| import OAI.Geometry.Immersion.ClosedSurface.QuadraticCorrection | |
| import OAI.Geometry.Immersion.ClosedSurface.ProjectionStability | |
| namespace OAI | |
| noncomputable section | |
| open Set Complex Bundle Manifold | |
| open scoped ContDiff Matrix Topology Manifold BigOperators | |
| namespace ClosedSurfaceR4.PhaseGeometry | |
| open SmallModes RealModes | |
| lemma secondQuadratic_sub_form {n : ℕ} (B C : Fin 3 → RVec n) (v : Base) : | |
| secondQuadratic (B-C) v = secondQuadratic B v-secondQuadratic C v := by | |
| simp only [secondQuadratic,Pi.sub_apply,smul_sub] | |
| abel | |
| lemma norm_secondQuadratic_le {n : ℕ} (B : Fin 3 → RVec n) (v : Base) : | |
| ‖secondQuadratic B v‖ ≤ 4*‖B‖*‖v‖^2 := by | |
| have h1 : |v.1| ≤ ‖v‖ := by simpa only [Real.norm_eq_abs] using norm_fst_le v | |
| have h2 : |v.2| ≤ ‖v‖ := by simpa only [Real.norm_eq_abs] using norm_snd_le v | |
| have hq1 : |v.1^2| ≤ ‖v‖^2 := by | |
| rw [abs_pow] | |
| exact pow_le_pow_left₀ (abs_nonneg _) h1 2 | |
| have hq2 : |v.2^2| ≤ ‖v‖^2 := by | |
| rw [abs_pow] | |
| exact pow_le_pow_left₀ (abs_nonneg _) h2 2 | |
| have hm : |2*v.1*v.2| ≤ 2*‖v‖^2 := by | |
| simp only [abs_mul,abs_of_pos (by norm_num : (0:ℝ)<2)] | |
| nlinarith [mul_le_mul h1 h2 (abs_nonneg _) (norm_nonneg v)] | |
| unfold secondQuadratic | |
| calc | |
| _ ≤ (‖v.1^2 • B 0‖+‖(2*v.1*v.2) • B 1‖)+‖v.2^2 • B 2‖ := | |
| (norm_add_le _ _).trans (add_le_add (norm_add_le _ _) le_rfl) | |
| _ ≤ (‖v‖^2*‖B‖+(2*‖v‖^2)*‖B‖)+‖v‖^2*‖B‖ := by | |
| simp only [norm_smul,Real.norm_eq_abs] | |
| gcongr | |
| · exact norm_le_pi_norm B 0 | |
| · exact norm_le_pi_norm B 1 | |
| · exact norm_le_pi_norm B 2 | |
| _ = _ := by ring | |
| lemma norm_rotated_covector (ξ : Base) : ‖(-ξ.2,ξ.1)‖ = ‖ξ‖ := by | |
| simp only [Prod.norm_def,norm_neg] | |
| exact max_comm _ _ | |
| theorem second_form_direction_persists {n : ℕ} (B C : Fin 3 → RVec n) (ξ : Base) | |
| {κ : ℝ} (hκ : 0 < κ) (hB : κ ≤ ‖secondQuadratic B (-ξ.2,ξ.1)‖) | |
| (hBC : 4*‖B-C‖*‖ξ‖^2 ≤ κ/2) : | |
| κ/2 ≤ ‖secondQuadratic C (-ξ.2,ξ.1)‖ ∧ Good C ξ := by | |
| have he : ‖secondQuadratic B (-ξ.2,ξ.1)-secondQuadratic C (-ξ.2,ξ.1)‖ ≤ κ/2 := by | |
| rw [← secondQuadratic_sub_form] | |
| exact (norm_secondQuadratic_le _ _).trans (by simpa only [norm_rotated_covector] using hBC) | |
| have hb' : κ/2 ≤ ‖secondQuadratic C (-ξ.2,ξ.1)‖ := by | |
| have hh := norm_sub_norm_le (secondQuadratic B (-ξ.2,ξ.1)) (secondQuadratic C (-ξ.2,ξ.1)) | |
| linarith | |
| have hn := norm_pos_iff.mp ((half_pos hκ).trans_le hb') | |
| refine ⟨hb',?_,hn⟩ | |
| intro hz | |
| apply hn | |
| simp [hz,secondQuadratic] | |
