Volume Comparison and Bishop-Gromov

Volume comparison theorems relate the volume growth of geodesic balls to curvature bounds, providing fundamental tools for controlling the geometry and topology of Riemannian manifolds.

Volume of Geodesic Balls

Definition9.3Geodesic ball and volume

The geodesic ball of radius $r$ centered at $p$ is $B_r(p) = \{q \in M : d(p,q) < r\}$ . Using the exponential map and polar coordinates:

\mathrm{Vol}(B_r(p)) = \int_{S^{n-1}} \int_0^{\min(r, c(\theta))} \theta(t)^{n-1} \, dt \, d\sigma(\theta),

where $\theta(t)$ encodes the Jacobian of $\exp_p$ in the direction $\theta \in S^{n-1}$ and $c(\theta)$ is the cut distance.

The Bishop-Gromov Theorem

Theorem9.3Bishop-Gromov volume comparison

Let $(M^n, g)$ be a complete Riemannian manifold with $\mathrm{Ric} \geq (n-1)\kappa$ for some $\kappa \in \mathbb{R}$ . Then the ratio

\frac{\mathrm{Vol}(B_r(p))}{\mathrm{Vol}_\kappa(r)}

is non-increasing in $r$ , where $\mathrm{Vol}_\kappa(r)$ is the volume of a ball of radius $r$ in the model space $M^n_\kappa$ . In particular:

\mathrm{Vol}(B_r(p)) \leq \mathrm{Vol}_\kappa(r) \quad \text{for all } r > 0.

Proof

The Jacobian of the exponential map in polar coordinates is $J(t, \theta) = \det(d\exp_p|_{t\theta})$ . The volume element is $J(t, \theta) \, dt \, d\theta$ . By the Rauch/Riccati comparison (applied to the mean curvature of geodesic spheres), the Ricci bound $\mathrm{Ric} \geq (n-1)\kappa$ implies

\frac{J(t, \theta)}{J_\kappa(t)} \text{ is non-increasing in } t,

where $J_\kappa(t) = \mathrm{sn}_\kappa(t)^{n-1}$ is the Jacobian for the model space. Integrating over $S^{n-1}$ gives the monotonicity of $\mathrm{Vol}(B_r)/\mathrm{Vol}_\kappa(r)$ . $\blacksquare$

■

Applications

ExampleVolume growth rates

$\mathrm{Ric} \geq 0$ ( $\kappa = 0$ ): $\mathrm{Vol}(B_r) \leq \omega_n r^n$ (at most Euclidean growth), where $\omega_n = \mathrm{Vol}(B_1 \subset \mathbb{R}^n)$ .
$\mathrm{Ric} \geq (n-1)\kappa > 0$ : $\mathrm{Vol}(B_r) \leq \mathrm{Vol}(B_r(S^n(\kappa)))$ , and the total volume is bounded.
$\mathrm{Ric} \geq -(n-1)\kappa$ ( $\kappa > 0$ ): $\mathrm{Vol}(B_r) \leq C e^{(n-1)\sqrt{\kappa} r}$ (at most exponential growth).

Theorem9.4Volume doubling and Gromov's compactness

The Bishop-Gromov theorem implies a volume doubling property: $\mathrm{Vol}(B_{2r}) \leq C \cdot \mathrm{Vol}(B_r)$ for manifolds with $\mathrm{Ric} \geq (n-1)\kappa$ . This leads to Gromov's precompactness theorem: the space of complete $n$ -manifolds with $\mathrm{Ric} \geq (n-1)\kappa$ and $\mathrm{diam} \leq D$ is precompact in the Gromov-Hausdorff topology.

RemarkRigidity

If $\mathrm{Ric} \geq (n-1)\kappa$ and $\mathrm{Vol}(B_R(p)) = \mathrm{Vol}_\kappa(R)$ for some $R > 0$ , then $B_R(p)$ is isometric to the ball of radius $R$ in $M^n_\kappa$ . This volume rigidity gives a powerful tool for characterizing spaces with equality in the volume comparison.