File size: 5,736 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import OAI.Geometry.IsometricImmersion.Caps.EllipticCutoff
import OAI.Geometry.IsometricImmersion.Flows.FlowTransport
import OAI.Geometry.IsometricImmersion.Coordinates.PatchGeometry
import Mathlib.Analysis.Calculus.LocalExtr.Basic
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology
namespace SmoothLocal.Weighted
open SmoothLocal.Geometry
def capOuterSpatialCutoff (L R A : ℝ) : Coord → ℝ :=
capSpatialCutoff (capOuterEdge L) (-capOuterEdge (-R)) (capOuterEdge A)
theorem capOuterSpatialCutoff_contDiff (L R A : ℝ) :
ContDiff ℝ ∞ (capOuterSpatialCutoff L R A) :=
capSpatialCutoff_contDiff _ _ _
theorem capOuterSpatialCutoff_range (L R A : ℝ) (p : Coord) :
0 ≤ capOuterSpatialCutoff L R A p ∧ capOuterSpatialCutoff L R A p ≤ 1 :=
capSpatialCutoff_range _ _ _ p
theorem capSpatialCutoff_nonzero_bounds {L R A : ℝ}
(hL : -2 < L) (hR : R < 2) (hA : -2 < A) {p : Coord}
(hp : capSpatialCutoff L R A p ≠ 0) :
capOuterEdge L < p 0 ∧ p 0 < -capOuterEdge (-R) ∧ capOuterEdge A < p 1 := by
obtain ⟨hlo, hli, hlu, hlx⟩ := capEdges_bounds hL
obtain ⟨hro, hri, hru, hrx⟩ := capEdges_bounds (x := -R) (by linarith)
obtain ⟨hao, hai, hau, hax⟩ := capEdges_bounds hA
have hz (h : p 0 ≤ capOuterEdge L ∨ -capOuterEdge (-R) ≤ p 0 ∨
p 1 ≤ capOuterEdge A) : capSpatialCutoff L R A p = 0 :=
threeEdgeCutoff_zero hli (neg_lt_neg hri) hai p h
refine ⟨?_, ?_, ?_⟩
· by_contra h
exact hp (hz (Or.inl (le_of_not_gt h)))
· by_contra h
exact hp (hz (Or.inr (Or.inl (le_of_not_gt h))))
· by_contra h
exact hp (hz (Or.inr (Or.inr (le_of_not_gt h))))
theorem capOuterSpatialCutoff_one_of_nonzero {L R A : ℝ}
(hL : -2 < L) (hR : R < 2) (hA : -2 < A) {p : Coord}
(hp : capSpatialCutoff L R A p ≠ 0) : capOuterSpatialCutoff L R A p = 1 := by
have hlo := (capEdges_bounds hL).1
have hro := (capEdges_bounds (x := -R) (by linarith)).1
have hao := (capEdges_bounds hA).1
obtain ⟨htl, htr, hsb⟩ := capSpatialCutoff_nonzero_bounds hL hR hA hp
exact capSpatialCutoff_one hlo (by linarith) hao p ⟨htl.le, htr.le, hsb.le⟩
theorem capSpatialCutoff_nonzero_of_partial {L R A : ℝ} {p : Coord} {i : Fin 2}
(hp : coordPartial i (capSpatialCutoff L R A) p ≠ 0) :
capSpatialCutoff L R A p ≠ 0 := by
intro hz
have hmin : IsLocalMin (capSpatialCutoff L R A) p := by
change ∀ᶠ q in 𝓝 p, capSpatialCutoff L R A p ≤ capSpatialCutoff L R A q
exact Filter.Eventually.of_forall (fun q => by
simpa only [hz] using (capSpatialCutoff_range L R A q).1)
have hd := hmin.fderiv_eq_zero
exact hp (by simp [coordPartial, hd])
theorem capOuterSpatialCutoff_one_of_partial {L R A : ℝ}
(hL : -2 < L) (hR : R < 2) (hA : -2 < A) {p : Coord} {i : Fin 2}
