An improved volume bound under Ricci and scalar curvature lower bounds

Abstract: We study volume comparison for closed Riemannian manifolds satisfying a positive Ricci curvature lower bound together with an improved scalar curvature lower bound. Using an integral extension of the finite shuffling comparison for scalar Jacobi solutions due to Brown and Freedman \cite{BrownFreedman2022}, we prove that if a closed Riemannian manifold $(N^n, g)$ satisfies $\operatorname{Ric}_g\ge (n-1)g$ and the scalar curvature $R_g\ge n(n-1)(1+\varepsilon)$, then its volume satisfies $$\lvert N\rvert_g \le \frac{1}{\sqrt{1+n\varepsilon}}\lvert\mathbb S^n\rvert.$$ In fact, assuming only $\mathrm{Ric}_g\ge(n-1)g$, we can prove that $$\frac{|N|_g}{\left|\mathbb{S}^n\right|} \le \frac{1}{|N|_g} \int_N\left(\frac{R_g}{n-1}-(n-1)\right)^{{-\frac{1}{2}}} d \mathrm{vol}_g.$$ The equality holds if and only if $N$ is isometric to the unit sphere. The proof combines a coefficient-adapted Jacobian comparison with the integral shuffling comparison. This yields an estimate that retains the full Ricci spectrum. The resulting volume bound agrees to first order in $\varepsilon$ with the factor predicted by Bray's conjecture. A further consequence of the argument is an averaged volume comparison for metric balls involving the scalar curvature.

• The paper proves that a closed n-manifold with Ricci curvature at least (n−1)g and scalar curvature at least n(n−1)(1+ε) has volume ratio at most (1+nε)^−1/2, with equality only for the unit sphere when ε=0.
• The paper develops coefficient-adapted Jacobian comparison and an integral shuffling theorem over the geodesic flow to retain directional Ricci information and derive sharper scalar-curvature and Ricci-spectrum volume estimates.
• The paper improves Bishop’s bound in every dimension n≥3 but does not resolve Bray’s conjecture for n≥4, leaving an order-ε² gap between its bound and Bray’s predicted factor.
• follow_up_questions

Overview and main results

This paper by Kwok-Kun Kwong establishes a quantitative improvement of Bishop's volume comparison theorem for closed Riemannian manifolds satisfying a Ricci curvature lower bound together with an improved scalar curvature lower bound. The main theorem states that if a closed $n$-manifold ($n \ge 3$) satisfies $\operatorname{Ric}_g \ge (n-1)g$ and $R_g \ge n(n-1)(1+\varepsilon)$ for some $\varepsilon \ge 0$, then

$$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$

with equality if and only if $N$ is isometric to the unit sphere and $\varepsilon = 0$. For every $\varepsilon > 0$, the right-hand side is strictly less than one, so the result gives a strict improvement over Bishop's bound in all dimensions, without any symmetry, $C^2$-closeness, or upper Ricci bound assumptions.

The uniform estimate follows from a stronger, pointwise-in-scalar-curvature statement: assuming only $n \ge 3$0,

$n \ge 3$1

This averaged form retains the full scalar curvature distribution rather than only its infimum. A still sharper intermediate result bounds the volume by the average of $n \ge 3$2, which involves the full spectrum of the Ricci tensor.

Relation to Bray's conjecture

The motivation is a conjecture of Bray from his thesis on the Riemannian Penrose inequality: for each $n \ge 3$3 there should exist $n \ge 3$4 such that $n \ge 3$5 and $n \ge 3$6 imply $n \ge 3$7. In dimension three this is essentially settled: Bray's football theorem provides a sharp function $n \ge 3$8, Gursky–Viaclovsky proved rigorously that $n \ge 3$9, and Brendle established the rigidity case using the Gauss equation and Gauss–Bonnet on two-dimensional isoperimetric surfaces — a step that does not extend to higher dimensions.

The paper's estimate compares to Bray's predicted factor as follows. By Bernoulli's inequality, $\operatorname{Ric}_g \ge (n-1)g$0 for $\operatorname{Ric}_g \ge (n-1)g$1, so the new bound does not reach Bray's conjectured volume factor. However, binomial expansion shows

$\operatorname{Ric}_g \ge (n-1)g$2

so the two factors agree to first order in $\operatorname{Ric}_g \ge (n-1)g$3 and differ only at order $\operatorname{Ric}_g \ge (n-1)g$4. The paper is explicit that Bray's conjecture under only its two lower curvature bounds remains open in dimensions $\operatorname{Ric}_g \ge (n-1)g$5; the present result is a step toward it rather than a resolution.

Proof ingredients

The argument combines three components.

