| import OAI.Geometry.Kahler.BaseCircleLog |
|
|
| namespace OAI |
|
|
| 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 |
| open scoped ContDiff ENNReal Pointwise |
| open scoped ContDiff |
| open Set Filter Topology MeasureTheory |
| noncomputable section |
|
|
| open Set Filter Topology MeasureTheory |
| namespace PinchedHartogs.BaseConstruction |
|
|
| def logFactor (q ε : ℝ) : ℝ := |
| (1+q^2+ε^2+Real.sqrt ((1+q^2+ε^2)^2-4*q^2))/2 |
|
|
| lemma logFactor_discriminant (q ε : ℝ) : |
| (1-q^2)^2 ≤ (1+q^2+ε^2)^2-4*q^2 := by |
| nlinarith [sq_nonneg ε,mul_nonneg (sq_nonneg ε) (sq_nonneg q),sq_nonneg (ε^2)] |
|
|
| lemma logFactor_one_le (q ε : ℝ) : 1 ≤ logFactor q ε := by |
| have hh := Real.sqrt_le_sqrt (logFactor_discriminant q ε) |
| rw [Real.sqrt_sq_eq_abs] at hh |
| have hl := le_abs_self (1-q^2) |
| unfold logFactor |
| nlinarith [sq_nonneg ε] |
|
|
| lemma logFactor_pos (q ε : ℝ) : 0 < logFactor q ε := lt_of_lt_of_le zero_lt_one (logFactor_one_le q ε) |
|
|
| lemma logFactor_gt {q ε : ℝ} (hε : ε ≠ 0) : q < logFactor q ε := by |
| have he := sq_pos_of_ne_zero hε |
| have hs := Real.sqrt_nonneg ((1+q^2+ε^2)^2-4*q^2) |
| unfold logFactor |
| nlinarith [sq_nonneg (q-1)] |
|
|
| lemma logFactor_equation (q ε : ℝ) : |
| logFactor q ε+q^2/logFactor q ε=1+q^2+ε^2 := by |
| have hd : 0 ≤ (1+q^2+ε^2)^2-4*q^2 := (sq_nonneg _).trans (logFactor_discriminant q ε) |
| have hs := Real.sq_sqrt hd |
| have hl := (logFactor_pos q ε).ne' |
| field_simp |
| unfold logFactor |
| nlinarith |
| |
| lemma logFactor_norm_identity (q ε : ℝ) (z : Circle) : |
| ‖1-(q:ℂ)*(z:ℂ)‖^2+ε^2 = logFactor q ε*‖1-(q/logFactor q ε:ℝ)*(z:ℂ)‖^2 := by |
| have hz : (z:ℂ).re^2+(z:ℂ).im^2=1 := by |
| have hh := Complex.sq_norm (z:ℂ) |
| rw [z.norm_coe] at hh |
| simp only [Complex.normSq_apply] at hh |
| nlinarith |
| have he := logFactor_equation q ε |
| have hl := (logFactor_pos q ε).ne' |
| simp only [Complex.sq_norm,Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.one_re,Complex.one_im, |
| Complex.mul_re,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im,zero_mul,sub_zero,zero_sub,add_zero] |
| field_simp at he ⊢ |
| nlinarith [congrArg (fun t => q^2*t) hz,congrArg (fun t => logFactor q ε*q^2*t) hz] |
|
|
| lemma logFactor_zero (q : ℝ) : logFactor q 0 = max 1 (q^2) := by |
| have he : (1+q^2+(0:ℝ)^2)^2-4*q^2=(1-q^2)^2 := by ring |
| unfold logFactor |
| rw [he,Real.sqrt_sq_eq_abs] |
| by_cases hq : q^2 ≤ 1 |
| · rw [abs_of_nonneg (by linarith),max_eq_left hq]; ring |
| · rw [abs_of_nonpos (by linarith),max_eq_right (by linarith)]; ring |
|
|
| lemma logFactor_continuous : Continuous (fun p : ℝ × ℝ => logFactor p.1 p.2) := by |
| unfold logFactor |
| fun_prop |
|
|
| lemma circle_log_continuous {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) : |
