| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Constants |
| import OAI.Geometry.CAT0Fillings.Scaling.HighCoefficient |
| import OAI.Geometry.CAT0Fillings.Density.RadialBound |
| import OAI.Geometry.CAT0Fillings.Variational.Extremizer |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
|
|
| lemma extremal_coefficient_two (hX : IsCAT0 X) |
| {T : Functional X 2} (hT : IsIntegral 2 T) (hz : boundarySucc T = 0) |
| {c : ℝ} (hc : 0 < c) (hm : 0 < mass T) |
| (hext : ∀ B : Functional X 2, IsIntegral 2 B → boundarySucc B = 0 → |
| c*((mass T)^((3:ℝ)/2)-(mass B)^((3:ℝ)/2)) ≤ fillingVolume (T-B)) : |
| c ≤ fillingCoefficient 2 := by |
| obtain ⟨q⟩ := exists_chartGeometry hX hT.1 hT.2.1 |
| let ell := c*((3:ℝ)/2)*(mass T)^((3:ℝ)/2-1) |
| have hell : 0 < ell := by dsimp [ell]; positivity |
| have hbound : 16*Real.pi*ell^2 ≤ mass T := by |
| apply q.radial_coefficient_bound_two hX hm hell |
| intro o r hr |
| exact radial_inequality_of_extremal hX hT hz q hc.le (by norm_num) hm hext o |
| ((inverseSquareCutoff_lipschitz hr).comp (LipschitzWith.dist_right o)) |
| (fun _ => inverseSquareCutoff_nonneg _) |
| (fun _ => inverseSquareCutoff_le_one hr _) |
| have hsq : ell^2 = ((9:ℝ)/4)*c^2*mass T := by |
| dsimp [ell] |
| rw [show (3:ℝ)/2-1 = 1/2 by norm_num,←Real.sqrt_eq_rpow] |
| nlinarith [Real.sq_sqrt hm.le] |
| rw [hsq] at hbound |
| have hcoef : 36*Real.pi*c^2 ≤ 1 := by |
| apply (mul_le_mul_iff_of_pos_right hm).mp |
| nlinarith |
| have hC := fillingCoefficient_two_pos |
| have hCsq := fillingCoefficient_two_sq |
| by_contra hn |
| have hh : (fillingCoefficient 2)^2 < c^2 := sq_lt_sq₀ hC.le hc.le |>.mpr (lt_of_not_ge hn) |
| have hp := mul_lt_mul_of_pos_left hh (show 0 < 36*Real.pi by positivity) |
| linarith |
|
|
| lemma fillingVolume_le_two_supercritical (hX : IsCAT0 X) |
| {T : Functional X 2} (hT : IsIntegral 2 T) (hz : boundarySucc T = 0) |
| {c : ℝ} (hc : fillingCoefficient 2 < c) : |
| fillingVolume T ≤ c*(mass T)^((3:ℝ)/2) := by |
| by_contra hnot |
| have hc0 : 0 < c := fillingCoefficient_two_pos.trans hc |
| let δ := (c-fillingCoefficient 2)/(c+fillingCoefficient 2) |
| have hsum : 0 < c+fillingCoefficient 2 := add_pos hc0 fillingCoefficient_two_pos |
| have hδ : 0 < δ := div_pos (sub_pos.mpr hc) hsum |
| obtain ⟨U,hU,hUz,hm,hdef,hext⟩ := exists_filling_extremizer hX hc0 |
| (by norm_num : (1:ℝ) < 3/2) hδ hT hz (show 0 < fillingDeficit c (3/2) T by |
| dsimp [fillingDeficit]; linarith) |
| have hden : 0 < 1+δ := by linarith |
| have hcoeff : c/(1+δ) = (c+fillingCoefficient 2)/2 := by |
| dsimp [δ] |
| field_simp [hsum.ne',hc0.ne'] |
| ring |
| have hb := extremal_coefficient_two hX hU hUz (div_pos hc0 hden) hm hext |
| rw [hcoeff] at hb |
| linarith |
|
|
| theorem fillingVolume_le_two (hX : IsCAT0 X) |
| {T : Functional X 2} (hT : IsIntegral 2 T) (hz : boundarySucc T = 0) : |
| fillingVolume T ≤ fillingCoefficient 2*(mass T)^fillingPower 2 := by |
| rw [fillingPower_two] |
| have hlim : Tendsto (fun j : ℕ => (fillingCoefficient 2+1/((j:ℝ)+1))*(mass T)^((3:ℝ)/2)) |
| atTop (𝓝 (fillingCoefficient 2*(mass T)^((3:ℝ)/2))) := by |
