---
tag: 82F5Q4F1S6
kind: statement
role: theorem
name: Bishop–Gromov inequality
msc: [53C21, 53C20, 53C23]
uses: [7CBT4SM83Y, 4GTNM0WHDH, B4QR306D8G, 1VM1AQT956]
about: [4GTNM0WHDH]
sources:
  - url: https://en.wikipedia.org/wiki/Bishop%E2%80%93Gromov_inequality
same_as:
  wikidata: Q4918129
  wikipedia: Bishop–Gromov inequality
---

## Statement

Let $(M, g)$ be a connected complete [[Riemannian manifold|7CBT4SM83Y]] of dimension $n \ge 2$, and let $k$ be a real number.
Assume that the [[Ricci curvature|4GTNM0WHDH]] satisfies $\operatorname{Ric}(v, v) \ge (n-1) k \, g(v, v)$ for every tangent vector $v$.
For $r > 0$ let $V_k(r)$ be the volume of a ball of radius $r$ in the simply connected $n$-dimensional [[space form|1VM1AQT956]] of curvature $k$.
Let $p \in M$ and let $B(p, r)$ be the ball of radius $r$ about $p$ for the [[Riemannian distance|B4QR306D8G]].
Let $R = \pi/\sqrt{k}$ if $k > 0$ and $R = \infty$ if $k \le 0$.
Then the function
$$r \mapsto \frac{\operatorname{vol} B(p, r)}{V_k(r)}$$
is nonincreasing on $(0, R)$.
Since it tends to $1$ as $r \to 0$, it follows that $\operatorname{vol} B(p, r) \le V_k(r)$ for $0 < r < R$.

## 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 $1$.

## Explanations

### L1 human

A lower bound on Ricci curvature means volumes grow no faster than in the model space.
Balls in $M$ are never larger than balls of the same radius in the model, and the ratio of the two volumes can only decrease as the radius grows.

### L2 human

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.