| import Mathlib |
| import OAI.Geometry.SmoothYau.Estimates.RectangularFactorization |
| import OAI.Geometry.SmoothYau.Smoothness.ChartJetBoundGlobalValue |
|
|
| 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 Manifold |
| open scoped Topology ContDiff |
| variable {E M : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
|
|
| lemma laplaceBeltrami_congr_nhds (g : SmoothMetric E M) {u v : M → ℝ} {x : M} |
| (hu : u =ᶠ[𝓝 x] v) : laplaceBeltrami g u x=laplaceBeltrami g v x := by |
| let c := chartAt E x |
| have hx := c.map_source (mem_chart_source E x) |
| have ht : Tendsto c.symm (𝓝 (c x)) (𝓝 x) := by |
| simpa only [c.left_inv (mem_chart_source E x)] using (c.continuousAt_symm hx).tendsto |
| have hu' := (hu.comp_tendsto ht).fderiv (𝕜:=ℝ) |
| have he (i : CoordIndex E) : |
| (fun y => Real.sqrt (metricCoefficients g x y).det * ∑ j, |
| (metricCoefficients g x y)⁻¹ i j * fderiv ℝ (u ∘ c.symm) y (Module.finBasis ℝ E j)) |
| =ᶠ[𝓝 (c x)] |
| (fun y => Real.sqrt (metricCoefficients g x y).det * ∑ j, |
| (metricCoefficients g x y)⁻¹ i j * fderiv ℝ (v ∘ c.symm) y (Module.finBasis ℝ E j)) := by |
| filter_upwards [hu'] with y hy |
| rw [hy] |
| unfold laplaceBeltrami |
| change (Real.sqrt (metricCoefficients g x (c x)).det)⁻¹ * ∑ i, _ = _ |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact congrArg (fun T : E →L[ℝ] ℝ => T (Module.finBasis ℝ E i)) (he i).fderiv_eq |
|
|
| lemma laplaceBeltrami_tsupport (g : SmoothMetric E M) (u : M → ℝ) : |
| tsupport (laplaceBeltrami g u) ⊆ tsupport u := by |
| apply closure_minimal _ isClosed_closure |
| intro x hx |
| by_contra hn |
| have he := laplaceBeltrami_congr_nhds g (notMem_tsupport_iff_eventuallyEq.mp hn) |
| have hz : laplaceBeltrami g (fun _ : M => (0:ℝ)) x=0 := by |
| simp [laplaceBeltrami,Function.comp_def] |
| exact hx (he.trans hz) |
|
|
| lemma laplaceResidual_tsupport (g : SmoothMetric E M) (u : M → ℝ) (Λ : ℝ) : |
| tsupport (fun x => laplaceBeltrami g u x+Λ*u x) ⊆ tsupport u := by |
| apply (tsupport_add _ _).trans |
| apply union_subset (laplaceBeltrami_tsupport g u) |
| exact tsupport_mul_subset_right (f := fun _ : M => Λ) (g := u) |
|
|
| variable [T2Space M] |
|
|
| theorem compact_chart_weighted_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 → ℝ) (f : E → ℝ), ContDiff ℝ ∞ f → |
| tsupport U ⊆ L → (∀ z ∈ (chartAt E p).target, U ((chartAt E p).symm z)=f z) → |
| ∀ (B : ℕ → M → ℝ), (∀ j x, 0 ≤ B j x) → |
| (∀ x ∈ L, ∀ j ≤ m, ∀ i ≤ j, |
| ‖iteratedFDeriv ℝ i f (chartAt E p x)‖ ≤ B j x) → |
| ∀ y ∈ K, ∀ j ≤ m, |
| ‖iteratedFDeriv ℝ j (U ∘ (chartAt E q).symm) y‖ ≤ C*B j ((chartAt E q).symm y) := by |
