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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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 | import Mathlib
import OAI.Analysis.BiholderTransport.Regularity.FrameMetric
import OAI.Analysis.BiholderTransport.LinearAlgebra.OuterActiveMatrix
namespace OAI
section
noncomputable section
open Set Filter Manifold Bundle
open scoped Topology ContDiff BoundedContinuousFunction
namespace WeakMTWTransport
section ChartOuterMatrix
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]
local instance chartOuterFinite (x:M):FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x):=
inferInstanceAs (FiniteDimensional ℝ (Model n))
local instance chartOuterComplete (x:M):CompleteSpace (TangentSpace 𝓘(ℝ,Model n) x):=
FiniteDimensional.complete ℝ _
lemma WeakMTW.exists_uniform_chart_outer_matrix
(hmtw:WeakMTW (n:=n) (M:=M)) {lam cap:ℝ} (hlam:0 < lam) (hcap:0 ≤ cap) :
∃K>0,∀ (x0:M) (uv:(M →ᵇ ℝ)×(M →ᵇ ℝ)) (φ:ℝ → ℝ),
uv∈densityDualClass (metricVolume n) lam cap x0 → Continuous φ → StrictMono φ →
∀ (a:M) (b p:Model n),b∈(extChartAt 𝓘(ℝ,Model n) a).target →
let z:TangentBundle 𝓘(ℝ,Model n) M:=⟨(extChartAt 𝓘(ℝ,Model n) a).symm b,chartFiberInverse a b p⟩
∀ l:ℝ,z.2∈activeLogs (φ ∘ uv.2) z.1 →
ContDiffAt ℝ 2 φ (uv.2 (riemannianExp z.1 z.2)) →
HasDerivAt φ l (uv.2 (riemannianExp z.1 z.2)) → 1 < l →
iteratedDeriv 2 φ (uv.2 (riemannianExp z.1 z.2)) ≤ 0 →
∀ B:Model n →L[ℝ] Model n →L[ℝ] ℝ,
(∀d e,B d e=B e d) → (∀d,0 ≤ B d d) →
HasLowerSecondTaylor (fun d=>φ (uv.2 (movingNormal a (b,p+d)))+
‖chartFiberInverse a b (p+d)‖^2/2) 0 B →
∃V:Model n →L[ℝ] Model n,
(∀d e,inner ℝ (V d) e=inner ℝ d (V e)) ∧
(∀d,d≠0 → 0 < inner ℝ (V d) d) ∧
(∀d,B d d-(iteratedDeriv 2 φ (uv.2 (riemannianExp z.1 z.2))/l^2)*
(frameMetric a b p d)^2 ≤ inner ℝ (V d) d) ∧
0 < expJacobian (n:=n) z.1 z.2 ∧
0 < expJacobian (n:=n) (reverseRay z).1 (l⁻¹ • (reverseRay z).2) ∧
expJacobian (n:=n) (reverseRay z).1 (l⁻¹ • (reverseRay z).2)*V.det ≤
(chartFiberInverse a b).toLinearMap.normDet^2*l^n*K*(expJacobian (n:=n) z.1 z.2)^2 := by
obtain ⟨K,hK,HK⟩:=hmtw.exists_uniform_active_outer_matrix hlam hcap
refine ⟨K,hK,?_⟩
intro x0 uv φ huv hφ hm a b p hb z l ha hφ2 hd hl hconc B hBs hB hjet
let x:M:=(extChartAt 𝓘(ℝ,Model n) a).symm b
let T:=trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a
have hx:x∈T.baseSet:=by
simpa only [x,T,TangentBundle.trivializationAt_baseSet,extChartAt_source] using
(extChartAt 𝓘(ℝ,Model n) a).map_target hb
let e:=T.continuousLinearEquivAt ℝ x hx
let A:=chartFiberInverse a b
have hAe:(A:Model n → TangentSpace 𝓘(ℝ,Model n) x)=e.symm:=
(T.symm_continuousLinearEquivAt_eq hx).symm
have hEA (d:Model n):e (A d)=d:=by rw [hAe]; exact e.apply_symm_apply d
have hAE (d:TangentSpace 𝓘(ℝ,Model n) x):A (e d)=d:=by rw [hAe]; exact e.symm_apply_apply d
let C:=pullBilinear B e.toContinuousLinearMap
have hC:HasLowerSecondTaylor (fun d:TangentSpace 𝓘(ℝ,Model n) x=>
φ (uv.2 (riemannianExp x (A p+d)))+‖A p+d‖^2/2) 0 C:=by
convert hjet.comp_stationary e.hasFDerivAt e.map_zero using 1
funext d
simp only [movingNormal_eq hb,map_add]
change φ (uv.2 (riemannianExp x (A p+d)))+‖A p+d‖^2/2=
φ (uv.2 (riemannianExp x (A p+A (e d))))+‖A p+A (e d)‖^2/2
rw [hAE]
obtain ⟨V,hVs,hVp,hgain,hσ,hσs,hdet⟩:=HK x0 uv φ huv hφ hm z l ha hφ2 hd hl hconc
C (fun d f=>hBs _ _) (fun d=>hB _) hC
let W:=A.adjoint.comp (V.comp A)
have hdim:Module.finrank ℝ (Model n)=Module.finrank ℝ (TangentSpace 𝓘(ℝ,Model n) x):=by
rw [tangent_finrank,finrank_euclideanSpace_fin]
have hP:=det_positive_hessian_pullback (B:=A) (W:=V) hdim
(chartFiberInverse_injective hb) hVs hVp (show (0:ℝ)<1 by norm_num)
simp only [one_smul,one_pow,one_mul] at hP
refine ⟨W,?_,hP.1,?_,hσ,hσs,?_⟩
· intro d f
simp only [W,ContinuousLinearMap.comp_apply,ContinuousLinearMap.adjoint_inner_left,
ContinuousLinearMap.adjoint_inner_right,hVs]
· intro d
have H:=hgain (A d)
simpa only [C,pullBilinear_apply,ContinuousLinearEquiv.coe_coe,hEA,
frameMetric_apply hb,W,ContinuousLinearMap.comp_apply,ContinuousLinearMap.adjoint_inner_left] using H
· change _*(A.adjoint.comp (V.comp A)).det≤_
rw [hP.2]
have H:=mul_le_mul_of_nonneg_left hdet (sq_nonneg A.toLinearMap.normDet)
nlinarith only [H]
end ChartOuterMatrix
end WeakMTWTransport
end
end
end OAI
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