| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.CompactMetricInverseJets |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace YauCounterexamples |
| section |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Matrix |
| open scoped Topology Matrix.Norms.Elementwise |
| open Set Filter Function |
| open scoped Topology ContDiff Manifold |
| variable {E F M : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] |
| [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] [T2Space M] |
|
|
| omit [FiniteDimensional ℝ E] [T2Space M] in |
| lemma contDiffOn_chartTransition_for_jets (p q : M) : |
| ContDiffOn ℝ ∞ ((chartAt E p).symm.trans (chartAt E q)) |
| ((chartAt E p).symm.trans (chartAt E q)).source := by |
| intro y hy |
| have hy' : y ∈ (chartAt E p).target ∧ (chartAt E p).symm y ∈ (chartAt E q).source := hy |
| exact (contMDiffAt_iff_contDiffAt.mp |
| ((contMDiffAt_of_mem_maximalAtlas (IsManifold.chart_mem_maximalAtlas q) hy'.2).comp y |
| (contMDiffAt_symm_of_mem_maximalAtlas (IsManifold.chart_mem_maximalAtlas p) hy'.1))).contDiffWithinAt |
|
|
| theorem compact_chart_jet_transfer (p q : M) {L : Set M} (hL : IsCompact L) |
| (hLp : L ⊆ (chartAt E p).source) {K : Set E} (hK : IsCompact K) |
| (hKq : K ⊆ (chartAt E q).target) (m : ℕ) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (U : M → F) (f : E → F), ContDiff ℝ ∞ f → |
| tsupport U ⊆ L → (∀ z ∈ (chartAt E p).target, U ((chartAt E p).symm z)=f z) → |
| ∀ y ∈ K, ∀ j ≤ m, ∀ B : ℝ, 0 ≤ B → |
| (∀ i ≤ j, ‖iteratedFDeriv ℝ i f (chartAt E p ((chartAt E q).symm y))‖ ≤ B) → |
| ‖iteratedFDeriv ℝ j (U ∘ (chartAt E q).symm) y‖ ≤ C*B := by |
| let c := chartAt E p |
| let d := chartAt E q |
| let T := (d.symm '' K) ∩ L |
| have hT : IsCompact T := (hK.image_of_continuousOn (d.continuousOn_symm.mono hKq)).inter_right hL.isClosed |
| have hTd : T ⊆ d.source := by |
| rintro x ⟨⟨y,hy,rfl⟩,_⟩ |
| exact d.map_target (hKq hy) |
| let S := d '' T |
| have hS : IsCompact S := hT.image_of_continuousOn (d.continuousOn.mono hTd) |
| let e := d.symm.trans c |
| have hSe : S ⊆ e.source := by |
| rintro y ⟨x,hx,rfl⟩ |
| refine ⟨d.map_source (hTd hx),?_⟩ |
| change d.symm (d x) ∈ c.source |
| rw [d.left_inv (hTd hx)] |
| exact hLp hx.2 |
| obtain ⟨ψ,hψ,heψ⟩ := smooth_extension_near_compact hS e.open_source hSe |
| (contDiffOn_chartTransition_for_jets q p) |
| obtain ⟨D,hD,hDj⟩ := compact_raw_jet_bound isOpen_univ hK (subset_univ _) |
| hψ.contDiffOn m |
| let C : ℝ := (m.factorial : ℝ)*D^m+1 |
| have hCp : 0 < C := by dsimp [C]; positivity |
| refine ⟨C,hCp,?_⟩ |
| intro U f hf hUs hUf y hy j hj B hB hfj |
| by_cases hx : d.symm y ∈ L |
| · have hyS : y ∈ S := ⟨d.symm y,⟨⟨y,hy,rfl⟩,hx⟩,d.right_inv (hKq hy)⟩ |
| have hey := hSe hyS |
| have heψy : (e : E → E) =ᶠ[𝓝 y] ψ := heψ.filter_mono (nhds_le_nhdsSet hyS) |
| have heq : U ∘ d.symm =ᶠ[𝓝 y] f ∘ ψ := by |
| filter_upwards [e.open_source.mem_nhds hey,heψy] with z hz hzz |
| have hcz : d.symm z ∈ c.source := hz.2 |
| change U (d.symm z)=f (ψ z) |
| rw [←hzz] |
| change U (d.symm z)=f (c (d.symm z)) |
| rw [←hUf _ (c.map_source hcz),c.left_inv hcz] |
| rw [(heq.iteratedFDeriv ℝ j).eq_of_nhds] |
| have hb : ∀ i ≤ j, ‖iteratedFDeriv ℝ i f (ψ y)‖ ≤ B := by |
| intro i hi |
| rw [←heψy.eq_of_nhds] |
| exact hfj i hi |
| have hd : ∀ i, 1 ≤ i → i ≤ j → ‖iteratedFDeriv ℝ i ψ y‖ ≤ D^i := by |
| intro i hi hij |
