| import OAI.Probability.FirstPassage.SphereCharts |
| import OAI.Probability.FirstPassage.NormBoundary |
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| namespace OAI |
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| namespace PlanarFPP |
| open Set MeasureTheory |
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|
| lemma HasC1CurveChart.preimage_isometry (S : Set Plane) (v : Plane) |
| (e : Plane ≃ₗᵢ[ℝ] Plane) (h : HasC1CurveChart S (e v)) : |
| HasC1CurveChart (e ⁻¹' S) v := by |
| obtain ⟨U,hU,hv,hvS,γ,θ,hγ,hθ,hmem,hleft,hright,hreg⟩ := h |
| refine ⟨e ⁻¹' U,hU.preimage e.continuous,hv,hvS,e.symm ∘ γ,θ ∘ e,?_,?_,?_,?_,?_,?_⟩ |
| · exact e.symm.toContinuousLinearEquiv.contDiff.comp hγ |
| · exact hθ.comp e.continuous.continuousOn (fun _ hz => hz) |
| · intro t |
| simpa only [Function.comp_apply,mem_inter_iff,mem_preimage,e.apply_symm_apply] using hmem t |
| · intro t |
| simpa only [Function.comp_apply,e.apply_symm_apply] using hleft t |
| · intro z hz |
| change e.symm (γ (θ (e z)))=z |
| rw [hright (e z) hz,e.symm_apply_apply] |
| · intro t |
| obtain ⟨d,hd,hne⟩ := hreg t |
| refine ⟨e.symm d,e.symm.toContinuousLinearEquiv.hasFDerivAt.comp_hasDerivAt t hd,?_⟩ |
| intro hz |
| exact hne (e.symm.injective (by simpa using hz)) |
|
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| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ} |
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| theorem actual_sphere_curve_chart (henv : ExponentialEnvironment P τ) (v : Plane) |
| (hv : actualSourceTimeConstant v=1) : HasC1CurveChart {z | actualSourceTimeConstant z=1} v := by |
| let μ := actualSourceTimeConstant |
| have horizontal (z : Plane) (hz : μ z=1) (hx : z 0 ≠ 0) : HasC1CurveChart {w | μ w=1} z := by |
| rcases lt_or_gt_of_ne hx with hneg | hpos |
| · exact sphere_horizontal_chart μ actualSourceTimeConstant_norm (actual_profile_contDiff henv) (-1) |
| (by norm_num) z hz (by linarith) |
| · exact sphere_horizontal_chart μ actualSourceTimeConstant_norm (actual_profile_contDiff henv) 1 |
| (by norm_num) z hz (by simpa using hpos) |
| by_cases hx : v 0 ≠ 0 |
| · exact horizontal v hv hx |
| · have hvne : v ≠ 0 := by |
| intro he |
| have hz := (actualSourceTimeConstant_norm.zero_iff v).mpr he |
| linarith |
| have hy : v 1 ≠ 0 := by |
| intro hy |
| apply hvne |
| ext i |
| fin_cases i <;> simp_all |
| have hsv : μ (swapPlane v)=1 := by dsimp [μ]; rw [actualSourceTimeConstant_swap henv,hv] |
| have hs := horizontal (swapPlane v) hsv (by simpa using hy) |
| have hh := HasC1CurveChart.preimage_isometry {z | μ z=1} v swapPlane hs |
| have he : swapPlane ⁻¹' {z | μ z=1}={z | μ z=1} := by |
| ext z |
| simp only [mem_preimage,mem_ofPred_eq,μ,actualSourceTimeConstant_swap henv] |
| rwa [he] at hh |
|
|
| theorem actual_boundary_curve (henv : ExponentialEnvironment P τ) (v : Plane) |
| (hv : v ∈ frontier {z | actualSourceTimeConstant z ≤ 1}) : |
| HasC1CurveChart (frontier {z | actualSourceTimeConstant z ≤ 1}) v := by |
| rw [norm_ball_frontier _ actualSourceTimeConstant_norm] at hv ⊢ |
| exact actual_sphere_curve_chart henv v hv |
|
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| theorem manuscriptMain (Ω : Type) [MeasurableSpace Ω] (P : Measure Ω) (τ : Edge → Ω → ℝ) |
| (henv : ExponentialEnvironment P τ) : |
| ∃ μ : Plane → ℝ, TimeConstantNorm μ ∧ IsTimeConstant P τ μ ∧ |
| DifferentiableAwayFromOrigin μ ∧ UniqueNormalizedSupports μ ∧ C1UnitSphere μ ∧ |
| (∀ v ∈ frontier {z | μ z ≤ 1}, ∃! L : Set Plane, |
| ∃ g : Plane →L[ℝ] ℝ, g ≠ 0 ∧ (∀ z, μ z ≤ 1 → g z ≤ g v) ∧ L={z | g z=g v}) ∧ |
| (∀ v ∈ frontier {z | μ z ≤ 1}, HasC1CurveChart (frontier {z | μ z ≤ 1}) v) := by |
| exact ⟨actualSourceTimeConstant,actualSourceTimeConstant_norm,actualSourceTimeConstant_isTimeConstant henv, |
| actual_differentiable_away henv, |
| unique_supports_of_differentiable _ actualSourceTimeConstant_norm (actual_differentiable_away henv), |
| actual_C1_unit_sphere henv, |
| norm_supporting_line_unique _ actualSourceTimeConstant_norm (actual_differentiable_away henv), |
| actual_boundary_curve henv⟩ |
|
|
| end PlanarFPP |
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| end OAI |
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