| import OAI.Analysis.AncientFlat.Weighted.WeightedScaling |
| import OAI.Analysis.AncientFlat.Evolution.CoreProfile |
| import OAI.Analysis.AncientFlat.Geometry.AnnularCore |
|
|
| namespace OAI |
| noncomputable section |
| open Set Filter |
| open scoped ContDiff |
| namespace FlatWeighted |
| open ParabolicInterior ParabolicLocalization |
| variable {V : Type} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] |
| def CoreCenter (p : V × ℝ) : Prop := ‖p.1‖≤1 ∧ p.2=-1 |
| def AnnularCenter (p : V × ℝ) : Prop := ‖p.1‖=1 ∧ -1≤p.2 ∧ p.2<0 |
| omit [FiniteDimensional ℝ V] in |
| lemma normalized_center (p : V × ℝ) (hp : p.2<0) : |
| ∃ r : ℝ, 0<r ∧ ∃ q : V × ℝ, scale r q=p ∧ (CoreCenter q ∨ AnnularCenter q) := by |
| let τ : ℝ := (-p.2)^((1:ℝ)/4) |
| have hτ : 0<τ := Real.rpow_pos_of_pos (neg_pos.mpr hp) _ |
| have hτ4 : τ^4=-p.2 := by |
| dsimp [τ] |
| rw [←Real.rpow_natCast,←Real.rpow_mul (neg_pos.mpr hp).le] |
| norm_num |
| by_cases hx : ‖p.1‖≤τ |
| · refine ⟨τ,hτ,(τ⁻¹ • p.1,-1),?_,Or.inl ?_⟩ |
| · simp only [scale,smul_smul,mul_inv_cancel₀ hτ.ne',one_smul,hτ4,mul_neg_one,neg_neg,Prod.mk.eta] |
| · constructor |
| · simp only [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hτ)] |
| exact (inv_mul_le_iff₀ hτ).mpr (by simpa using hx) |
| · rfl |
| · have hr : 0<‖p.1‖ := hτ.trans (not_le.mp hx) |
| refine ⟨‖p.1‖,hr,(‖p.1‖⁻¹ • p.1,p.2/‖p.1‖^4),?_,Or.inr ?_⟩ |
| · simp only [scale,smul_smul,mul_inv_cancel₀ hr.ne',one_smul,mul_div_cancel₀ _ (pow_ne_zero 4 hr.ne'),Prod.mk.eta] |
| · refine ⟨?_,?_,div_neg_of_neg_of_pos hp (pow_pos hr _)⟩ |
| · simp only [norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hr),inv_mul_cancel₀ hr.ne'] |
| · apply (le_div_iff₀ (pow_pos hr 4)).mpr |
| have hh := pow_le_pow_left₀ hτ.le (not_le.mp hx).le 4 |
| rw [hτ4] at hh |
| linarith |
| omit [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] in |
| lemma normalized_sigma {p : V × ℝ} (hp : CoreCenter p ∨ AnnularCenter p) : |
| p.2<0 ∧ 1 ≤ sigma p ∧ sigma p≤2 := by |
| have ht : p.2<0 := by |
| rcases hp with h|h |
| · rw [h.2]; norm_num |
| · exact h.2.2 |
| have hs : 0≤Real.sqrt (-p.2) := Real.sqrt_nonneg _ |
| have hsq : ‖p.1‖^2+Real.sqrt (-p.2)≥1 ∧ ‖p.1‖^2+Real.sqrt (-p.2)≤4 := by |
| rcases hp with h|h |
| · rw [h.2] |
| norm_num |
| nlinarith [norm_nonneg p.1,h.1] |
| · rw [h.1] |
| have hb : Real.sqrt (-p.2)≤1 := (Real.sqrt_le_one).mpr (by linarith [h.2.1]) |
| constructor <;> nlinarith |
| refine ⟨ht,?_,?_⟩ |
| · unfold sigma |
| exact (Real.le_sqrt (by norm_num) (by positivity)).mpr (by norm_num; exact hsq.1) |
| · unfold sigma |
| exact (Real.sqrt_le_left (by norm_num)).mpr (by norm_num; exact hsq.2) |
| lemma normalized_chart_maps {a : ℝ} (ha : 0<a) (ha1 : a≤1/16) (p : V × ℝ) |
| (hp : CoreCenter p ∨ AnnularCenter p) : |
| (CoreCenter p → MapsTo (chart p) (localDomain a p) ParabolicCore.inner) ∧ |
| (AnnularCenter p → MapsTo (chart p) (localDomain a p) ParabolicAnnular.inner) := by |
| obtain ⟨ht,hs1,hs2⟩ := normalized_sigma hp |
| have hs := sigma_pos p ht |
| have hsp : ∀ q∈localDomain a p, ‖sigma p • q.1‖<1/8 := by |
| intro q hq |
| rw [norm_smul,Real.norm_of_nonneg hs.le] |
| have hqa : ‖q.1‖<1/16 := hq.1.trans_le ha1 |
| nlinarith [norm_nonneg q.1] |
| have htm : ∀ q∈localDomain a p, |sigma p^4*q.2|<1/2 := by |
| intro q hq |
| have hqa : |q.2|<a^4 := abs_lt.mpr ⟨hq.2.1,hq.2.2.1⟩ |
| have ha4 : a^4≤(1:ℝ)/65536 := by |
| have hh := pow_le_pow_left₀ ha.le ha1 4 |
| norm_num at hh |
| exact hh |
| have hs4 : sigma p^4≤16 := by |
| have hh := pow_le_pow_left₀ hs.le hs2 4 |
| norm_num at hh |
| exact hh |
| rw [abs_mul,abs_of_nonneg (pow_nonneg hs.le 4)] |
| nlinarith [abs_nonneg q.2,pow_nonneg hs.le 4] |
| constructor |
| · intro hc q hq |
| have hn := (norm_add_le p.1 (sigma p • q.1)).trans_lt (by linarith [hsp q hq,hc.1] : ‖p.1‖+‖sigma p • q.1‖<2) |
| have hu := abs_lt.mp (htm q hq) |
| exact ⟨hn,by dsimp [chart]; rw [hc.2]; linarith,by dsimp [chart]; rw [hc.2]; linarith⟩ |
| · intro hc q hq |
| have hn : ‖p.1+sigma p • q.1‖<5/4 := (norm_add_le _ _).trans_lt (by linarith [hsp q hq,hc.1]) |
| have hlo : 3/4<‖p.1+sigma p • q.1‖ := by |
| have hh := norm_sub_norm_le p.1 (-(sigma p • q.1)) |
| simp only [sub_neg_eq_add,norm_neg] at hh |
| linarith [hsp q hq,hc.1] |
| have hu := abs_lt.mp (htm q hq) |
| exact ⟨hlo,hn,by dsimp [chart]; linarith [hc.2.1],hq.2.2.2⟩ |
| end FlatWeighted |
|
|
| end |
|
|
| end OAI |
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