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Volume doubling under a nonnegative ricci lower 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 $p \in M$ , and let $V_{k}^{⋆}$ be the saturated model ball volume of Model space radial area and ball volume. Then for every $r > 0$ , ${vol}_{g} (B (p, 2 r)) \leq \frac{V_{k}^{⋆} (2 r)}{V_{k}^{⋆} (r)} {vol}_{g} (B (p, r)),$ so the ball volume is doubled at the cost of the explicit model factor. When $k = 0$ this factor is exactly $\frac{V_{0}^{⋆} (2 r)}{V_{0}^{⋆} (r)} = 2^{n},$ while for $k < 0$ it is the scale-dependent model ratio $\frac{V_{k}^{⋆} (2 r)}{V_{k}^{⋆} (r)} = \frac{\int_{0}^{2 r} \sinh^{n - 1} (\sqrt{- k} t) d t}{\int_{0}^{r} \sinh^{n - 1} (\sqrt{- k} t) d t},$ a function of the product $\sqrt{- k} r$ alone; no uniform bound independent of $r$ is asserted for $k < 0$ . For $k > 0$ each model volume is saturated once its own radius reaches the model pole, so the factor is $\int_{0}^{\min {2 r, π / \sqrt{k}}} {sn}_{k}^{n - 1} / \int_{0}^{\min {r, π / \sqrt{k}}} {sn}_{k}^{n - 1}$ . 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 volumes ${vol}_{g} (B (p, \cdot))$ ; 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 and change-of-variables suppliers below.

[F1]

Bishop–Gromov volume comparison (Bishop gromov volume comparison): the ratio $R_{p} (r) = {vol}_{g} (B (p, r)) / V_{k}^{⋆} (r)$ is well defined, nonincreasing on $(0, \infty)$ , tends to $1$ as $r ↓ 0$ , and satisfies ${vol}_{g} (B (p, r)) \leq V_{k}^{⋆} (r) < + \infty$ for every $r > 0$ .

[F2]

Model volume (Model space radial area and ball volume, Comparison sine, cosine and cotangent functions): $V_{k}^{⋆} (r) = V_{k} (\min {r, π / \sqrt{k}}) = ω_{n - 1} \int_{0}^{\min {r, π / \sqrt{k}}} {sn}_{k} (t)^{n - 1} d t$ for $k > 0$ and $V_{k}^{⋆} (r) = V_{k} (r) = ω_{n - 1} \int_{0}^{r} {sn}_{k} (t)^{n - 1} d t$ for $k \leq 0$ , where $ω_{n - 1} > 0$ ; moreover ${sn}_{0} (t) = t$ and, for $k < 0$ , ${sn}_{k} (t)^{n - 1} = \frac{\sinh^{n - 1} (\sqrt{- k} t)}{(\sqrt{- k})^{n - 1}}, V_{k} (r) = \frac{ω_{n - 1}}{(\sqrt{- k})^{n - 1}} \int_{0}^{r} \sinh^{n - 1} (\sqrt{- k} t) d t .$

[F3]

Change of variables (A C^1 diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions): for a $C^{1}$ diffeomorphism $T : U \to V$ of open subsets of $R$ and every nonnegative Lebesgue measurable $f : V \to [0, \infty]$ , $\int_{V} f (y) d λ_{1} (y) = \int_{U} f (T (x)) ∣ T^{'} (x) ∣ d λ_{1} (x) .$

[F4]

Powers (Laws of integer exponents, Integer powers $a^{m}$ ): for real $a, b$ and natural $m$ , $(a b)^{m} = a^{m} b^{m}$ , and $a^{0} = 1$ .

Proof

technique · direct: the Bishop–Gromov ratio is nonincreasing, and for the two radii $r<2r$ this rearrangement gives the doubling inequality; the model factors are then evaluated from the explicit comparison-sine formulas, with the homogeneous substitution $t\mapsto 2t$ for the Lebesgue integral

1.1F1given

The doubling inequality. [F1, given] Fix $r > 0$ . By monotonicity of the Bishop–Gromov ratio in [F1], $\frac{{vol}_{g} (B (p, 2 r))}{V_{k}^{⋆} (2 r)} = R_{p} (2 r) \leq R_{p} (r) = \frac{{vol}_{g} (B (p, r))}{V_{k}^{⋆} (r)} .$ Multiplying by the positive number $V_{k}^{⋆} (r) V_{k}^{⋆} (2 r)$ [F1, F2] gives ${vol}_{g} (B (p, 2 r)) \leq \frac{V_{k}^{⋆} (2 r)}{V_{k}^{⋆} (r)} {vol}_{g} (B (p, r)),$ the asserted inequality, both volumes being finite by [F1].