| Continuous (fun z : Circle => Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2)) := by |
| apply Continuous.log |
| · fun_prop |
| · intro z |
| apply pow_ne_zero |
| apply norm_ne_zero_iff.mpr |
| intro hz |
| have hh := congrArg norm (sub_eq_zero.mp hz) |
| rw [norm_one,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hr,z.norm_coe,mul_one] at hh |
| linarith |
|
|
| lemma regularized_circle_moments {q ε : ℝ} (hq : 0 ≤ q) (hε : ε ≠ 0) (f : ℝ) : |
| (∫ z : Circle, regularizedLog 1 ε ((q:ℂ)*(z:ℂ))*(1+f*(z:ℂ).re) ∂circleMeasure) = |
| Real.log (logFactor q ε)-Real.log (1+ε^2)-f*q/logFactor q ε := by |
| let r := q/logFactor q ε |
| have hr : 0 ≤ r := div_nonneg hq (logFactor_pos q ε).le |
| have hr1 : r < 1 := (div_lt_one (logFactor_pos q ε)).mpr (logFactor_gt hε) |
| have hmom := circle_log_moments hr hr1 |
| have hi := compact_continuous_integrable (μ := circleMeasure) (circle_log_continuous hr hr1) |
| have hic : Integrable (fun z : Circle => Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2)*(z:ℂ).re) circleMeasure := by |
| apply compact_continuous_integrable |
| exact (circle_log_continuous hr hr1).mul (by fun_prop) |
| have he : ∀ z : Circle, regularizedLog 1 ε ((q:ℂ)*(z:ℂ)) = |
| Real.log (logFactor q ε)-Real.log (1+ε^2)+Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2) := by |
| intro z |
| rw [regularizedLog_eq 1 ε _ hε] |
| simp only [Complex.ofReal_one,one_mul] |
| rw [logFactor_norm_identity,Real.log_mul (logFactor_pos q ε).ne' (by |
| have hh := logFactor_norm_identity q ε z |
| have hp : 0 < ‖1-(q:ℂ)*(z:ℂ)‖^2+ε^2 := by positivity |
| exact ne_of_gt (pos_of_mul_pos_right (hh ▸ hp) (logFactor_pos q ε).le))] |
| dsimp [r] |
| ring |
| let A := Real.log (logFactor q ε)-Real.log (1+ε^2) |
| let g : Circle → ℝ := fun z => Real.log (‖1-(r:ℂ)*(z:ℂ)‖^2) |
| have hz : (∫ z : Circle, (z:ℂ).re ∂circleMeasure)=0 := by |
| simpa [phaseChar] using circle_real_char_mean (by norm_num : (1:ℤ) ≠ 0) |
| have he' : ∀ z : Circle, regularizedLog 1 ε ((q:ℂ)*(z:ℂ))*(1+f*(z:ℂ).re) = |
| A+(A*f)*(z:ℂ).re+g z+f*(g z*(z:ℂ).re) := by |
| intro z |
| rw [he] |
| dsimp [A,g] |
| ring |
| simp_rw [he'] |
| have hiz : Integrable (fun z : Circle => (z:ℂ).re) circleMeasure := |
| compact_continuous_integrable (by fun_prop) |
| erw [integral_add (((integrable_const A).add (hiz.const_mul _)).add hi) (hic.const_mul _), |
| integral_add ((integrable_const A).add (hiz.const_mul _)) hi, |
| integral_add (integrable_const A) (hiz.const_mul _),integral_const_mul,integral_const_mul] |
| change (∫ z : Circle, A ∂circleMeasure)+(A*f)*_+_+f*_=_ |
| rw [hmom.1,hmom.2,hz] |
| simp [A,r] |
| ring |
| |
| end PinchedHartogs.BaseConstruction |
| |
| end |
| |
| end OAI |
| |