| end ClosedSurfaceR4.PhaseGeometry | |
| namespace ClosedSurfaceR4.WeightedEstimates | |
| open Set | |
| lemma WeightedBound.convex_variation {E G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [NormedAddCommGroup G] [NormedSpace ℝ G] {U : Set E} (hU : IsOpen U) | |
| (hconv : Convex ℝ U) {s A : ℝ} {f : E → G} (hs : 0 < s) | |
| (hf : ContDiffOn ℝ ∞ f U) (hb : WeightedBound U s 1 A f) | |
| {x y : E} (hx : x ∈ U) (hy : y ∈ U) : | |
| ‖f x-f y‖ ≤ (A/s)*‖x-y‖ := by | |
| apply Convex.norm_image_sub_le_of_norm_fderivWithin_le | |
| (hf.differentiableOn (by simp)) _ hconv hy hx | |
| intro p hp | |
| have hh := hb.deriv_le hs (show 1 ≤ 1 by rfl) hp | |
| simpa only [norm_iteratedFDerivWithin_one (𝕜 := ℝ) f (hU.uniqueDiffOn p hp),pow_one] using hh | |
| end ClosedSurfaceR4.WeightedEstimates | |
| namespace ClosedSurfaceR4.RealModes | |
| open SmallModes WeightedEstimates Set | |
| lemma weighted_coordinate_second_jet {F : RField 4} {U : Set Base} (hU : IsOpen U) | |
| (hF : ContDiffOn ℝ ∞ F U) {z A : ℝ} (hz : 0 < z) | |
| (hb : WeightedBound U z 3 A F) {v w : Base} (hv : ‖v‖ = 1) (hw : ‖w‖ = 1) : | |
| WeightedBound U z 1 (A/z^2) (coordDeriv v (coordDeriv w F)) := by | |
| have h1 : WeightedBound U z 2 (A/z) (coordDeriv w F) := by | |
| have hh := hb.directional hU hz hF w | |
| simp only [hw,one_mul] at hh | |
| exact hh.congr (fun p hp => rfl) | |
| have h2 := h1.directional hU hz (contDiffOn_coordDeriv_vector hU hF w) v | |
| simp only [hv,one_mul,div_div,← pow_two] at h2 | |
| exact h2.congr (fun p hp => rfl) | |
| lemma coordinate_jets_variation {F : RField 4} {U : Set Base} (hU : IsOpen U) | |
| (hconv : Convex ℝ U) (hF : ContDiffOn ℝ ∞ F U) {z A : ℝ} (hz : 0 < z) | |
| (hA : 0 ≤ A) (hb : WeightedBound U z 3 A F) {x y : Base} (hx : x ∈ U) (hy : y ∈ U) : | |
| ‖firstJetPair F x-firstJetPair F y‖ ≤ (A/z^2)*‖x-y‖ ∧ | |
| ‖secondJetTriple F x‖ ≤ A/z^2 ∧ | |
| ‖secondJetTriple F x-secondJetTriple F y‖ ≤ (A/z^3)*‖x-y‖ := by | |
| have hdx : ‖dx‖ = 1 := by simp [dx,Prod.norm_def] | |
| have hdy : ‖dy‖ = 1 := by simp [dy,Prod.norm_def] | |
| have h1 (v : Base) (hv : ‖v‖ = 1) : | |
| ‖coordDeriv v F x-coordDeriv v F y‖ ≤ (A/z^2)*‖x-y‖ := by | |
| have hh : WeightedBound U z 1 (A/z) (coordDeriv v F) := by | |
| have hh := (hb.mono_order (by omega : 2 ≤ 3)).directional hU hz hF v | |
| simp only [hv,one_mul] at hh | |
| exact hh.congr (fun p hp => rfl) | |
| simpa only [div_div,pow_two] using hh.convex_variation hU hconv hz | |
| (contDiffOn_coordDeriv_vector hU hF v) hx hy | |
| have h2 (v w : Base) (hv : ‖v‖ = 1) (hw : ‖w‖ = 1) : | |
| ‖coordDeriv v (coordDeriv w F) x‖ ≤ A/z^2 ∧ | |
