| import Mathlib |
| import OAI.Analysis.BiholderTransport.Volume.RadialNull |
| import OAI.Analysis.BiholderTransport.Regularity.TrialBlowup |
| import OAI.Analysis.BiholderTransport.LinearAlgebra.BranchHessian |
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|
| namespace OAI |
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| noncomputable section |
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| namespace WeakMTWTransport |
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| open Set MeasureTheory Manifold Bundle |
| open scoped ContDiff Manifold ENNReal NNReal Topology |
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| open Set Filter |
| open scoped Topology NNReal |
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| open Set Filter |
| open scoped Topology |
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| open Set Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set |
| open scoped Topology |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter ContinuousLinearMap InnerProductSpace |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
| open scoped NNReal |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology ContDiff |
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| open Set Filter ContinuousLinearMap |
| open scoped Topology |
| open MeasureTheory |
| open scoped ContDiff ENNReal |
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| open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory |
| open scoped Topology ContDiff Manifold Bundle ENNReal |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology Manifold |
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| open Set Filter Manifold Bundle ContinuousLinearMap |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff Manifold Bundle |
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| open Set Filter Bundle |
| open scoped Topology Bundle |
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| open scoped Topology |
| open Function Manifold Set |
| open Manifold Bundle |
| open scoped Manifold Bundle |
| open Set |
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| open Set Filter |
| open scoped Topology ContDiff |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
|
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter |
| open scoped ContDiff Topology |
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| open Filter Set |
| open scoped Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
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| section |
| 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] |
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| lemma WeakMTW.contracted_segment_concaveOn (hmtw : WeakMTW (n := n) (M := M)) |
| {x : M} {p e xi : TangentSpace 𝓘(ℝ,Model n) x} {a b r : ℝ} |
| (hseg : ∀ s ∈ Icc a b, p+s • e ∈ minimizingVectors x) |
| (hr : 0<r) (hr1 : r<1) (hortho : inner ℝ xi e=0) : |
| ConcaveOn ℝ (Icc a b) (fun s => hessianValue x (r • (p+s • e)) xi) := by |
| have heq (s : ℝ) : r • p+s • (r • e)=r • (p+s • e) := by |
| simp only [smul_add,smul_smul,mul_comm] |
| have H := hmtw.transverse_concaveOn_injectivityDomain x (r • p) xi (r • e) (convex_Icc a b) |
| (fun s hs => by rw [heq]; exact contracted_minimizer_mem_injectivityDomain (hseg s hs) hr hr1) |
| (by simp only [real_inner_smul_right,hortho,mul_zero]) |
| simpa only [heq] using H |
|
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| lemma WeakMTW.firstcut_transverse_lower (hmtw : WeakMTW (n := n) (M := M)) |
| {x : M} {p e xi : TangentSpace 𝓘(ℝ,Model n) x} {a b : ℝ} |
| (ha : a<0) (hb : 0<b) |
| (hseg : ∀ s ∈ Icc a b, p+s • e ∈ minimizingVectors x) |
| (hleft : Function.Injective (fderiv ℝ (fun v => extChartAt 𝓘(ℝ,Model n) |
| (riemannianExp x (p+a • e)) (riemannianExp x v)) (p+a • e))) |
| (hright : Function.Injective (fderiv ℝ (fun v => extChartAt 𝓘(ℝ,Model n) |
| (riemannianExp x (p+b • e)) (riemannianExp x v)) (p+b • e))) |
| (hortho : inner ℝ xi e=0) : |
