Volume Growth under Positive Intermediate Curvature and a Ricci Lower Bound

Abstract: Let $(M<sup>n,g)$ be a complete Riemannian manifold with $\Ric\ge-kg$ and uniformly positive $m$-intermediate curvature in the sense of Brendle--Hirsch--Johne. We prove that the Fisher eigenvalue $λ_{n-m+1}$ is small, after heat averaging, below the curvature scale $k<sup>{-1}$. Consequently, balls have polynomial volume growth of order $R<sup>{m-1}$ for $R\le k<sup>{-1/2}$. At larger scales we obtain the corresponding estimate with an exponential factor $\exp(C\sqrt k R)$, and an example shows that this factor is necessary.

• The paper proves that complete manifolds with cC_m ge kappa and Ric ge -kg have local volume growth bounded by C R^n(1+kappa R^2)^{-(n-m+1)/2} and global growth bounded by Ckappa^{-(n-m+1)/2}R^{m-1}e^{C_nsqrt{k}R}.
• The method combines Fisher-metric eigenvalue decay, exterior powers, the HodgeWeitzenbock formula, and Nash entropy to show that heat flow effectively loses at least n-m+1 macroscopic directions below the Ricci scale k^{-1/2}.
• Explicit hyperbolic and spherehyperbolic product examples show that both curvature hypotheses are necessary and that the large-scale exponential factor cannot generally be removed, while the scalar-curvature case recovers the recent Gromov volume conjecture.

Setting and main result

This paper, by Robert Koirala, establishes a volume growth estimate for complete Riemannian manifolds satisfying two curvature hypotheses: a Ricci lower bound $\Ric \ge -kg$ and a uniformly positive lower bound $\cC_m \ge \kappa > 0$ on the $m$-intermediate curvature in the sense of Brendle–Hirsch–Johne. The intermediate curvature is defined for an orthonormal $m$-frame by

$\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$

with endpoint identities $\cC_1 = \Ric$ and $2\cC_{n-1} = \scal$, so the condition interpolates between Ricci and scalar curvature positivity. The main theorem states that under these assumptions there are constants depending only on $(n,m)$ such that, for every $x\in M$:

• Local regime ($kR^2 \le 1$): $\cC_m \ge \kappa > 0$0;
• Global regime: $\cC_m \ge \kappa > 0$1.

The result quantifies the codimension-two loss predicted by Gromov's conjectures in the intermediate-curvature setting: below the scale set by the negative Ricci part, the heat flow sees at most $\cC_m \ge \kappa > 0$2 macroscopic directions. In the scalar-curvature case ($\cC_m \ge \kappa > 0$3, $\cC_m \ge \kappa > 0$4) this recovers the recently proved Gromov volume conjecture, established independently by Antonelli (Antonelli, 14 Aug 2026), Ge (Ge, 13 Aug 2026), and Kong–Zhu (Kong et al., 14 Aug 2026); Antonelli also treated positive intermediate curvature with $\cC_m \ge \kappa > 0$5. The contribution here is the extension to arbitrary Ricci lower bounds, at the cost of an exponential factor at large scales.

Sharpness of the hypotheses

The paper includes explicit examples showing both hypotheses are necessary and that the exponential factor cannot be removed:

• Without $\cC_m \ge \kappa > 0$6: hyperbolic space $\cC_m \ge \kappa > 0$7 has $\cC_m \ge \kappa > 0$8 and exponential volume growth.
• Without the Ricci lower bound: for $\cC_m \ge \kappa > 0$9, the product $m$0 satisfies $m$1, so choosing $m$2 small makes $m$3 while volume growth carries a factor $m$4.
• Scale dependence: taking $m$5 yields $m$6 with exponential factor $m$7, showing the dependence on $m$8 in the theorem is necessary.

These counterexamples make clear that the theorem's form — polynomial growth of order $m$9 up to scale $m$0, then exponential — is essentially optimal within this framework.

Method: Fisher metric and Nash entropy

The proof combines heat-kernel analysis with information geometry. The central object is the Fisher metric of the minimal heat kernel,

$m$1

where $m$2 and $m$3 is the potential. Its eigenvalues $m$4 record which directions the heat flow has averaged out at scale $m$5; on Euclidean space all eigenvalues equal one, while on $m$6 the sphere-direction eigenvalues tend to zero as $m$7.

