| import OAI.Geometry.Kahler.BasePhaseDifferential |
| |
| namespace OAI |
|
|
| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open Set Filter Topology |
| open scoped ContDiff ENNReal |
| open scoped ContDiff |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| noncomputable section |
|
|
| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
| |
| def Horizontal (ξ : Sphere) (v : Base) : Prop := bracket v (ξ:Base)=0 |
| |
| lemma orthogonal_residual_norm (ξ p : Sphere) : |
| ‖(p:Base)-bracket (p:Base) (ξ:Base) • (ξ:Base)‖ = projectiveDistance ξ p := by |
| apply (sq_eq_sq₀ (norm_nonneg _) (projectiveDistance_nonneg _ _)).mp |
| rw [norm_sub_sq (𝕜 := ℂ),norm_smul,sphere_norm,sphere_norm,mul_one,projectiveDistance_sq] |
| simp only [bracket,inner_smul_right] |
| have hconj : inner ℂ (ξ:Base) (p:Base) = star (inner ℂ (p:Base) (ξ:Base)) := (inner_conj_symm _ _).symm |
| have he : (inner ℂ (ξ:Base) (p:Base)*(inner ℂ (p:Base) (ξ:Base))).re = ‖inner ℂ (p:Base) (ξ:Base)‖^2 := by |
| rw [hconj] |
| rw [← Complex.normSq_eq_norm_sq] |
| simp only [Complex.star_def,Complex.mul_re,Complex.conj_re,Complex.conj_im,Complex.normSq_apply] |
| ring |
| have hn : ‖inner ℂ (ξ:Base) (p:Base)‖=‖inner ℂ (p:Base) (ξ:Base)‖ := norm_inner_symm _ _ |
| change 1^2-2*(inner ℂ (ξ:Base) (p:Base)*inner ℂ (p:Base) (ξ:Base)).re+_=_ |
| rw [he,hn] |
| ring |
|
|
| lemma horizontal_bracket_bound (ξ p : Sphere) (v : Base) (hv : Horizontal ξ v) : |
| ‖bracket v (p:Base)‖ ≤ projectiveDistance ξ p*‖v‖ := by |
| have he : bracket v ((p:Base)-bracket (p:Base) (ξ:Base) • (ξ:Base)) = bracket v (p:Base) := by |
| dsimp [bracket,Horizontal] at hv ⊢ |
| rw [inner_sub_left,inner_smul_left,hv,mul_zero,sub_zero] |
| calc |
| ‖bracket v (p:Base)‖ = ‖bracket v ((p:Base)-bracket (p:Base) (ξ:Base) • (ξ:Base))‖ := by rw [he] |
| _ ≤ ‖(p:Base)-bracket (p:Base) (ξ:Base) • (ξ:Base)‖*‖v‖ := norm_inner_le_norm _ _ |
| _ = _ := by rw [orthogonal_residual_norm] |
| |
| lemma patch_bracket_lower {k : ℕ} {R : ℝ} (hk : 0 < k) (hR : 0 ≤ R) (hkR : 2*R ≤ k) |
| (p ξ : Sphere) (hξ : ξ ∈ peakPatch k R p) : |
| (1:ℝ)/2 < ‖bracket (ξ:Base) (p:Base)‖ := by |
| have hk0 : (0:ℝ) < k := by exact_mod_cast hk |
| have radiusBound : |R/k| ≤ (1:ℝ)/2 := by |
| rw [abs_of_nonneg (div_nonneg hR hk0.le)] |
| exact (div_le_iff₀ hk0).mpr (by nlinarith) |
| have hh : -(1:ℝ)/2 ≤ -R/k := by |
| simpa only [neg_div] using neg_le_neg ((le_abs_self (R/k)).trans radiusBound) |
| have he := Real.add_one_le_exp (-R/k) |
| have hc := hξ |
| change Real.exp (-R/k) < ‖bracket (ξ:Base) (p:Base)‖ at hc |
| linarith |
|
|
| lemma horizontal_patch_log_derivative_bound {k : ℕ} {R : ℝ} (hk : 0 < k) (hR : 0 < R) |
| (hkR : 2*R ≤ k) (p ξ : Sphere) (hξ : ξ ∈ peakPatch k R p) |
| (v : Base) (hv : Horizontal ξ v) : |
| ‖bracket v (p:Base)/bracket (ξ:Base) (p:Base)‖ ≤ 2*Real.sqrt (2*R/(k:ℝ))*‖v‖ := by |
| have hn := patch_bracket_lower hk hR.le hkR p ξ hξ |
| have hd0 := peakPatch_radius hk hR hξ |
| have hk0 : (0:ℝ) < k := by exact_mod_cast hk |
| have hd : projectiveDistance ξ p < Real.sqrt (2*R/(k:ℝ)) := by |
| rw [Real.sqrt_div (by positivity)] |
| apply (lt_div_iff₀ (Real.sqrt_pos.mpr hk0)).mpr |
| nlinarith |
| rw [norm_div] |
| apply (div_le_iff₀ (by linarith : 0 < ‖bracket (ξ:Base) (p:Base)‖)).mpr |
| have hb := horizontal_bracket_bound ξ p v hv |
| have hm := mul_le_mul_of_nonneg_right hd.le (norm_nonneg v) |
| have hnn : 0 ≤ Real.sqrt (2*R/(k:ℝ))*‖v‖ := mul_nonneg (Real.sqrt_nonneg _) (norm_nonneg _) |
| nlinarith |
|
|
| end PinchedHartogs.BaseConstruction |
| |
| end |
|
|
| end OAI |
|
|
