File size: 6,534 Bytes
8e4d533 | 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 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 | 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
|