| import OAI.Geometry.Kahler.BaseAnisotropic |
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| namespace OAI |
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| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology |
| open scoped ContDiff |
| noncomputable section |
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| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
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| def anisotropicChart (k : ℕ) (r : ℝ) (z : Base) : Base := |
| WithLp.toLp 2 ![(k:ℂ)*(((r:ℂ)+z 0)/(1+(r:ℂ)*z 0)-(r:ℂ)), |
| (Real.sqrt (k:ℝ):ℂ)*(Real.sqrt (1-r^2):ℂ)*z 1/(1+(r:ℂ)*z 0)] |
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| lemma anisotropicChart_identity {k : ℕ} (hk : 1 ≤ k) (r : ℝ) |
| (U : Base ≃ₗᵢ[ℂ] Base) (z : Base) : |
| U (anisotropicMap k r (anisotropicChart k r z))=centeredChart r U z := by |
| have hk0 : (k:ℂ) ≠ 0 := by exact_mod_cast (show k ≠ 0 by omega) |
| have hs0 : (Real.sqrt (k:ℝ):ℂ) ≠ 0 := by |
| exact_mod_cast (Real.sqrt_pos.mpr (by exact_mod_cast (show 0 < k by omega))).ne' |
| congr 1 |
| ext i |
| fin_cases i <;> dsimp [anisotropicMap,anisotropicChart,centeredChart] |
| all_goals |
| by_cases hd : 1+(r:ℂ)*z 0=0 |
| · simp [hd] |
| <;> field_simp [hk0,hs0] |
| <;> ring |
| · field_simp [hk0,hs0,hd] |
| <;> ring |
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| lemma anisotropicChart_norm {k : ℕ} (hk : 1 ≤ k) {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) |
| (hkr : (k:ℝ)*(1-r) ≤ 1) {z : Base} (hz : ‖z‖ < (1/32:ℝ)) : |
| ‖anisotropicChart k r z‖ ≤ Real.sqrt ((k:ℝ)*(1-r))/4 := by |
| have hkpos : 0 < (k:ℝ) := by exact_mod_cast (show 0 < k by omega) |
| have hδ : 0 < 1-r := by linarith |
| have hs : 0 < Real.sqrt ((k:ℝ)*(1-r)) := Real.sqrt_pos.mpr (mul_pos hkpos hδ) |
| have hs2 := Real.sq_sqrt (mul_nonneg hkpos.le hδ.le) |
| have hs1 : Real.sqrt ((k:ℝ)*(1-r)) ≤ 1 := (Real.sqrt_le_one).2 hkr |
| have hkst : (k:ℝ)*(1-r) ≤ Real.sqrt ((k:ℝ)*(1-r)) := by nlinarith |
| have hden := chart_small_ball hr hr1 hz |
| have hdenpos : 0 < ‖1+(r:ℂ)*z 0‖ := by linarith |
| have hd : 1+(r:ℂ)*z 0 ≠ 0 := norm_ne_zero_iff.mp hdenpos.ne' |
| have hz0 := PiLp.norm_apply_le (p := (2:ENNReal)) z 0 |
| have hz1 := PiLp.norm_apply_le (p := (2:ENNReal)) z 1 |
| have hrr : 0 ≤ 1-r^2 := by nlinarith |
| have hid : ((r:ℂ)+z 0)/(1+(r:ℂ)*z 0)-(r:ℂ) = ((1-r^2:ℝ):ℂ)*z 0/(1+(r:ℂ)*z 0) := by |
| push_cast |
| field_simp |
| ring |
| have hfirst : ‖(k:ℂ)*(((r:ℂ)+z 0)/(1+(r:ℂ)*z 0)-(r:ℂ))‖ ≤ Real.sqrt ((k:ℝ)*(1-r))/8 := by |
| rw [hid,norm_mul,norm_div,norm_mul,Complex.norm_natCast,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hrr] |
| rw [← mul_div_assoc] |
| apply (div_le_iff₀ hdenpos).2 |
| have hn : (k:ℝ)*(1-r^2) ≤ 2*((k:ℝ)*(1-r)) := by nlinarith |
