| import OAI.Geometry.Relativity.CKS.SurfaceNormalization | |
| import OAI.Geometry.Relativity.CKS.RadialTail | |
| namespace OAI | |
| noncomputable section | |
| namespace CKSADM | |
| noncomputable section | |
| open Set Filter MeasureTheory CKSSphericalHarmonics CKSSphericalChart CKSBending CKSInducedSphere | |
| open scoped Topology ContDiff | |
| lemma surfaceMeasure_real : surfaceMeasure.real univ = 4*Real.pi := by | |
| rw [← sphereArea_eq_four_pi,sphereArea,sphereIntegral_apply] | |
| simp | |
| lemma canonicalF_mean (f₀ : C(Sphere,ℝ)) (hf₀ : SmoothSphere f₀) (m t : ℝ) : | |
| (∫ n : Sphere, canonicalF f₀ m t n ∂surfaceMeasure) = m*(4*Real.pi) := by | |
| let p := heatPolynomials f₀ | |
| have hp := smoothDatum_rapid f₀ hf₀ | |
| let q : C(Sphere,ℝ) := sphereEval (MvPolynomial.C m) + sphereHeatJet p (0,[]) t | |
| have he : (fun x : Sphere => canonicalF f₀ m t x) = fun x => q x := by | |
| funext x | |
| change m + jointHeatSeries p (0,[]) (t,x) = sphereEval (MvPolynomial.C m) x + sphereHeatJet p (0,[]) t x | |
| rw [sphereHeatJet_eq p hp] | |
| simp [sphereEval] | |
| rw [he,← sphereIntegral_apply] | |
| change sphereIntegral (sphereEval (MvPolynomial.C m) + sphereHeatJet p (0,[]) t) = _ | |
| rw [map_add,sphereIntegral_constant,sphereHeatJet_mean_zero p hp (heatPolynomials_harmonic f₀), | |
| add_zero,sphereArea_eq_four_pi] | |
| lemma canonicalF_continuous_sphere (f₀ : C(Sphere,ℝ)) (hf₀ : SmoothSphere f₀) (m t : ℝ) : | |
| Continuous (fun n : Sphere => canonicalF f₀ m t n) := by | |
| apply ((joint_slice (canonicalF_joint f₀ hf₀ m) t).continuousOn).comp_continuous continuous_subtype_val | |
| intro n | |
| exact ne_of_apply_ne norm (by simp) | |
| def tailSphereMass (f₀ : C(Sphere,ℝ)) (m R r : ℝ) (n : Sphere) : ℝ := | |
| paddingMass R r + canonicalF f₀ m (heatTime R r) n | |
| lemma tailSphereMass_continuous (f₀ : C(Sphere,ℝ)) (hf₀ : SmoothSphere f₀) (m R r : ℝ) : | |
| Continuous (tailSphereMass f₀ m R r) := | |
| continuous_const.add (canonicalF_continuous_sphere f₀ hf₀ m (heatTime R r)) | |
| lemma tailSphereMass_mean (f₀ : C(Sphere,ℝ)) (hf₀ : SmoothSphere f₀) (m R r : ℝ) : | |
| (∫ n, tailSphereMass f₀ m R r n ∂surfaceMeasure) = (m+paddingMass R r)*(4*Real.pi) := by | |
| unfold tailSphereMass | |
| rw [integral_add (integrable_const _) ((canonicalF_continuous_sphere f₀ hf₀ m _).integrable_of_hasCompactSupport | |
| (HasCompactSupport.of_compactSpace _)),integral_const,canonicalF_mean f₀ hf₀ m, | |
| smul_eq_mul,surfaceMeasure_real] | |
| ring | |
| lemma tailSphereMass_bound (f₀ : C(Sphere,ℝ)) (hf₀ : SmoothSphere f₀) (m : ℝ) : | |
| ∃ B : ℝ, 0 < B ∧ ∀ R r : ℝ, 1 ≤ R → 3*R ≤ r → ∀ n : Sphere, | |
| |tailSphereMass f₀ m R r n| ≤ B := by | |
| obtain ⟨K,hK,hbound⟩ := canonical_uniform_data f₀ hf₀ m | |
| refine ⟨|m|+K+3,by positivity,?_⟩ | |
| intro R r hR hr n | |
| have hRp : 0 < R := by linarith | |
| have hrp : 0 < r := by linarith | |
| have hc := paddingMass_bound hRp (by linarith : R ≤ r) | |
| have hs : 1 ≤ Real.sqrt R := by simpa using Real.sqrt_le_sqrt hR | |
| have ht := heatTime_nonneg hRp hrp | |
| have hf := (hbound (heatTime R r) ht n (by simp)).1 | |
| have hc' : paddingMass R r ≤ 2 := hc.trans ((div_le_iff₀ (Real.sqrt_pos.mpr hRp)).mpr (by linarith)) | |
| apply (abs_add_le _ _).trans | |
| rw [abs_of_nonneg (paddingMass_nonneg hRp hrp)] | |
| linarith | |
| lemma denominator_lower {F B r : ℝ} (hB : 0 ≤ B) (hF : |F| ≤ B) (hr : 4*B+1 ≤ r) : | |
| (1:ℝ)/2 ≤ 1-2*F/r := by | |
| have hrp : 0 < r := by linarith | |
| have hf := (le_abs_self F).trans hF | |
| have hh : 2*F/r ≤ (1:ℝ)/2 := (div_le_iff₀ hrp).mpr (by linarith) | |
| linarith | |
| lemma rational_mass_error {F B r : ℝ} (hB : 0 ≤ B) (hF : |F| ≤ B) (hr : 4*B+1 ≤ r) : | |
| |F/(1-2*F/r)-F| ≤ 4*B^2/r := by | |
| have hrp : 0 < r := by linarith | |
