Universal Volume Growth Bounds from Positive Intermediate Curvature

Abstract: Let $n,m$ be integers such that $n\geq 2$ and $0\leq m\leq n-2$. Let $(M<sup>n,g)$ be a complete Riemannian manifold, and let $C_{m+1}$ be the $(m+1)$-intermediate curvature introduced by Brendle--Hirsch--Johne. We prove [ \mathrm{Ric}\geq0,\qquad C_{m+1}\geq 1 \quad\Longrightarrow\quad \mathrm{Vol} B_R(p)\leq C(n,m)R^m, ] for every $p\in M$ and every $R&gt;0$. In particular, when $m=n-2$, we deduce that [ \mathrm{Ric}\geq0,\qquad \mathrm{Scal}\geq 1 \quad\Longrightarrow\quad \mathrm{Vol} B_R(p)\leq C(n)R^{n-2}, ] for every $p\in M$ and every $R&gt;0$.

• The paper proves that complete manifolds with nonnegative Ricci curvature and normalized intermediate curvature mathfrak K_mgeqsigma^2>0 have volume growth bounded by C(n,m)sigma^{-(n-m)}R^m, without any noncollapsing assumption.
• The paper combines a quantitative Hodge-theoretic obstruction to high-rank splitting with an effective rank-improvement argument based on harmonic functions, almost-splitting theory, and volume-ratio control.
• The paper establishes the sharpness of the exponent using mathbb R^mtimesmathbb S^{n-m}, resolves Gromov’s scalar-curvature growth question, and derives consequences for Urysohn width, Ricci limits, diameter bounds, and first Betti numbers.

This paper by Gioacchino Antonelli establishes a universal volume growth estimate for complete Riemannian manifolds with nonnegative Ricci curvature and a uniformly positive intermediate curvature in the sense of Brendle–Hirsch–Johne. The scalar-curvature case resolves affirmatively a question posed by Gromov in the 1980s, and the general statement covers, uniformly and without any noncollapsing hypothesis, all members of the curvature family interpolating between Ricci and scalar curvature (2608.14507).

Statement of the main results

Let $\mathfrak K_m$ denote the normalized $m$-intermediate curvature: for an $(m+1)$-plane $W$, $\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$ where $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$. This family satisfies $\mathfrak K_0 = 2\,\mathrm{Ric}$, $\mathfrak K_1 = 2\,\mathrm{biRic}$, and $\mathfrak K_{n-2} = \mathrm{Scal}$. The main theorem states that if $(M^n,g)$ is complete with

$m$0

then for every $m$1 and $m$2,

$m$3

The exponent $m$4 is sharp: the product $m$5 has $m$6, uniformly positive $m$7, and exact $m$8 volume growth. The case $m$9 gives the headline corollary: $(m+1)$0 and $(m+1)$1 imply at most $(m+1)$2 volume growth, answering Gromov's question from [GromovLarge]. Notably, the constant is universal — it depends only on $(m+1)$3 (and $(m+1)$4), not on the manifold or basepoint, and no lower bound on unit-ball volumes is required.

Context and prior work

Prior to this result, the corresponding statements were known only under additional hypotheses or in low dimension. Under $(m+1)$5 the bound follows from Petrunin's local scalar-curvature integral estimate. In dimension three, Munteanu–Wang established linear volume growth with a universal constant; Chodosh–Li–Stryker gave an alternative proof and extended the decay range of admissible scalar-curvature bounds; Wei–Xu–Zhang obtained the sharp asymptotic ratio; Y. Wang proved an effective local version. In higher dimensions, B. Zhu obtained $(m+1)$6 estimates only under a uniform injectivity-radius lower bound. In the Ricci-limit setting, Wang–Xie–Zhu–Zhu excluded an $(m+1)$7-splitting for noncollapsed limits, and X. Zhu showed that asymptotic cones of noncollapsed manifolds with $(m+1)$8, $(m+1)$9 have essential dimension at most $W$0. The present theorem removes the noncollapsing assumption from both of these results, via its quantitative splitting obstruction (Corollary on bounded splitting). For biRicci curvature ($W$1), Antonelli–Xu had proven linear growth in dimensions $W$2 under a spectral condition outside a compact set, extended to $W$3 by Zhou–Zhu; the present theorem settles the global pointwise version in all dimensions $W$4, showing in particular that the hoped-for bounded-isoperimetry conclusion does not follow from pointwise positive biRicci alone.