| have hmul : (k:ℝ)*(1-r^2)*‖z 0‖ ≤ 2*Real.sqrt ((k:ℝ)*(1-r))*(1/32) := by |
| calc _ ≤ (2*((k:ℝ)*(1-r)))*‖z 0‖ := mul_le_mul_of_nonneg_right hn (norm_nonneg _) |
| _ ≤ (2*Real.sqrt ((k:ℝ)*(1-r)))*(1/32) := |
| mul_le_mul (by nlinarith) (by linarith) (norm_nonneg _) (by positivity) |
| nlinarith |
| have hsprod : Real.sqrt (k:ℝ)*Real.sqrt (1-r^2) ≤ 2*Real.sqrt ((k:ℝ)*(1-r)) := by |
| have h1 := Real.sq_sqrt hkpos.le |
| have h2 := Real.sq_sqrt hrr |
| have he : (Real.sqrt (k:ℝ)*Real.sqrt (1-r^2))^2=(k:ℝ)*(1-r^2) := by rw [mul_pow,h1,h2] |
| have hn : (k:ℝ)*(1-r^2) ≤ 4*((k:ℝ)*(1-r)) := by nlinarith |
| nlinarith [mul_nonneg (Real.sqrt_nonneg (k:ℝ)) (Real.sqrt_nonneg (1-r^2))] |
| have hsecond : ‖(Real.sqrt (k:ℝ):ℂ)*(Real.sqrt (1-r^2):ℂ)*z 1/(1+(r:ℂ)*z 0)‖ ≤ Real.sqrt ((k:ℝ)*(1-r))/8 := by |
| rw [norm_div,norm_mul,norm_mul,Complex.norm_real,Complex.norm_real,Real.norm_eq_abs,Real.norm_eq_abs, |
| abs_of_nonneg (Real.sqrt_nonneg _),abs_of_nonneg (Real.sqrt_nonneg _)] |
| apply (div_le_iff₀ hdenpos).2 |
| have hm : Real.sqrt (k:ℝ)*Real.sqrt (1-r^2)*‖z 1‖ ≤ 2*Real.sqrt ((k:ℝ)*(1-r))*(1/32) := |
| mul_le_mul hsprod (by linarith) (norm_nonneg _) (by positivity) |
| nlinarith |
| have hnorm : ‖anisotropicChart k r z‖^2 = |
| ‖(k:ℂ)*(((r:ℂ)+z 0)/(1+(r:ℂ)*z 0)-(r:ℂ))‖^2+ |
| ‖(Real.sqrt (k:ℝ):ℂ)*(Real.sqrt (1-r^2):ℂ)*z 1/(1+(r:ℂ)*z 0)‖^2 := by |
| simp [anisotropicChart,EuclideanSpace.norm_sq_eq,Fin.sum_univ_two] |
| have hn0 := norm_nonneg ((k:ℂ)*(((r:ℂ)+z 0)/(1+(r:ℂ)*z 0)-(r:ℂ))) |
| have hn1 := norm_nonneg ((Real.sqrt (k:ℝ):ℂ)*(Real.sqrt (1-r^2):ℂ)*z 1/(1+(r:ℂ)*z 0)) |
| nlinarith [norm_nonneg (anisotropicChart k r z)] |
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| lemma peak_chart_low : ∃ B : ℝ, 0 ≤ B ∧ ∀ {D : ℝ}, 1 ≤ D → ∀ {k : ℕ}, 1 ≤ k → |
| ∀ {P : Finset Sphere}, ProjectivelySeparated (D/Real.sqrt k) P → ∀ {r : ℝ}, 0 ≤ r → r < 1 → |
| (k:ℝ)*(1-r) ≤ 1 → ∀ U : Base ≃ₗᵢ[ℂ] Base, ∀ z : Base, ‖z‖ < 1/32 → |
| ‖peakPolynomial P k (centeredChart r U z)-peakPolynomial P k (centeredChart r U 0)‖ ≤ B*Real.sqrt ((k:ℝ)*(1-r)) := by |
| obtain ⟨B,hB,hBb⟩ := peak_anisotropic_lipschitz |
| refine ⟨B,hB,?_⟩ |
| intro D hD k hk P hP r hr hr1 hkr U z hz |
| have hn := anisotropicChart_norm hk hr hr1 hkr hz |
| have hs1 : Real.sqrt ((k:ℝ)*(1-r)) ≤ 1 := Real.sqrt_le_one.mpr hkr |
| have hn1 : ‖anisotropicChart k r z‖ ≤ 1/2 := by linarith |
| have hh := hBb hD hk hP hr hr1.le U _ hn1 |
| rw [anisotropicChart_identity hk] at hh |
| have hzero : U (anisotropicMap k r 0)=centeredChart r U 0 := by |
| unfold anisotropicMap centeredChart |
| simp |
| rw [hzero] at hh |
| exact hh.trans ((mul_le_mul_of_nonneg_left hn hB).trans (by nlinarith [Real.sqrt_nonneg ((k:ℝ)*(1-r))])) |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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