| import Mathlib |
| import OAI.Geometry.SmoothYau.Estimates.LpInverseGramFstContraction |
| import OAI.Geometry.SmoothYau.Model |
| import OAI.Geometry.SmoothYau.Smoothness.LinearPowerDirectional |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace YauCounterexamples |
| section |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology InnerProductSpace |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Function Manifold Module |
| open scoped Topology ContDiff InnerProductSpace |
|
|
| variable {n : ℕ} |
| local instance : Fact (Module.finrank ℝ (Euclidean (n+1)) = n+1) := ⟨by simp [Euclidean]⟩ |
|
|
| def sphereFrame (p : Sphere n) : Euclidean n →ₗᵢ[ℝ] Euclidean (n+1) := |
| (ℝ ∙ (-(p : Euclidean (n+1))))ᗮ.subtypeₗᵢ.comp |
| (OrthonormalBasis.fromOrthogonalSpanSingleton n (ne_zero_of_mem_unit_sphere (-p))).repr.symm.toLinearIsometry |
|
|
| lemma sphereFrame_orthogonal (p : Sphere n) (w : Euclidean n) : |
| inner ℝ (p : Euclidean (n+1)) (sphereFrame p w) = 0 := by |
| have hh := ((OrthonormalBasis.fromOrthogonalSpanSingleton n |
| (ne_zero_of_mem_unit_sphere (-p))).repr.symm w).property |
| rw [Submodule.mem_orthogonal_singleton_iff_inner_left] at hh |
| change inner ℝ (p : Euclidean (n+1)) _ = 0 |
| rw [real_inner_comm] |
| change inner ℝ _ (-(p : Euclidean (n+1))) = 0 at hh |
| rw [inner_neg_right,neg_eq_zero] at hh |
| exact hh |
|
|
| lemma sphere_chart_symm (p : Sphere n) : |
| (fun y => ((chartAt (Euclidean n) p).symm y : Euclidean (n+1))) = |
| roundChart (p : Euclidean (n+1)) (sphereFrame p) := by |
| funext y |
| change ((stereographic' n (-p)).symm y : Euclidean (n+1)) = _ |
| rw [stereographic'_symm_apply] |
| simp only [roundChart,sphereFrame,stereoInvFunAux,smul_add,smul_smul] |
| rfl |
|
|
| lemma sphere_chart_center (p : Sphere n) : chartAt (Euclidean n) p p = 0 := by |
| have hh : (chartAt (Euclidean n) p).symm 0 = p := by |
| apply Subtype.ext |
| change (fun y => ((chartAt (Euclidean n) p).symm y : Euclidean (n+1))) 0 = _ |
| rw [sphere_chart_symm,roundChart_zero] |
| have hz : (0 : Euclidean n) ∈ (chartAt (Euclidean n) p).target := by |
| change 0 ∈ (stereographic' n (-p)).target |
| simp |
| exact (congrArg (chartAt (Euclidean n) p) hh).symm.trans ((chartAt (Euclidean n) p).right_inv hz) |
| |
| lemma sphere_chart_target (p : Sphere n) : (chartAt (Euclidean n) p).target = univ := by |
| change (stereographic' n (-p)).target = _ |
| simp |
|
|
| lemma sphere_coordinate_embedding (p : Sphere n) (y : Euclidean n) (v : Euclidean n) : |
| mfderiv 𝓘(ℝ,Euclidean n) 𝓘(ℝ,Euclidean (n+1)) |
| (fun q : Sphere n => (q : Euclidean (n+1))) ((chartAt (Euclidean n) p).symm y) |
| (mfderiv 𝓘(ℝ,Euclidean n) 𝓘(ℝ,Euclidean n) (chartAt (Euclidean n) p).symm y v) = |
| fderiv ℝ (roundChart (p : Euclidean (n+1)) (sphereFrame p)) y v := by |
| have hy : y ∈ (chartAt (Euclidean n) p).target := by rw [sphere_chart_target]; trivial |
| have hh := mfderiv_comp y |
| ((contMDiff_coe_sphere (m := ∞)).mdifferentiable (by simp)).mdifferentiableAt |