| exact (hDj i (hij.trans hj) y hy).trans (by simpa using pow_le_pow_right₀ hD hi) |
| have hbound := norm_iteratedFDeriv_comp_le hf hψ |
| (le_of_lt (WithTop.coe_lt_coe.mpr (ENat.natCast_lt_top j))) y hb hd |
| apply hbound.trans |
| have hfact : (j.factorial : ℝ) ≤ m.factorial := by exact_mod_cast Nat.factorial_le hj |
| have hp := pow_le_pow_right₀ hD hj |
| have hmul := mul_le_mul hfact hp (pow_nonneg (zero_le_one.trans hD) _) (Nat.cast_nonneg _) |
| calc |
| _ = ((j.factorial : ℝ)*D^j)*B := by ring |
| _ ≤ ((m.factorial : ℝ)*D^m)*B := mul_le_mul_of_nonneg_right hmul hB |
| _ ≤ C*B := mul_le_mul_of_nonneg_right (by dsimp [C]; linarith) hB |
| · have hn : d.symm y ∉ tsupport U := fun h => hx (hUs h) |
| have he : U ∘ d.symm =ᶠ[𝓝 y] (fun _ => (0:F)) := |
| (notMem_tsupport_iff_eventuallyEq.mp hn).comp_tendsto (d.continuousAt_symm (hKq hy)).tendsto |
| rw [(he.iteratedFDeriv ℝ j).eq_of_nhds] |
| simpa using mul_nonneg hCp.le hB |
|
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|
|
| end |
|
|
| open Set Filter Function Manifold |
| open scoped Topology ContDiff |
| variable {E M : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] |
| [T2Space M] [CompactSpace M] |
| namespace CompactMetricAtlas |
| variable {g : SmoothMetric E M} {k : ℕ} {hs : Module.finrank ℝ E < 2*(2*(k:ℝ))} |
| (A : CompactMetricAtlas g k hs) |
|
|
| omit [T2Space M] [CompactSpace M] in |
| lemma chartJetBound_global_value {m : ℕ} {u : M → ℂ} {B : ℝ} |
| (hB : 0 ≤ B) (hj : A.ChartJetBound m u B) (x : M) : |
| ‖u x‖ ≤ (Fintype.card A.t:ℝ)*B := by |
| classical |
| have hb (i : A.t) : ‖A.η i x*u x‖ ≤ B := by |
| by_cases hx : x ∈ tsupport (A.η i) |
| · have hxs := A.chart_η i hx |
| have h := hj i 0 (Nat.zero_le _) (chartAt E (A.p i) x) |
| simpa only [norm_iteratedFDeriv_zero,chartLocalize_eq _ _ _ ((chartAt E (A.p i)).map_source hxs), |
| (chartAt E (A.p i)).left_inv hxs] using h |
| · rw [image_eq_zero_of_notMem_tsupport hx,zero_mul,norm_zero] |
| exact hB |
| calc |
| ‖u x‖ = ‖∑ i : A.t, A.η i x*u x‖ := by rw [←Finset.sum_mul,A.sum_η,one_mul] |
| _ ≤ ∑ i : A.t, ‖A.η i x*u x‖ := norm_sum_le _ _ |
| _ ≤ ∑ _i : A.t, B := Finset.sum_le_sum (fun i _ => hb i) |
| _ = _ := by simp |
|
|
| theorem chartJetBound_controls_compact_raw (q : M) {K : Set E} |
| (hK : IsCompact K) (hKt : K ⊆ (chartAt E q).target) (m : ℕ) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (u : M → ℂ), ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℂ) ∞ u → |
| ∀ B : ℝ, 0 ≤ B → A.ChartJetBound m u B → |
| ∀ y ∈ K, ∀ j ≤ m, ‖iteratedFDeriv ℝ j (u ∘ (chartAt E q).symm) y‖ ≤ C*B := by |
| classical |
| choose Cs hCs htrans using fun i : A.t => compact_chart_jet_transfer (F := ℂ) (A.p i) q |
| (HasCompactSupport.of_compactSpace (A.η i)) (A.chart_η i) hK hKt m |
| let C : ℝ := 1+∑ i : A.t, Cs i |
| have hC : 0 < C := by |
| have hsum : 0 ≤ ∑ i : A.t, Cs i := Finset.sum_nonneg (fun i _ => (hCs i).le) |
| exact add_pos_of_pos_of_nonneg zero_lt_one hsum |
| refine ⟨C,hC,?_⟩ |
| intro u hu B hB hjet y hy j hj |
| let U : A.t → M → ℂ := fun i x => A.η i x*u x |
| have hU (i : A.t) : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℂ) ∞ (U i) := contMDiff_complex_mul (A.smooth_η i) hu |
| have hsum : u ∘ (chartAt E q).symm=fun z => ∑ i : A.t, U i ((chartAt E q).symm z) := by |
| funext z |
| simp only [U,←Finset.sum_mul,A.sum_η,one_mul,Function.comp_apply] |
| have hdiff (i : A.t) : ContDiffAt ℝ j (U i ∘ (chartAt E q).symm) y := |
