| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.UniformJetBoundsFiberComp |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff |
| namespace YauCounterexamples |
|
|
| lemma compact_normal_forward_bound (g : SmoothMetric NormalWaveSpace NormalWaveSpace) |
| {K : Set NormalWaveSpace} (hK : IsCompact K) {R : ℝ} (hR : 0 ≤ R) : |
| ∃ C : ℝ, 1 ≤ C ∧ ∀ q ∈ metricFrameSet g K, ∀ x : Fin 3 → ℝ, ‖x‖ ≤ R → |
| ‖normalJetMap q.1 q.2 ((metricChristoffel g q.1).bilinearComp q.2 q.2) |
| (normalWaveEquiv x)-q.1‖ ≤ C*‖x‖ := by |
| let F := fun w : NormalWaveParameter × (Fin 3 → ℝ) => |
| normalJetMap w.1.1 w.1.2 ((metricChristoffel g w.1.1).bilinearComp w.1.2 w.1.2) (normalWaveEquiv w.2) |
| have hF : ContDiff ℝ ∞ F := (contDiff_normalJetFamily g).comp |
| (contDiff_fst.prodMk (normalWaveEquiv.contDiff.comp contDiff_snd)) |
| have hL : IsCompact (metricFrameSet g K ×ˢ Metric.closedBall (0 : Fin 3 → ℝ) R) := |
| (metricFrameSet_isCompact g hK).prod (isCompact_closedBall 0 R) |
| obtain ⟨C,hC,hb⟩ := compact_fiber_jet_bound F hF hL 1 |
| refine ⟨C,hC,?_⟩ |
| intro q hq x hx |
| have hFq : ContDiff ℝ ∞ (fun x => F (q,x)) := hF.comp (contDiff_const.prodMk contDiff_id) |
| have hd (v) (hv : v ∈ Metric.closedBall (0 : Fin 3 → ℝ) R) : |
| ‖fderiv ℝ (fun x => F (q,x)) v‖ ≤ C := by |
| rw [←norm_iteratedFDeriv_one] |
| exact hb (q,v) ⟨hq,hv⟩ 1 le_rfl |
| have hh := (convex_closedBall (0 : Fin 3 → ℝ) R).norm_image_sub_le_of_norm_fderiv_le |
| (fun v _ => hFq.differentiable (by simp) v) hd (Metric.mem_closedBall_self hR) |
| (show x ∈ Metric.closedBall (0 : Fin 3 → ℝ) R by simpa only [Metric.mem_closedBall,dist_zero_right] using hx) |
| simpa only [F,map_zero,normalJetMap_zero,sub_zero] using hh |
|
|
| lemma gaussian_coordinate_to_physical {c C n : ℝ} (hc : 0 ≤ c) (hC : 0 < C) (hn : 0 ≤ n) |
| {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hba : b ≤ C*a) : |
| Real.exp (-c*n*a^2) ≤ Real.exp (-(c/C^2)*n*b^2) := by |
| apply Real.exp_le_exp.mpr |
| have hsq : b^2 ≤ C^2*a^2 := by |
| calc |
| b^2 ≤ (C*a)^2 := by |
| simpa only [pow_two] using mul_le_mul hba hba hb (mul_nonneg hC.le ha) |
| _ = C^2*a^2 := mul_pow C a 2 |
| have hm := mul_le_mul_of_nonneg_left hsq (div_nonneg (mul_nonneg hc hn) (sq_nonneg C)) |
| have hnC : C^2 ≠ 0 := pow_ne_zero _ hC.ne' |
| have he : (c*n/C^2)*(C^2*a^2) = c*n*a^2 := by field_simp |
| rw [he] at hm |
| calc |
| -c*n*a^2 = -(c*n*a^2) := by ring |
| _ ≤ -((c*n/C^2)*b^2) := neg_le_neg hm |
| _ = -(c/C^2)*n*b^2 := by ring |
| end YauCounterexamples |
| end |
| |
| end OAI |
| |
