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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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Geometry.IsometricImmersion.Caps.PulseReflection
import OAI.Geometry.IsometricImmersion.Immersions.BoundedHeightClasses
import OAI.Geometry.IsometricImmersion.Metrics.MetricPatchOpen
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology Matrix
namespace SmoothLocal.Geometry
open SmoothLocal.Flow SmoothLocal.Flow.Reflection
theorem PatchAdmissibleHeight.reflected {g : MetricField} {z : Coord → ℝ}
(h : PatchAdmissibleHeight g z) :
PatchAdmissibleHeight (reflectedMetric g) (reflectedScalar z) := by
obtain ⟨U,hU,hSU,hg,hz,hD,hE,hyy,hq⟩ := h
have hmem {p : Coord} (hp : p ∈ modelSquare) : reflectPoint p ∈ U :=
hSU ((reflectPoint_mem_modelSquare p).mpr hp)
refine ⟨reflectPoint ⁻¹' U, reflectedDomain_isOpen hU,
reflectedDomain_contains_modelSquare hSU, reflectedMetric_smoothPositive hg,
reflectedScalar_contDiffOn hz, ?_, ?_, ?_, ?_⟩
· intro p hp
rw [reflectedMetric_hessian_det hg hz hU (hmem hp),
reflectedMetric_curvature hg hU (hmem hp), reflectedMetric_energy hz hU (hmem hp)]
exact hD _ ((reflectPoint_mem_modelSquare p).mpr hp)
· intro p hp
rw [reflectedMetric_energy hz hU (hmem hp)]
exact hE _ ((reflectPoint_mem_modelSquare p).mpr hp)
· intro p hp
rw [reflectedMetric_hessian_entry hg hz hU (hmem hp)]
simpa only [ite_true] using hyy _ ((reflectPoint_mem_modelSquare p).mpr hp)
· intro p hp
rw [reflectedMetric_quotient hg hz hU (hmem hp),abs_neg]
exact hq _ ((reflectPoint_mem_modelSquare p).mpr hp)
theorem BoundedAdmissibleHeight.reflected {g : MetricField} {z : Coord → ℝ} {M : ℕ}
(h : BoundedAdmissibleHeight g M z) :
BoundedAdmissibleHeight (reflectedMetric g) M (reflectedScalar z) := by
obtain ⟨hM,ha,hB,hyy,hE⟩ := h
have har := ha.reflected
obtain ⟨U,hU,hSU,hg,hz,hD,hpos,hne,hq⟩ := ha
refine ⟨hM,har,reflectedScalar_C8 hB,?_,?_⟩
· intro p hp
have hpr := (reflectPoint_mem_modelSquare p).mpr hp
rw [reflectedMetric_hessian_entry hg hz hU (hSU hpr)]
simpa only [ite_true] using hyy _ hpr
· intro p hp
have hpr := (reflectPoint_mem_modelSquare p).mpr hp
rw [reflectedMetric_energy hz hU (hSU hpr)]
exact hE _ hpr
theorem bounded_class_reflected_center {g : MetricField} {z : Coord → ℝ} {M : ℕ}
(h : BoundedAdmissibleHeight g M z) (q0 : ℝ) :
|hessianQuotient (reflectedMetric g) (reflectedScalar z) 0-(-q0)| =
|hessianQuotient g z 0-q0| := by
obtain ⟨U,hU,hSU,hg,hz,hD,hE,hyy,hq⟩ := h.2.1
have hzero : (0 : Coord) ∈ modelSquare := by constructor <;> intro i <;> norm_num
have hrzero : reflectPoint (0 : Coord) = 0 := by ext i; fin_cases i <;> simp
have hzeroU : reflectPoint (0 : Coord) ∈ U := by
simpa only [hrzero] using hSU hzero
rw [reflectedMetric_quotient hg hz hU (p := 0) hzeroU,hrzero]
have he : -hessianQuotient g z 0-(-q0) = -(hessianQuotient g z 0-q0) := by ring
rw [he,abs_neg]
end SmoothLocal.Geometry
namespace SmoothLocal.Perturbation
open SmoothLocal.Geometry SmoothLocal.Flow.Reflection
theorem reflected_actual_metric_zero_patch {g0 : MetricField} {kappa : ℝ}
(eta : metricPatchSet g0 kappa) {z : Coord → ℝ} {M : ℕ}
(hclass : BoundedAdmissibleHeight (perturbedMetric g0 eta.val) M z) :
(0 : SymmetricPerturbation) ∈
metricPatchSet (reflectedMetric (perturbedMetric g0 eta.val)) kappa := by
obtain ⟨U,hU,hSU,hg,hz,hD,hE,hyy,hq⟩ := hclass.2.1
apply zero_mem_metricPatchSet
· intro p hp
exact (reflectedMetric_smoothPositive hg).2 p
(hSU ((reflectPoint_mem_modelSquare p).mpr hp))
· intro p hp
have hpr : reflectPoint p ∈ centralBox :=
(reflectPoint_mem_symmetricSquare (1/5) p).mpr hp
rw [reflectedMetric_curvature hg hU (hSU (centralBox_subset_modelSquare hpr))]
exact eta.property.2 _ hpr
end SmoothLocal.Perturbation
end
end OAI
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