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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 | import Mathlib
import OAI.Geometry.CAT0Fillings.Currents.PushBoundary
import OAI.Geometry.CAT0Fillings.Charts.Postcompose
namespace OAI
section
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
open Filter Set
open scoped Topology NNReal
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
open Set MeasureTheory
open scoped RealInnerProductSpace
open Matrix
open scoped RealInnerProductSpace MatrixOrder
namespace CAT0Fillings
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 CAT0Fillings
end
end
end OAI
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