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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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | import OAI.Geometry.IsometricImmersion.Calculus.AffineHessian
import OAI.Geometry.IsometricImmersion.Immersions.RankOneOrientation
import OAI.Geometry.IsometricImmersion.Curvature.NegativeCurvatureHeight
import OAI.Geometry.IsometricImmersion.Calculus.HessianCommutator
namespace OAI
noncomputable section
open scoped ContDiff Topology BigOperators Matrix
open Filter
namespace SmoothLocal.Geometry
theorem exists_open_admissibility_neighborhood
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord}
(hg : SmoothPositiveOn g U) (hz : ContDiffOn ℝ ∞ z U) (hU : IsOpen U)
(hp : p ∈ U) (hE : 0 < heightEnergy g z p)
(hyy : covHessian g z p 1 1 ≠ 0)
(hq : |covHessian g z p 0 1 / covHessian g z p 1 1| < (1 / 100 : ℝ)) :
∃ V : Set Coord, IsOpen V ∧ p ∈ V ∧ V ⊆ U ∧
∀ q ∈ V, 0 < heightEnergy g z q ∧ covHessian g z q 1 1 ≠ 0 ∧
|covHessian g z q 0 1 / covHessian g z q 1 1| < (1 / 100 : ℝ) := by
have hEc : ContinuousAt (heightEnergy g z) p :=
(heightEnergy_contDiffOn hg hU hz).continuousOn.continuousAt (hU.mem_nhds hp)
have hHc (i j : Fin 2) : ContinuousAt (fun q => covHessian g z q i j) p :=
(covHessian_contDiffOn hg hU hz i j).continuousOn.continuousAt (hU.mem_nhds hp)
have hqc : ContinuousAt
(fun q => |covHessian g z q 0 1 / covHessian g z q 1 1|) p :=
((hHc 0 1).div (hHc 1 1) hyy).abs
have hgood : {q | q ∈ U ∧ 0 < heightEnergy g z q ∧
covHessian g z q 1 1 ≠ 0 ∧
|covHessian g z q 0 1 / covHessian g z q 1 1| < (1 / 100 : ℝ)} ∈ 𝓝 p := by
filter_upwards [hU.mem_nhds hp,
continuousAt_const.eventually_lt hEc hE,
(hHc 1 1).eventually_ne hyy,
hqc.eventually_lt continuousAt_const hq] with q hqU hqE hqyy hqq
exact ⟨hqU, hqE, hqyy, hqq⟩
obtain ⟨V, hVsub, hVo, hpV⟩ := mem_nhds_iff.mp hgood
exact ⟨V, hVo, hpV, fun q hqV => (hVsub hqV).1,
fun q hqV => (hVsub hqV).2⟩
theorem exists_open_admissibility_at_critical
{g : MetricField} {z : Coord → ℝ} {U : Set Coord} {p : Coord}
(hg : SmoothPositiveOn g U) (hz : ContDiffOn ℝ ∞ z U) (hU : IsOpen U)
(hp : p ∈ U) (hcrit : fderiv ℝ z p = 0)
(hyy : covHessian g z p 1 1 ≠ 0)
(hq : |covHessian g z p 0 1 / covHessian g z p 1 1| < (1 / 100 : ℝ)) :
∃ V : Set Coord, IsOpen V ∧ p ∈ V ∧ V ⊆ U ∧
∀ q ∈ V, 0 < heightEnergy g z q ∧ covHessian g z q 1 1 ≠ 0 ∧
|covHessian g z q 0 1 / covHessian g z q 1 1| < (1 / 100 : ℝ) := by
apply exists_open_admissibility_neighborhood hg hz hU hp _ hyy hq
rw [heightEnergy_at_critical g z p hcrit]
exact (hg.2 p hp).det_pos
theorem exists_oriented_immersion_admissibility
{g : MetricField} {F : Coord → Ambient} {U : Set Coord} {b : Coord}
(hg : SmoothPositiveOn g U) (hF : IsometricOn g F U) (hU : IsOpen U)
(hb : b ∈ U) (n : Ambient) (hn : IsUnitNormalAt F n b)
(hrank : (secondFundamental F n b).rank = 1) :