| obtain ⟨C,hC,hT⟩ := compact_chart_jet_transfer (F := ℝ) p q hL hLp hK hKq m |
| refine ⟨C,hC,?_⟩ |
| intro U f hf hs he B hB hb y hy j hj |
| by_cases hx : (chartAt E q).symm y ∈ L |
| · exact hT U f hf hs he y hy j hj _ (hB j _) (fun i hi => hb _ hx j hj i hi) |
| · have hn : (chartAt E q).symm y ∉ tsupport U := fun h => hx (hs h) |
| have hzero : U ∘ (chartAt E q).symm =ᶠ[𝓝 y] (fun _ => (0:ℝ)) := |
| (notMem_tsupport_iff_eventuallyEq.mp hn).comp_tendsto |
| ((chartAt E q).continuousAt_symm (hKq hy)).tendsto |
| rw [(hzero.iteratedFDeriv ℝ j).eq_of_nhds] |
| simpa using mul_nonneg hC.le (hB j ((chartAt E q).symm y)) |
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|
|
| end |
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|
|
| section |
| open Set Filter Function Manifold Metric |
| open scoped Topology ContDiff |
|
|
| lemma pinning_factor_squared (F : ℝ → ℝ) {I : Set ℝ} (hIo : IsOpen I) |
| (hIc : IsPreconnected I) (hF : ContDiffOn ℝ 2 F I) |
| {n : ℕ} (hn : 2 ≤ n) {r₀ r₁ : ℝ} (hr₀ : 0 < r₀) (hr : r₀ < r₁) |
| (he : ∀ s ∈ I, ∀ r ∈ ({r₀,r₁} : Set ℝ), |
| (n : ℝ)^2 * deriv (deriv F) s * (r^(n-1) - s^2) + |
| ((n : ℝ)*((n : ℝ)+2)) * (F s - s * deriv F s) = 0) : |
| ∃ c : ℝ, ∀ s ∈ I, F s = c * s := by |
| have hn₀ : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega) |
| have hp : r₀^(n-1) < r₁^(n-1) := pow_lt_pow_left₀ hr hr₀.le (by omega) |
| have hsecond (s : ℝ) (hs : s ∈ I) : deriv (deriv F) s = 0 := by |
| have h0 := he s hs r₀ (by simp) |
| have h1 := he s hs r₁ (by simp) |
| have hx : (n : ℝ)^2 * (r₁^(n-1)-r₀^(n-1)) ≠ 0 := |
| (mul_pos (sq_pos_of_pos hn₀) (sub_pos.mpr hp)).ne' |
| have hz : ((n : ℝ)^2 * (r₁^(n-1)-r₀^(n-1))) * deriv (deriv F) s = 0 := by |
| nlinarith [h0,h1] |
| exact (mul_eq_zero.mp hz).resolve_left hx |
| have hd : DifferentiableOn ℝ (deriv F) I := |
| (hF.deriv_of_isOpen (m := 1) hIo (by norm_num)).differentiableOn (by norm_num) |
| obtain ⟨c,hc⟩ := hIo.exists_is_const_of_deriv_eq_zero hIc hd (fun s hs => hsecond s hs) |
| refine ⟨c,fun s hs => ?_⟩ |
| have h0 := he s hs r₀ (by simp) |
| rw [hsecond s hs,hc s hs] at h0 |
| have hlambda : (n : ℝ)*((n : ℝ)+2) ≠ 0 := (mul_pos hn₀ (by linarith)).ne' |
| have hz : F s - s * c = 0 := (mul_eq_zero.mp (by simpa using h0)).resolve_left hlambda |
| linarith |
| variable {E M : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
| omit [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] |
| [IsManifold 𝓘(ℝ,E) ∞ M] in |
| lemma chart_germ_of_pullback {u v : M → ℝ} (p : M) {y : E} |
| (hy : y ∈ (chartAt E p).target) |
| (he : (u ∘ (chartAt E p).symm) =ᶠ[𝓝 y] (v ∘ (chartAt E p).symm)) : |
