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import OAI.Analysis.BiholderTransport.Volume.RadialNull
import OAI.Analysis.BiholderTransport.Regularity.TrialBlowup
import OAI.Analysis.BiholderTransport.LinearAlgebra.BranchHessian
namespace OAI
noncomputable section
namespace WeakMTWTransport
open Set MeasureTheory Manifold Bundle
open scoped ContDiff Manifold ENNReal NNReal Topology
open Set Filter
open scoped Topology NNReal
open Set Filter
open scoped Topology
open Set Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set
open scoped Topology
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter ContinuousLinearMap InnerProductSpace
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open scoped NNReal
open Set Filter ContinuousLinearMap
open scoped Topology ContDiff
open Set Filter ContinuousLinearMap
open scoped Topology
open MeasureTheory
open scoped ContDiff ENNReal
open Set Filter Manifold Bundle ContinuousLinearMap MeasureTheory
open scoped Topology ContDiff Manifold Bundle ENNReal
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology Manifold
open Set Filter Manifold Bundle ContinuousLinearMap
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Manifold Bundle
open scoped Topology ContDiff Manifold Bundle
open Set Filter Bundle
open scoped Topology Bundle
open scoped Topology
open Function Manifold Set
open Manifold Bundle
open scoped Manifold Bundle
open Set
open Set Filter
open scoped Topology ContDiff
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter
open scoped ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter
open scoped ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter
open scoped ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter
open scoped ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter
open scoped ContDiff Topology
open Filter Set
open scoped Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
open Set Filter Manifold MeasureTheory Bundle
open scoped ENNReal ContDiff Topology
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]
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
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
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]
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
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