| ((contMDiffAt_symm_of_mem_maximalAtlas (IsManifold.chart_mem_maximalAtlas (n := ∞) p) hy).mdifferentiableAt (by simp)) |
| have he := congrArg (fun L => L v) hh |
| rw [mfderiv_eq_fderiv] at he |
| rw [← sphere_chart_symm] |
| exact he.symm |
|
|
|
|
| end |
|
|
| section |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology InnerProductSpace |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Manifold BoxIntegral Metric |
| open scoped Topology ContDiff InnerProductSpace |
|
|
| def sphericalRadius (a b : Euclidean 4) (q : Sphere 3) : ℝ := |
| Real.sqrt ((inner ℝ a (q : Euclidean 4))^2+(inner ℝ b (q : Euclidean 4))^2) |
|
|
| lemma sphericalRadius_nonneg (a b : Euclidean 4) (q : Sphere 3) : |
| 0 ≤ sphericalRadius a b q := Real.sqrt_nonneg _ |
|
|
| lemma sphericalRadius_sq (a b : Euclidean 4) (q : Sphere 3) : |
| (sphericalRadius a b q)^2 = |
| (inner ℝ a (q : Euclidean 4))^2+(inner ℝ b (q : Euclidean 4))^2 := |
| Real.sq_sqrt (add_nonneg (sq_nonneg _) (sq_nonneg _)) |
|
|
| lemma sphericalRadius_continuous (a b : Euclidean 4) : Continuous (sphericalRadius a b) := |
| (((continuous_const.inner continuous_subtype_val).pow 2).add |
| ((continuous_const.inner continuous_subtype_val).pow 2)).sqrt |
|
|
| lemma orthogonal_pair_sq_le (a b q : Euclidean 4) |
| (ha : inner ℝ a a = 1) (hb : inner ℝ b b = 1) (hab : inner ℝ a b = 0) : |
| (inner ℝ a q)^2+(inner ℝ b q)^2 ≤ inner ℝ q q := by |
| have hba : inner ℝ b a = 0 := (real_inner_comm a b).trans hab |
| have hqa : inner ℝ q a = inner ℝ a q := real_inner_comm a q |
| have hqb : inner ℝ q b = inner ℝ b q := real_inner_comm b q |
| have h := real_inner_self_nonneg (x := q-(inner ℝ a q) • a-(inner ℝ b q) • b) |
| simp only [inner_sub_left,inner_sub_right,inner_smul_left,inner_smul_right, |
| ha,hb,hab,hba,hqa,hqb] at h |
| nlinarith |
|
|
| lemma sphericalRadius_le_one (a b : Euclidean 4) |
| (ha : inner ℝ a a = 1) (hb : inner ℝ b b = 1) (hab : inner ℝ a b = 0) |
| (q : Sphere 3) : sphericalRadius a b q ≤ 1 := by |
| have h := orthogonal_pair_sq_le a b q ha hb hab |
| have hq : ‖(q : Euclidean 4)‖ = 1 := by simpa only [Metric.mem_sphere,dist_zero_right] using q.property |
| rw [real_inner_self_eq_norm_sq,hq,one_pow,← sphericalRadius_sq] at h |
| nlinarith [sphericalRadius_nonneg a b q] |
|
|
| lemma sphere_chart_source (p : Sphere 3) : (chartAt (Euclidean 3) p).source = {-p}ᶜ := by |
| let : Fact (Module.finrank ℝ (Euclidean 4) = 3+1) := ⟨by simp [Euclidean]⟩ |
| change (stereographic' 3 (-p)).source = _ |
| simp |
| |
| abbrev sourceAxisOne : Euclidean 4 := EuclideanSpace.single 0 1 |
| abbrev sourceAxisTwo : Euclidean 4 := EuclideanSpace.single 1 1 |
| |
| lemma source_axes_orthonormal : inner ℝ sourceAxisOne sourceAxisOne = 1 ∧ |
| inner ℝ sourceAxisTwo sourceAxisTwo = 1 ∧ inner ℝ sourceAxisOne sourceAxisTwo = 0 := by |
| norm_num [sourceAxisOne,sourceAxisTwo,EuclideanSpace.inner_single_left,PiLp.single_apply] |
| |
| def sourcePole : Sphere 3 := ⟨sourceAxisOne,by simp [sourceAxisOne]⟩ |
