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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 | import OAI.Geometry.IsometricImmersion.Immersions.HeightCoefficientRegularity
import OAI.Geometry.IsometricImmersion.Coordinates.PatchGeometry
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model
variable {g0 eta : MetricField} {z : Coord → ℝ} {U : Set Coord}
variable {Y : ℝ → ℝ → ℝ} {p : Coord} {kappa : ℝ}
theorem patch_capCurvature_eventually_model
(hY : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hp : p ∈ capChartDomain) (hk : 0 < kappa) (hU : IsOpen U)
(hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p)
(hsupport : tsupport eta ⊆ patchBox)
(hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0 + eta) p < -kappa / 2)
(himage : capChart Y p ∈ U)
(hsmall : |capPullback Y (gaussianCurvature (g0 + eta)) p| ≤ kappa / 8) :
capPullback Y (gaussianCurvature (g0 + eta)) =ᶠ[𝓝 p] capModelCurvature kappa Y := by
have heq := patch_smallCurvature_eventually_model hk hU hbackground hsupport hcentral himage hsmall
have hc : ContinuousAt (capChart Y) p :=
(capChart_contDiffOn hY).continuousOn.continuousAt (capChartDomain_isOpen.mem_nhds hp)
exact heq.comp_tendsto hc.tendsto
theorem patch_heightChartA_eventually_model
(hY : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hp : p ∈ capChartDomain) (hk : 0 < kappa) (hU : IsOpen U)
(hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p)
(hsupport : tsupport eta ⊆ patchBox)
(hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0 + eta) p < -kappa / 2)
(himage : capChart Y p ∈ U)
(hsmall : |capPullback Y (gaussianCurvature (g0 + eta)) p| ≤ kappa / 8) :
heightChartA (g0 + eta) z Y =ᶠ[𝓝 p]
(fun q => heightChartG1 (g0 + eta) z Y q * capModelCurvature kappa Y q) := by
have heq := patch_capCurvature_eventually_model hY hp hk hU hbackground hsupport hcentral himage hsmall
filter_upwards [heq] with q hq
rw [heightChartA_eq, hq]
theorem patch_heightChart_near_principal_lower
(hg : SmoothPositiveOn (g0 + eta) U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
(hyy : ∀ p ∈ U, covHessian (g0 + eta) z p 1 1 ≠ 0)
(hY : ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => Y p.2 p.1) (pairRectangle 2 (-2) 2))
(hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-hessianQuotient (g0 + eta) z (coordinatePoint t (Y s t)))
(Icc (-2 : ℝ) 2) t)
(hvar : ∀ s ∈ Ioo (-2 : ℝ) 2, ∀ t ∈ Ioo (-2 : ℝ) 2,
0 < deriv (fun r => Y r t) s)
(hdisp : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
|Y s t - s| ≤ (1 : ℝ) / 50)
(hqsmall : ∀ p ∈ modelSquare, |hessianQuotient (g0 + eta) z p| ≤ (1 : ℝ) / 100)
(hmap : MapsTo (capChart Y) capChartDomain U)
(hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p)
(hsupport : tsupport eta ⊆ patchBox)
(hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0 + eta) p < -kappa / 2)
(hp : p ∈ capChartDomain) (hs : p 1 ≤ 0) (ell : ℕ)
{cG rhoMax epsilon MG : ℝ}
(hkappa : 0 < kappa) (hcG : 0 < cG) (hG : cG ≤ heightChartG1 (g0 + eta) z Y p)
(hrhoMax : capChartRho Y p ≤ rhoMax) (heps : 0 < epsilon) (heps1 : epsilon ≤ 1)
(hsmall : epsilon * 2 * ((1 : ℝ) / 100) ≤ ((ell : ℝ) + 1 / 2) / rhoMax)
(hGs : |coordPartial 1 (heightChartG1 (g0 + eta) z Y) p| ≤ MG)
(hGt : |coordPartial 0 (heightChartG1 (g0 + eta) z Y) p| ≤ MG)
(hKsmall : |capPullback Y (gaussianCurvature (g0 + eta)) p| ≤
modelPrincipalRadius cG kappa rhoMax epsilon 2 MG ell) :
modelCoercivityMargin cG kappa rhoMax epsilon ell / 2 ≤
directedPrincipal (heightChartA (g0 + eta) z Y) ell epsilon p := by
have hK8 := hKsmall.trans (min_le_left _ _)
have hKeq := patch_capCurvature_eventually_model hY hp hkappa hU hbackground hsupport hcentral
(hmap hp) hK8
have hAeq := patch_heightChartA_eventually_model (z := z) hY hp hkappa hU hbackground hsupport hcentral
(hmap hp) hK8
have hGsm : ContDiffOn ℝ ∞ (heightChartG1 (g0 + eta) z Y) capChartDomain :=
(capPullback_contDiffOn hY (darbouxG_contDiffOn hg hU hz hyy) hmap).mul
((capChartRho_contDiffOn hY hvar).pow 2)
have hGdiff := (hGsm.contDiffAt (capChartDomain_isOpen.mem_nhds hp)).differentiableAt (by simp)
have hKmodel : |capModelCurvature kappa Y p| ≤
modelPrincipalRadius cG kappa rhoMax epsilon 2 MG ell := by
rw [← hKeq.self_of_nhds]
exact hKsmall
have hmodel := cap_model_full_principal_lower hY hode hvar hdisp hqsmall hp hs ell hGdiff
hkappa hcG hG hrhoMax heps heps1 hsmall hGs hGt hKmodel
have hprincipal : directedPrincipal (heightChartA (g0 + eta) z Y) ell epsilon p =
directedPrincipal (fun q => heightChartG1 (g0 + eta) z Y q * capModelCurvature kappa Y q)
ell epsilon p := by
unfold directedPrincipal
rw [coordPartial_eq_of_eventuallyEq hAeq 1, coordPartial_eq_of_eventuallyEq hAeq 0]
rw [hprincipal]
exact hmodel
theorem actual_positive_curvature_strip_budget
{G K d D c1 lambda M mu : ℝ}
(hd : 0 < d) (hD : d ≤ D) (hc1 : 0 ≤ c1) (hlambda : 0 ≤ lambda)
(hG : 0 ≤ G) (hGM : G ≤ M) (hK : K ≤ c1 * d)
(hbudget : M * c1 * (lambda * D + 8) ≤ mu / 3) :
(lambda + 8 / d) * (G * K) ≤ mu / 3 := by
have hM : 0 ≤ M := hG.trans hGM
have hH : 0 ≤ lambda + 8 / d := add_nonneg hlambda (div_nonneg (by norm_num) hd.le)
have hGK : G * K ≤ M * (c1 * d) :=
(mul_le_mul_of_nonneg_left hK hG).trans
(mul_le_mul_of_nonneg_right hGM (mul_nonneg hc1 hd.le))
calc
_ ≤ (lambda + 8 / d) * (M * (c1 * d)) := mul_le_mul_of_nonneg_left hGK hH
_ = M * c1 * (lambda * d + 8) := by field_simp [hd.ne']
_ ≤ M * c1 * (lambda * D + 8) :=
mul_le_mul_of_nonneg_left
(add_le_add (mul_le_mul_of_nonneg_left hD hlambda) le_rfl) (mul_nonneg hM hc1)
_ ≤ mu / 3 := hbudget
end SmoothLocal.Flow
end
end OAI
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