Universal Volume Growth Bounds from Positive Intermediate Curvature

Let $n,m$ be integers such that $n\ge 2$ and $0\le m\le n-2$. Let $(M^n,g)$ be a complete Riemannian manifold, and let $C_{m+1}$ be the $(m+1)$-intermediate curvature introduced by Brendle–Hirsch–Johne. We prove $$\mathrm{R}\mathrm{i}\mathrm{c}\ge 0,C_{m+1}\ge 1⟹\mathrm{V}\mathrm{o}\mathrm{l}{B}_R(p)\le C(n,m)R^m,$$ for every $p\in M$ and every $R>0$. In particular, when $m=n-2$, we deduce that $$\mathrm{R}\mathrm{i}\mathrm{c}\ge 0,\mathrm{S}\mathrm{c}\mathrm{a}\mathrm{l}\ge 1⟹\mathrm{V}\mathrm{o}\mathrm{l}{B}_R(p)\le C(n)R^{n-2},$$ for every $p\in M$ and every $R>0$.