| u =ᶠ[𝓝 ((chartAt E p).symm y)] v := by |
| let c := chartAt E p |
| have hs : c.symm y ∈ c.source := c.map_target hy |
| have ht : Tendsto c (𝓝 (c.symm y)) (𝓝 y) := by |
| simpa only [c.right_inv hy] using (c.continuousAt hs).tendsto |
| filter_upwards [he.comp_tendsto ht,c.open_source.mem_nhds hs] with z hz hzs |
| change u (c.symm (c z)) = v (c.symm (c z)) at hz |
| simpa only [c.left_inv hzs] using hz |
|
|
| theorem local_eigen_proportional_of_annular_rank |
| (g : SmoothMetric E M) {u v R : M → ℝ} (p : M) {U : Set E} {x : E} |
| (hU : IsOpen U) (hx : x ∈ U) (hUt : U ⊆ (chartAt E p).target) |
| (hu : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) 2 u) |
| (hv : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) 2 v) |
| (hdu : fderiv ℝ (u ∘ (chartAt E p).symm) x ≠ 0) |
| (hker : ∀ y ∈ U, ∀ w, fderiv ℝ (u ∘ (chartAt E p).symm) y w = 0 → |
| fderiv ℝ (v ∘ (chartAt E p).symm) y w = 0) |
| (hpair : ContDiffAt ℝ 1 (fun y => (u ((chartAt E p).symm y), R ((chartAt E p).symm y))) x) |
| (hrank : Function.Surjective (fderiv ℝ |
| (fun y => (u ((chartAt E p).symm y), R ((chartAt E p).symm y))) x)) |
| (hR : 0 < R ((chartAt E p).symm x)) {n : ℕ} (hn : 2 ≤ n) |
| (hue : ∀ y ∈ U, laplaceBeltrami g u ((chartAt E p).symm y) = |
| -((n:ℝ)*((n:ℝ)+2)) * u ((chartAt E p).symm y)) |
| (hve : ∀ y ∈ U, laplaceBeltrami g v ((chartAt E p).symm y) = |
| -((n:ℝ)*((n:ℝ)+2)) * v ((chartAt E p).symm y)) |
| (hquad : ∀ y ∈ U, coordinateGradientPair g u u ((chartAt E p).symm y) = |
| (n:ℝ)^2 * ((R ((chartAt E p).symm y))^(n-1) - (u ((chartAt E p).symm y))^2)) : |
| ∃ c : ℝ, v =ᶠ[𝓝 ((chartAt E p).symm x)] (fun y => c*u y) := by |
| let chart := chartAt E p |
| let f := u ∘ chart.symm |
| let h := v ∘ chart.symm |
| let H : E → ℝ × ℝ := fun y => (f y, R (chart.symm y)) |
| have hf : ContDiffOn ℝ 2 f U := fun y hy => |
| (contDiffAt_inChart_C2 hu p (hUt hy)).contDiffWithinAt |
| have hh : ContDiffOn ℝ 2 h U := fun y hy => |
| (contDiffAt_inChart_C2 hv p (hUt hy)).contDiffWithinAt |
| obtain ⟨V,I,F,hVo,hxV,hVU,hIo,hIc,hfxI,hF,hfV,hVF⟩ := |
| local_factor_of_kernel hU hx hf hh hdu hker |
| have hlocal (y : E) (hy : y ∈ V) : |
| v =ᶠ[𝓝 (chart.symm y)] (F ∘ u) := by |
| apply chart_germ_of_pullback p (hUt (hVU hy)) |
| filter_upwards [hVo.mem_nhds hy] with z hz |
| exact hVF hz |
| have hPDE (y : E) (hy : y ∈ V) : |
| (n:ℝ)^2 * deriv (deriv F) (f y) * ((R (chart.symm y))^(n-1) - (f y)^2) + |
| ((n:ℝ)*((n:ℝ)+2)) * (F (f y) - f y * deriv F (f y)) = 0 := by |
| have hcomp := laplaceBeltrami_comp_at g (chart.symm y) |
| ((hF _ (hfV hy)).contDiffAt (hIo.mem_nhds (hfV hy))) hu |
