p∈injectivityDomain x) | |
| (hright : (1-h) • (sprayFlow h (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).2∈ | |
| injectivityDomain (sprayFlow h (⟨x,p⟩ : TangentBundle 𝓘(ℝ,Model n) M)).1) | |
| (v : TangentSpace 𝓘(ℝ,Model n) x) : | |
| 0 ≤ fixedJoinMiddle x h 1 p v v := by | |
| have HB := (splitNormalAction_contDiffAt hleft hright).comp | |
| (f := fun z : TangentSpace 𝓘(ℝ,Model n) x × TangentSpace 𝓘(ℝ,Model n) x => (p,z)) | |
| (0,p) (contDiffAt_const.prodMk contDiffAt_id) | |
| have HB2 := HB.of_le (m := 2) (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2) | |
| have H := fixedJoinMiddle_rescale (x := x) (h := h) (p := p) (T := 1) | |
| (by norm_num) (by simpa only [Function.comp_def,div_one,one_smul] using HB2) v v | |
| simp only [div_one,one_smul,one_mul] at H | |
| rw [H,←splitTrialValue_eq_hessian hleft hright] | |
| exact splitTrialValue_nonneg hh hh1 hp hleft hright | |
| end WeakMTWTransport | |
| end | |
| end OAI | |