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59d1c80 | 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 | import OAI.Geometry.IsometricImmersion.Darboux.HeightPrincipalBounds
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
def heightCapCurvatureBound (G Z d c : ℝ) : ℝ :=
curvatureFirstBound G d*(1+flowCoordinateBound (heightQuotientJetBound G Z d c))
def heightCapCValueBound (G Z d c e0 : ℝ) (ell : ℕ) : ℝ :=
((ell : ℝ)+1)*heightAJetBound G Z d c +
heightAJetBound G Z d c*heightRemainderBound G Z d c e0 ell
def heightCapCoefficientBound (G Z d c e0 : ℝ) (ell : ℕ) : ℝ :=
heightAJetBound G Z d c + 2*heightBJetBound G Z d c ell + heightCapCValueBound G Z d c e0 ell
theorem heightCapCurvatureBound_nonneg {G Z d c : ℝ}
(hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) :
0 ≤ heightCapCurvatureBound G Z d c := by
have hK := curvatureFirstBound_nonneg hG hd
have hF := flowCoordinateBound_nonneg (heightQuotientJetBound_nonneg hG hZ hd hc)
unfold heightCapCurvatureBound
positivity
theorem heightCapCoefficientBound_dominates {G Z d c e0 : ℝ}
(hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) (he0 : 0 < e0) (ell : ℕ) :
0 ≤ heightCapCoefficientBound G Z d c e0 ell ∧
heightAJetBound G Z d c ≤ heightCapCoefficientBound G Z d c e0 ell ∧
2*heightBJetBound G Z d c ell ≤ heightCapCoefficientBound G Z d c e0 ell ∧
heightBJetBound G Z d c ell ≤ heightCapCoefficientBound G Z d c e0 ell ∧
heightCapCValueBound G Z d c e0 ell ≤ heightCapCoefficientBound G Z d c e0 ell := by
have hA := heightAJetBound_nonneg hG hZ hd hc
have hB := heightBJetBound_nonneg hG hZ hd hc ell
have hr := heightRemainderBound_nonneg hG hZ hd hc he0 ell
have hC : 0 ≤ heightCapCValueBound G Z d c e0 ell := by
unfold heightCapCValueBound
positivity
unfold heightCapCoefficientBound
constructor
· positivity
constructor
· linarith only [hB,hC]
constructor
· linarith only [hA,hC]
constructor
· linarith only [hA,hB,hC]
· linarith only [hA,hB]
theorem heightChartC_value_bound
{g : MetricField} {z : Coord → ℝ} {Y : ℝ → ℝ → ℝ} {G Z d c e0 : ℝ}
(hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) (he0 : 0 < e0) (ell : ℕ)
(hAB : CoordinateBound (heightChartA g z Y) capChartDomain 1 (heightAJetBound G Z d c))
(hrB : ∀ p ∈ capChartDomain,
|heightChartRemainder g z Y ell p| ≤ heightRemainderBound G Z d c e0 ell ∧
heightChartC g z Y ell p = ((ell : ℝ)+1)*coordPartial 1 (heightChartA g z Y) p +
heightChartA g z Y p*heightChartRemainder g z Y ell p)
{p : Coord} (hp : p ∈ capChartDomain) :
|heightChartC g z Y ell p| ≤ heightCapCValueBound G Z d c e0 ell := by
have hel : 0 ≤ (ell : ℝ)+1 := by positivity
have hfirst := mul_le_mul_of_nonneg_left (hAB [1] (by norm_num) p hp) hel
have hsecond :=
(mul_le_mul_of_nonneg_left (hrB p hp).1 (abs_nonneg (heightChartA g z Y p))).trans
(mul_le_mul_of_nonneg_right (hAB [] (by norm_num) p hp)
(heightRemainderBound_nonneg hG hZ hd hc he0 ell))
rw [(hrB p hp).2]
calc
_ ≤ |((ell : ℝ)+1)*coordPartial 1 (heightChartA g z Y) p| +
|heightChartA g z Y p*heightChartRemainder g z Y ell p| := abs_add_le _ _
_ ≤ _ := by
rw [abs_mul,abs_of_nonneg hel,abs_mul]
exact add_le_add hfirst hsecond
theorem heightChartB_primitive_first_bounds
{g : MetricField} {z : Coord → ℝ} {Y : ℝ → ℝ → ℝ} {MB : ℝ} (ell : ℕ)
(hMB : 0 ≤ MB) (hB : ContDiffOn ℝ ∞ (heightChartB g z Y ell) capChartDomain)
(hBB : CoordinateBound (heightChartB g z Y ell) capChartDomain 1 MB)
{p : Coord} (hp : p ∈ capChartDomain) :
|coordinatePrimitive (heightChartB g z Y ell) p| ≤ 2*MB ∧
∀ i : Fin 2, |coordPartial i (coordinatePrimitive (heightChartB g z Y ell)) p| ≤ 2*MB := by
obtain ⟨hIv,hIs⟩ := heightChartB_primitive_bounds ell hB hBB hp
refine ⟨hIv,?_⟩
intro i
fin_cases i
· simp only [Fin.mk_zero]
rw [coordinatePrimitive_partial_t hB (by norm_num : (0 : ℝ) < 2) hp]
exact (hBB [] (by norm_num) p hp).trans (by linarith only [hMB])
· simpa only [Fin.mk_one] using hIs
theorem heightCurvaturePullback_bound_one
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} {G Z d c : ℝ}
(hg : SmoothPositiveOn g U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
(hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c)
(hgB : ∀ i j : Fin 2, CoordinateBound (fun p => g p i j) modelSquare 4 G)
(hdet : ∀ p ∈ modelSquare, d ≤ |(g p).det|)
(hY : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2, 0 < deriv (fun r => Y r t) s)
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-hessianQuotient g z (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
(hmapS : MapsTo (capChart Y) capChartDomain modelSquare)
(hsmall : ∀ p ∈ modelSquare, |hessianQuotient g z p| ≤ (1 : ℝ)/100)
(hrho : ∀ p ∈ capChartDomain, 1/Real.exp (2*heightQuotientJetBound G Z d c) ≤ capChartRho Y p) :
CoordinateBound (capPullback Y (gaussianCurvature g)) capChartDomain 1 (heightCapCurvatureBound G Z d c) := by
have hM := heightQuotientJetBound_nonneg hG hZ hd hc
have hF := flowCoordinateBound_nonneg hM
have hK := curvatureFirstBound_nonneg hG hd
exact capPullback_bound_one_from_partials hY (gaussianCurvature_contDiffOn hg hU) hU
(fun p hp => hSU (hmapS hp)) hmapS hK hF
(gaussianCurvature_coordinate_bound_one_closed hg hU hSU hG hd hgB hdet)
(fun i p hp => capFlowHeight_partial_bound_of_rho hY hvar hode hmapS hsmall hM hrho hp i)
end SmoothLocal.Flow
end
end OAI
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