| |
| lemma sphericalRadius_neg_sourcePole : sphericalRadius sourceAxisOne sourceAxisTwo (-sourcePole) = 1 := by |
| norm_num [sphericalRadius,sourcePole,sourceAxisOne,sourceAxisTwo, |
| EuclideanSpace.inner_single_left,PiLp.single_apply] |
| |
| def sourceSignPoint : Sphere 3 := |
| ⟨EuclideanSpace.single 0 (40/401)+EuclideanSpace.single (2 : Fin 4) (399/401),by |
| rw [Metric.mem_sphere,dist_zero_right] |
| have hi : inner ℝ |
| (EuclideanSpace.single (0 : Fin 4) (40/401 : ℝ)+EuclideanSpace.single (2 : Fin 4) (399/401)) |
| (EuclideanSpace.single (0 : Fin 4) (40/401 : ℝ)+EuclideanSpace.single (2 : Fin 4) (399/401)) = 1 := by |
| simp only [inner_add_left,inner_add_right,EuclideanSpace.inner_single_left,PiLp.single_apply] |
| norm_num [Fin.ext_iff] |
| rw [real_inner_self_eq_norm_sq] at hi |
| nlinarith [norm_nonneg (EuclideanSpace.single (0 : Fin 4) (40/401 : ℝ)+EuclideanSpace.single (2 : Fin 4) (399/401))]⟩ |
| |
| lemma sphericalRadius_sourceSignPoint : |
| sphericalRadius sourceAxisOne sourceAxisTwo sourceSignPoint = 40/401 := by |
| unfold sphericalRadius |
| have he : (inner ℝ sourceAxisOne (sourceSignPoint : Euclidean 4))^2+ |
| (inner ℝ sourceAxisTwo (sourceSignPoint : Euclidean 4))^2 = (40/401 : ℝ)^2 := by |
| simp only [sourceAxisOne,sourceAxisTwo,sourceSignPoint,inner_add_right, |
| EuclideanSpace.inner_single_left,PiLp.single_apply] |
| norm_num [Fin.ext_iff] |
| rw [he,Real.sqrt_sq_eq_abs] |
| norm_num |
| |
| lemma sphericalRadius_sublevel_in_source (p : Sphere 3) (a b : Euclidean 4) |
| (hp : sphericalRadius a b (-p) = 1) {R : ℝ} (hR : R < 1) : |
| {q | sphericalRadius a b q ≤ R} ⊆ (chartAt (Euclidean 3) p).source := by |
| intro q hq |
| rw [sphere_chart_source] |
| intro hneg |
| have he : q = -p := hneg |
| change sphericalRadius a b q ≤ R at hq |
| rw [he,hp] at hq |
| exact (not_le_of_gt hR) hq |
| |
| def sphericalCoverage (p : Sphere 3) (a b : Euclidean 4) (R : ℝ) : Set (Euclidean 3) := |
| chartAt (Euclidean 3) p '' {q | sphericalRadius a b q ≤ R} |
| |
| lemma sphericalCoverage_compact (p : Sphere 3) (a b : Euclidean 4) |
| (hp : sphericalRadius a b (-p) = 1) {R : ℝ} (hR : R < 1) : |
| IsCompact (sphericalCoverage p a b R) := by |
| have hc : IsCompact {q : Sphere 3 | sphericalRadius a b q ≤ R} := |
| (isClosed_le (sphericalRadius_continuous a b) continuous_const).isCompact |
| exact hc.image_of_continuousOn ((chartAt (Euclidean 3) p).continuousOn.mono |
| (sphericalRadius_sublevel_in_source p a b hp hR)) |
| |
| lemma mem_sphericalCoverage_iff (p : Sphere 3) (a b : Euclidean 4) |
| (hp : sphericalRadius a b (-p) = 1) {R : ℝ} (hR : R < 1) (y : Euclidean 3) : |
| y ∈ sphericalCoverage p a b R ↔ |
| sphericalRadius a b ((chartAt (Euclidean 3) p).symm y) ≤ R := by |
| constructor |
| · rintro ⟨q,hq,rfl⟩ |
| rw [(chartAt (Euclidean 3) p).left_inv (sphericalRadius_sublevel_in_source p a b hp hR hq)] |
| exact hq |
| · intro hy |
| exact ⟨(chartAt (Euclidean 3) p).symm y,hy, |
