| import OAI.Geometry.IsometricImmersion.Immersions.HeightCoefficientRegularity |
| import OAI.Geometry.IsometricImmersion.Coordinates.PatchGeometry |
|
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| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted SmoothLocal.Model |
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| 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 |
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|
| end |
|
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| end OAI |
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