| import OAI.Geometry.NodalSets.Spectral.MetricProjectionBounds |
|
|
| namespace OAI |
|
|
| namespace Yau.Geometry |
| open Yau.Jets |
| noncomputable section |
|
|
| theorem metric_projection_uniform_stability {M K : ℝ} (hM : 0 ≤ M) (hK : 0 ≤ K) : |
| ∃ ε₀ C : ℝ, 0 < ε₀ ∧ 0 < C ∧ |
| ∀ (g h : Coord →L[ℝ] Coord →L[ℝ] ℝ) (e e₀ t : Coord) (ε : ℝ), |
| ‖g‖ ≤ M → ‖e‖ ≤ K → ‖e₀‖ ≤ K → ‖t‖ ≤ K → |
| g e e = 1 → h t t = 1 → h e₀ t = 0 → |
| ‖e-e₀‖ ≤ ε → ‖g-h‖ ≤ ε → ε ≤ ε₀ → |
| let q := metricPerpProjection g e t |
| q ≠ 0 ∧ g (metricNormalize g q) (metricNormalize g q) = 1 ∧ |
| g e (metricNormalize g q) = 0 ∧ ‖metricNormalize g q-t‖ ≤ C*ε := by |
| let P : ℝ := (M*K+K^2)*K |
| let H : ℝ := 2*M*(K+1)*P+(K+1)^2 |
| have hP : 0 ≤ P := by dsimp [P]; positivity |
| have hH : 0 ≤ H := by dsimp [H]; positivity |
| let ε₀ : ℝ := min (1/(P+1)) (1/(2*(H+1))) |
| let C : ℝ := 2*P+8*(K+1)*H+1 |
| have hε₀ : 0 < ε₀ := lt_min (by positivity) (by positivity) |
| have hC : 0 < C := by dsimp [C]; positivity |
| refine ⟨ε₀,C,hε₀,hC,?_⟩ |
| intro g h e e₀ t ε hg he he₀ ht heunit htunit horth hd hm hε |
| have hε0 : 0 ≤ ε := (norm_nonneg _).trans hd |
| have hε1 := hε.trans (min_le_left _ _) |
| have hε2 := hε.trans (min_le_right _ _) |
| have hsmall1 : P*ε ≤ 1 := by |
| have hh := (le_div_iff₀ (by positivity : 0 < P+1)).mp hε1 |
| nlinarith |
| have hsmall2 : H*ε ≤ 1/2 := by |
| have hh := (le_div_iff₀ (by positivity : 0 < 2*(H+1))).mp hε2 |
| nlinarith |
| have h1 : (M*ε*K+ε*K^2)*K ≤ 1 := by convert hsmall1 using 1; dsimp [P]; ring |
| have h2 : 2*M*(K+1)*((M*ε*K+ε*K^2)*K)+ε*(K+1)^2 ≤ 1/2 := by |
| convert hsmall2 using 1 |
| dsimp [H,P] |
| ring |
| obtain ⟨hn,hu,ho,hb⟩ := normalized_projection_stability g h e e₀ t hM hK hg he he₀ ht |
| heunit htunit horth hd hm h1 h2 |
| refine ⟨hn,hu,ho,hb.trans ?_⟩ |
| dsimp [C,H,P] |
| nlinarith only [hε0] |
|
|
| end |
| end Yau.Geometry |
|
|
| end OAI |
|
|