| ∃ L : ℝ, ∀ᶠ r : ℝ in 𝓝[<] 1, ∀ s ∈ Icc a b, |
| L≤hessianValue x (r • (p+s • e)) xi := by |
| have hab : a≤b := (ha.trans hb).le |
| obtain ⟨C,hC⟩ := nonconjugate_inward_hessian_bounded (hseg a ⟨le_rfl,hab⟩) hleft xi |
| obtain ⟨D,hD⟩ := nonconjugate_inward_hessian_bounded (hseg b ⟨hab,le_rfl⟩) hright xi |
| refine ⟨min (-C) (-D),?_⟩ |
| filter_upwards [hC,hD, |
| (eventually_gt_nhds (show (0:ℝ)<1 by norm_num)).filter_mono nhdsWithin_le_nhds, |
| self_mem_nhdsWithin] with r hrC hrD hr0 hr1 |
| intro s hs |
| have H := (hmtw.contracted_segment_concaveOn hseg hr0 hr1 hortho).min_le_of_mem_Icc |
| (show a ∈ Icc a b from ⟨le_rfl,hab⟩) (show b ∈ Icc a b from ⟨hab,le_rfl⟩) hs |
| exact (min_le_min (neg_le_of_abs_le hrC) (neg_le_of_abs_le hrD)).trans H |
|
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| lemma WeakMTW.firstcut_kernel_transverse_orthogonal (hmtw : WeakMTW (n := n) (M := M)) |
| {x : M} {p e k : TangentSpace 𝓘(ℝ,Model n) x} {a b : ℝ} |
| (ha : a<0) (hb : 0<b) |
| (hseg : ∀ s ∈ Icc a b, p+s • e ∈ minimizingVectors x) |
| (hleft : Function.Injective (fderiv ℝ (fun v => extChartAt 𝓘(ℝ,Model n) |
| (riemannianExp x (p+a • e)) (riemannianExp x v)) (p+a • e))) |
| (hright : Function.Injective (fderiv ℝ (fun v => extChartAt 𝓘(ℝ,Model n) |
| (riemannianExp x (p+b • e)) (riemannianExp x v)) (p+b • e))) |
| (hk : mfderiv 𝓘(ℝ,TangentSpace 𝓘(ℝ,Model n) x) 𝓘(ℝ,Model n) (riemannianExp x) p k=0) |
| (xi : TangentSpace 𝓘(ℝ,Model n) x) (hortho : inner ℝ xi e=0) : inner ℝ xi k=0 := by |
| by_contra hpair |
| have hp : p ∈ minimizingVectors x := by simpa using hseg 0 ⟨ha.le,hb.le⟩ |
| have hneg := conjugate_radial_hessian_tendsto_atBot hp hk |
| (by rwa [real_inner_comm] : inner ℝ k xi≠0) |
| obtain ⟨L,hL⟩ := hmtw.firstcut_transverse_lower ha hb hseg hleft hright hortho |
| have hlo : ∀ᶠ r : ℝ in 𝓝[<] 1, L≤hessianValue x (r • p) xi := |
| hL.mono (fun r hr => by simpa using hr 0 ⟨ha.le,hb.le⟩) |
| obtain ⟨r,hr,hrr⟩ := (hlo.and (hneg.eventually (eventually_lt_atBot L))).exists |
| exact (not_lt_of_ge hr) hrr |
| end |
|
|
| open Set Filter Manifold MeasureTheory Bundle |
| open scoped ENNReal ContDiff Topology |
|
|
| 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] |
|
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| lemma WeakMTW.firstcut_radial_lower (hmtw : WeakMTW (n := n) (M := M)) |
| {x : M} {p e : TangentSpace 𝓘(ℝ,Model n) x} {b : ℝ} |
| (hb : 0<b) (hseg : ∀ s ∈ Icc 0 b, p+s • e ∈ minimizingVectors x) |
| (hright : Function.Injective (fderiv ℝ (fun v => extChartAt 𝓘(ℝ,Model n) |
| (riemannianExp x (p+b • e)) (riemannianExp x v)) (p+b • e))) |
| (hortho : inner ℝ p e=0) : |
| ∃ D : ℝ, ∀ᶠ r : ℝ in 𝓝[<] 1, ∀ s ∈ Icc 0 b, |
| -D * s ≤ s^2 * hessianValue x (r • (p+s • e)) e := by |
| have hp : p ∈ minimizingVectors x := by simpa using hseg 0 ⟨le_rfl,hb.le⟩ |
| obtain ⟨C,hC⟩ := nonconjugate_inward_hessian_bounded (hseg b ⟨hb.le,le_rfl⟩) hright p |
| let D := (C+‖p‖^2)/b |
| refine ⟨D,?_⟩ |
| filter_upwards [hC, |
| (eventually_gt_nhds (show (0:ℝ)<1 by norm_num)).filter_mono nhdsWithin_le_nhds, |
| self_mem_nhdsWithin] with r hrC hr0 hr1 |
| have hconc := hmtw.contracted_segment_concaveOn hseg hr0 hr1 hortho |
| have hconv := hconc.neg.add_const (‖p‖^2) |
| have hrad0 := hessianValue_contracted_radial hr0.ne' |
| (contracted_minimizer_mem_injectivityDomain hp hr0 hr1) |
| have hz : -hessianValue x (r • (p+(0:ℝ) • e)) p+‖p‖^2=0 := by |
| simp only [zero_smul,add_zero,hrad0,neg_add_cancel] |
| have hend : -hessianValue x (r • (p+b • e)) p+‖p‖^2≤D*b := by |
| have hd : D*b=C+‖p‖^2 := div_mul_cancel₀ _ hb.ne' |
| rw [hd] |
| linarith [neg_le_of_abs_le hrC] |
| intro s hs |
| have hlin := convex_linear_bound hb hconv hz hend hs |
| have hrad := hessianValue_transverse_radial_identity hr0.ne' |
| (contracted_minimizer_mem_injectivityDomain (hseg s hs) hr0 hr1) hortho |
| dsimp only [Pi.neg_apply,Pi.add_apply] at hlin |
| nlinarith [mul_nonneg (sq_nonneg s) (sq_nonneg ‖e‖)] |
| |
| end WeakMTWTransport |
| |
| end |
| |
| end OAI |
| |
