Current state
bishopGromov_volume_comparison (in OpenGALib/Comparison/BishopGromov/VolumeComparison.lean) has its final signature in place (commit 9fd1435):
```lean
theorem bishopGromov_volume_comparison
(hRic : ∀ x v, ((n_M : ℝ) - 1) * K * ⟪v, v⟫_g ≤ Ric_g(v, v) x)
(p : M) {r R : ℝ} (hr : r ∈ 𝒟_K) (hR : R ∈ 𝒟_K) (hrR : r ≤ R) :
dV_g[(HasMetric.metric : RiemannianMetric I M)].real B(p, R) / V_K^n_M(R) ≤
dV_g[(HasMetric.metric : RiemannianMetric I M)].real B(p, r) / V_K^n_M(r) := by
sorry
```
The sorry is the only thing remaining. After #9 (explicit-g refactor) the signature becomes cleaner.
Proof outline (do Carmo Ch.10 §2 / Petersen Ch.9 §27)
-
Exponential chart polar decomposition: in normal coordinates centered at p,
```
vol_g(B(p, r)) = ∫_0^r ∫_{S_p^{n-1}} J_g(s, v) ds dσ(v)
```
where J_g(s, v) = det(d_{sv} exp_p) is the Jacobian of the exponential map and dσ is the round measure on the unit sphere.
-
Jacobi field equation: J_g(s, v) = ‖J_1 ∧ ... ∧ J_{n-1}‖ where J_i are Jacobi fields along t ↦ exp_p(tv).
-
Comparison via Riccati: the Ricci lower bound Ric ≥ (n-1)K implies J_g(s, v) ≤ J_K(s) where J_K is the constant-curvature reference Jacobian. By Rauch comparison.
-
Integrate to get monotonicity:
```
vol_g(B(p, R)) / V_K^n(R) ≤ vol_g(B(p, r)) / V_K^n(r)
```
via the integral comparison: if f/F is monotone in s for s ≤ s', s' ≤ s'', then ratio of integrals is monotone.
Dependencies
- Blocked-upstream / framework-self-build: Mathlib's
Riemannian.expMap is gappy (Phase 4 of riemannian-volume). For OpenGA to prove BG, either:
- Mathlib
expMap matures, OR
- OpenGA self-builds an exponential-chart polar-coordinate decomposition lemma.
- Jacobi field infrastructure: OpenGA doesn't have Jacobi field machinery yet. Needs to be built (or imported from Mathlib if available).
- Rauch comparison: Mathlib status unknown.
Estimated scope
~500 LOC for the proof itself if all prerequisites are in place. Probably 2-3x that with prerequisites.
Acceptance
bishopGromov_volume_comparison 0 sorries.
docs/SORRY_CATALOG.md updated.
- CI
EXPECTED decreased by 1.
See also
Current state
bishopGromov_volume_comparison(inOpenGALib/Comparison/BishopGromov/VolumeComparison.lean) has its final signature in place (commit9fd1435):```lean
theorem bishopGromov_volume_comparison
(hRic : ∀ x v, ((n_M : ℝ) - 1) * K * ⟪v, v⟫_g ≤ Ric_g(v, v) x)
(p : M) {r R : ℝ} (hr : r ∈ 𝒟_K) (hR : R ∈ 𝒟_K) (hrR : r ≤ R) :
dV_g[(HasMetric.metric : RiemannianMetric I M)].real B(p, R) / V_K^n_M(R) ≤
dV_g[(HasMetric.metric : RiemannianMetric I M)].real B(p, r) / V_K^n_M(r) := by
sorry
```
The
sorryis the only thing remaining. After #9 (explicit-g refactor) the signature becomes cleaner.Proof outline (do Carmo Ch.10 §2 / Petersen Ch.9 §27)
Exponential chart polar decomposition: in normal coordinates centered at
p,```
vol_g(B(p, r)) = ∫_0^r ∫_{S_p^{n-1}} J_g(s, v) ds dσ(v)
```
where
J_g(s, v) = det(d_{sv} exp_p)is the Jacobian of the exponential map anddσis the round measure on the unit sphere.Jacobi field equation:
J_g(s, v) = ‖J_1 ∧ ... ∧ J_{n-1}‖whereJ_iare Jacobi fields alongt ↦ exp_p(tv).Comparison via Riccati: the Ricci lower bound
Ric ≥ (n-1)KimpliesJ_g(s, v) ≤ J_K(s)whereJ_Kis the constant-curvature reference Jacobian. By Rauch comparison.Integrate to get monotonicity:
```
vol_g(B(p, R)) / V_K^n(R) ≤ vol_g(B(p, r)) / V_K^n(r)
```
via the integral comparison: if
f/Fis monotone insfors ≤ s', s' ≤ s'', then ratio of integrals is monotone.Dependencies
Riemannian.expMapis gappy (Phase 4 of riemannian-volume). For OpenGA to prove BG, either:expMapmatures, OREstimated scope
~500 LOC for the proof itself if all prerequisites are in place. Probably 2-3x that with prerequisites.
Acceptance
bishopGromov_volume_comparison0 sorries.docs/SORRY_CATALOG.mdupdated.EXPECTEDdecreased by 1.See also
OpenGALib/Riemannian/Volume/Exponential.lean— Phase 4 of riemannian-volume, sorry'd, related.