RegularBilinearData.lean`

GitHub source file

3 declarations, 0 contain sorry, 0 declared as axioms

The constructed common harmonic and heat data

structure OAI.RegularPointCoordinates.RegularBilinearData line 18
Source documentation
A record of the already constructed heat and harmonic objects. No chart, directional pinching, or metric-iteration conclusion is a field.
```
structure RegularBilinearData {K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X)
    [(hausdorff n X).Regular]
    (p : X) {ker : ℝ → X → X → ℝ}
    (hk : IsContinuousHeatKernel (hausdorff n X) hX.quadratic ker)
    (enlargement zeta : ℝ) where
…
```
Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.IsContinuousHeatKernel
theorem OAI.RegularPointCoordinates.exists_regular_bilinear_data_uniform_degree line 105
Source documentation
This bundles the proved capped construction, retaining the same actual coordinate functions, Gram operators and positive measure.
```
theorem exists_regular_bilinear_data_uniform_degree {K : ℝ} (m : ℕ)
    (hX : NoncollapsedRCD K (m + 2) X) [(hausdorff (m + 2) X).Regular]
    {p : X} {a : ℝ} (ha : 0 ≤ a)
    (hcompact : PointedRescalingPrecompactness p) (hregular : IsRegular (m + 2) p)
    (hvolume : EuclideanRescalingVolumeConvergence (hausdorff (m + 2) X) (m + 2) p
      (euclideanUnitVolume (m + 2)))
    (hmeasured : EuclideanMeasuredRescalingConvergence (hausdorff (m + 2) X) (m + 2) p)
    (hBG : BallVolumeComparison (hausdorff (m + 2) X) (comparisonModelVolume (m + 1) a))
    (hdensity : ∀ y, SmallScaleVolumeUpper (hausdorff (m + 2) X)
      (comparisonModelVolume (m + 1) a) y (euclideanUnitVolume (m + 2)))
    (hinitial : LocalInitialSplittingInput hX) (htrans : LocalTransformationInput hX)
    (hsplit : LocalFlatSplittingInput hX) (hrigidity : UniformRegularFlatnessInput hX)
…
```
Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.PointedRescalingPrecompactness, OAI.RegularPointCoordinates.IsRegular, OAI.RegularPointCoordinates.EuclideanRescalingVolumeConvergence, OAI.RegularPointCoordinates.euclideanUnitVolume, OAI.RegularPointCoordinates.EuclideanMeasuredRescalingConvergence, OAI.RegularPointCoordinates.BallVolumeComparison, OAI.RegularPointCoordinates.comparisonModelVolume, OAI.RegularPointCoordinates.SmallScaleVolumeUpper, OAI.RegularPointCoordinates.LocalInitialSplittingInput, OAI.RegularPointCoordinates.LocalTransformationInput, OAI.RegularPointCoordinates.LocalFlatSplittingInput, OAI.RegularPointCoordinates.UniformRegularFlatnessInput
theorem OAI.RegularPointCoordinates.exists_regular_bilinear_data line 154
Source documentation
Forget the fixed degree when only a single neighborhood is needed.
```
theorem exists_regular_bilinear_data {K : ℝ} (m : ℕ)
    (hX : NoncollapsedRCD K (m + 2) X) [(hausdorff (m + 2) X).Regular]
    {p : X} {a : ℝ} (ha : 0 ≤ a)
    (hcompact : PointedRescalingPrecompactness p) (hregular : IsRegular (m + 2) p)
    (hvolume : EuclideanRescalingVolumeConvergence (hausdorff (m + 2) X) (m + 2) p
      (euclideanUnitVolume (m + 2)))
    (hmeasured : EuclideanMeasuredRescalingConvergence (hausdorff (m + 2) X) (m + 2) p)
    (hBG : BallVolumeComparison (hausdorff (m + 2) X) (comparisonModelVolume (m + 1) a))
    (hdensity : ∀ y, SmallScaleVolumeUpper (hausdorff (m + 2) X)
      (comparisonModelVolume (m + 1) a) y (euclideanUnitVolume (m + 2)))
    (hinitial : LocalInitialSplittingInput hX) (htrans : LocalTransformationInput hX)
    (hsplit : LocalFlatSplittingInput hX) (hrigidity : UniformRegularFlatnessInput hX)
…
```
Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.PointedRescalingPrecompactness, OAI.RegularPointCoordinates.IsRegular, OAI.RegularPointCoordinates.EuclideanRescalingVolumeConvergence, OAI.RegularPointCoordinates.euclideanUnitVolume, OAI.RegularPointCoordinates.EuclideanMeasuredRescalingConvergence, OAI.RegularPointCoordinates.BallVolumeComparison, OAI.RegularPointCoordinates.comparisonModelVolume, OAI.RegularPointCoordinates.SmallScaleVolumeUpper, OAI.RegularPointCoordinates.LocalInitialSplittingInput, OAI.RegularPointCoordinates.LocalTransformationInput, OAI.RegularPointCoordinates.LocalFlatSplittingInput, OAI.RegularPointCoordinates.UniformRegularFlatnessInput