Coefficient-adapted Jacobian comparison. For $\operatorname{Ric}_g \ge (n-1)g$6, let $\operatorname{Ric}_g \ge (n-1)g$7 solve the scalar Jacobi equation $\operatorname{Ric}_g \ge (n-1)g$8 with $\operatorname{Ric}_g \ge (n-1)g$9, $R_g \ge n(n-1)(1+\varepsilon)$0. Via a refined Riccati analysis tracking the traceless shape operator $R_g \ge n(n-1)(1+\varepsilon)$1, the paper proves that before the first conjugate time,

$R_g \ge n(n-1)(1+\varepsilon)$2

where $R_g \ge n(n-1)(1+\varepsilon)$3 is the polar Jacobian. Notably, this comparison requires no curvature lower bound. An exact identity expresses $R_g \ge n(n-1)(1+\varepsilon)$4 as a double integral of $R_g \ge n(n-1)(1+\varepsilon)$5, giving both the inequality and the equality characterization.

Integral shuffling comparison. Brown and Freedman proved a finite-family comparison: rearranging coefficients of stopped Jacobi solutions in nondecreasing order at each time can only increase the sum of $R_g \ge n(n-1)(1+\varepsilon)$6-th powers of terminal values. The paper extends this to a continuum via measure-preserving flows. The key technical work is a continuity lemma showing that stopped solutions depend continuously on their coefficients in $R_g \ge n(n-1)(1+\varepsilon)$7 (handled case-by-case according to whether the first zero lies beyond, before, or exactly at time $R_g \ge n(n-1)(1+\varepsilon)$8), followed by finite-rational approximation and dominated convergence. The resulting theorem states that for a continuous measure-preserving flow $R_g \ge n(n-1)(1+\varepsilon)$9 on a compact probability space $\varepsilon \ge 0$0 and continuous $\varepsilon \ge 0$1,

$\varepsilon \ge 0$2

where $\varepsilon \ge 0$3 solves $\varepsilon \ge 0$4 and $\varepsilon \ge 0$5 is the stopped constant-coefficient solution.

Liouville measure preservation. Applying the integral shuffling theorem to the geodesic flow $\varepsilon \ge 0$6 on $\varepsilon \ge 0$7 with normalized Liouville measure $\varepsilon \ge 0$8 and $\varepsilon \ge 0$9 converts estimates along individual geodesics into averaged estimates over the unit tangent bundle. Measure preservation ensures that although $$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$0 varies along each geodesic, the distribution of its values over $$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$1 is fixed in time — precisely the continuous analogue of the equal-level-measure condition in the finite lemma.

Consequences for metric balls and the Ricci spectrum

Two further results follow from the same machinery. First, an averaged volume comparison for metric balls incorporating scalar-curvature excess: under $$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$2, writing $$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$3,

$$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$4

where $$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$5 is the double-integral kernel introduced in Kwong's earlier work and $$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$6 is a decreasing function defined by a spherical integral. This refines the Bishop–Gromov bound quantitatively whenever the scalar excess is not identically zero. The proof uses a convexity argument showing that spherical averages of exponentials of quadratic forms are maximized when the form has rank one.

Second, when $$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$7, letting $$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$8 yields the spectral bound

$$\frac{|N|_g}{|\mathbb S^n|} \le (1+n\varepsilon)^{-1/2},$$9

The exponent $N$0 is critical here: Jensen's inequality would give only a lower bound for general exponents, but for this special exponent the spherical average evaluates exactly, via a Gaussian integral computation, to $N$1 where $N$2. This yields the determinant bound

$N$3

Equality forces umbilic small geodesic spheres, hence $N$4 on $N$5 for all unit $N$6; combined with Schur's lemma and a quotient-manifold volume count, this gives rigidity to round spheres. The final step to the main theorem uses the elementary inequality $N$7 for $N$8, applied to eigenvalues of $N$9, replacing the determinant by the scalar-curvature expression.

Limitations and open questions

Several qualifications are stated plainly in the paper. The uniform bound $\varepsilon = 0$0 falls short of Bray's conjectured factor $\varepsilon = 0$1 by a term of order $\varepsilon = 0$2, so Bray's conjecture remains open in dimensions $\varepsilon = 0$3 under only the two curvature lower bounds. The averaged ball comparison holds for almost every radius, since metric spheres may meet cut loci in positive $\varepsilon = 0$4-measure at exceptional radii, though the volume estimate itself holds for every radius. The total-volume estimate requires positive Ricci curvature; the passage from the determinant bound to the scalar bound discards information from higher elementary symmetric functions $\varepsilon = 0$5 of the nonnegative tensor $\varepsilon = 0$6, and whether exploiting these could close the order-$\varepsilon = 0$7 gap to Bray's factor is a natural open question left unaddressed.

Conclusion

The paper proves that a scalar curvature excess above $\varepsilon = 0$8 yields an explicit, quantitative improvement of Bishop's volume bound in all dimensions $\varepsilon = 0$9, without auxiliary hypotheses, and identifies the equality cases completely. Methodologically, it demonstrates that the shuffling principle of Brown and Freedman, extended to integral form over the geodesic flow, allows volume comparison to retain directional Ricci data rather than collapsing to a common lower bound. The first-order agreement with Bray's conjectured volume factor makes the remaining second-order gap a concrete target for further work.