| rw [← laplaceBeltrami_congr_nhds g (hlocal y hy),hue y (hVU hy),hve y (hVU hy),hquad y (hVU hy)] at hcomp |
| have hval : v (chart.symm y) = F (f y) := hVF hy |
| rw [hval] at hcomp |
| change -((n:ℝ)*((n:ℝ)+2))*F (f y) = |
| deriv F (f y) * (-((n:ℝ)*((n:ℝ)+2))*f y) + |
| deriv (deriv F) (f y) * ((n:ℝ)^2 * ((R (chart.symm y))^(n-1) - (f y)^2)) at hcomp |
| linear_combination -hcomp |
| have hmap : Filter.map H (𝓝 x) = 𝓝 (H x) := |
| (hpair.hasStrictFDerivAt one_ne_zero).map_nhds_eq_of_surj (LinearMap.range_eq_top.mpr hrank) |
| have himage : H '' V ∈ 𝓝 (H x) := by |
| rw [←hmap] |
| exact Filter.image_mem_map (hVo.mem_nhds hxV) |
| have hbox : I ×ˢ Ioi (0:ℝ) ∈ 𝓝 (H x) := |
| (hIo.prod isOpen_Ioi).mem_nhds ⟨hfxI,hR⟩ |
| obtain ⟨ε,hε,hεsub⟩ := Metric.mem_nhds_iff.mp (inter_mem himage hbox) |
| let K := ball (f x) ε |
| let r₀ := R (chart.symm x) - ε/2 |
| let r₁ := R (chart.symm x) + ε/2 |
| have hr₀B : r₀ ∈ ball (R (chart.symm x)) ε := by |
| rw [mem_ball,Real.dist_eq,abs_lt]; dsimp [r₀]; constructor <;> linarith |
| have hr₁B : r₁ ∈ ball (R (chart.symm x)) ε := by |
| rw [mem_ball,Real.dist_eq,abs_lt]; dsimp [r₁]; constructor <;> linarith |
| have hpairmem (s : ℝ) (hs : s ∈ K) (r : ℝ) (hr : r ∈ ({r₀,r₁}:Set ℝ)) : |
| (s,r) ∈ H '' V ∩ (I ×ˢ Ioi (0:ℝ)) := by |
| apply hεsub |
| change (s,r) ∈ ball (f x,R (chart.symm x)) ε |
| rw [←ball_prod_same] |
| refine ⟨hs,?_⟩ |
| simp only [mem_insert_iff,mem_singleton_iff] at hr |
| rcases hr with rfl | rfl <;> assumption |
| have hKI : K ⊆ I := fun s hs => (hpairmem s hs r₀ (by simp)).2.1 |
| have hr₀pos : 0 < r₀ := (hpairmem (f x) (mem_ball_self hε) r₀ (by simp)).2.2 |
| have hrr : r₀ < r₁ := by dsimp [r₀,r₁]; linarith |
| obtain ⟨a,ha⟩ := pinning_factor_squared F isOpen_ball (convex_ball (f x) ε).isPreconnected |
| (hF.mono hKI) hn hr₀pos hrr (fun s hs r hr => by |
| obtain ⟨y,hy,hyr⟩ := (hpairmem s hs r hr).1 |
| have hs' : f y = s := congrArg Prod.fst hyr |
| have hr' : R (chart.symm y) = r := congrArg Prod.snd hyr |
| simpa only [hs',hr'] using hPDE y hy) |
| refine ⟨a,chart_germ_of_pullback p (hUt hx) ?_⟩ |
| have hnx : f ⁻¹' K ∈ 𝓝 x := |
| ((hf x hx).contDiffAt (hU.mem_nhds hx)).continuousAt (isOpen_ball.mem_nhds (mem_ball_self hε)) |
| filter_upwards [hVo.mem_nhds hxV,hnx] with y hy hyK |
| exact (hVF hy).trans (ha (f y) hyK) |
|
|
|
|
| end |
|
|
| open Set Filter Function Manifold Metric |
| open scoped Topology ContDiff |
| lemma pinning_factor_gradient (F : ℝ → ℝ) {I : Set ℝ} (hIo : IsOpen I) |
| (hIc : IsPreconnected I) (hF : ContDiffOn ℝ 2 F I) |
| {Λ : ℝ} (hΛ : 0<Λ) {r₀ r₁ : ℝ} (hr : r₀<r₁) |