An addendum records independent work of Jian Ge, who proves the scalar-curvature theorem by a different method based on heat-kernel Fisher metric and Nash entropy.

Proof strategy

The argument combines two core ingredients into an effective induction on splitting rank.

Quantitative Hodge obstruction. The first ingredient shows that a ball which is quantitatively close to splitting $W$5 Euclidean directions has radius controlled by the curvature scale. Specifically, if $W$6, $W$7 holds on $W$8, and harmonic functions $W$9 on $\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$0 satisfy the Cheeger–Colding harmonic-replacement bounds (Gram matrix close to identity, Hessian small in $\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$1), then $\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$2.

The proof is a Weitzenböck/Bochner computation on differential forms. On the set where the Gram matrix $\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$3 is close to the identity, one orthonormalizes the coframe $\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$4 to obtain forms $\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$5 and their wedge product $\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$6. The key algebraic identity is

$\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$7

which is precisely what makes the intermediate curvature appear: the wedge term contributes the "mixed" sectional curvatures $\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$8 with $\mathfrak K_m(W) = 2\,\mathfrak C_{m+1}(W)$9, while the one-form terms contribute the remaining pieces. Integrating the Weitzenböck formula against the compactly supported forms $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$0 and $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$1, using the integrated Bochner inequality and the derivative bounds for the orthonormalized coframe, yields an upper bound of order $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$2, while Markov's inequality plus Bishop–Gromov give a lower bound of order $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$3. Combining these forces $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$4 to be bounded. The author notes that the addition of the wedge contribution is the new feature relative to earlier arguments in Wang–Xie–Zhu–Zhu and Cucinotta–Mondino.

Effective rank improvement. The second ingredient, inspired by the Kapovitch–Wilking rescaling theorem and Jansen's finite formulation thereof, is a self-contained proposition stating: if $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$5 is $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$6-split (i.e., $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$7-Gromov–Hausdorff close to a ball in $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$8), then there exists a ball $$\mathfrak C_{m+1}(W)=\sum_{a=1}^{m+1}\sum_{b=m+2}^{n}K_{ab}$$9 that is $\mathfrak K_0 = 2\,\mathrm{Ric}$0-split while losing only a multiplicative factor in the $\mathfrak K_0 = 2\,\mathrm{Ric}$1-volume ratio:

$\mathfrak K_0 = 2\,\mathrm{Ric}$2

The proof splits into two cases. If the residual factor $\mathfrak K_0 = 2\,\mathrm{Ric}$3 contains a sufficiently long segment through the basepoint, the Cheeger–Colding almost-splitting theorem directly promotes the segment to an additional Euclidean direction. If instead $\mathfrak K_0 = 2\,\mathrm{Ric}$4 is macroscopically small, the ball is $\mathfrak K_0 = 2\,\mathrm{Ric}$5-close to $\mathfrak K_0 = 2\,\mathrm{Ric}$6 itself, and a critical-scale decomposition — built from harmonic replacement maps, a maximal-function argument identifying good scales, Vitali covering, and anchoring cells via distortion control — produces finitely many disjoint cells $\mathfrak K_0 = 2\,\mathrm{Ric}$7 at scales $\mathfrak K_0 = 2\,\mathrm{Ric}$8 with $\mathfrak K_0 = 2\,\mathrm{Ric}$9 and total volume a definite fraction of $\mathfrak K_1 = 2\,\mathrm{biRic}$0. Since $\mathfrak K_1 = 2\,\mathrm{biRic}$1, some cell carries a definite fraction of the $\mathfrak K_1 = 2\,\mathrm{biRic}$2-volume ratio, and a segment in its residual factor supplies the extra direction.