| (chartAt (Euclidean 3) p).right_inv (by rw [sphere_chart_target]; trivial)⟩ |
| |
| lemma sphericalCoverage_mono (p : Sphere 3) (a b : Euclidean 4) {R S : ℝ} (h : R ≤ S) : |
| sphericalCoverage p a b R ⊆ sphericalCoverage p a b S := |
| image_mono (fun _ hx => hx.trans h) |
| |
| lemma sphere_chart_symm_continuous (p : Sphere 3) : Continuous (chartAt (Euclidean 3) p).symm := by |
| exact continuousOn_univ.mp (by simpa only [sphere_chart_target] using |
| (chartAt (Euclidean 3) p).continuousOn_symm) |
| |
| lemma sphericalCoverage_subset_open (p : Sphere 3) (a b : Euclidean 4) |
| (hp : sphericalRadius a b (-p) = 1) {R S : ℝ} (hR : R < 1) (hRS : R < S) : |
| sphericalCoverage p a b R ⊆ |
| {y | sphericalRadius a b ((chartAt (Euclidean 3) p).symm y) < S} := by |
| intro y hy |
| exact ((mem_sphericalCoverage_iff p a b hp hR y).mp hy).trans_lt hRS |
| |
| lemma sphericalCoverage_open (p : Sphere 3) (a b : Euclidean 4) (S : ℝ) : |
| IsOpen {y | sphericalRadius a b ((chartAt (Euclidean 3) p).symm y) < S} := |
| isOpen_lt ((sphericalRadius_continuous a b).comp (sphere_chart_symm_continuous p)) continuous_const |
| |
| |
| end |
| |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology InnerProductSpace |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Manifold |
| open scoped Topology ContDiff |
| open Set Filter MeasureTheory Metric |
| open scoped Topology ENNReal NNReal |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology InnerProductSpace |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Function Metric |
| open scoped Topology |
| open Set Filter Function Module Manifold |
| open scoped Topology ContDiff InnerProductSpace Matrix |
| variable {n : ℕ} |
| local instance : Fact (Module.finrank ℝ (Euclidean (n+1)) = n+1) := ⟨by simp [Euclidean]⟩ |
| |
| def sphereProjection (p : Sphere n) (a : Euclidean (n+1)) : Euclidean (n+1) := |
| a - inner ℝ (p : Euclidean (n+1)) a • (p : Euclidean (n+1)) |
| |
| lemma sphereProjection_orthogonal (p : Sphere n) (a : Euclidean (n+1)) : |
| sphereProjection p a ∈ (ℝ ∙ (-(p : Euclidean (n+1))))ᗮ := by |
| rw [Submodule.mem_orthogonal_singleton_iff_inner_left] |
| simp [sphereProjection,inner_sub_left,inner_smul_left,real_inner_comm a (p : Euclidean (n+1))] |
| |
| def sphereCoordinates (p : Sphere n) (a : Euclidean (n+1)) : Euclidean n := |
| (OrthonormalBasis.fromOrthogonalSpanSingleton n (ne_zero_of_mem_unit_sphere (-p))).repr |
| ⟨sphereProjection p a,sphereProjection_orthogonal p a⟩ |
| |
| lemma sphereFrame_coordinates (p : Sphere n) (a : Euclidean (n+1)) : |
| sphereFrame p (sphereCoordinates p a) = sphereProjection p a := by |
| change (((OrthonormalBasis.fromOrthogonalSpanSingleton n |
| (ne_zero_of_mem_unit_sphere (-p))).repr.symm |
| ((OrthonormalBasis.fromOrthogonalSpanSingleton n |
| (ne_zero_of_mem_unit_sphere (-p))).repr ⟨sphereProjection p a,_⟩)) : Euclidean (n+1)) = _ |
| rw [LinearIsometryEquiv.symm_apply_apply] |
| |
| lemma sphereCoordinates_inner (p : Sphere n) (a b : Euclidean (n+1)) : |
