Bishop–Gromov inequality

Remarks

The idea of the proof: write the volume in geodesic polar coordinates about $p$, with density $\mathcal{A}(t, \theta)$ along the geodesic in direction $\theta$, set to $0$ beyond the cut locus. The Ricci bound and the Riccati equation for the mean curvature of distance spheres give that $\mathcal{A}(t, \theta) / \bar{\mathcal{A}}_k(t)$ is nonincreasing in $t$, where $\bar{\mathcal{A}}_k$ is the density of the model. Integrating over $\theta$ and $t$ gives the monotonicity of the ratio of volumes. The inequality $\operatorname{vol} B(p, r) \le V_k(r)$ is due to Bishop; the monotonicity is due to Gromov.

Example: for $\operatorname{Ric} \ge 0$ the comparison is with $\R^n$, so $\operatorname{vol} B(p, r) / (\omega_n r^n)$ is nonincreasing, where $\omega_n$ is the volume of the unit ball of $\R^n$. For $M = \R^n$ the ratio is constant equal to $1$.

Going deeper

For a student

The inequality is the basic volume estimate under a Ricci lower bound. Its monotonicity gives the doubling property $\operatorname{vol} B(p, 2r) \le C \operatorname{vol} B(p, r)$ with $C$ depending only on $n$, $k$ and an upper bound for $r$. This is the input for Gromov's precompactness theorem for the Gromov–Hausdorff distance and for many estimates in geometric analysis.