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67aee1d | 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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | 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
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