| inner ℝ (sphereCoordinates p a) (sphereCoordinates p b) = |
| inner ℝ a b - inner ℝ (p : Euclidean (n+1)) a * inner ℝ (p : Euclidean (n+1)) b := by |
| rw [← (sphereFrame p).inner_map_map,sphereFrame_coordinates,sphereFrame_coordinates] |
| simp [sphereProjection,inner_sub_left,inner_sub_right,inner_smul_left,inner_smul_right, |
| real_inner_comm a (p : Euclidean (n+1))] |
| |
| lemma sphereFrame_inner (p : Sphere n) (a : Euclidean (n+1)) (w : Euclidean n) : |
| inner ℝ (sphereFrame p w) a = inner ℝ w (sphereCoordinates p a) := by |
| rw [← (sphereFrame p).inner_map_map,sphereFrame_coordinates] |
| have hh := sphereFrame_orthogonal p w |
| rw [real_inner_comm] at hh |
| simp [sphereProjection,inner_sub_right,inner_smul_right,hh] |
| |
| lemma sphere_inverse_gram_contraction (p : Sphere n) (a b : Euclidean (n+1)) : |
| ∑ i, ∑ j, (Matrix.gram ℝ (Module.finBasis ℝ (Euclidean n)))⁻¹ i j * |
| inner ℝ (sphereFrame p (Module.finBasis ℝ (Euclidean n) j)) a * |
| inner ℝ (sphereFrame p (Module.finBasis ℝ (Euclidean n) i)) b = |
| inner ℝ a b - inner ℝ (p : Euclidean (n+1)) a * inner ℝ (p : Euclidean (n+1)) b := by |
| simp_rw [sphereFrame_inner] |
| rw [inverse_gram_contraction,sphereCoordinates_inner] |
| |
| section Planar |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| def planarLinear (a b : E) : E →L[ℝ] ℂ := |
| Complex.ofRealCLM.comp (innerSL ℝ a) + Complex.I • (Complex.ofRealCLM.comp (innerSL ℝ b)) |
| |
| lemma planarLinear_apply (a b x : E) : |
| planarLinear a b x = (inner ℝ a x : ℂ) + Complex.I * (inner ℝ b x : ℂ) := by |
| rfl |
| lemma planarLinear_apply_flip (a b x : E) : |
| planarLinear a b x = (inner ℝ x a : ℂ) + Complex.I * (inner ℝ x b : ℂ) := by |
| rw [planarLinear_apply,real_inner_comm a x,real_inner_comm b x] |
| end Planar |
| |
| lemma complex_bilinear_expand (r a b c d : ℝ) : |
| (r:ℂ) * ((a:ℂ)+Complex.I*b) * ((c:ℂ)+Complex.I*d) = |
| ((r*a*c-r*b*d:ℝ):ℂ) + Complex.I * ((r*a*d+r*b*c:ℝ):ℂ) := by |
| apply Complex.ext <;> simp [Complex.mul_re,Complex.mul_im] |
| |
| |
| theorem sphere_planar_contraction (p : Sphere n) (a b : Euclidean (n+1)) |
| (ha : inner ℝ a a = 1) (hb : inner ℝ b b = 1) (hab : inner ℝ a b = 0) : |
| ∑ i, ∑ j, ((Matrix.gram ℝ (Module.finBasis ℝ (Euclidean n)))⁻¹ i j : ℂ) * |
| planarLinear a b (sphereFrame p (Module.finBasis ℝ (Euclidean n) j)) * |
| planarLinear a b (sphereFrame p (Module.finBasis ℝ (Euclidean n) i)) = |
| -(planarLinear a b (p : Euclidean (n+1)))^2 := by |
| simp_rw [planarLinear_apply_flip,complex_bilinear_expand] |
| simp_rw [Finset.sum_add_distrib,← Finset.mul_sum,← Complex.ofReal_sum, |
| Finset.sum_sub_distrib] |
| simp only [Finset.sum_add_distrib] |
| rw [sphere_inverse_gram_contraction,sphere_inverse_gram_contraction, |
| sphere_inverse_gram_contraction,sphere_inverse_gram_contraction] |
| rw [ha,hb,hab,real_inner_comm a b,hab] |
| apply Complex.ext <;> simp [Complex.mul_re,Complex.mul_im,pow_two]; ring |
| |
| |
| |
| end YauCounterexamples |
| end |
| |
| end OAI |
| |