1.2F2F3F4

The factor for $k = 0$ is $2^{n}$ . [F2, F3, F4] By [F2], $V_{0}^{⋆} (r) = ω_{n - 1} \int_{0}^{r} t^{n - 1} d t$ for every $r > 0$ . The map $T : (0, r) \to (0, 2 r)$ , $T (x) = 2 x$ , is a $C^{1}$ diffeomorphism with $T^{'} (x) = 2$ , and $t \mapsto t^{n - 1}$ is nonnegative and continuous, hence Lebesgue measurable; [F3] applied to it gives, using [F4] to expand $(2 x)^{n - 1} = 2^{n - 1} x^{n - 1}$ , $\int_{0}^{2 r} t^{n - 1} d t = \int_{0}^{r} (2 x)^{n - 1} \cdot 2 d x = 2^{n - 1} \cdot 2 \int_{0}^{r} x^{n - 1} d x = 2^{n} \int_{0}^{r} t^{n - 1} d t .$ The common factor $ω_{n - 1}$ cancels in the quotient, so $V_{0}^{⋆} (2 r) / V_{0}^{⋆} (r) = 2^{n}$ .

1.3F2F3

The factor for $k < 0$ . [F2, F3] Write $σ : = \sqrt{- k} > 0$ . By the substituted formula of [F2], the common factor $ω_{n - 1} σ^{1 - n}$ cancels in the quotient and $\frac{V_{k}^{⋆} (2 r)}{V_{k}^{⋆} (r)} = \frac{\int_{0}^{2 r} \sinh^{n - 1} (σ t) d t}{\int_{0}^{r} \sinh^{n - 1} (σ t) d t} .$ Applying the substitution $T (x) = 2 x$ of [F3] to the numerator, $\int_{0}^{2 r} \sinh^{n - 1} (σ t) d t = 2 \int_{0}^{r} \sinh^{n - 1} (2 σ x) d x$ ; applying it once more with $x = r s$ to both numerator and denominator shows that the quotient is a function of the product $σ r = \sqrt{- k} r$ alone.

2.1F2step 1.1step 1.2step 1.3∎

Conclusion. [F2, step 1.1, step 1.2, step 1.3] Step 1.1 is the doubling inequality for every real $k$ and every $r > 0$ . Its factor is $2^{n}$ when $k = 0$ by step 1.2 and is the displayed function of $\sqrt{- k} r$ when $k < 0$ by step 1.3; in particular the factor for $k < 0$ is scale-dependent and no constant independent of $r$ is asserted. For $k > 0$ , $V_{k}^{⋆} (2 r) = V_{k} (π / \sqrt{k})$ once $2 r \geq π / \sqrt{k}$ ; the denominator reaches this value only when $r \geq π / \sqrt{k}$ . In general the saturated formula of [F2] gives the displayed quotient of integrals over $(0, \min {2 r, π / \sqrt{k}})$ and $(0, \min {r, π / \sqrt{k}})$ , which is $1$ for $r \geq π / \sqrt{k}$ . The case $n = 2$ is included with exponent $n - 1 = 1$ ; the endpoint $r = 0$ is excluded. Only the single substitution $t \mapsto 2 t$ and the inherited [A1] are used, so no additional choice is made.

Source locator

Datar §§27.2 and 28.1, pp.200–209, and Eschenburg §§4–5, pp.15–20, record the Bishop–Gromov ratio and its model comparison; the doubling form is the rearrangement of the monotonicity at the two radii $r < 2 r$ . The Euclidean factor $2^{n}$ is the homogeneity of the model density $t^{n - 1}$ , and the negative-curvature factor is the corresponding hyperbolic-sine quotient of the model volume definition.

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Sources

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