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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 | import OAI.Geometry.IsometricImmersion.Metrics.ModelProfile
import OAI.Geometry.IsometricImmersion.Curvature.PrescribedCurvature
import OAI.Geometry.IsometricImmersion.Curvature.AffineCurvature
namespace OAI
noncomputable section
open Set
open scoped ContDiff Matrix
namespace SmoothLocal.Model
open SmoothLocal.Geometry SmoothLocal.ODE
def quarterMatrix : Matrix (Fin 2) (Fin 2) ℝ := !![(1 / 4 : ℝ), 0; 0, 1 / 4]
theorem quarterMatrix_isUnit : IsUnit quarterMatrix := by
apply (Matrix.isUnit_iff_isUnit_det _).2
apply isUnit_iff_ne_zero.mpr
norm_num [quarterMatrix, Matrix.det_fin_two]
theorem quarterCoordinates_eq (p : Coord) :
affineCoordinates 0 quarterMatrix p = (1 / 4 : ℝ) • p := by
ext i
fin_cases i <;> simp [affineCoordinates, quarterMatrix,
dotProduct, Fin.sum_univ_two]
def scaledModelCurvature (kappa : ℝ) (p : Coord) : ℝ :=
modelCurvature kappa ((4 : ℝ) • p)
theorem scaledModelCurvature_contDiff (kappa : ℝ) :
ContDiff ℝ ∞ (scaledModelCurvature kappa) := by
have hd : ContDiff ℝ ∞ (fun p : Coord => (4 : ℝ) • p) :=
contDiff_const_smul (4 : ℝ)
exact (modelCurvature_contDiff kappa).comp hd
theorem scaledModelCurvature_bound {kappa : ℝ} (hk : 0 ≤ kappa)
{p : Coord} (hp : p ∈ square) : |scaledModelCurvature kappa p| ≤ 31 * kappa := by
have hx : (4 * p 0) ^ 2 ≤ (4 : ℝ) ^ 2 := sq_le_sq'
(by linarith [(hp 0).1]) (by linarith [(hp 0).2])
have hy : (4 * p 1) ^ 2 ≤ (4 : ℝ) ^ 2 := sq_le_sq'
(by linarith [(hp 1).1]) (by linarith [(hp 1).2])
have hh := modelProfile_bounds (4 * p 1)
have hb : |(4 * p 0) ^ 2 - modelProfile (4 * p 1)| ≤ 31 := by
apply abs_le.mpr
constructor <;> nlinarith [sq_nonneg (4 * p 0)]
change |kappa * ((4 * p 0) ^ 2 - modelProfile (4 * p 1))| ≤ _
rw [abs_mul, abs_of_nonneg hk]
nlinarith only [mul_le_mul_of_nonneg_left hb hk]
def modelMetricDomain : Set Coord := affineCoordinates 0 quarterMatrix ⁻¹' square
theorem modelMetricDomain_isOpen : IsOpen modelMetricDomain :=
isOpen_affine_preimage coordinate_square_isOpen 0 quarterMatrix
theorem modelMetricDomain_contains_model_square :
Icc (fun _ : Fin 2 => (-3 : ℝ)) (fun _ => (3 : ℝ)) ⊆ modelMetricDomain := by
intro p hp
change affineCoordinates 0 quarterMatrix p ∈ square
rw [quarterCoordinates_eq]
intro i
change -1 < (1 / 4 : ℝ) * p i ∧ (1 / 4 : ℝ) * p i < 1
constructor <;> linarith [hp.1 i, hp.2 i]
def constructedModelMetric (kappa : ℝ) : MetricField :=
affinePullbackMetric (prescribedCurvatureMetric (scaledModelCurvature kappa))
0 quarterMatrix
theorem constructedModelMetric_spec {kappa : ℝ}
(hk : 0 < kappa) (hkSmall : kappa ≤ (1 : ℝ) / 31000) :
SmoothPositiveOn (constructedModelMetric kappa) modelMetricDomain ∧
(∀ p ∈ modelMetricDomain,
gaussianCurvature (constructedModelMetric kappa) p = modelCurvature kappa p) := by
have hK := (scaledModelCurvature_contDiff kappa).contDiffOn (s := square)
have hbound : ∀ p ∈ square, |scaledModelCurvature kappa p| ≤ (1 : ℝ) / 1000 := by
intro p hp
exact (scaledModelCurvature_bound hk.le hp).trans (by linarith)
have hg := prescribedCurvatureMetric_smoothPositiveOn hK hbound
refine ⟨affinePullbackMetric_smoothPositive hg 0 quarterMatrix
(Matrix.mulVec_injective_of_isUnit quarterMatrix_isUnit), ?_⟩
intro p hp
change gaussianCurvature
(affinePullbackMetric (prescribedCurvatureMetric (scaledModelCurvature kappa)) 0 quarterMatrix) p = _
rw [gaussianCurvature_affinePullback hg coordinate_square_isOpen 0 quarterMatrix
quarterMatrix_isUnit p hp, prescribedCurvatureMetric_gaussianCurvature hK hbound hp]
unfold scaledModelCurvature
rw [quarterCoordinates_eq, smul_smul]
norm_num
theorem constructedModelMetric_central_curvature {kappa : ℝ}
(hk : 0 < kappa) (hkSmall : kappa ≤ (1 : ℝ) / 31000) {p : Coord}
(hp : ∀ i : Fin 2, p i ∈ Icc (-(1 / 5 : ℝ)) (1 / 5)) :
gaussianCurvature (constructedModelMetric kappa) p < -kappa / 2 := by
have hS : p ∈ Icc (fun _ : Fin 2 => (-3 : ℝ)) (fun _ => (3 : ℝ)) := by
constructor <;> intro i <;> linarith [(hp i).1, (hp i).2]
rw [(constructedModelMetric_spec hk hkSmall).2 p (modelMetricDomain_contains_model_square hS)]
exact modelCurvature_central_square hk hp
end SmoothLocal.Model
end
end OAI
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