| import Mathlib |
| import OAI.Analysis.FourierExtension.SmoothCharts |
| import OAI.Analysis.FourierExtension.ConvexExtension |
|
|
| namespace OAI |
|
|
| /-! Local convex graph coordinates for curved surfaces. -/ |
|
|
| open MeasureTheory |
| open scoped NNReal ENNReal ContDiff |
| noncomputable section |
| open Filter |
| open scoped Topology ContDiff |
| open MeasureTheory Set |
| open scoped ContDiff FourierTransform InnerProductSpace ENNReal |
| open MeasureTheory Set Filter Metric |
| open scoped ContDiff Topology |
| open Set Filter |
|
|
| namespace DiagonalExtension.SmoothLevel |
| abbrev Ambient := EuclideanSpace ℝ (Fin 3) |
| abbrev Plane := EuclideanSpace ℝ (Fin 2) |
|
|
| lemma tangent_finrank (L : Ambient →L[ℝ] ℝ) (hL : L ≠ 0) : |
| Module.finrank ℝ L.ker = 2 := by |
| have hL' : L.toLinearMap ≠ 0 := by |
| intro he; apply hL; ext x; exact DFunLike.congr_fun he x |
| have hh := Module.Dual.finrank_ker_add_one_of_ne_zero hL' |
| have hdim : Module.finrank ℝ Ambient = 3 := by simp [Ambient] |
| rw [hdim] at hh |
| omega |
|
|
| def tangentPlaneEquiv (L : Ambient →L[ℝ] ℝ) (hL : L ≠ 0) : L.ker ≃ₗᵢ[ℝ] Plane := |
| ((stdOrthonormalBasis ℝ L.ker).reindex (finCongr (tangent_finrank L hL))).repr |
|
|
| end DiagonalExtension.SmoothLevel |
|
|
| namespace DiagonalExtension.LocalGraph |
| open SmoothLevel Set Filter |
| open scoped Topology |
| abbrev Plane := EuclideanSpace ℝ (Fin 2) |
| abbrev Space := WithLp 2 (Plane × ℝ) |
|
|
| def graph (φ : Plane → ℝ) (u : Plane) : Space := WithLp.toLp 2 (u,φ u) |
|
|
| lemma hessian_comp_linear {V W : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] |
| [NormedAddCommGroup W] [NormedSpace ℝ W] {f : W → ℝ} (hf : ContDiff ℝ ∞ f) |
| (L : V →L[ℝ] W) (x v : V) : |
| (fderiv ℝ (fderiv ℝ (f ∘ L)) x v) v = |
| (fderiv ℝ (fderiv ℝ f) (L x) (L v)) (L v) := by |
| have h := congrArg (fun B => B ![v,v]) |
| (L.iteratedFDeriv_comp_right hf x (i := 2) (by exact WithTop.coe_le_coe.mpr le_top)) |
| simpa only [iteratedFDeriv_two_apply, ContinuousMultilinearMap.compContinuousLinearMap_apply, |
| Matrix.cons_val_zero,Matrix.cons_val_one] using h |
|
|
| theorem exists_local_graph {S : Set E3} (hS : IsSurface S) {a : E3} (ha : a ∈ S) : |
| ∃ (A : Space ≃L[ℝ] E3) (φ : Plane → ℝ) (c R : ℝ) (V : Set E3), |
| 0 < c ∧ 0 < R ∧ ContDiff ℝ ∞ φ ∧ |
| (∀ x v, c * ‖v‖^2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) ∧ |
| IsOpen V ∧ a ∈ V ∧ |
| S ∩ V ⊆ (fun u => a + A (graph φ u)) '' Metric.closedBall 0 R := by |
| obtain ⟨U,F,G,n,h,c,r,hU,haU,hF,hG,hreg,hF0,hG0,hn,hc,hr,hh0,hd0, |
| hh,hlower,hiff,hbd⟩ := hS.exists_convex_chart ha |
| let K := (fderiv ℝ F a).ker |
| let e : K ≃ₗᵢ[ℝ] Plane := tangentPlaneEquiv (fderiv ℝ F a) hreg |
| let L : (K × ℝ) ≃L[ℝ] E3 := kernelNormalEquiv (fderiv ℝ F a) n hn |