| ‖coordDeriv v (coordDeriv w F) x-coordDeriv v (coordDeriv w F) y‖ ≤ (A/z^3)*‖x-y‖ := by | |
| have hh := weighted_coordinate_second_jet hU hF hz hb hv hw | |
| refine ⟨hh.norm_le hx,?_⟩ | |
| have hvw := hh.convex_variation hU hconv hz | |
| (contDiffOn_coordDeriv_vector hU (contDiffOn_coordDeriv_vector hU hF w) v) hx hy | |
| simpa only [div_div,← pow_succ] using hvw | |
| refine ⟨?_,?_,?_⟩ | |
| · change max ‖coordDeriv dx F x-coordDeriv dx F y‖ ‖coordDeriv dy F x-coordDeriv dy F y‖ ≤ _ | |
| exact max_le (h1 dx hdx) (h1 dy hdy) | |
| · apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr | |
| intro i | |
| fin_cases i | |
| · exact (h2 dx dx hdx hdx).1 | |
| · exact (h2 dx dy hdx hdy).1 | |
| · exact (h2 dy dy hdy hdy).1 | |
| · apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr | |
| intro i | |
| fin_cases i | |
| · exact (h2 dx dx hdx hdx).2 | |
| · exact (h2 dx dy hdx hdy).2 | |
| · exact (h2 dy dy hdy hdy).2 | |
| theorem second_form_cell_variation (R c A : ℝ) (hc : 0 < c) (hA : 0 ≤ A) : | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ (z : ℝ), 0 < z → z ≤ 1 → | |
| ∀ (F : RField 4) (U : Set Base), IsOpen U → Convex ℝ U → ContDiffOn ℝ ∞ F U → | |
| WeightedBound U z 3 A F → | |
| (∀ p ∈ U, ‖firstJetPair F p‖ ≤ R ∧ c ≤ NormalFrame.gramDet (firstJetPair F p).1 (firstJetPair F p).2) → | |
| ∀ x ∈ U, ∀ y ∈ U, ‖realSecondTensor F x-realSecondTensor F y‖ ≤ (C/z^4)*‖x-y‖ := by | |
| obtain ⟨L,D,hL,hD,hb⟩ := compact_normal_projection_bounds R c hc | |
| refine ⟨L*A^2+D*A,by positivity,fun z hz hz1 F U hU hconv hF hFbound hmetric x hx y hy => ?_⟩ | |
| obtain ⟨h1,h2,h3⟩ := coordinate_jets_variation hU hconv hF hz hA hFbound hx hy | |
| have hp4 : z^4 ≤ z^3 := pow_le_pow_of_le_one hz.le hz1 (by omega : 3 ≤ 4) | |
| have h43 : A/z^3 ≤ A/z^4 := div_le_div_of_nonneg_left hA (pow_pos hz 4) hp4 | |
| apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr | |
| intro i | |
| simp only [Pi.sub_apply,realSecondTensor_eq_projection] | |
| have hh := hb (firstJetPair F x) (firstJetPair F y) | |
| (hmetric x hx).1 (hmetric x hx).2 (hmetric y hy).1 (hmetric y hy).2 | |
| (secondJetTriple F x i) (secondJetTriple F y i) | |
| calc | |
| _ ≤ L*‖firstJetPair F x-firstJetPair F y‖*‖secondJetTriple F x i‖+ | |
| D*‖secondJetTriple F x i-secondJetTriple F y i‖ := hh | |
| _ ≤ L*((A/z^2)*‖x-y‖)*(A/z^2)+D*((A/z^3)*‖x-y‖) := by | |
| gcongr | |
| · exact (norm_le_pi_norm _ i).trans h2 | |
| · exact (norm_le_pi_norm _ i).trans h3 | |
| _ ≤ L*((A/z^2)*‖x-y‖)*(A/z^2)+D*((A/z^4)*‖x-y‖) := by gcongr | |
| _ = _ := by field_simp [hz.ne'] | |
| end ClosedSurfaceR4.RealModes | |
| end | |
| end OAI | |