Bishop-Gromov volume comparison
/ BISH-up GROH-mof /
Blow a balloon up to radius r in flat space and its volume grows like r^n. In a positively curved space it grows more slowly — the space 'runs out of room' because curvature pulls geodesics back — and in a negatively curved space it grows faster. Bishop-Gromov volume comparison turns this into a precise, global inequality: under a Ricci lower bound, the volume of a ball can never exceed the volume of the same-radius ball in the constant-curvature model, and a crucial ratio is monotone in the radius. It is the volume-side companion to the Rauch and Toponogov comparisons.
Precisely: let M be a complete Riemannian n-manifold with Ric >= (n-1) k g for a constant k. Let V(p, r) be the volume of the geodesic ball of radius r about p, and let V_k(r) be the volume of a ball of radius r in the simply connected model space of constant curvature k (the sphere, Euclidean space, or hyperbolic space). The Bishop part says V(p, r) <= V_k(r). The Gromov refinement is sharper and more useful: the ratio V(p, r) / V_k(r) is nonincreasing in r — it starts at 1 for small r and only decreases. The mechanism is to write the volume in geodesic polar coordinates as an integral of the Jacobian of exp_p, bound that Jacobian's logarithmic derivative by a Riccati comparison driven by the Ricci bound (the same Sturm comparison behind Rauch, traced on the volume element), and integrate.
Bishop-Gromov is arguably the single most-used inequality in modern Riemannian geometry. The monotone ratio is exactly what makes Gromov's precompactness theorem work — families of manifolds with a Ricci lower bound and a diameter bound are precompact in the Gromov-Hausdorff topology — which launched the whole convergence theory of metric measure spaces and underlies Cheeger-Colding theory and the analysis behind Perelman's no-local-collapsing. Honest caveats. First, the hypothesis is a Ricci lower bound only — remarkably, volume comparison does NOT need a sectional bound, which is why it is so powerful and so robust under limits. Second, the inequality is one-directional: Ricci bounded below controls volume from above, not below (you cannot conclude a lower volume bound from Ric >= k). Third, equality in Bishop forces the ball to be isometric to the model ball — the rigidity case — but generic strict inequality says nothing about the metric being close to the model.
On the unit sphere S^n (Ric = (n-1) g, so k = 1) the volume of a ball of radius r is V_1(r), which grows then peaks and shrinks as r approaches pi (the sphere is finite). Bishop-Gromov says any manifold with Ric >= (n-1) g has ball volumes V(p,r) <= V_1(r) and ratio V/V_1 nonincreasing — in particular total volume <= vol(S^n), recovering a quantitative Bonnet-Myers. For Ric >= 0 the model is flat (k = 0, V_0(r) = c_n r^n), so V(p,r)/r^n is nonincreasing: a nonnegatively-Ricci manifold has at most Euclidean volume growth, the key input to many splitting and rigidity arguments.
Ricci lower bound caps ball volume by the model and makes V/V_k decrease in r — the engine of Gromov precompactness.
Volume comparison needs only a RICCI lower bound, not a sectional one — its robustness under Gromov-Hausdorff limits flows from this. And it bounds volume from ABOVE only; Ric >= k gives no lower volume bound.