(hp : coordPartial i (capSpatialCutoff L R A) p ≠ 0) :
capOuterSpatialCutoff L R A p = 1 :=
capOuterSpatialCutoff_one_of_nonzero hL hR hA (capSpatialCutoff_nonzero_of_partial hp)
end SmoothLocal.Weighted
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.Weighted SmoothLocal.Model
theorem modelCurvature_lower_flow_bottom {kappa : ℝ} (hk : 0 < kappa)
{Y : ℝ → ℝ → ℝ} {p : Coord}
(hdisp : |capFlowHeight Y p - p 1| ≤ (1 : ℝ) / 50) (hs : p 1 < -3 / 2) :
kappa < modelCurvature kappa (capChart Y p) := by
have hy : capFlowHeight Y p < -(37 / 25 : ℝ) := by
linarith [(abs_le.mp hdisp).2]
have hy2 : 2 < (capFlowHeight Y p)^2 := by
nlinarith [sq_nonneg (capFlowHeight Y p + 37 / 25)]
have hh := modelProfile_eq_one_sub_sq (y := capFlowHeight Y p) (by linarith)
unfold modelCurvature
rw [capChart_zero, capChart_one, hh]
have hbase : 1 < (p 0)^2 - (1 - (capFlowHeight Y p)^2) := by
nlinarith [sq_nonneg (p 0)]
simpa only [mul_one] using mul_lt_mul_of_pos_left hbase hk
theorem capChart_not_mem_patchBox_of_cutoff_band
{Y : ℝ → ℝ → ℝ} {p : Coord}
(hdisp : |capFlowHeight Y p - p 1| ≤ (1 : ℝ) / 50)
(hband : (3 / 2 < |p 0| ∧ |p 0| < 2) ∨ (-2 < p 1 ∧ p 1 < -3 / 2)) :
capChart Y p ∉ patchBox := by
intro hin
rcases hband with ht | hs
· have hxlo := hin.1 0
have hxhi := hin.2 0
simp only [capChart_zero] at hxlo hxhi
have habs : |p 0| ≤ (1 : ℝ) / 10 := abs_le.mpr ⟨hxlo, hxhi⟩
linarith [ht.1]
· have hylo := hin.1 1
simp only [capChart_one] at hylo
linarith [hs.2, (abs_le.mp hdisp).2]
theorem capSpatialCutoff_partial_actual_curvature_lower
{g0 eta : MetricField} {kappa L R A : ℝ} {U : Set Coord}
{Y : ℝ → ℝ → ℝ} {p : Coord} {i : Fin 2}
(hk : 0 < kappa) (hL : -2 < L) (hR : R < 2) (hA : -2 < A)
(hbackground : ∀ q ∈ U, gaussianCurvature g0 q = modelCurvature kappa q)
(hsupport : tsupport eta ⊆ patchBox) (hpU : capChart Y p ∈ U)
(hdisp : |capFlowHeight Y p - p 1| ≤ (1 : ℝ) / 50)
(hpartial : coordPartial i (capSpatialCutoff L R A) p ≠ 0) :
kappa < capPullback Y (gaussianCurvature (g0 + eta)) p := by
have hband := capSpatialCutoff_derivative_bands hL hR hA p i hpartial
have hnot := capChart_not_mem_patchBox_of_cutoff_band hdisp hband
have hts : capChart Y p ∉ tsupport eta := fun h => hnot (hsupport h)
have hsame := gaussianCurvature_eq_of_eventuallyEq
(metric_add_eventuallyEq_of_not_mem_tsupport g0 eta hts)
change kappa < gaussianCurvature (g0 + eta) (capChart Y p)
rw [hsame, hbackground _ hpU]
rcases hband with ht | hs
· have hh := modelCurvature_lateral_band_lower hk (p := capChart Y p)
(by simpa only [capChart_zero] using ht.1)
linarith
· exact modelCurvature_lower_flow_bottom hk hdisp hs.2
end SmoothLocal.Flow
end
end OAI
|