| import Mathlib |
| import OAI.Geometry.IntegralFillings.Charts.Extensions |
| import OAI.Geometry.IntegralFillings.Charts.CurrentAxioms |
|
|
| namespace OAI |
|
|
| section |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open Set MeasureTheory Filter |
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
|
|
| namespace SharpIntegralFillings |
| open Set MeasureTheory Filter |
| |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| (μ : Measure E) [Measure.IsAddHaarMeasure μ] |
| lemma LipschitzWith.ae_fderiv_zero_on_level {F : E → ℝ} {K : ℝ≥0} |
| (hF : LipschitzWith K F) (c : ℝ) : |
| ∀ᵐ x ∂μ, F x = c → fderiv ℝ F x = 0 := by |
| have hset : MeasurableSet {x | F x = c} := isClosed_eq hF.continuous continuous_const |>.measurableSet |
| apply (ae_restrict_iff' hset).mp |
| have heq : EqOn F (fun _ => c) {x | F x = c} := fun _ hx => hx |
| filter_upwards [ae_uniqueDiffWithinAt μ {x | F x = c},ae_restrict_mem hset, |
| (hF.ae_differentiableAt (μ := μ)).filter_mono (ae_mono Measure.restrict_le_self)] |
| with x hx hxc hd |
| rw [← fderivWithin_eq_fderiv hx hd, fderivWithin_congr' heq hxc] |
| simpa only [Function.const_def,Pi.zero_apply] using |
| congrFun (fderivWithin_const (𝕜 := ℝ) (E := E) (s := {x | F x = c}) c) x |
|
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| lemma ae_fderivWithin_zero_on_level {s : Set E} (hs : MeasurableSet s) |
| {f : E → ℝ} {K : ℝ≥0} (hf : LipschitzOnWith K f s) (c : ℝ) : |
| ∀ᵐ x ∂μ.restrict s, f x = c → fderivWithin ℝ f s x = 0 := by |
| obtain ⟨F,hF,heq⟩ := hf.extend_real |
| filter_upwards [ae_fderivWithin_eq_fderiv_extension μ hs hF heq, |
| (LipschitzWith.ae_fderiv_zero_on_level μ hF c).filter_mono |
| (ae_mono Measure.restrict_le_self),ae_restrict_mem hs] with x hx hz hxs |
| intro hfc |
| rw [hx] |
| exact hz ((heq hxs).symm.trans hfc) |
|
|
| end SharpIntegralFillings |
|
|
| namespace SharpIntegralFillings.IntegerChart |
| variable {X : Type*} [MetricSpace X] [Nonempty X] {k : ℕ} |
| (C : IntegerChart X (k+1)) |
|
|
| lemma boundary_action_locality (b : X → ℝ) (π : Fin k → X → ℝ) (h : Admissible b π) |
| (hloc : ∃ (i : Fin k) (U : Set X) (c : ℝ), IsOpen U ∧ Function.support b ⊆ U ∧ |
| ∀ x ∈ U, π i x = c) : C.action (fun _ => 1) (Matrix.vecCons b π) = 0 := by |
| obtain ⟨i,U,c,hU,hb,hπ⟩ := hloc |
| obtain ⟨L,J,hφ,_⟩ := C.bilipschitz |
| obtain ⟨K,hK⟩ := h.1.1 |
| have hadm : Admissible (fun _ : X => (1:ℝ)) (Matrix.vecCons b π) := |
| ⟨BoundedLip.const 1,fun j => Fin.cases h.1.1 (fun l => h.2 l) j⟩ |
| rw [action,ite_eq_left hadm] |
| apply integral_eq_zero_of_ae |
| filter_upwards [ae_restrict_mem C.borel, |
| ae_fderivWithin_zero_on_level volume C.borel (C.scalar_lipschitzOn hφ hK) 0] |
| with z hz hzero |
| have hdet : C.jacobian (Matrix.vecCons b π) z = 0 := by |
| by_cases hbz : C.scalar b z = 0 |
| · apply Matrix.det_eq_zero_of_row_eq_zero 0 |
| intro j |
| change fderivWithin ℝ (C.scalar b) C.domain z (EuclideanSpace.single j 1) = 0 |
| rw [hzero hbz]; rfl |
| · have hext : C.paramExtended z = C.param ⟨z,hz⟩ := by simp [paramExtended,hz] |
| have hzU : C.paramExtended z ∈ U := by |
| rw [hext] |
| apply hb |
| simpa [Function.mem_support,C.scalar_eq hz] using hbz |
| have hh := (C.continuousOn_paramExtended z hz).preimage_mem_nhdsWithin (hU.mem_nhds hzU) |
| have heq : C.scalar (π i) =ᶠ[𝓝[C.domain] z] fun _ => c := by |
| filter_upwards [hh,eventually_mem_nhdsWithin] with y hy hys |
| rw [C.scalar_eq hys] |
| have hye : C.paramExtended y = C.param ⟨y,hys⟩ := by simp [paramExtended,hys] |
| exact hπ _ (hye ▸ hy) |
| have heqz : C.scalar (π i) z = c := by |
| rw [C.scalar_eq hz] |
| exact hπ _ (hext ▸ hzU) |
| have hd0 : fderivWithin ℝ (C.scalar (π i)) C.domain z = 0 := |
| (heq.fderivWithin_eq (𝕜 := ℝ) heqz).trans (by |
| simpa only [Function.const_def,Pi.zero_apply] using |
| congrFun (fderivWithin_const (𝕜 := ℝ) (E := Euc (k+1)) (s := C.domain) c) z) |
| apply Matrix.det_eq_zero_of_row_eq_zero i.succ |
| intro j |
| change fderivWithin ℝ (C.scalar (π i)) C.domain z (EuclideanSpace.single j 1) = 0 |
| rw [hd0]; rfl |
| rw [hdet,mul_zero]; rfl |
|
|
| end SharpIntegralFillings.IntegerChart |
|
|
| namespace SharpIntegralFillings |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] {k : ℕ} |
|
|
| omit [BorelSpace X] in |
| lemma IntegerRectifiable.boundary_locality {T : Functional X (k+1)} |
| (hT : IntegerRectifiable T) (b : X → ℝ) (π : Fin k → X → ℝ) (hab : Admissible b π) |
| (hloc : ∃ (i : Fin k) (U : Set X) (c : ℝ), IsOpen U ∧ Function.support b ⊆ U ∧ |
| ∀ x ∈ U, π i x = c) : boundarySucc T b π = 0 := by |
| classical |
| rw [boundarySucc,ite_eq_left hab] |
| obtain ⟨C,hd,hC,hs,heq⟩ := hT |
| rw [heq] |
| suffices hall : ∀ i, (C i).action (fun _ => 1) (Matrix.vecCons b π) = 0 by |
| simp only [hall,tsum_zero] |
| intro i |
| rcases isEmpty_or_nonempty X with hX | hX |
| · have hb0 : (fun _ : X => (1:ℝ)) = fun _ => 0 := funext (fun x => isEmptyElim x) |
| rw [hb0] |
| simpa using (hC i).linearFirst (fun _ => 0) (fun _ => 0) (Matrix.vecCons b π) 0 0 |
| (BoundedLip.const 0) (BoundedLip.const 0) (fun j => Fin.cases hab.1.1 (fun l => hab.2 l) j) |
| · exact (C i).boundary_action_locality b π hab hloc |
|
|
| end SharpIntegralFillings |
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| end |
|
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| end OAI |
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