Preprint

Inserted: 14 aug 2026
Last Updated: 14 aug 2026

Year: 2026

Abstract:

Let $n,m$ be integers such that $n\geq 2$ and $0\leq m\leq n-2$. Let $(M^n,g)$ be a complete Riemannian manifold, and let $\mathcal{C}_{m+1}$ be the $(m+1)$-intermediate curvature introduced by Brendle--Hirsch--Johne. We prove \[ \mathrm{Ric}\geq0,\qquad \mathcal{C}_{m+1}\geq 1 \quad\Longrightarrow\quad \mathrm{Vol} B_R(p)\leq C(n,m)R^m, \] for every $p\in M$ and every $R>0$. 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 $R>0$.

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• VolumeGrowth.pdf