Positive Scalar Curvature and Volume Growth

Abstract

For a complete Riemannian manifold with nonnegative Ricci curvature, we prove two sharp volume growth order estimates, thereby resolve a conjecture of Gromov in 1986. There first is that a uniform deficit in the volume of unit balls, an analog of positive macroscopic scalar curvature, forces codimension one volume growth, and the second one is that a uniformly positive scalar curvature lower bound forces codimension two growth known as the codimension two volume growth conjecture.