Volume Growth under Positive Intermediate Curvature and a Ricci Lower Bound

Abstract

Let $(M^n,g)$ 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 ${\lambda }_{n-m+1}$ is small, after heat averaging, below the curvature scale $k^{-1}$. Consequently, balls have polynomial volume growth of order $R^{m-1}$ for $R\le k^{-1/2}$. 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.

arXiv comment15 pages. Comments are welcome