| import OAI.Geometry.Kahler.BaseProfileBounds |
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| namespace OAI |
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| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology |
| open scoped ContDiff |
| noncomputable section |
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| open Set Filter Topology |
| namespace PinchedHartogs.BaseConstruction |
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| lemma density_conditions_eventually {F B D K c C₁ C₂ R : ℝ} |
| (hgap : F < 1-2*c) (hder : C₂ < K*c) : |
| ∀ᶠ q : ℝ in atTop, |
| Real.exp (K*D/Real.sqrt q+2*R/q) ≤ 2 ∧ |
| Real.exp (K*D/Real.sqrt q+2*R/q)*(F+B*(K/q)) ≤ 1-2*c ∧ |
| K/q ≤ 1 ∧ C₁/q^2 ≤ 1 ∧ C₂+K/Real.sqrt q ≤ K*c ∧ C₂+K/q ≤ K*c := by |
| have hq : Tendsto (fun q : ℝ => K/q) atTop (𝓝 0) := tendsto_const_nhds.div_atTop tendsto_id |
| have hqs : Tendsto (fun q : ℝ => K/Real.sqrt q) atTop (𝓝 0) := tendsto_const_nhds.div_atTop Real.tendsto_sqrt_atTop |
| have hqd : Tendsto (fun q : ℝ => K*D/Real.sqrt q) atTop (𝓝 0) := tendsto_const_nhds.div_atTop Real.tendsto_sqrt_atTop |
| have hqr : Tendsto (fun q : ℝ => 2*R/q) atTop (𝓝 0) := tendsto_const_nhds.div_atTop tendsto_id |
| have hq₂ : Tendsto (fun q : ℝ => C₁/q^2) atTop (𝓝 0) := tendsto_const_nhds.div_atTop (tendsto_pow_atTop (by norm_num : 2 ≠ 0)) |
| have hH : Tendsto (fun q : ℝ => Real.exp (K*D/Real.sqrt q+2*R/q)) atTop (𝓝 1) := by |
| have hadd : Tendsto (fun q : ℝ => K*D/Real.sqrt q+2*R/q) atTop (𝓝 0) := by simpa using hqd.add hqr |
| simpa only [Function.comp_def, Real.exp_zero] using (Real.continuous_exp.tendsto 0).comp hadd |
| have hprod : Tendsto (fun q : ℝ => Real.exp (K*D/Real.sqrt q+2*R/q)*(F+B*(K/q))) atTop (𝓝 F) := by |
| simpa using hH.mul (tendsto_const_nhds.add (tendsto_const_nhds.mul hq)) |
| have hdH : Tendsto (fun q : ℝ => C₂+K/Real.sqrt q) atTop (𝓝 C₂) := by simpa using tendsto_const_nhds.add hqs |
| have hdT : Tendsto (fun q : ℝ => C₂+K/q) atTop (𝓝 C₂) := by simpa using tendsto_const_nhds.add hq |
| filter_upwards [hH.eventually (gt_mem_nhds (by norm_num : (1:ℝ)<2)),hprod.eventually (gt_mem_nhds hgap), |
| hq.eventually (gt_mem_nhds zero_lt_one),hq₂.eventually (gt_mem_nhds zero_lt_one), |
| hdH.eventually (gt_mem_nhds hder),hdT.eventually (gt_mem_nhds hder)] with q h1 h2 h3 h4 h5 h6 |
| exact ⟨h1.le,h2.le,h3.le,h4.le,h5.le,h6.le⟩ |
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| lemma exists_density_constants {a : ℝ} (p : RadialProfiles a) {B F₁ B₁ : ℝ} |
| (hB : 0 ≤ B) (hF₁ : 0 ≤ F₁) (hB₁ : 0 ≤ B₁) : |
| ∃ c K : ℝ, ∃ N : ℕ, 0 < c ∧ 2*c ≤ 1 ∧ 0 < K ∧ 2/c ≤ (2:ℝ)^N ∧ |
| p.F<1-2*c ∧ 2*(1+2*Real.sqrt (2*p.R))*(4*(p.F+B)+F₁+B₁) < K*c := by |
| let c := (1-p.F)/4 |
| have hc : 0 < c := by dsimp [c]; linarith [p.F_lt] |
| let C₂ := 2*(1+2*Real.sqrt (2*p.R))*(4*(p.F+B)+F₁+B₁) |
| have hC : 0 ≤ C₂ := by dsimp [C₂]; have hF := p.F_nonneg; positivity |
| let K := C₂/c+1 |
| have hK : 0 < K := by dsimp [K]; positivity |
| have hc2 : c ≤ 2 := by dsimp [c]; linarith [p.F_nonneg] |
| obtain ⟨N,hN⟩ := exists_nat_pow_near ((one_le_div hc).mpr hc2) (by norm_num : (1:ℝ)<2) |
| refine ⟨c,K,N+1,hc,?_,hK,?_,?_,?_⟩ |
| · dsimp [c]; linarith [p.F_nonneg] |
| · exact hN.2.le |
| · dsimp [c]; linarith [p.F_lt] |
| · dsimp [K] |
| have he : (C₂/c+1)*c=C₂+c := by field_simp |
| rw [he] |
| linarith |
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| end PinchedHartogs.BaseConstruction |
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| end |
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| end OAI |
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