File size: 3,613 Bytes
59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import OAI.Geometry.IsometricImmersion.Coordinates.ActualShearHyperbolicity
import OAI.Geometry.IsometricImmersion.Immersions.BoundedHeightClasses
namespace OAI
noncomputable section
open Set
open scoped ContDiff Matrix
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Pulse SmoothLocal.HighEquation
theorem boundedClassSpeed_sq {kappa : ℝ} (hkappa : 0 < kappa)
{M : ℕ} (hM : 0 < M) :
(boundedClassSpeed kappa M) ^ 2 = kappa / (2 * (M : ℝ) ^ 3) := by
have hMp : (0 : ℝ) < (M : ℝ) := Nat.cast_pos.mpr hM
have he : ((M : ℝ) ^ (-(3 : ℝ) / 2)) ^ (2 : ℕ) = ((M : ℝ) ^ (3 : ℕ))⁻¹ := by
rw [← Real.rpow_mul_natCast hMp.le]
norm_num [Real.rpow_neg, Real.rpow_natCast]
unfold boundedClassSpeed
rw [mul_pow, Real.sq_sqrt (by positivity), he]
ring
theorem boundedClassSpeed_sq_energy {kappa : ℝ} (hkappa : 0 < kappa)
{M : ℕ} (hM : 0 < M) :
(boundedClassSpeed kappa M) ^ 2 =
kappa * (1 / (M : ℝ)) / (2 * (M : ℝ) ^ 2) := by
rw [boundedClassSpeed_sq hkappa hM]
field_simp
theorem bounded_class_shear_estimates {g : MetricField} {z : Coord → ℝ}
{M : ℕ} (h : BoundedAdmissibleHeight g M z) {kappa q0 : ℝ}
(hkappa : 0 < kappa) {p : Coord}
(hp : inverseShearCoordinates q0 p ∈ modelSquare)
(hK : gaussianCurvature g (inverseShearCoordinates q0 p) ≤ -kappa / 2)
(hq : |hessianQuotient g z (inverseShearCoordinates q0 p) - q0| ≤
1 / (10 * boundedClassWidth kappa M)) :
(boundedClassSpeed kappa M) ^ 2 / 2 ≤ -shearTimeFactor g z q0 p ∧
(boundedClassSpeed kappa M) ^ 2 / (2 * (M : ℝ)) ≤
|covHessian (metricInShearCoordinates g q0) (heightInShearCoordinates z q0) p 0 0| ∧
0 < qFirstCoefficient (metricInShearCoordinates g q0) 5
(qSolutionJet (heightInShearCoordinates z q0) p) ∧
|qFirstCoefficient (metricInShearCoordinates g q0) 4
(qSolutionJet (heightInShearCoordinates z q0) p)| +
Real.sqrt (qFirstCoefficient (metricInShearCoordinates g q0) 5
(qSolutionJet (heightInShearCoordinates z q0) p)) < boundedClassWidth kappa M / 4 := by
obtain ⟨hM, hadm, hjet, hHb, hEb⟩ := h
obtain ⟨U, hU, hSU, hg, hz, hD, hE, hyy, hsmall⟩ := hadm
have hMp : (0 : ℝ) < (M : ℝ) := Nat.cast_pos.mpr hM
have hv := boundedClassSpeed_pos hkappa hM
obtain ⟨hd, hxx, hS, hchar⟩ := actual_shear_hyperbolicity hg hz hU (hSU hp)
hkappa (one_div_pos.mpr hMp) hv (boundedClassSpeed_sq_energy hkappa hM).le
(hyy _ hp) (hHb _ hp).2 (hEb _ hp) hK hq (hD _ hp)
have hdneg : shearTimeFactor g z q0 p < 0 := by
have hv2 : 0 < (boundedClassSpeed kappa M) ^ 2 / 2 := half_pos (sq_pos_of_pos hv)
linarith
have hfac : covHessian (metricInShearCoordinates g q0) (heightInShearCoordinates z q0) p 0 0 =
shearTimeFactor g z q0 p * covHessian g z (inverseShearCoordinates q0 p) 1 1 :=
sheared_Hxx_factor hg hz hU q0 (hSU hp) (hyy _ hp) (hD _ hp)
refine ⟨hd, ?_, hS, hchar⟩
calc
_ = ((boundedClassSpeed kappa M) ^ 2 / 2) * (1 / (M : ℝ)) := by ring
_ ≤ (-shearTimeFactor g z q0 p) * |covHessian g z (inverseShearCoordinates q0 p) 1 1| :=
mul_le_mul hd (hHb _ hp).1 (one_div_nonneg.mpr hMp.le) (neg_pos.mpr hdneg).le
_ = _ := by rw [hfac, abs_mul, abs_of_neg hdneg]
theorem boundedClassShearHxxFloor_pos {kappa : ℝ} (hkappa : 0 < kappa)
{M : ℕ} (hM : 0 < M) :
0 < (boundedClassSpeed kappa M) ^ 2 / (2 * (M : ℝ)) :=
div_pos (sq_pos_of_pos (boundedClassSpeed_pos hkappa hM))
(mul_pos (by norm_num) (Nat.cast_pos.mpr hM))
end SmoothLocal.Perturbation
end
end OAI
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