File size: 2,763 Bytes
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 | import OAI.Geometry.IsometricImmersion.Metrics.MetricLocality
import OAI.Geometry.IsometricImmersion.Metrics.ModelProfile
namespace OAI
noncomputable section
open Set Filter
open scoped ContDiff Topology
namespace SmoothLocal.Geometry
open SmoothLocal.Model
def patchBox : Set Coord :=
Icc (fun _ => (-(1 / 10 : ℝ))) (fun _ => (1 / 10 : ℝ))
def centralBox : Set Coord :=
Icc (fun _ => (-(1 / 5 : ℝ))) (fun _ => (1 / 5 : ℝ))
theorem patchBox_subset_centralBox : patchBox ⊆ centralBox := by
intro p hp
constructor <;> intro i <;> linarith [hp.1 i, hp.2 i]
theorem gaussianCurvature_eventuallyEq_of_metric_eventuallyEq
{g h : MetricField} {p : Coord} (heq : g =ᶠ[𝓝 p] h) :
gaussianCurvature g =ᶠ[𝓝 p] gaussianCurvature h := by
filter_upwards [heq.eventuallyEq_nhds] with q hq
exact gaussianCurvature_eq_of_eventuallyEq hq
theorem patch_smallCurvature_eventually_model
{g0 eta : MetricField} {kappa : ℝ} {U : Set Coord}
(hk : 0 < kappa) (hU : IsOpen U)
(hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p)
(hsupport : tsupport eta ⊆ patchBox)
(hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0 + eta) p < -kappa / 2)
{p : Coord} (hp : p ∈ U)
(hsmall : |gaussianCurvature (g0 + eta) p| ≤ kappa / 8) :
gaussianCurvature (g0 + eta) =ᶠ[𝓝 p] modelCurvature kappa := by
have hout : p ∉ patchBox := by
intro hin
have hneg := hcentral p (patchBox_subset_centralBox hin)
have hlo := (abs_le.mp hsmall).1
linarith
have hts : p ∉ tsupport eta := fun hin => hout (hsupport hin)
have hsame := gaussianCurvature_eventuallyEq_of_metric_eventuallyEq
(metric_add_eventuallyEq_of_not_mem_tsupport g0 eta hts)
have hmodel : gaussianCurvature g0 =ᶠ[𝓝 p] modelCurvature kappa := by
filter_upwards [hU.mem_nhds hp] with q hq
exact hbackground q hq
exact hsame.trans hmodel
theorem patch_smallCurvature_model_firstJet
{g0 eta : MetricField} {kappa : ℝ} {U : Set Coord}
(hk : 0 < kappa) (hU : IsOpen U)
(hbackground : ∀ p ∈ U, gaussianCurvature g0 p = modelCurvature kappa p)
(hsupport : tsupport eta ⊆ patchBox)
(hcentral : ∀ p ∈ centralBox, gaussianCurvature (g0 + eta) p < -kappa / 2)
{p : Coord} (hp : p ∈ U)
(hsmall : |gaussianCurvature (g0 + eta) p| ≤ kappa / 8) :
gaussianCurvature (g0 + eta) p = modelCurvature kappa p ∧
∀ i : Fin 2, coordPartial i (gaussianCurvature (g0 + eta)) p =
coordPartial i (modelCurvature kappa) p := by
have heq := patch_smallCurvature_eventually_model hk hU hbackground hsupport hcentral hp hsmall
exact ⟨heq.self_of_nhds, fun i => coordPartial_eq_of_eventuallyEq heq i⟩
end SmoothLocal.Geometry
end
end OAI
|