| obtain ⟨c',hc',ψ,hψ,hψh,hψout,hψcomp,hψlower⟩ := |
| ConvexNeighborhood.local_convex_extension Metric.isOpen_ball |
| (Metric.mem_ball_self hr) hh (fun v hv => |
| lt_of_lt_of_le (mul_pos hc (sq_pos_of_ne_zero (norm_ne_zero_iff.mpr hv))) |
| (hlower 0 (Metric.mem_ball_self hr) v)) (by norm_num : (0:ℝ)<1) |
| let φ : Plane → ℝ := ψ ∘ e.symm |
| have hφ : ContDiff ℝ ∞ φ := hψ.comp e.symm.toContinuousLinearEquiv.contDiff |
| have hφlower (x v : Plane) : c' * ‖v‖^2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v := by |
| rw [show φ = ψ ∘ e.symm.toContinuousLinearEquiv.toContinuousLinearMap from rfl, |
| hessian_comp_linear hψ] |
| simpa only [ContinuousLinearEquiv.coe_coe, LinearIsometryEquiv.coe_toContinuousLinearEquiv, |
| LinearIsometryEquiv.norm_map] using hψlower (e.symm x) (e.symm v) |
| let A : Space ≃L[ℝ] E3 := |
| ((WithLp.prodContinuousLinearEquiv 2 ℝ Plane ℝ).trans |
| (e.symm.toContinuousLinearEquiv.prodCongr (ContinuousLinearEquiv.refl ℝ ℝ))).trans L |
| obtain ⟨s,hs,hse⟩ := Metric.eventually_nhds_iff.mp hψh |
| let R : ℝ := min r s / 2 |
| have hR : 0 < R := half_pos (lt_min hr hs) |
| have hRr : R < r := (half_lt_self (lt_min hr hs)).trans_le (min_le_left _ _) |
| have hRs : R < s := (half_lt_self (lt_min hr hs)).trans_le (min_le_right _ _) |
| let V : Set E3 := {y | ‖L.symm (y-a)‖ < R} |
| have hV : IsOpen V := isOpen_lt (by fun_prop) continuous_const |
| have haV : a ∈ V := by simpa [V] using hR |
| refine ⟨A,φ,c',R,V,hc',hR,hφ,hφlower,hV,haV,?_⟩ |
| rintro y ⟨hyS,hyV⟩ |
| let z : K × ℝ := L.symm (y-a) |
| have hzR : ‖z‖ < R := hyV |
| have hzr : z ∈ Metric.ball 0 r := by simpa only [Metric.mem_ball,dist_zero_right] using hzR.trans hRr |
| have hyEq : a + (z.1 : E3) + z.2 • n = y := by |
| have heq := L.apply_symm_apply (y-a) |
| change (z.1 : E3) + z.2 • n = y-a at heq |
| rw [add_assoc,heq] |
| abel |
| have hheight : z.2 = h z.1 := ((hiff z hzr).2.mp (by rwa [hyEq])).1 |
| have hz1 : ‖z.1‖ < R := (norm_fst_le z).trans_lt hzR |
| have hpatch : ψ z.1 = h z.1 := hse (by simpa only [dist_zero_right] using hz1.trans hRs) |
| refine ⟨e z.1,?_,?_⟩ |
| · simpa only [Metric.mem_closedBall,dist_zero_right,LinearIsometryEquiv.norm_map] |
| using hz1.le |
| · have heval : A (graph φ (e z.1)) = L z := by |
| simp only [A,graph,ContinuousLinearEquiv.trans_apply, |
| WithLp.prodContinuousLinearEquiv_apply,ContinuousLinearEquiv.prodCongr_apply, |
| ContinuousLinearEquiv.refl_apply,φ,Function.comp_apply, |
| LinearIsometryEquiv.coe_toContinuousLinearEquiv,LinearIsometryEquiv.symm_apply_apply] |
| rw [hpatch, ← hheight] |
| change a + A (graph φ (e z.1)) = y |
| rw [heval] |
| change a + ((z.1 : E3) + z.2 • n) = y |
| rwa [← add_assoc] |
| |
| end DiagonalExtension.LocalGraph |
| |
| end |
| |
| end OAI |
| |
