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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 | 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
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