Induction. Starting from the trivial $\mathfrak K_1 = 2\,\mathrm{biRic}$3-splitting of any ball and iterating rank improvement $\mathfrak K_1 = 2\,\mathrm{biRic}$4 times, one obtains an $\mathfrak K_1 = 2\,\mathrm{biRic}$5-split ball whose $\mathfrak K_1 = 2\,\mathrm{biRic}$6-volume ratio exceeds $\mathfrak K_1 = 2\,\mathrm{biRic}$7. But Corollary (bounded splitting) caps the radius of such a ball at $\mathfrak K_1 = 2\,\mathrm{biRic}$8, so Bishop–Gromov bounds its $\mathfrak K_1 = 2\,\mathrm{biRic}$9-volume ratio by $\mathfrak K_{n-2} = \mathrm{Scal}$0. This yields the theorem. The whole argument is effective: constants are explicit in principle, and no contradiction-based compactness is used in the final proof.

Applications

Beyond the volume estimate itself, the paper derives several consequences. First, combining the main theorem with Papasoglu's Urysohn-width/volume criterion gives a noncollapsed Urysohn width bound: if $\mathfrak K_{n-2} = \mathrm{Scal}$1, $\mathfrak K_{n-2} = \mathrm{Scal}$2, and $\mathfrak K_{n-2} = \mathrm{Scal}$3, then $\mathfrak K_{n-2} = \mathrm{Scal}$4. In particular, under $\mathfrak K_{n-2} = \mathrm{Scal}$5 one obtains $\mathfrak K_{n-2} = \mathrm{Scal}$6, a noncollapsed form of Gromov's conjectured width estimate whose three-dimensional case was known through work of Liokumovich–Maximo and Liokumovich–Wang. Second, the bounded-splitting corollary implies a diameter bound $\mathfrak K_{n-2} = \mathrm{Scal}$7 for the residual factor in any Gromov–Hausdorff limit $\mathfrak K_{n-2} = \mathrm{Scal}$8 of manifolds with $\mathfrak K_{n-2} = \mathrm{Scal}$9, $(M^n,g)$0 — again without noncollapsing, extending diameter results of B. Zhu–X. Zhu and Wang–Xie–Zhu–Zhu. Third, applying the volume bound to universal covers and invoking Anderson's topology theorem together with Huang–Huang's splitting result for slow-growth manifolds yields $(M^n,g)$1 for noncompact manifolds with $(M^n,g)$2, $(M^n,g)$3, with rigidity: if $(M^n,g)$4, the universal cover splits off isometrically an $(M^n,g)$5 factor. This answers a question of X. Zhu without any uniform noncollapsing assumption.

Limitations and open questions

The paper is explicit about what remains unresolved. The optimal constants $(M^n,g)$6 in the main volume estimate are unknown except in a few cases, as is the optimal diameter constant in the residual-factor bound. In the Urysohn-width corollary, the noncollapsing assumption $(M^n,g)$7 has not been removed; the author flags this as an interesting open problem. More broadly, the volume theorem would follow from a solution of Yau's conjecture — that complete noncompact manifolds with $(M^n,g)$8 satisfy $(M^n,g)$9 — which remains open in every dimension $m$00 even under $m$01. Finally, the proof relies on the full strength of Cheeger–Colding theory only through specific quantitative inputs (harmonic replacement, almost splitting, and a lemma from Kapovitch–Wilking); whether the argument extends beyond the Brendle–Hirsch–Johne family or to integral curvature hypotheses is not addressed here.

Conclusion

The paper proves that nonnegative Ricci curvature combined with any uniformly positive intermediate curvature $m$02 forces at most $m$03 volume growth, with a universal constant and sharp exponent. The method — a quantitative Hodge-theoretic obstruction to high-rank splitting, coupled with an effective Kapovitch–Wilking-type rank-improvement procedure — is new in this generality, requires no noncollapsing assumptions, and immediately strengthens several recent results in the scalar-curvature and Ricci-limit literature. It settles Gromov's volume growth question affirmatively and provides the sharpest currently available structural information about manifolds with positive scalar curvature and nonnegative Ricci curvature.