File size: 3,376 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.Curvature.InitialCurvature
import OAI.Geometry.IsometricImmersion.Curvature.PrescribedCurvature
import OAI.Geometry.IsometricImmersion.Curvature.AffineCurvature
namespace OAI
noncomputable section
open scoped ContDiff Topology BigOperators Matrix
open Set
namespace SmoothLocal.Geometry
open SmoothLocal.ODE SmoothLocal.Model
def initialAssemblyMetric : MetricField := prescribedCurvatureMetric initialCurvature
theorem initialAssemblyMetric_smoothPositiveOn :
SmoothPositiveOn initialAssemblyMetric square :=
prescribedCurvatureMetric_smoothPositiveOn initialCurvature_contDiff.contDiffOn
(fun p _ => initialCurvature_small p)
theorem initialAssemblyMetric_gaussianCurvature {p : Coord} (hp : p ∈ square) :
gaussianCurvature initialAssemblyMetric p = initialCurvature p :=
prescribedCurvatureMetric_gaussianCurvature initialCurvature_contDiff.contDiffOn
(fun p _ => initialCurvature_small p) hp
theorem initialAssemblyMetric_negative_on_disk_interior (n : ℕ) {p : Coord}
(hp : p ∈ interior (accumulatingDisk n)) : gaussianCurvature initialAssemblyMetric p < 0 := by
rw [initialAssemblyMetric_gaussianCurvature
(accumulatingDisk_subset_square n (interior_subset hp))]
exact initialCurvature_negative_on_disk_interior n hp
theorem initialAssemblyMetric_zero_on_disk_frontier (n : ℕ) {p : Coord}
(hp : p ∈ frontier (accumulatingDisk n)) : gaussianCurvature initialAssemblyMetric p = 0 := by
have hdisk : p ∈ accumulatingDisk n := by
simpa only [(accumulatingDisk_isCompact n).isClosed.closure_eq] using frontier_subset_closure hp
rw [initialAssemblyMetric_gaussianCurvature (accumulatingDisk_subset_square n hdisk)]
exact initialCurvature_zero_on_disk_frontier n hp
theorem patch_core_image_mem_square (n k : ℕ) (R : OrientationLabel) (c : Coord)
(hc : c ∈ orientedExteriorCenters n k R) {q : Coord} (hq : ‖q‖ ≤ 13 / 4) :
affineCoordinates c (orientedPatchScale n k R • R.val) q ∈ square := by
apply orientedClosedPatch_subset_square n k R hc
exact ⟨q, (mem_closedPatchBox_iff_norm_le q).mpr (by linarith), rfl⟩
theorem initialAssemblyMetric_model_in_physical_coordinates (n k : ℕ)
(R : OrientationLabel) (c : Coord) (hc : c ∈ orientedExteriorCenters n k R)
{q : Coord} (hq : ‖q‖ ≤ 13 / 4) :
gaussianCurvature initialAssemblyMetric
(affineCoordinates c (orientedPatchScale n k R • R.val) q) =
initialPatchAmplitude n k R c hc * ((q 0) ^ 2 - modelProfile (q 1)) := by
rw [initialAssemblyMetric_gaussianCurvature (patch_core_image_mem_square n k R c hc hq)]
exact initialCurvature_model_on_patch n k R c hc hq
theorem initialAssemblyMetric_model_in_patch_coordinates (n k : ℕ)
(R : OrientationLabel) (c : Coord) (hc : c ∈ orientedExteriorCenters n k R)
{q : Coord} (hq : ‖q‖ ≤ 13 / 4) :
gaussianCurvature (affinePullbackMetric initialAssemblyMetric c
(orientedPatchScale n k R • R.val)) q =
modelCurvature (initialPatchAmplitude n k R c hc) q := by
rw [gaussianCurvature_affinePullback initialAssemblyMetric_smoothPositiveOn
coordinate_square_isOpen c _ (orientedPatchMatrix_isUnit n k R) q
(patch_core_image_mem_square n k R c hc hq)]
exact initialAssemblyMetric_model_in_physical_coordinates n k R c hc hq
end SmoothLocal.Geometry
end
end OAI
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