File size: 7,895 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 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 | import Mathlib
import OAI.Analysis.BiholderTransport.Coordinates.ManifoldLipschitz
import OAI.Analysis.BiholderTransport.Coordinates.ChartCostJets
namespace OAI
noncomputable section
namespace WeakMTWTransport
section
open Set Filter
open scoped Topology ContDiff
variable {Q E : Type*} [NormedAddCommGroup Q] [NormedSpace ℝ Q]
[NormedAddCommGroup E] [NormedSpace ℝ E]
lemma compact_parameter_second_jet {K : Set Q} (hK : IsCompact K)
{F : (ℝ×Q) → E → ℝ} {t : ℝ}
(hF : ∀ q∈K, ContDiffAt ℝ ∞ (Function.uncurry F) ((t,q),0)) :
∃ B>0, ∀ᶠ s in 𝓝 t, ∀ q∈K,
ContDiffAt ℝ 2 (F (s,q)) 0 ∧ ‖fderiv ℝ (fderiv ℝ (F (s,q))) 0‖≤B := by
let H := fun z : ℝ×Q => fderiv ℝ (fderiv ℝ (F z)) 0
have hH : ∀ q∈K, ContinuousAt H (t,q) := by
intro q hq
exact (ContDiffAt.partial_snd_fderiv_two (hF q hq)).continuousAt.comp
(x := (t,q)) (f := fun z : ℝ×Q => (z,(0:E)))
(continuousAt_id.prodMk continuousAt_const)
obtain ⟨B,hB,HB⟩ := compact_eventually_fiberwise_bound hK
(f := fun _ : Q => ()) continuous_const (g := fun _ : ℝ => ())
(continuousAt_const (x := t)) (h := H) (fun q hq _ => hH q hq)
have HS : ∀ᶠ s in 𝓝 t, ∀ q∈K, ()=() → ContDiffAt ℝ 2 (F (s,q)) 0 := by
apply compact_eventually_fiberwise hK (f := fun _ : Q => ()) continuous_const
(g := fun _ : ℝ => ()) (continuousAt_const (x := t))
intro q hq _
have He := ((hF q hq).of_le
(m := 2) (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)).eventually (by simp)
have hc : ContinuousAt (fun z : ℝ×Q => (z,(0:E))) (t,q) :=
continuousAt_id.prodMk continuousAt_const
filter_upwards [hc.tendsto.eventually He] with z hz
exact hz.comp (0:E) (contDiffAt_const.prodMk contDiffAt_id)
refine ⟨B,hB,?_⟩
filter_upwards [HB,HS] with s hb hs
exact fun q hq => ⟨hs q hq rfl,hb q hq rfl⟩
end
open Set Filter Manifold Bundle Metric
open scoped Topology ContDiff NNReal
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]
def shortLegCost (x : M) (t : ℝ) (p h : TangentSpace 𝓘(ℝ,Model n) x) : ℝ :=
cost (riemannianExp x (t • (p+h))) (riemannianExp x p)
def movingShortLegCost (a : M) (z : ℝ × (Model n × Model n)) (h : Model n) : ℝ :=
cost (movingNormal a (z.2.1,z.1 • (z.2.2+h))) (movingNormal a z.2)
lemma movingShortLegCost_contDiffAt {a : M} {b p : Model n}
(hb : b∈(extChartAt 𝓘(ℝ,Model n) a).target) :
ContDiffAt ℝ ∞ (Function.uncurry (movingShortLegCost a)) ((1,(b,p)),0) := by
let Z := (ℝ × (Model n × Model n)) × Model n
have hA : ContMDiffAt 𝓘(ℝ,Z) 𝓘(ℝ,Model n) ∞
(fun q : Z => movingNormal a (q.1.2.1,q.1.1 • (q.1.2.2+q.2))) ((1,(b,p)),0) := by
refine (show ContMDiffAt 𝓘(ℝ,Model n × Model n) 𝓘(ℝ,Model n) ∞
(movingNormal a) (b,(1:ℝ) • (p+0)) from by simpa using movingNormal_contMDiffAt hb).comp
(f := fun q : Z => (q.1.2.1,q.1.1 • (q.1.2.2+q.2))) ((1,(b,p)),0) ?_
exact (contDiffAt_fst.snd.fst.prodMk
(contDiffAt_fst.fst.smul (contDiffAt_fst.snd.snd.add contDiffAt_snd))).contMDiffAt
have hB : ContMDiffAt 𝓘(ℝ,Z) 𝓘(ℝ,Model n) ∞
(fun q : Z => movingNormal a q.1.2) ((1,(b,p)),0) :=
(movingNormal_contMDiffAt (q := (b,p)) hb).comp ((1,(b,p)),0)
contDiffAt_fst.snd.contMDiffAt
have hC : ContMDiffAt (𝓘(ℝ,Model n).prod 𝓘(ℝ,Model n)) 𝓘(ℝ,ℝ) ∞
(fun q : M × M => cost q.1 q.2)
(movingNormal a (b,(1:ℝ) • (p+0)),movingNormal a (b,p)) := by
simpa only [add_zero,one_smul,riemannianExp_zero] using
cost_contMDiffAt_of_injectivityDomain
