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59d1c80 | 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 | import Mathlib
import OAI.Geometry.IntegralFillings.Geometry.Comparison
import OAI.Geometry.IntegralFillings.Currents.PushBoundary
import OAI.Geometry.IntegralFillings.Charts.Postcompose
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace SharpIntegralFillings
open Set MeasureTheory
open CurrentOperations
variable {X Y : Type*} [MetricSpace X] [MetricSpace Y]
[MeasurableSpace X] [MeasurableSpace Y] [BorelSpace X] [BorelSpace Y]
[CompactSpace X]
lemma integerRectifiable_push_of_bilipschitz {k : ℕ} {T : Functional X k}
(hT : IntegerRectifiable T) {f : X → Y} {K J : ℝ≥0}
(hf : LipschitzWith K f) (hfJ : AntilipschitzWith J f) :
IntegerRectifiable (pushCurrent f T) := by
classical
obtain ⟨C,hdis,hC,hm,heq⟩ := hT
choose L U hL hU using fun i => (C i).bilipschitz
let D : ℕ → IntegerChart Y k := fun i => (C i).postcompose hf (hfJ.comp (hU i))
have hDaction (i : ℕ) : (D i).action = pushCurrent f (C i).action :=
(C i).postcompose_action hf (hfJ.comp (hU i))
have hD (i : ℕ) : IsMetricCurrent (D i).action := by
rw [hDaction]
exact pushCurrent_isMetricCurrent (hC i) hf
refine ⟨D,?_,hD,?_,?_⟩
· intro i j hij
change Disjoint ((C i).postcompose hf _).image ((C j).postcompose hf _).image
rw [(C i).postcompose_image,(C j).postcompose_image]
exact (Set.disjoint_image_iff hfJ.injective).mpr (hdis hij)
· apply (hm.mul_left ((K : ℝ)^k)).of_nonneg_of_le
· intro i
exact mass_nonneg (D i).action
· intro i
rw [hDaction]
exact mass_pushCurrent_le (hC i) hf
· intro b π
by_cases h : Admissible b π
· rw [pushCurrent_apply f T h,heq]
exact tsum_congr (fun i => by rw [hDaction,pushCurrent_apply f (C i).action h])
· rw [show pushCurrent f T b π = 0 from ite_eq_right h]
symm
calc
(∑' i, (D i).action b π) = ∑' (_i : ℕ), (0 : ℝ) :=
tsum_congr (fun i => (hD i).offDomain b π h)
_ = 0 := tsum_zero
lemma integral_push_of_bilipschitz {k : ℕ} {T : Functional X k}
(hT : IsIntegral k T) {f : X → Y} {K J : ℝ≥0}
(hf : LipschitzWith K f) (hfJ : AntilipschitzWith J f) :
IsIntegral k (pushCurrent f T) := by
cases k with
| zero => exact ⟨pushCurrent_isMetricCurrent hT.1 hf,
integerRectifiable_push_of_bilipschitz hT.2 hf hfJ⟩
| succ k =>
refine ⟨pushCurrent_isMetricCurrent hT.1 hf,
integerRectifiable_push_of_bilipschitz hT.2.1 hf hfJ,?_,?_⟩
· rw [pushCurrent_boundarySucc T hf]
exact pushCurrent_isMetricCurrent hT.2.2.1 hf
· rw [pushCurrent_boundarySucc T hf]
exact integerRectifiable_push_of_bilipschitz hT.2.2.2 hf hfJ
end SharpIntegralFillings
namespace SharpIntegralFillings
universe u
theorem IsCAT0.closedBall {X : Type u} [MetricSpace X] (hX : IsCAT0 X)
(o : X) {R : ℝ} (hR : 0 ≤ R) : IsCAT0 (Metric.closedBall o R) := by
obtain ⟨s, hs, hd, hc⟩ := hX
have hconv (x y : Metric.closedBall o R) (t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) :
s x.val y.val t ∈ Metric.closedBall o R := by
have hh := hc x.val y.val o t 1 ht (by simp)
rw [(hs x.val o).2] at hh
have he : (t * dist x.val y.val - 1 * dist x.val o) ^ 2 +
t * 1 * (dist y.val o ^ 2 - (dist x.val y.val - dist x.val o) ^ 2) =
(1 - t) * dist x.val o ^ 2 + t * dist y.val o ^ 2 -
t * (1 - t) * dist x.val y.val ^ 2 := by ring
rw [he] at hh
have hx := (sq_le_sq₀ dist_nonneg hR).mpr x.property
have hy := (sq_le_sq₀ dist_nonneg hR).mpr y.property
have hx' := mul_le_mul_of_nonneg_left hx (sub_nonneg.mpr ht.2)
have hy' := mul_le_mul_of_nonneg_left hy ht.1
have hh' : dist (s x.val y.val t) o ^ 2 ≤ R ^ 2 := by
nlinarith [mul_nonneg (mul_nonneg ht.1 (sub_nonneg.mpr ht.2))
(sq_nonneg (dist x.val y.val))]
exact (sq_le_sq₀ dist_nonneg hR).mp hh'
let clamp : ℝ → ℝ := fun t => max 0 (min 1 t)
have hclamp (t : ℝ) : clamp t ∈ Icc (0 : ℝ) 1 := by
exact ⟨le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩
have hclamp_eq (t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) : clamp t = t := by
simp only [clamp, min_eq_right ht.2, max_eq_right ht.1]
let sB : Metric.closedBall o R → Metric.closedBall o R → ℝ →
Metric.closedBall o R := fun x y t => ⟨s x.val y.val (clamp t), hconv x y _ (hclamp t)⟩
refine ⟨sB, ?_, ?_, ?_⟩
· intro x y
constructor <;> apply Subtype.ext
· change s x.val y.val (clamp 0) = x.val
rw [hclamp_eq _ (by simp), (hs x.val y.val).1]
· change s x.val y.val (clamp 1) = y.val
rw [hclamp_eq _ (by simp), (hs x.val y.val).2]
· intro x y u v hu hv
change dist (s x.val y.val (clamp u)) (s x.val y.val (clamp v)) = _
rw [hclamp_eq _ hu, hclamp_eq _ hv]
exact hd x.val y.val u v hu hv
· intro p x y u v hu hv
change dist (s p.val x.val (clamp u)) (s p.val y.val (clamp v)) ^ 2 ≤ _
rw [hclamp_eq _ hu, hclamp_eq _ hv]
exact hc p.val x.val y.val u v hu hv
end SharpIntegralFillings
end
end OAI
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