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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | import Mathlib
import OAI.Geometry.CAT0Fillings.Variational.Ekeland
import OAI.Geometry.CAT0Fillings.Variational.LowerContinuity
namespace OAI
section
open Set Filter MeasureTheory Metric
open scoped Topology NNReal
namespace CAT0Fillings
open Foundations Slicing CurrentOperations
lemma exists_mass_bound_of_positive_deficit {A c p : ℝ} (hc : 0 < c) (hp : 1 < p) :
∃ M : ℝ≥0, ∀ m : ℝ, 0 ≤ m → 0 ≤ A*m-c*m^p → m ≤ M := by
obtain ⟨R,hR⟩ := eventually_atTop.mp ((tendsto_rpow_atTop (sub_pos.mpr hp)).eventually
(eventually_ge_atTop ((A+1)/c)))
refine ⟨⟨max R 1,le_trans zero_le_one (le_max_right _ _)⟩,fun m hm hpos => ?_⟩
by_contra hnot
have hlarge : max R 1 < m := lt_of_not_ge hnot
have hm0 : 0 < m := lt_of_lt_of_le zero_lt_one ((le_max_right R 1).trans hlarge.le)
have hh := hR m ((le_max_left R 1).trans hlarge.le)
have hpow : m^p = m^(p-1)*m := by
conv_lhs => rw [show p = (p-1)+1 by ring]
rw [Real.rpow_add hm0,Real.rpow_one]
have hmul : A+1 ≤ c*m^(p-1) := by
exact (div_le_iff₀ hc).mp hh |>.trans_eq (mul_comm _ _)
rw [hpow] at hpos
nlinarith
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[CompactSpace X] [Nonempty X]
noncomputable def fillingDeficit {k : ℕ} (c p : ℝ) (T : Functional X (k+1)) : ℝ :=
fillingVolume T-c*(mass T)^p
theorem exists_filling_extremizer (hX : IsCAT0 X) {k : ℕ} {c p δ : ℝ}
(hc : 0 < c) (hp : 1 < p) (hδ : 0 < δ)
{T0 : Functional X (k+2)} (hT0 : IsIntegral (k+2) T0) (hz0 : boundarySucc T0 = 0)
(hpositive : 0 < fillingDeficit c p T0) :
∃ T : Functional X (k+2), IsIntegral (k+2) T ∧ boundarySucc T = 0 ∧
0 < mass T ∧ fillingDeficit c p T0 ≤ fillingDeficit c p T ∧
∀ B : Functional X (k+2), IsIntegral (k+2) B → boundarySucc B = 0 →
c/(1+δ)*((mass T)^p-(mass B)^p) ≤ fillingVolume (T-B) := by
classical
have hpos : 0 < fillingVolume T0-c*(mass T0)^p := hpositive
let A : ℝ := (k+3:ℝ)*diam (univ : Set X)
have hA : 0 ≤ A := mul_nonneg (by positivity) diam_nonneg
obtain ⟨M,hM⟩ := exists_mass_bound_of_positive_deficit (A := A) hc hp
let C := {T : Functional X (k+2) // IsIntegral (k+2) T ∧ boundarySucc T = 0 ∧
fillingDeficit c p T0 ≤ fillingDeficit c p T}
let F : C → ℝ := fun T => fillingDeficit c p T.val
let d : C → C → ℝ := fun T U => fillingVolume (T.val-U.val)
have hlin (T : C) : fillingVolume T.val ≤ A*mass T.val := by
simpa only [A,Nat.cast_add,Nat.cast_one,show (1:ℝ)+2 = 3 by norm_num,add_assoc] using
hX.fillingVolume_le_linear T.property.1 T.property.2.1
have hmass (T : C) : mass T.val ≤ M := by
apply hM _ (mass_nonneg _)
have hh := T.property.2.2
have hl := hlin T
dsimp only [fillingDeficit] at hh
linarith
have hbound : BddAbove (range F) := by
refine ⟨A*M,?_⟩
rintro v ⟨T,rfl⟩
have hl := hlin T
have hm := mul_le_mul_of_nonneg_left (hmass T) hA
have hn := mul_nonneg hc.le (Real.rpow_nonneg (mass_nonneg T.val) p)
dsimp only [F,fillingDeficit]
linarith
have hcluster : ∀ u : ℕ → C, ∀ L : ℝ,
Tendsto (fun j => F (u j)) atTop (𝓝 L) →
(∀ ε > 0, ∃ N : ℕ, ∀ i ≥ N, ∀ j ≥ N, d (u i) (u j) < ε) →
∃ x : C, Tendsto (fun j => d (u j) x) atTop (𝓝 0) ∧ L ≤ F x := by
intro u L hL hdc
obtain ⟨m,hmI,σ,hσ,hm⟩ := isCompact_Icc.isSeqCompact