| (he : ∀ s ∈ I, ∀ r ∈ ({r₀,r₁}:Set ℝ), |
| deriv (deriv F) s*r + Λ*(F s-s*deriv F s)=0) : |
| ∃ c : ℝ, ∀ s ∈ I, F s=c*s := by |
| have hsecond (s : ℝ) (hs : s∈I) : deriv (deriv F) s=0 := by |
| have h0 := he s hs r₀ (by simp) |
| have h1 := he s hs r₁ (by simp) |
| have hz : (r₁-r₀)*deriv (deriv F) s=0 := by nlinarith [h0,h1] |
| exact (mul_eq_zero.mp hz).resolve_left (sub_pos.mpr hr).ne' |
| have hd : DifferentiableOn ℝ (deriv F) I := |
| (hF.deriv_of_isOpen (m:=1) hIo (by norm_num)).differentiableOn (by norm_num) |
| obtain ⟨c,hc⟩ := hIo.exists_is_const_of_deriv_eq_zero hIc hd hsecond |
| refine ⟨c,fun s hs => ?_⟩ |
| have h0 := he s hs r₀ (by simp) |
| rw [hsecond s hs,hc s hs] at h0 |
| have hz : F s-s*c=0 := (mul_eq_zero.mp (by simpa using h0)).resolve_left hΛ.ne' |
| linarith |
| variable {E M : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [TopologicalSpace M] [ChartedSpace E M] |
| [IsManifold 𝓘(ℝ,E) ∞ M] |
|
|
| theorem local_eigen_proportional_of_gradient_rank |
| (g : SmoothMetric E M) {u v R : M → ℝ} (p : M) {U : Set E} {x : E} |
| (hU : IsOpen U) (hx : x ∈ U) (hUt : U ⊆ (chartAt E p).target) |
| (hu : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) 2 u) |
| (hv : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) 2 v) |
| (hdu : fderiv ℝ (u ∘ (chartAt E p).symm) x ≠ 0) |
| (hker : ∀ y ∈ U, ∀ w, fderiv ℝ (u ∘ (chartAt E p).symm) y w = 0 → |
| fderiv ℝ (v ∘ (chartAt E p).symm) y w = 0) |
| (hpair : ContDiffAt ℝ 1 (fun y => (u ((chartAt E p).symm y), R ((chartAt E p).symm y))) x) |
| (hrank : Function.Surjective (fderiv ℝ |
| (fun y => (u ((chartAt E p).symm y), R ((chartAt E p).symm y))) x)) |
| (hR : 0 < R ((chartAt E p).symm x)) {Λ : ℝ} (hΛ : 0 < Λ) |
| (hue : ∀ y ∈ U, laplaceBeltrami g u ((chartAt E p).symm y) = |
| -Λ * u ((chartAt E p).symm y)) |
| (hve : ∀ y ∈ U, laplaceBeltrami g v ((chartAt E p).symm y) = |
| -Λ * v ((chartAt E p).symm y)) |
| (hquad : ∀ y ∈ U, coordinateGradientPair g u u ((chartAt E p).symm y) = |
| R ((chartAt E p).symm y)) : |
| ∃ c : ℝ, v =ᶠ[𝓝 ((chartAt E p).symm x)] (fun y => c*u y) := by |
| let chart := chartAt E p |
| let f := u ∘ chart.symm |
| let h := v ∘ chart.symm |
| let H : E → ℝ × ℝ := fun y => (f y, R (chart.symm y)) |
| have hf : ContDiffOn ℝ 2 f U := fun y hy => |
| (contDiffAt_inChart_C2 hu p (hUt hy)).contDiffWithinAt |
| have hh : ContDiffOn ℝ 2 h U := fun y hy => |
| (contDiffAt_inChart_C2 hv p (hUt hy)).contDiffWithinAt |
| obtain ⟨V,I,F,hVo,hxV,hVU,hIo,hIc,hfxI,hF,hfV,hVF⟩ := |
| local_factor_of_kernel hU hx hf hh hdu hker |
| have hlocal (y : E) (hy : y ∈ V) : |
| v =ᶠ[𝓝 (chart.symm y)] (F ∘ u) := by |
