| 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 |
|
|
