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Bishop volume upper bound

Statement

Assume the inherited Axiom of Countable Choice ${A C}_{ω}$ . Let $(M, g)$ be a complete, connected, boundaryless Riemannian manifold of dimension $n \geq 2$ with $Ric \geq (n - 1) k g$ for a real number $k$ , let $p \in M$ , let $B (p, r) = {q \in M : d_{g} (p, q) < r}$ be the open metric ball and let $V_{k}^{⋆}$ be the saturated model ball volume of Model space radial area and ball volume. Then ${vol}_{g} (B (p, r)) \leq V_{k}^{⋆} (r) for every r > 0.$ No compactness of $M$ is assumed, and no choice beyond the inherited ${A C}_{ω}$ is used.

Facts & Assumptions

Given: The inherited ${A C}_{ω}$ of [A1]; a complete, connected, boundaryless Riemannian manifold $(M, g)$ of dimension $n \geq 2$ with $Ric \geq (n - 1) k g$ ; a point $p \in M$ ; the ball volume function $r \mapsto {vol}_{g} (B (p, r))$ ; and the saturated model volume $V_{k}^{⋆}$ .

[A1]

The countable-choice premise is the inherited ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), carried by the Bishop–Gromov comparison cited below.

[F1]

Bishop–Gromov volume comparison (Bishop gromov volume comparison): under the stated hypotheses the Bishop–Gromov ratio $R_{p} (r) = \frac{{vol}_{g} (B (p, r))}{V_{k}^{⋆} (r)}, r > 0,$ is well defined, nonincreasing on $(0, \infty)$ and satisfies $\lim_{r ↓ 0} R_{p} (r) = 1$ ; in particular ${vol}_{g} (B (p, r)) \leq V_{k}^{⋆} (r)$ for every $r > 0$ .

[F2]

Positivity of the model volume (Model space radial area and ball volume, Comparison sine, cosine and cotangent functions): the model radial area is $A_{k} (r) = ω_{n - 1} {sn}_{k} (r)^{n - 1}$ and $V_{k} (r) = \int_{0}^{r} A_{k} (t) d t$ ; the comparison sine is positive on its positive domain $(0, π / \sqrt{k})$ for $k > 0$ and $(0, \infty)$ for $k \leq 0$ , and $ω_{n - 1} > 0$ , so $A_{k}$ is positive there and $V_{k}^{⋆} (r) > 0$ for every $r > 0$ (for $k > 0$ by saturation at $π / \sqrt{k}$ ).

Proof

Proof technique: direct: evaluate the Bishop–Gromov ratio against its limit one at the origin, using monotonicity along a sequence of radii decreasing to zero.

1.1F1given

The ratio is at most one. [F1, given] Fix $r > 0$ . By the monotonicity in [F1], $0 < s < r$ implies $R_{p} (r) \leq R_{p} (s)$ . Choosing any sequence $s_{j} ↓ 0$ and using $\lim_{s ↓ 0} R_{p} (s) = 1$ from [F1], $R_{p} (r) \leq \lim_{s ↓ 0} R_{p} (s) = 1.$

2.1F2step 1.1∎

Conclusion. [F2, step 1.1] Multiplying the inequality of step 1.1 by the positive number $V_{k}^{⋆} (r)$ [F2] gives ${vol}_{g} (B (p, r)) = R_{p} (r) V_{k}^{⋆} (r) \leq V_{k}^{⋆} (r),$ which is the asserted bound. The value $r = 0$ is excluded, where $B (p, 0) = \emptyset$ and $V_{k}^{⋆} (0) = 0$ ; for $k > 0$ the bound is constant from the model pole onward, and for $k \leq 0$ it is the genuine model volume. No step selects a direction, a sequence of directions, or a family of balls beyond the single radius sequence already carried by the limit in [F1], so no choice beyond [A1] is used.

Source locator

Datar §§27.2 and 28.1, pp.200–209, and Eschenburg §§4–5, pp.15–20, record the Bishop–Gromov ratio as nonincreasing with limit one at the origin, which is exactly the statement that the ball volume is bounded by the model volume. The proof above is the one-line unpacking of the in-run Bishop–Gromov theorem.

Depends on

Used by

Complete noncompact manifolds with nonnegative ricci curvature have at most euclidean volume growth Corollary

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ved Datar, Lectures on Riemannian Geometry (2025) (standard reference, not scraped)
J.-H. Eschenburg, Comparison Theorems in Riemannian Geometry (standard reference, not scraped)