File size: 2,821 Bytes
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 | 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
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