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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 | import OAI.Geometry.IsometricImmersion.Flows.PatchFlowCoercivity
namespace OAI
noncomputable section
open Set Function
open scoped ContDiff
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model
theorem patch_actual_directed_near_coercivity
{g0 eta : MetricField} {z : Coord → ℝ} {U : Set Coord}
{Y : ℝ → ℝ → ℝ} {p : Coord} {kappa : ℝ}
(hg : SmoothPositiveOn (g0 + eta) U) (hU : IsOpen U) (hz : ContDiffOn ℝ ∞ z U)
(hDarboux : ∀ p ∈ U,
(covHessian (g0 + eta) z p).det = gaussianCurvature (g0 + eta) p * heightEnergy (g0 + eta) z p)
(hyy : ∀ p ∈ U, covHessian (g0 + eta) z p 1 1 ≠ 0)
(hE : ∀ p ∈ U, 0 < heightEnergy (g0 + eta) z p)
(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 MI MB Mr MA MAs MG1 c1 D edge lambda : 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)
(hqbudget : 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)
(hedge : p 1 < edge) (hlambda : 0 < lambda)
(hIs : |coordPartial 1 (coordinatePrimitive (heightChartB (g0 + eta) z Y ell)) p| ≤ MI)
(hBp : |heightChartB (g0 + eta) z Y ell p| ≤ MB)
(hr : |heightChartRemainder (g0 + eta) z Y ell p| ≤ Mr)
(hAp : |heightChartA (g0 + eta) z Y p| ≤ MA)
(hAs : |coordPartial 1 (heightChartA (g0 + eta) z Y) p| ≤ MAs)
(hTbudget : 2 * (MI + epsilon + epsilon * 2 * MB) ≤ lambda)
(hEbudget : (MI + 1 + 2 * MB) / 2 + Mr ≤ lambda / 8)
(hcrossbudget : 4 * (2 * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1))^2 *
epsilon^2 * MA ≤ 1 / 8)
(hloss : (4 * (2 * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1))^2 * epsilon^2) / lambda ≤
modelCoercivityMargin cG kappa rhoMax epsilon ell / 8)
(hGmax : heightChartG1 (g0 + eta) z Y p ≤ MG1)
(hc1 : 0 ≤ c1) (hdist : edge - p 1 ≤ D)
(hKstrip : capPullback Y (gaussianCurvature (g0 + eta)) p ≤ c1 * (edge - p 1))
(hstripbudget : MG1 * c1 * (lambda * D + 8) ≤
modelCoercivityMargin cG kappa rhoMax epsilon ell / 6)
(x y : ℝ) :
let A := heightChartA (g0 + eta) z Y
let B := heightChartB (g0 + eta) z Y ell
let C := heightChartC (g0 + eta) z Y ell
let I := coordinatePrimitive B
min (lambda / 8) (modelCoercivityMargin cG kappa rhoMax epsilon ell / 4) *
directedWeight edge lambda I p * (x^2 + y^2) ≤
multiplierT B (directedM edge lambda I) (directedN edge lambda epsilon I) p * x^2 +
multiplierS A C (directedM edge lambda I) (directedN edge lambda epsilon I) p * y^2 +
multiplierJ A B C (directedM edge lambda I) (directedN edge lambda epsilon I) p * x * y := by
let mu := modelCoercivityMargin cG kappa rhoMax epsilon ell / 2
have hmu : 0 < mu := div_pos
(modelCoercivityMargin_pos ell hcG hkappa ((capChartRho_pos hY hvar hp).trans_le hrhoMax) heps)
(by norm_num)
have hA : DifferentiableAt ℝ (heightChartA (g0 + eta) z Y) p :=
((heightChartA_contDiffOn hg hU hz hyy hY hvar hmap).contDiffAt
(capChartDomain_isOpen.mem_nhds hp)).differentiableAt (by simp)
have hB : ContDiffOn ℝ ∞ (heightChartB (g0 + eta) z Y ell) (coordinateRectangle 2 (-2) 2) :=
heightChartB_contDiffOn hg hU hz hyy hY hmap ell
have hC := heightChartC_factor hg hU hz hDarboux hyy hE hY hvar hmap hp ell
have hprincipal : mu ≤ directedPrincipal (heightChartA (g0 + eta) z Y) ell epsilon p :=
patch_heightChart_near_principal_lower hg hU hz hyy hY hode hvar hdisp hqsmall hmap
hbackground hsupport hcentral hp hs ell hkappa hcG hG hrhoMax heps heps1 hqbudget hGs hGt hKsmall
have hstrip : 0 < heightChartA (g0 + eta) z Y p →
weightH0 edge lambda p * heightChartA (g0 + eta) z Y p ≤ mu / 3 := by
intro hpositive
rw [heightChartA_eq]
apply actual_positive_curvature_strip_budget (sub_pos.mpr hedge) hdist hc1 hlambda.le
(hcG.trans_le hG).le hGmax hKstrip
dsimp only [mu]
linarith
have hloss' : (4 * (2 * ((ell : ℝ) * MAs + MA * Mr + MA * MI + 1))^2 * epsilon^2) / lambda ≤
mu / 4 := by dsimp only [mu]; linarith
have hh := primitive_directed_near_coercivity hB (by norm_num : (0 : ℝ) < 2) hp
(heightChartA (g0 + eta) z Y) (heightChartC (g0 + eta) z Y ell)
(heightChartRemainder (g0 + eta) z Y ell) hA edge lambda epsilon ell hedge hlambda
heps.le heps1 hC MI MB Mr MA MAs mu hmu hIs hBp hr hAp hAs hTbudget hEbudget hcrossbudget
hloss' hprincipal hstrip x y
have hm : mu / 2 = modelCoercivityMargin cG kappa rhoMax epsilon ell / 4 := by
dsimp only [mu]
ring
rw [hm] at hh
exact hh
end SmoothLocal.Flow
end
end OAI
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