| import OAI.Analysis.CircleDomains.Sobolev.LipschitzLengthFiberMajorant |
| import OAI.Analysis.CircleDomains.Sobolev.LocalDerivativeCover |
| import Mathlib.MeasureTheory.Measure.Lebesgue.Complex |
|
|
| namespace OAI |
|
|
| /-! Critical points of a C1 scalar planar map contribute zero fiber length |
| for almost every level. No assertion that C1 critical values have measure |
| zero is used. -/ |
| open Set Function Filter MeasureTheory Metric |
| open scoped ENNReal NNReal Topology Classical |
| noncomputable section |
| namespace CircleDomainRigidity |
|
|
| lemma critical_fiber_majorant |
| {D : Set ℂ} (hD : IsOpen D) {f : ℂ → ℝ} (hf : ContDiffOn ℝ 1 f D) |
| {A : Set ℂ} (hA : MeasurableSet A) (hAD : A ⊆ D) |
| (hAfin : (μH[2] : Measure ℂ) A < ∞) |
| (hcrit : ∀ x ∈ A, fderiv ℝ f x = 0) (ε : ℝ≥0) (hε : 0 < ε) : |
| ∃ G : ℝ → ℝ≥0∞, Measurable G ∧ |
| (∀ y, (μH[1] : Measure ℂ) (A ∩ f ⁻¹' {y}) ≤ G y) ∧ |
| (∫⁻ y, G y) ≤ (ε : ℝ≥0∞) * (μH[2] : Measure ℂ) A := by |
| obtain ⟨U, L, hU, hcover, hUD, hLip, hnorm⟩ := |
| exists_countable_derivative_cover hD hf (show (0 : ℝ) < ε from hε) |
| let P : ℕ → Set ℂ := fun n => A ∩ disjointed U n |
| have hP : ∀ n, MeasurableSet (P n) := fun n => |
| hA.inter (MeasurableSet.disjointed (fun j => (hU j).measurableSet) n) |
| have hPA : ∀ n, P n ⊆ A := fun _ => inter_subset_left |
| have hPU : ∀ n, P n ⊆ U n := fun n => inter_subset_right.trans (disjointed_subset U n) |
| have hdj : Pairwise (Disjoint on P) := fun i j hij => |
| (disjoint_disjointed U hij).mono inter_subset_right inter_subset_right |
| have heqA : (⋃ n, P n) = A := by |
| dsimp only [P] |
| rw [← inter_iUnion, iUnion_disjointed, inter_eq_left.mpr (hAD.trans hcover)] |
| have hLP : ∀ n, LipschitzOnWith ε f (P n) := by |
| intro n |
| apply lipschitzOnWith_iff_dist_le_mul.mpr |
| intro a ha b hb |
| have hL : (L n : ℝ) ≤ ε := by |
| have h := hnorm n a (hPU n ha) |
| simpa only [hcrit a (hPA n ha), norm_zero, zero_add] using h |
| exact ((hLip n).dist_le_mul a (hPU n ha) b (hPU n hb)).trans |
| (mul_le_mul_of_nonneg_right hL (dist_nonneg)) |
| choose g hg heq using fun n => (hLP n).extend_real |
| have hfin : ∀ n, (μH[2] : Measure ℂ) (P n) < ∞ := fun n => |
| (measure_mono (hPA n)).trans_lt hAfin |
| choose G hG hbound hint using fun n => |
| exists_exact_lipschitz_length_fiber_majorant (P n) (hfin n) ε (g n) (hg n) |
| have heqfiber : ∀ n y, P n ∩ (g n) ⁻¹' {y} = P n ∩ f ⁻¹' {y} := by |
| intro n y |
| ext x |
| by_cases hx : x ∈ P n |
| · simp only [mem_inter_iff, hx, true_and, mem_preimage, mem_singleton_iff, ← heq n hx] |
| · simp only [mem_inter_iff, hx, false_and] |
| refine ⟨fun y => ∑' n, G n y, Measurable.tsum hG, ?_, ?_⟩ |
| · intro y |
| calc |
| (μH[1] : Measure ℂ) (A ∩ f ⁻¹' {y}) = |
| (μH[1] : Measure ℂ) (⋃ n, P n ∩ f ⁻¹' {y}) := by rw [← iUnion_inter, heqA] |
| _ ≤ ∑' n, (μH[1] : Measure ℂ) (P n ∩ f ⁻¹' {y}) := measure_iUnion_le _ |
| _ ≤ ∑' n, G n y := ENNReal.tsum_le_tsum (fun n => by rw [← heqfiber]; exact hbound n y) |
| · rw [lintegral_tsum (fun n => (hG n).aemeasurable)] |
| calc |
| (∑' n, ∫⁻ y, G n y) ≤ ∑' n, (ε : ℝ≥0∞) * (μH[2] : Measure ℂ) (P n) := |
| ENNReal.tsum_le_tsum hint |
| _ = (ε : ℝ≥0∞) * (μH[2] : Measure ℂ) A := by |
| rw [ENNReal.tsum_mul_left, ← measure_iUnion hdj hP, heqA] |
|
|
| lemma ae_zero_critical_fiber_length |
| {D : Set ℂ} (hD : IsOpen D) {f : ℂ → ℝ} (hf : ContDiffOn ℝ 1 f D) |
| {A : Set ℂ} (hA : MeasurableSet A) (hAD : A ⊆ D) |
| (hAfin : (μH[2] : Measure ℂ) A < ∞) |
| (hcrit : ∀ x ∈ A, fderiv ℝ f x = 0) : |
| ∀ᵐ y, (μH[1] : Measure ℂ) (A ∩ f ⁻¹' {y}) = 0 := by |
| let ε : ℕ → ℝ≥0 := fun n => ⟨1 / (n + 1 : ℝ), by positivity⟩ |
| have hε : ∀ n, 0 < ε n := fun n => by change (0 : ℝ) < 1 / (n + 1 : ℝ); positivity |
| choose G hG hbound hint using fun n => critical_fiber_majorant hD hf hA hAD hAfin hcrit (ε n) (hε n) |
| let H : ℝ → ℝ≥0∞ := fun y => ⨅ n, G n y |
| have hH : Measurable H := Measurable.iInf hG |
| have hεlim : Tendsto (fun n => (ε n : ℝ≥0∞)) atTop (𝓝 0) := by |
| have h := ENNReal.continuous_ofReal.continuousAt.tendsto.comp |
| (tendsto_one_div_add_atTop_nhds_zero_nat (𝕜 := ℝ)) |
| have heq : (fun n => (ε n : ℝ≥0∞)) = fun n : ℕ => ENNReal.ofReal (1 / (n + 1 : ℝ)) := by |
| funext n |
| exact (ENNReal.ofReal_eq_coe_nnreal (by positivity)).symm |
| rw [heq] |
| simpa only [ENNReal.ofReal_zero, Function.comp_def] using h |
| have hintH : (∫⁻ y, H y) = 0 := by |
| apply le_antisymm _ bot_le |
| have ht := ENNReal.Tendsto.mul_const hεlim (Or.inr hAfin.ne) |
| simp only [zero_mul] at ht |
| apply ge_of_tendsto' ht |
| intro n |
| exact (lintegral_mono (fun y => iInf_le (fun k => G k y) n)).trans (hint n) |
| filter_upwards [(lintegral_eq_zero_iff hH).mp hintH] with y hy |
| apply le_antisymm _ bot_le |
| change H y = 0 at hy |
| exact (le_iInf (fun n => hbound n y)).trans_eq hy |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
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|
