Whether a complete non-compact non-flat Riemannian $n$-manifold $M$ with non-negative sectional curvature has Euclidean volume growth?


That is, whether there is a constant $C>0$ such that $\mathrm{Vol}(B\_x(r))\geq Cr^n$ for all $r>0$ and $x\in M$? Here $\mathrm{Vol}(B\_x(r))$ is the volume of the $r$-ball in $M$.


Since manifolds with non-negative Ricci curvature and Euclidean volume growth are studied a lot, I am curious about the non-negative sectional curvature case.



Non-negatively curved manifolds and the volume of balls
It is certainly not true that every complete nonflat open manifold of nonnegative curvature has Euclidean volume growth. Counterexamples are trivial to construct. Say, a capped cylinder. More generally any nonnegatively curved manifold $M^n$ with nontrivial soul has slower than Euclidean volume growth. Because its asymptotic cone at infinity has dimension strictly smaller than $n$ if the soul is not a point while manifolds with Euclidean volume growth have asymptotic cones of dimension $n$.


This also implies that any nonnegatively curved manifold with Euclidean volume growth is diffeomorphic to $\mathbb R^n$.


