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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 | import Mathlib
import OAI.Geometry.CAT0Fillings.Fillings.WeightedCone
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology NNReal
namespace CAT0Fillings
open Foundations Slicing CurrentOperations
variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
[CompactSpace X] [Nonempty X]
theorem fillingVolume_le_of_bounded_weak_limit (hX : IsCAT0 X) {k : ℕ}
{Ts : ℕ → Functional X (k+1)} {T : Functional X (k+1)}
(hTs : ∀ j, IsIntegral (k+1) (Ts j)) (hz : ∀ j, boundarySucc (Ts j) = 0)
(M B : ℝ≥0) (hM : ∀ j, mass (Ts j) ≤ M) (hB : ∀ j, fillingVolume (Ts j) ≤ B)
(hlim : ∀ b π, Tendsto (fun j => Ts j b π) atTop (𝓝 (T b π))) :
fillingVolume T ≤ B := by
choose S hS hb hm using fun j => hX.exists_minimal_integral_filling (hTs j) (hz j)
obtain ⟨ψ,hψ,U,hU,hUm,_,hweak⟩ := exists_integral_weak_subsequence hS B M
(fun j => by rw [hm]; exact hB j) (fun j => by rw [hb]; exact hM j) hX
have hboundary : boundarySucc U = T := by
funext b π
apply tendsto_nhds_unique (boundarySucc_weak_limit hweak b π)
simpa only [hb,Function.comp_def] using (hlim b π).comp hψ.tendsto_atTop
exact (fillingVolume_le hU hboundary).trans hUm
theorem fillingVolume_le_of_eventually_bounded_weak_limit (hX : IsCAT0 X) {k : ℕ}
{Ts : ℕ → Functional X (k+1)} {T : Functional X (k+1)}
(hTs : ∀ j, IsIntegral (k+1) (Ts j)) (hz : ∀ j, boundarySucc (Ts j) = 0)
(M B : ℝ≥0) (hM : ∀ j, mass (Ts j) ≤ M)
(hB : ∀ᶠ j in atTop, fillingVolume (Ts j) ≤ B)
(hlim : ∀ b π, Tendsto (fun j => Ts j b π) atTop (𝓝 (T b π))) :
fillingVolume T ≤ B := by
obtain ⟨N,hN⟩ := eventually_atTop.mp hB
exact fillingVolume_le_of_bounded_weak_limit hX (fun j => hTs (j+N))
(fun j => hz (j+N)) M B (fun j => hM (j+N))
(fun j => hN (j+N) (Nat.le_add_left N j))
(fun b π => (hlim b π).comp (tendsto_add_atTop_nat N))
theorem fillingVolume_sub_tendsto_zero (hX : IsCAT0 X) {k : ℕ}
{Ts : ℕ → Functional X (k+1)} {T : Functional X (k+1)}
(hTs : ∀ j, IsIntegral (k+1) (Ts j)) (hz : ∀ j, boundarySucc (Ts j) = 0)
(M : ℝ≥0) (hM : ∀ j, mass (Ts j) ≤ M)
(hcauchy : ∀ ε > 0, ∃ N : ℕ, ∀ i ≥ N, ∀ j ≥ N,
fillingVolume (Ts i-Ts j) < ε)
{ψ : ℕ → ℕ} (hψ : StrictMono ψ)
(hlim : ∀ b π, Tendsto (fun j => Ts (ψ j) b π) atTop (𝓝 (T b π))) :
Tendsto (fun j => fillingVolume (Ts j-T)) atTop (𝓝 0) := by
apply tendsto_order.mpr
constructor
· intro a ha
exact Eventually.of_forall fun j => ha.trans_le (fillingVolume_nonneg _)
· intro ε hε
obtain ⟨N,hN⟩ := hcauchy (ε/2) (half_pos hε)
apply eventually_atTop.mpr
refine ⟨N,fun i hi => ?_⟩
have hb : fillingVolume (Ts i-T) ≤ ε/2 := by
apply fillingVolume_le_of_eventually_bounded_weak_limit hX
(fun j => (hTs i).sub (hTs (ψ j)))
(fun j => by rw [boundarySucc_sub,hz,hz,sub_self])
(M+M) (⟨ε/2,(half_pos hε).le⟩ : ℝ≥0)
· intro j
exact (mass_sub_le (hTs i).1 (hTs (ψ j)).1).trans (add_le_add (hM i) (hM (ψ j)))
· filter_upwards [hψ.tendsto_atTop.eventually (eventually_ge_atTop N)] with j hj
exact (hN i hi (ψ j) hj).le
· intro b π
exact tendsto_const_nhds.sub (hlim b π)
exact hb.trans_lt (half_lt_self hε)
lemma fillingVolume_tendsto_of_sub_tendsto_zero (hX : IsCAT0 X) {k : ℕ}
{Ts : ℕ → Functional X (k+1)} {T : Functional X (k+1)}
(hTs : ∀ j, IsIntegral (k+1) (Ts j)) (hzs : ∀ j, boundarySucc (Ts j) = 0)
(hT : IsIntegral (k+1) T) (hz : boundarySucc T = 0)
(hlim : Tendsto (fun j => fillingVolume (Ts j-T)) atTop (𝓝 0)) :
Tendsto (fun j => fillingVolume (Ts j)) atTop (𝓝 (fillingVolume T)) := by
apply tendsto_iff_norm_sub_tendsto_zero.mpr
exact squeeze_zero (fun _ => norm_nonneg _) (fun j => by
simpa only [Real.norm_eq_abs] using abs_fillingVolume_sub_le hX (hTs j) (hzs j) hT hz) hlim
end CAT0Fillings
end
end OAI
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