| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Fillings.WeightedCone |
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| namespace OAI |
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| section |
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| open Set Filter MeasureTheory |
| open scoped Topology NNReal |
|
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| namespace CAT0Fillings |
| open Foundations Slicing CurrentOperations |
|
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| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
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| 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] |
| 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 |
|
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| 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)) |
|
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| 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) |
|
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| 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 |
|
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| end CAT0Fillings |
| end |
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| end OAI |
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