Volume growth and integral curvature bound for non-negatively curved three-manifolds

Abstract

Motivated by results in K\"ahler geometry, in this work, we are interested in understanding the relation between integral curvature bounds and volume growth, under non-negative curvature in dimension three. In case of non-negative sectional curvature, we show that for metric on Euclidean space, it is of Euclidean volume growth if and only if it has average quadratic curvature decay. This is based on showing that metrics on three-dimensional Euclidean space with non-negative sectional curvature is of Euclidean volume growth if its asymptotic scaling invariant integral of scalar curvature is smaller than the sharp constant $8\pi$. We also show a gap Theorem if the curvature decay fast enough in the average sense, under non-negative Ricci curvature.