(⟨movingNormal a (b,p),0⟩ : TangentBundle 𝓘(ℝ,Model n) M)
(zero_mem_injectivityDomain (movingNormal a (b,p)))
exact (hC.comp ((1,(b,p)),0) (hA.prodMk hB)).contDiffAt
lemma shortLeg_second_bound_near_base (R : ℝ) (a : M) :
∃ C : ℝ≥0, ∃ K∈𝓝 a, ∀ᶠ t in 𝓝 (1:ℝ), ∀ x∈K,
∀ p : TangentSpace 𝓘(ℝ,Model n) x, ‖p‖≤R →
ContDiffAt ℝ 2 (shortLegCost x t p) 0 ∧
∀ k : TangentSpace 𝓘(ℝ,Model n) x,
‖fderiv ℝ (fderiv ℝ (shortLegCost x t p)) 0 k k‖≤C*‖k‖^2 := by
have : IsContinuousRiemannianBundle (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x) :=
continuousRiemannianBundle_of_smooth (IB := 𝓘(ℝ,Model n))
obtain ⟨B,hBpos,hB⟩ := eventually_norm_trivializationAt_lt (Model n)
(fun x : M => TangentSpace 𝓘(ℝ,Model n) x) a
obtain ⟨K,hKn,hKs,hKc⟩ := local_compact_nhds
(inter_mem (extChartAt_source_mem_nhds (I := 𝓘(ℝ,Model n)) a) hB)
let χ := extChartAt 𝓘(ℝ,Model n) a
let P := (χ '' K) ×ˢ closedBall (0 : Model n) (B*R)
have hPc : IsCompact P :=
(hKc.image_of_continuousOn ((continuousOn_extChartAt a).mono
(fun x hx => (hKs hx).1))).prod (isCompact_closedBall _ _)
obtain ⟨D,hDpos,HD⟩ := compact_parameter_second_jet hPc
(F := movingShortLegCost a) (t := 1) (fun q hq => movingShortLegCost_contDiffAt (by
obtain ⟨x,hx,hxq⟩ := hq.1
rw [←hxq]
exact χ.map_source (hKs hx).1))
refine ⟨⟨D*B^2,mul_nonneg hDpos.le (sq_nonneg B)⟩,K,hKn,?_⟩
filter_upwards [HD] with t ht
intro x hx p hp
let A := (trivializationAt (Model n) (TangentSpace 𝓘(ℝ,Model n)) a).continuousLinearMapAt ℝ x
have hpP : (χ x,A p)∈P := by
refine ⟨mem_image_of_mem χ hx,?_⟩
rw [mem_closedBall,dist_zero_right]
exact (A.le_opNorm p).trans (mul_le_mul (hKs hx).2.le hp (norm_nonneg _) hBpos.le)
have he : shortLegCost x t p = (fun h => movingShortLegCost a (t,(χ x,A p)) (A h)) := by
funext h
dsimp only [shortLegCost,movingShortLegCost]
rw [←map_add,←map_smul,movingNormal_trivialization_exp (hKs hx).1,
movingNormal_trivialization_exp (hKs hx).1]
rw [he]
have HD' := ht (χ x,A p) hpP
refine ⟨(show ContDiffAt ℝ 2 (movingShortLegCost a (t,(χ x,A p))) (A 0) from by
simpa only [map_zero] using HD'.1).comp 0 A.contDiff.contDiffAt,?_⟩
intro k
rw [second_fderiv_comp_linear_zero HD'.1 A k]
let H := fderiv ℝ (fderiv ℝ (movingShortLegCost a (t,(χ x,A p)))) 0
have hAk : ‖A k‖≤B*‖k‖ :=
(A.le_opNorm k).trans (mul_le_mul_of_nonneg_right (hKs hx).2.le (norm_nonneg _))
calc
‖H (A k) (A k)‖≤‖H (A k)‖*‖A k‖ := (H (A k)).le_opNorm _
_≤(‖H‖*‖A k‖)*‖A k‖ := mul_le_mul_of_nonneg_right (H.le_opNorm _) (norm_nonneg _)
_≤(D*‖A k‖)*‖A k‖ := by gcongr; exact HD'.2
_≤(D*(B*‖k‖))*(B*‖k‖) := by gcongr
_=(D*B^2)*‖k‖^2 := by ring
lemma exists_uniform_shortLeg_second_bound (R : ℝ) :
∃ C : ℝ≥0, ∀ᶠ t in 𝓝 (1:ℝ), ∀ x : M,
∀ p : TangentSpace 𝓘(ℝ,Model n) x, ‖p‖≤R →
ContDiffAt ℝ 2 (shortLegCost x t p) 0 ∧
∀ k : TangentSpace 𝓘(ℝ,Model n) x,
‖fderiv ℝ (fderiv ℝ (shortLegCost x t p)) 0 k k‖≤C*‖k‖^2 := by
classical
choose C K hKn hK using shortLeg_second_bound_near_base (n := n) (M := M) R
obtain ⟨s,hs⟩ := finite_cover_nhds hKn
refine ⟨s.sup C,?_⟩
have HT := (eventually_all_finset s).mpr (fun a _ => hK a)
filter_upwards [HT] with t ht
intro x p hp
have hx : x∈ ⋃ a∈s,K a := by rw [hs]; trivial
obtain ⟨a,ha,hxa⟩ := mem_iUnion₂.mp hx
have H := ht a ha x hxa p hp
refine ⟨H.1,fun k => (H.2 k).trans ?_⟩
exact mul_le_mul_of_nonneg_right (by exact_mod_cast Finset.le_sup (f := C) ha) (sq_nonneg _)
end WeakMTWTransport
end
end OAI
|