| apply chart_germ_of_pullback p (hUt (hVU hy)) |
| filter_upwards [hVo.mem_nhds hy] with z hz |
| exact hVF hz |
| have hPDE (y : E) (hy : y ∈ V) : |
| deriv (deriv F) (f y) * R (chart.symm y) + |
| Λ * (F (f y) - f y * deriv F (f y)) = 0 := by |
| have hcomp := laplaceBeltrami_comp_at g (chart.symm y) |
| ((hF _ (hfV hy)).contDiffAt (hIo.mem_nhds (hfV hy))) hu |
| rw [← laplaceBeltrami_congr_nhds g (hlocal y hy),hue y (hVU hy),hve y (hVU hy),hquad y (hVU hy)] at hcomp |
| have hval : v (chart.symm y) = F (f y) := hVF hy |
| rw [hval] at hcomp |
| change -Λ*F (f y) = |
| deriv F (f y) * (-Λ*f y) + |
| deriv (deriv F) (f y) * R (chart.symm y) at hcomp |
| linear_combination -hcomp |
| have hmap : Filter.map H (𝓝 x) = 𝓝 (H x) := |
| (hpair.hasStrictFDerivAt one_ne_zero).map_nhds_eq_of_surj (LinearMap.range_eq_top.mpr hrank) |
| have himage : H '' V ∈ 𝓝 (H x) := by |
| rw [←hmap] |
| exact Filter.image_mem_map (hVo.mem_nhds hxV) |
| have hbox : I ×ˢ Ioi (0:ℝ) ∈ 𝓝 (H x) := |
| (hIo.prod isOpen_Ioi).mem_nhds ⟨hfxI,hR⟩ |
| obtain ⟨ε,hε,hεsub⟩ := Metric.mem_nhds_iff.mp (inter_mem himage hbox) |
| let K := ball (f x) ε |
| let r₀ := R (chart.symm x) - ε/2 |
| let r₁ := R (chart.symm x) + ε/2 |
| have hr₀B : r₀ ∈ ball (R (chart.symm x)) ε := by |
| rw [mem_ball,Real.dist_eq,abs_lt]; dsimp [r₀]; constructor <;> linarith |
| have hr₁B : r₁ ∈ ball (R (chart.symm x)) ε := by |
| rw [mem_ball,Real.dist_eq,abs_lt]; dsimp [r₁]; constructor <;> linarith |
| have hpairmem (s : ℝ) (hs : s ∈ K) (r : ℝ) (hr : r ∈ ({r₀,r₁}:Set ℝ)) : |
| (s,r) ∈ H '' V ∩ (I ×ˢ Ioi (0:ℝ)) := by |
| apply hεsub |
| change (s,r) ∈ ball (f x,R (chart.symm x)) ε |
| rw [←ball_prod_same] |
| refine ⟨hs,?_⟩ |
| simp only [mem_insert_iff,mem_singleton_iff] at hr |
| rcases hr with rfl | rfl <;> assumption |
| have hKI : K ⊆ I := fun s hs => (hpairmem s hs r₀ (by simp)).2.1 |
| have hrr : r₀ < r₁ := by dsimp [r₀,r₁]; linarith |
| obtain ⟨a,ha⟩ := pinning_factor_gradient F isOpen_ball (convex_ball (f x) ε).isPreconnected |
| (hF.mono hKI) hΛ hrr (fun s hs r hr => by |
| obtain ⟨y,hy,hyr⟩ := (hpairmem s hs r hr).1 |
| have hs' : f y = s := congrArg Prod.fst hyr |
| have hr' : R (chart.symm y) = r := congrArg Prod.snd hyr |
| simpa only [hs',hr'] using hPDE y hy) |
| refine ⟨a,chart_germ_of_pullback p (hUt hx) ?_⟩ |
| have hnx : f ⁻¹' K ∈ 𝓝 x := |
| ((hf x hx).contDiffAt (hU.mem_nhds hx)).continuousAt (isOpen_ball.mem_nhds (mem_ball_self hε)) |
| filter_upwards [hVo.mem_nhds hxV,hnx] with y hy hyK |
| exact (hVF hy).trans (ha (f y) hyK) |
|
|
| end YauCounterexamples |
| end |
|
|
| end OAI |
|
|
