ScaledRegularData.lean`

GitHub source file

3 declarations, 0 contain sorry, 0 declared as axioms

Fixed heat constants for arbitrarily small regular neighborhoods

structure OAI.RegularPointCoordinates.RegularHeatConstants line 16
Source documentation
The constants retained before choosing the pinching tolerance.
```
structure RegularHeatConstants (n : ℕ) where
…
```
def OAI.RegularPointCoordinates.RegularHeatConstants.Matches line 33
Source documentation
Exact heat parameters, dimensionless regularizer, and curvature smallness for the normalized harmonic data.
```
def RegularHeatConstants.Matches {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} (c : RegularHeatConstants n)
    (d : RegularBilinearData hX p hk enlargement zeta) : Prop :=
…
```
Mentions: OAI.RegularPointCoordinates.NoncollapsedRCD, OAI.RegularPointCoordinates.hausdorff, OAI.RegularPointCoordinates.IsContinuousHeatKernel, OAI.RegularPointCoordinates.RegularHeatConstants, OAI.RegularPointCoordinates.RegularBilinearData
theorem OAI.RegularPointCoordinates.exists_scaled_regular_bilinear_data line 45
Source documentation
The published inputs build one such fixed set of constants and then supply the actual harmonic data for any positive tolerance and radius cap.
```
theorem exists_scaled_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