∃ R ∈ orientationRotations, ∃ V : Set Coord,
IsOpen V ∧ (0 : Coord) ∈ V ∧ V ⊆ affineCoordinates b R ⁻¹' U ∧
SmoothPositiveOn (affinePullbackMetric g b R) V ∧
IsometricOn (affinePullbackMetric g b R) (affinePullbackImmersion F b R) V ∧
∀ p ∈ V,
let gR := affinePullbackMetric g b R
let zR := height (affinePullbackImmersion F b R) n
(covHessian gR zR p).det = gaussianCurvature gR p * heightEnergy gR zR p ∧
0 < heightEnergy gR zR p ∧ covHessian gR zR p 1 1 ≠ 0 ∧
|covHessian gR zR p 0 1 / covHessian gR zR p 1 1| ≤ (1 / 100 : ℝ) := by
have hz := height_smooth hF.1 n
have hnpoint := geometric_height_normal_point hF hU hb hn
have hsym : (covHessian g (height F n) b)ᵀ = covHessian g (height F n) b := by
ext i j
exact covHessian_symm hg hU hz hb j i
have hHrank : (covHessian g (height F n) b).rank = 1 := by
rw [hnpoint.2]
exact hrank
obtain ⟨R, hRnet, hRyy, hRq⟩ :=
exists_orientationRotation_for_rank_one (covHessian g (height F n) b) hsym hHrank
have hRinj := mulVec_injective_of_transpose_mul_eq_one R
(orientationRotations_properties hRnet).1
let W := affineCoordinates b R ⁻¹' U
let gR := affinePullbackMetric g b R
let FR := affinePullbackImmersion F b R
let zR := height FR n
have hWo : IsOpen W := isOpen_affine_preimage hU b R
have h0W : (0 : Coord) ∈ W := by
change affineCoordinates b R 0 ∈ U
simpa only [affineCoordinates_zero] using hb
have hgR : SmoothPositiveOn gR W := affinePullbackMetric_smoothPositive hg b R hRinj
have hFR : IsometricOn gR FR W := affinePullbackImmersion_isometric hF hU b R
have hzR : ContDiffOn ℝ ∞ zR W := height_smooth hFR.1 n
have hcritR : fderiv ℝ zR 0 = 0 := by
change fderiv ℝ ((height F n) ∘ affineCoordinates b R) 0 = 0
apply fderiv_affine_comp_eq_zero b R 0
· simpa only [affineCoordinates_zero] using
(((hz b hb).contDiffAt (hU.mem_nhds hb)).differentiableAt (by simp))
· simpa only [affineCoordinates_zero] using hnpoint.1
have hHR : covHessian gR zR 0 = Rᵀ * covHessian g (height F n) b * R := by
change covHessian (affinePullbackMetric g b R)
((height F n) ∘ affineCoordinates b R) 0 = _
exact covHessian_affinePullback_at_zero g hz hU b R hb hnpoint.1
have hyyR : covHessian gR zR 0 1 1 ≠ 0 := by rw [hHR]; exact hRyy
have hqR : |covHessian gR zR 0 0 1 / covHessian gR zR 0 1 1| < (1 / 100 : ℝ) := by
rw [hHR]
exact hRq
obtain ⟨V, hVo, h0V, hVW, hVad⟩ :=
exists_open_admissibility_at_critical hgR hzR hWo h0W hcritR hyyR hqR
refine ⟨R, hRnet, V, hVo, h0V, hVW, ?_, ?_, ?_⟩
· exact ⟨fun i j => (hgR.1 i j).mono hVW, fun p hp => hgR.2 p (hVW hp)⟩
· exact ⟨hFR.1.mono hVW, fun p hp => hFR.2 p (hVW hp)⟩
· intro p hp
have heq := geometric_height_equation hgR hFR hWo n hn.1 (hVW hp)
have had := hVad p hp
refine ⟨?_, had.1, had.2.1, had.2.2.le⟩
simpa only [heightEnergy, mul_assoc] using heq
end SmoothLocal.Geometry
end
end OAI
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