| simpa using ((tendsto_one_div_add_atTop_nhds_zero_nat : |
| Tendsto (fun j : ℕ => 1/((j:ℝ)+1)) atTop (𝓝 0)).const_add (fillingCoefficient 2)).mul_const |
| ((mass T)^((3:ℝ)/2)) |
| apply ge_of_tendsto hlim |
| exact Eventually.of_forall fun j => fillingVolume_le_two_supercritical hX hT hz |
| (lt_add_of_pos_right _ (by positivity)) |
|
|
| end CAT0Fillings |
| end |
|
|
| section |
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings |
| universe u |
| lemma fillingVolume_le_of_coefficient {X : Type u} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] (hX : IsCAT0 X) {k : ℕ} |
| (hcoef : ∀ U : Functional X (k+2), IsIntegral (k+2) U → IsCycle U → |
| ∀ c : ℝ, 0 < c → 0 < mass U → |
| (∀ B : Functional X (k+2), IsIntegral (k+2) B → IsCycle B → |
| c*((mass U)^(fillingPower (k+2))-(mass B)^(fillingPower (k+2))) ≤ fillingVolume (U-B)) → |
| c ≤ fillingCoefficient (k+2)) |
| {T : Functional X (k+2)} (hT : IsIntegral (k+2) T) (hz : IsCycle T) : |
| fillingVolume T ≤ fillingCoefficient (k+2)*(mass T)^(fillingPower (k+2)) := by |
| have hsuper (c : ℝ) (hc : fillingCoefficient (k+2) < c) : |
| fillingVolume T ≤ c*(mass T)^(fillingPower (k+2)) := by |
| by_contra hnot |
| have hc0 := (fillingCoefficient_pos (by omega : 0 < k+2)).trans hc |
| let δ := (c-fillingCoefficient (k+2))/(c+fillingCoefficient (k+2)) |
| have hsum : 0 < c+fillingCoefficient (k+2) := add_pos hc0 (fillingCoefficient_pos (by omega)) |
| have hδ : 0 < δ := div_pos (sub_pos.mpr hc) hsum |
| obtain ⟨U,hU,hUz,hm,hdef,hext⟩ := exists_filling_extremizer hX hc0 |
| (fillingPower_gt_one (by omega : 0 < k+2)) hδ hT hz |
| (show 0 < fillingDeficit c (fillingPower (k+2)) T by dsimp [fillingDeficit]; linarith) |
| have hden : 0 < 1+δ := by linarith |
| have hcoeff : c/(1+δ) = (c+fillingCoefficient (k+2))/2 := by |
| dsimp [δ] |
| field_simp [hsum.ne',hc0.ne'] |
| ring |
| have hb := hcoef U hU hUz (c/(1+δ)) (div_pos hc0 hden) hm hext |
| rw [hcoeff] at hb |
| linarith |
| have hlim : Tendsto (fun j : ℕ => (fillingCoefficient (k+2)+1/((j:ℝ)+1))*(mass T)^(fillingPower (k+2))) |
| atTop (𝓝 (fillingCoefficient (k+2)*(mass T)^(fillingPower (k+2)))) := by |
| simpa using ((tendsto_one_div_add_atTop_nhds_zero_nat : |
| Tendsto (fun j : ℕ => 1/((j:ℝ)+1)) atTop (𝓝 0)).const_add (fillingCoefficient (k+2))).mul_const |
| ((mass T)^(fillingPower (k+2))) |
| apply ge_of_tendsto hlim |
| exact Eventually.of_forall fun j => hsuper _ (lt_add_of_pos_right _ (by positivity)) |
|
|
| theorem compact_filling_bound : ∀ k : ℕ, |
| ∀ (X : Type u) [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] [Nonempty X], |
| IsCAT0 X → ∀ T : Functional X (k+2), IsIntegral (k+2) T → boundarySucc T = 0 → |
| fillingVolume T ≤ fillingCoefficient (k+2)*(mass T)^(fillingPower (k+2)) := by |
| intro k |
| induction k with |
| | zero => |
| intro X _ _ _ _ _ hX T hT hz |
| exact fillingVolume_le_two hX hT hz |
| | succ k ih => |
| intro X _ _ _ _ _ hX T hT hz |
| apply fillingVolume_le_of_coefficient hX ?_ hT hz |
| intro U hU hUz c hc hm hext |
| exact extremal_coefficient_high (by omega) hX hU hUz hc hm ih hext |
| end CAT0Fillings |
| end |
|
|
| end OAI |
|
|