(fun j => (show mass (u j).val ∈ Icc (0:ℝ) M from ⟨mass_nonneg _,hmass _⟩))
obtain ⟨ψ,hψ,T,hT,hTM,_,hweak⟩ := exists_integral_weak_subsequence
(fun j => (u (σ j)).property.1) M 0 (fun j => hmass _)
(fun j => by
rw [(u (σ j)).property.2.1]
change mass (fun _ _ => (0:ℝ)) ≤ (0:ℝ)
rw [mass_zero]) hX
let τ : ℕ → ℕ := σ ∘ ψ
have hτ : StrictMono τ := hσ.comp hψ
have hz : boundarySucc T = 0 := by
funext b π
apply tendsto_nhds_unique (boundarySucc_weak_limit hweak b π)
simpa only [(u _).property.2.1,Pi.zero_apply] using
(tendsto_const_nhds : Tendsto (fun _ : ℕ => (0:ℝ)) atTop (𝓝 0))
have hdlim : Tendsto (fun j => fillingVolume ((u j).val-T)) atTop (𝓝 0) :=
fillingVolume_sub_tendsto_zero hX (fun j => (u j).property.1)
(fun j => (u j).property.2.1) M (fun j => hmass _) hdc hτ hweak
have hV := fillingVolume_tendsto_of_sub_tendsto_zero hX
(fun j => (u j).property.1) (fun j => (u j).property.2.1) hT hz hdlim
have hmlim : Tendsto (fun j => mass (u (τ j)).val) atTop (𝓝 m) := hm.comp hψ.tendsto_atTop
have hmle : mass T ≤ m := mass_le_of_mass_tendsto
(fun j => (u (τ j)).property.1.1) hmlim hweak
have hFlim : Tendsto (fun j => F (u (τ j))) atTop (𝓝 (fillingVolume T-c*m^p)) :=
(hV.comp hτ.tendsto_atTop).sub ((hmlim.rpow_const (Or.inr (by linarith))).const_mul c)
have heq : L = fillingVolume T-c*m^p := tendsto_nhds_unique
(hL.comp hτ.tendsto_atTop) hFlim
have hLle : L ≤ fillingDeficit c p T := by
rw [heq]
have hpow := mul_le_mul_of_nonneg_left
(Real.rpow_le_rpow (mass_nonneg T) hmle (by linarith : 0 ≤ p)) hc.le
dsimp only [fillingDeficit]
linarith
have hT0le : fillingDeficit c p T0 ≤ L := ge_of_tendsto hL
(Eventually.of_forall fun j => (u j).property.2.2)
exact ⟨⟨T,hT,hz,hT0le.trans hLle⟩,hdlim,hLle⟩
let x0 : C := ⟨T0,hT0,hz0,le_refl _⟩
obtain ⟨x,hx0,hx⟩ := exists_variational_point d
(fun x => by dsimp only [d]; rw [sub_self,fillingVolume_zero])
(fun x y => fillingVolume_nonneg _)
(fun x y => fillingVolume_sub_symm hX x.property.1 x.property.2.1 y.property.1 y.property.2.1)
(fun x y z => fillingVolume_triangle hX x.property.1 x.property.2.1
y.property.1 y.property.2.1 z.property.1 z.property.2.1)
F hbound hcluster x0 hδ
have hx0' : fillingDeficit c p T0 ≤ fillingDeficit c p x.val := hx0
refine ⟨x.val,x.property.1,x.property.2.1,?_,hx0',?_⟩
· by_contra hnot
have hm0 : mass x.val = 0 := le_antisymm (le_of_not_gt hnot) (mass_nonneg _)
have hl := hlin x
have hp0 : p ≠ 0 := by linarith
simp only [hm0,Real.zero_rpow hp0,mul_zero,fillingDeficit] at hl hx0'
linarith
· intro B hB hzB
have hF : fillingDeficit c p B ≤ fillingDeficit c p x.val+δ*fillingVolume (x.val-B) := by
by_cases hb : fillingDeficit c p T0 ≤ fillingDeficit c p B
· exact hx ⟨B,hB,hzB,hb⟩
· have hn := mul_nonneg hδ.le (fillingVolume_nonneg (x.val-B))
linarith [lt_of_not_ge hb]
have hV := (le_abs_self (fillingVolume x.val-fillingVolume B)).trans
(abs_fillingVolume_sub_le hX x.property.1 x.property.2.1 hB hzB)
have hh : c*((mass x.val)^p-(mass B)^p) ≤ (1+δ)*fillingVolume (x.val-B) := by
dsimp only [fillingDeficit] at hF
nlinarith
have hden : 0 < 1+δ := by linarith
rw [div_mul_eq_mul_div,div_le_iff₀ hden]
nlinarith
end CAT0Fillings
end
end OAI
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