| 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 |
|
|