| have hd := denominator_lower hB hF hr | |
| have hdp : 0 < 1-2*F/r := by linarith | |
| have hs : F^2 ≤ B^2 := by simpa only [sq_abs] using (sq_le_sq₀ (abs_nonneg F) hB).mpr hF | |
| have hden : r-2*F ≠ 0 := by have hh := (le_abs_self F).trans hF; linarith | |
| have he : F/(1-2*F/r)-F = (2*F^2/r)/(1-2*F/r) := by | |
| apply (eq_div_iff hdp.ne').mpr | |
| rw [sub_mul,div_mul_cancel₀ _ hdp.ne'] | |
| field_simp [hrp.ne'] | |
| ring | |
| rw [he,abs_of_nonneg (by positivity)] | |
| apply (div_le_iff₀ hdp).mpr | |
| have hnn : 0 ≤ 4*B^2/r := by positivity | |
| have hh := mul_le_mul_of_nonneg_left hd hnn | |
| calc | |
| 2*F^2/r ≤ 2*B^2/r := div_le_div_of_nonneg_right (by nlinarith) hrp.le | |
| _ = (4*B^2/r)*((1:ℝ)/2) := by ring | |
| _ ≤ _ := hh | |
| def reducedEnergy (f₀ : C(Sphere,ℝ)) (m R r : ℝ) : ℝ := | |
| (4*Real.pi)⁻¹ * ∫ n : Sphere, | |
| tailSphereMass f₀ m R r n/(1-2*tailSphereMass f₀ m R r n/r) ∂surfaceMeasure | |
| lemma reducedEnergy_error (f₀ : C(Sphere,ℝ)) (hf₀ : SmoothSphere f₀) (m R r B : ℝ) | |
| (hB : 0 ≤ B) (hF : ∀ n : Sphere, |tailSphereMass f₀ m R r n| ≤ B) | |
| (hr : 4*B+1 ≤ r) : | |
| |reducedEnergy f₀ m R r-(m+paddingMass R r)| ≤ 4*B^2/r := by | |
| let F := tailSphereMass f₀ m R r | |
| have hf : Continuous F := tailSphereMass_continuous f₀ hf₀ m R r | |
| have hd : ∀ n : Sphere, 1-2*F n/r ≠ 0 := fun n => ne_of_gt (by | |
| have hh := denominator_lower hB (hF n) hr; linarith) | |
| have hq : Continuous (fun n => F n/(1-2*F n/r)) := | |
| hf.div (continuous_const.sub ((continuous_const.mul hf).div_const r)) hd | |
| have hfi : Integrable F surfaceMeasure := hf.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) | |
| have hqi : Integrable (fun n => F n/(1-2*F n/r)) surfaceMeasure := hq.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) | |
| have hn := norm_integral_le_of_norm_le_const (μ := surfaceMeasure) | |
| (f := fun n => F n/(1-2*F n/r)-F n) | |
| (Filter.Eventually.of_forall (fun n => by simpa only [Real.norm_eq_abs] using rational_mass_error hB (hF n) hr)) | |
| rw [integral_sub hqi hfi,surfaceMeasure_real,Real.norm_eq_abs] at hn | |
| have hm : (∫ n : Sphere, F n ∂surfaceMeasure) = (m+paddingMass R r)*(4*Real.pi) := tailSphereMass_mean f₀ hf₀ m R r | |
| rw [hm] at hn | |
| have hc : 0 < 4*Real.pi := by positivity | |
| have he : reducedEnergy f₀ m R r-(m+paddingMass R r) = | |
| (4*Real.pi)⁻¹*((∫ n : Sphere, F n/(1-2*F n/r) ∂surfaceMeasure)-(m+paddingMass R r)*(4*Real.pi)) := by | |
| unfold reducedEnergy | |
| dsimp only [F] | |
| rw [mul_sub] | |
| congr 1 | |
| field_simp | |
| rw [he,abs_mul,abs_of_pos (inv_pos.mpr hc)] | |
| calc | |
| _ ≤ (4*Real.pi)⁻¹*((4*B^2/r)*(4*Real.pi)) := mul_le_mul_of_nonneg_left hn (inv_nonneg.mpr hc.le) | |
| _ = _ := by field_simp | |
| theorem reducedEnergy_limit (f₀ : C(Sphere,ℝ)) (hf₀ : SmoothSphere f₀) (m : ℝ) | |
| {R : ℝ} (hR : 1 ≤ R) : | |
| Tendsto (reducedEnergy f₀ m R) atTop (𝓝 (m+paddingCharge R)) := by | |
| obtain ⟨B,hB,hbound⟩ := tailSphereMass_bound f₀ hf₀ m | |
| have herr : Tendsto (fun r => reducedEnergy f₀ m R r-(m+paddingMass R r)) atTop (𝓝 0) := by | |
| rw [tendsto_iff_norm_sub_tendsto_zero] | |
| simp only [sub_zero,Real.norm_eq_abs] | |
| apply squeeze_zero' (Filter.Eventually.of_forall (fun _ => abs_nonneg _)) | |
| (g := fun r : ℝ => 4*B^2/r) | |
| · filter_upwards [eventually_ge_atTop (3*R),eventually_ge_atTop (4*B+1)] with r hr hlarge | |
| exact reducedEnergy_error f₀ hf₀ m R r B hB.le (hbound R r hR hr) hlarge | |
| · exact tendsto_const_nhds.div_atTop tendsto_id | |
| have hmean := (tendsto_const_nhds (x := m)).add (paddingCharge_limit (by linarith : 0 < R)) | |
| have hh := herr.add hmean | |
| simpa only [sub_add_cancel,zero_add] using hh | |
| end | |
| end CKSADM | |
| end | |
| end OAI | |