| import OAI.Geometry.IsometricImmersion.Flows.PatchFlowCoercivity |
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| namespace OAI |
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| noncomputable section |
| open Set Function |
| open scoped ContDiff |
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| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model |
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| 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 |
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| end SmoothLocal.Flow |
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| end |
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| end OAI |
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