| (contMDiffAt_iff_contDiffAt.mp ((hU i).contMDiffAt.comp y |
| (contMDiffAt_symm_of_mem_maximalAtlas (IsManifold.chart_mem_maximalAtlas q) (hKt hy)))).of_le |
| (le_of_lt (WithTop.coe_lt_coe.mpr (ENat.natCast_lt_top j))) |
| have hbound (i : A.t) : |
| ‖iteratedFDeriv ℝ j (U i ∘ (chartAt E q).symm) y‖ ≤ Cs i*B := by |
| apply htrans i (U i) (chartLocalize (E := E) (A.p i) (A.η i) u) |
| (contDiff_chartLocalize _ _ _ (HasCompactSupport.of_compactSpace _) (A.chart_η i) (A.smooth_η i) hu) |
| tsupport_mul_subset_left |
| (fun z hz => (chartLocalize_eq _ _ _ hz).symm) y hy j hj B hB |
| intro l hl |
| exact hjet i l (hl.trans hj) _ |
| rw [hsum] |
| have hej := iteratedFDeriv_fun_sum_apply (u := Finset.univ) (fun i _ => hdiff i) |
| simp only [Function.comp_apply] at hej |
| rw [hej] |
| calc |
| _ ≤ ∑ i : A.t, ‖iteratedFDeriv ℝ j (U i ∘ (chartAt E q).symm) y‖ := norm_sum_le _ _ |
| _ ≤ ∑ i : A.t, Cs i*B := Finset.sum_le_sum (fun i _ => hbound i) |
| _ = (∑ i : A.t, Cs i)*B := (Finset.sum_mul ..).symm |
| _ ≤ C*B := mul_le_mul_of_nonneg_right (by change (∑ i : A.t, Cs i) ≤ 1+∑ i : A.t, Cs i; linarith) hB |
|
|
| theorem chartJetBound_controls_real_compact_raw (q : M) {K : Set E} |
| (hK : IsCompact K) (hKt : K ⊆ (chartAt E q).target) (m : ℕ) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (u : M → ℝ), ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u → |
| ∀ B : ℝ, 0 ≤ B → A.ChartJetBound m (fun x => (u x:ℂ)) B → |
| ∀ y ∈ K, ∀ j ≤ m, ‖iteratedFDeriv ℝ j (u ∘ (chartAt E q).symm) y‖ ≤ C*B := by |
| obtain ⟨C,hC,hj⟩ := A.chartJetBound_controls_compact_raw q hK hKt m |
| refine ⟨C,hC,?_⟩ |
| intro u hu B hB hjet y hy j hjm |
| have hu' : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℂ) ∞ (fun x => (u x:ℂ)) := |
| Complex.ofRealCLM.contMDiff.comp hu |
| have hh := hj _ hu' B hB hjet y hy j hjm |
| have hd : ContDiffAt ℝ ∞ (u ∘ (chartAt E q).symm) y := |
| contMDiffAt_iff_contDiffAt.mp (hu.contMDiffAt.comp y |
| (contMDiffAt_symm_of_mem_maximalAtlas (IsManifold.chart_mem_maximalAtlas q) (hKt hy))) |
| change ‖iteratedFDeriv ℝ j (Complex.ofRealLI ∘ (u ∘ (chartAt E q).symm)) y‖ ≤ C*B at hh |
| rw [Complex.ofRealLI.norm_iteratedFDeriv_comp_left hd |
| (le_of_lt (WithTop.coe_lt_coe.mpr (ENat.natCast_lt_top j)))] at hh |
| exact hh |
|
|
| theorem chartJetBound_controls_finite_real_raw (tests : List (CoordinateTest E M)) (m : ℕ) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (u : M → ℝ), ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u → |
| ∀ B : ℝ, 0 ≤ B → A.ChartJetBound m (fun x => (u x:ℂ)) B → |
| ∀ t ∈ tests, ∀ y ∈ t.compactSet, ∀ j ≤ m, |
| ‖iteratedFDeriv ℝ j (u ∘ (chartAt E t.center).symm) y‖ ≤ C*B := by |
| classical |
| choose Cs hCs hJ using fun t : CoordinateTest E M => |
| A.chartJetBound_controls_real_compact_raw t.center t.isCompact t.inTarget m |
| let C := 1+∑ t ∈ tests.toFinset, Cs t |
| have hC : 0 < C := by |
| have hh : 0 ≤ ∑ t ∈ tests.toFinset, Cs t := Finset.sum_nonneg (fun t _ => (hCs t).le) |
| dsimp [C]; linarith |
| refine ⟨C,hC,?_⟩ |
| intro u hu B hB hjet t ht y hy j hj |
| apply (hJ t u hu B hB hjet y hy j hj).trans |
| apply mul_le_mul_of_nonneg_right _ hB |
| have hh := Finset.single_le_sum (fun t (_ : t ∈ tests.toFinset) => (hCs t).le) |
| (List.mem_toFinset.mpr ht) |
| dsimp [C]; linarith |
| end CompactMetricAtlas |
|
|
| end YauCounterexamples |
| end |
|
|
| end OAI |
|
|