Two structural facts drive the argument. First, the pointed Nash entropy satisfies

$m$8

so entropy decay is governed by how many Fisher eigenvalues are small. Second, the key analytic estimate (the Fisher decay theorem) shows that whenever $m$9,

$\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$0

i.e., after heat averaging, at least $\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$1 eigenvalues are small. Integrating into the entropy evolution gives

$\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$2

and combining this with the Li–Yau Gaussian bound, which controls the volume ratio by the Nash entropy, yields the local volume estimate.

Exterior powers and the Weitzenböck mechanism

The Fisher decay estimate is proved via the $\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$3th exterior power of the Fisher metric. The trace $\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$4 dominates $\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$5 by retaining the top index set. Writing each one-form $\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$6, the Binet–Cauchy identity gives $\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$7 as the averaged wedge product $\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$8 over $\cC_m(e_1,\ldots,e_m) := \sum_{p=1}^m\sum_{q=p+1}^n \Rm(e_p,e_q,e_p,e_q),$9.

Positive intermediate curvature enters through the Hodge–Weitzenböck formula on $\cC_1 = \Ric$0-forms. For a simple unit $\cC_1 = \Ric$1-form $\cC_1 = \Ric$2 spanning $\cC_1 = \Ric$3, the identity

$\cC_1 = \Ric$4

converts the weighted trace of the Weitzenböck curvature endomorphism against $\cC_1 = \Ric$5 into a lower bound proportional to $\cC_1 = \Ric$6. The remaining terms — the gradient energy of the $\cC_1 = \Ric$7 and the Ricci contraction — are controlled by a curvature-corrected quantity

$\cC_1 = \Ric$8

where $\cC_1 = \Ric$9. The correction term compensates for the negative Ricci contribution so that $2\cC_{n-1} = \scal$0 and its heat-space-time integral satisfies $2\cC_{n-1} = \scal$1. Averaging the resulting fixed-time inequality over $2\cC_{n-1} = \scal$2, using the data-processing monotonicity $2\cC_{n-1} = \scal$3 (proved via Chapman–Kolmogorov and Jensen), a Li–Yau Harnack comparison across comparable times, and concavity of $2\cC_{n-1} = \scal$4, produces the decay estimate. All noncompact-manifold manipulations are justified by Greene–Wu cutoffs and Kotschwar's heat-kernel gradient estimate; the paper is careful about these exhaustion arguments throughout.

Limitations and open questions

Several restrictions are inherent to the result as stated. The global estimate carries the factor $2\cC_{n-1} = \scal$5, shown to be unavoidable by the product example, but the optimal constant in front of $2\cC_{n-1} = \scal$6 is not determined. The local polynomial exponent $2\cC_{n-1} = \scal$7 matches the expected codimension loss, yet no matching lower-bound example demonstrates sharpness of the power itself for general $2\cC_{n-1} = \scal$8. The method relies on the uniform pointwise lower bound $2\cC_{n-1} = \scal$9; integral or almost-everywhere versions of the hypothesis are not addressed. Finally, the paper proves only volume growth — the corresponding Urysohn-width and macroscopic-dimension conjectures of Gromov remain open in dimensions above three, and Kumar–Sen have shown the macroscopic Urysohn-width version is false in dimensions at least four, indicating that volume estimates do not automatically transfer to width statements.

Conclusion

The paper proves that uniformly positive $(n,m)$0-intermediate curvature together with $(n,m)$1 forces balls to grow like $(n,m)$2 up to the scale $(n,m)$3, with a necessary exponential factor beyond. Technically, it introduces a Fisher-metric/exterior-power framework in which the Brendle–Hirsch–Johne curvature condition couples to the heat flow through the weighted Weitzenböck formula, with the corrected nonnegative source $(n,m)$4 absorbing the negative Ricci contributions. The result extends the recently settled Gromov volume conjecture to the intermediate-curvature setting with arbitrary Ricci lower bounds, and its sharpness examples delineate precisely which hypotheses carry the conclusion.